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b/09E0T4oBgHgl3EQfuQEC/content/tmp_files/2301.02601v1.pdf.txt @@ -0,0 +1,804 @@ +SEQUENT: Towards Traceable Quantum Machine Learning using +Sequential Quantum Enhanced Training ∗ +Philipp Altmann1, Leo S¨unkel1, Jonas Stein1, Tobias M¨uller2, +Christoph Roch1 and Claudia Linnhoff-Popien1 +1LMU Munich +2SAP SE, Walldorf, Germany +philipp.altmann@ifi.lmu.de +Keywords: +Quantum Machine Learning, Transfer Learning, Supervised Learning, Hybrid Quantum Computing. +Abstract: +Applying new computing paradigms like quantum computing to the field of machine learning has recently +gained attention. However, as high-dimensional real-world applications are not yet feasible to be solved us- +ing purely quantum hardware, hybrid methods using both classical and quantum machine learning paradigms +have been proposed. For instance, transfer learning methods have been shown to be successfully applicable +to hybrid image classification tasks. Nevertheless, beneficial circuit architectures still need to be explored. +Therefore, tracing the impact of the chosen circuit architecture and parameterization is crucial for the devel- +opment of beneficially applicable hybrid methods. However, current methods include processes where both +parts are trained concurrently, therefore not allowing for a strict separability of classical and quantum impact. +Thus, those architectures might produce models that yield a superior prediction accuracy whilst employing the +least possible quantum impact. To tackle this issue, we propose Sequential Quantum Enhanced Training (SE- +QUENT) an improved architecture and training process for the traceable application of quantum computing +methods to hybrid machine learning. Furthermore, we provide formal evidence for the disadvantage of current +methods and preliminary experimental results as a proof-of-concept for the applicability of SEQUENT. +1 +INTRODUCTION +With classical computation evolving towards per- +formance saturation, new computing paradigms like +quantum computing arise, promising superior perfor- +mance in complex problem domains. However, cur- +rent architectures merely reach numbers of 100 quan- +tum bits (qubits), prone to noise, and classical com- +puters run out of resources simulating similar sized +systems (Preskill, 2018). Thus, most real world appli- +cations are not yet feasible solely relying on quantum +compute. Especially in the field of machine learn- +ing, where parameter spaces sized upwards of 50 mil- +lion are required for tasks like image classification, +the resources of current quantum hardware or simula- +tors is not yet sufficient for pure quantum approaches +(He et al., 2016). Therefore, hybrid approaches have +been proposed, where the power of both classical and +quantum computation are united for improved results +(Bergholm et al., 2018). By this, it is possible to lever- +age the advantages of quantum computing for tasks +with parameter spaces that cannot be computed solely +*accepted for publication at ICAART 2023 +by quantum computers due to hardware and simula- +tion limitations. Within those hybrid algorithms the +quantum part is, analogue to the classical deep neu- +ral networks (DNNs), represented by so called vari- +ational quantum circuits (VQCs), which are param- +eterized and can be trained in a supervised manner +using labeled data (Cerezo et al., 2021). For hybrid +machine learning, we will from hereon refer to VQCs +as quantum parts and to DNNs as classical parts. +To solve large-scale real-world tasks, like image +classification, the concept of transfer learning has +been applied for training such hybrid models (Gir- +shick et al., 2014; Pan and Yang, 2010). Given a com- +plex model, with high-dimensional input- and param- +eter spaces, the term transfer leaning classically refers +to the two-step procedures of first pre-training using a +large but generic dataset and secondly fine-tuning us- +ing a smaller but more specific dataset (Torrey and +Shavlik, 2010). +Usually, a subset of the model’s +weights are frozen for the fine-tuning to compensate +for insufficient amounts of fine-tuning data. +Applied to hybrid quantum machine learning +(QML), the pre-trained model is used as a feature ex- +arXiv:2301.02601v1 [quant-ph] 6 Jan 2023 + +tractor and the dense classifier is replaced by a hybrid +model referred to as dressed quantum circuit (DQC) +including classical pre- and post-processing layers, +and the central VQC (Mari et al., 2020). This archi- +tecture results in concurrent updates to both classical +and quantum weights. Even though, this produces up- +dates towards overall optimal classification results, it +does not allow for tracing the advantageousness of the +quantum part of the architecture. Thus, besides pro- +viding competitive classification results, such hybrid +approaches do not allow for valid judgment whether +the chosen quantum circuit benefits the classification. +The only arguable result is that it does not harm the +overall performance or that the introduced inaccura- +cies may be compensated by the classical layers in the +end. However, as we currently are still only exploring +VQCs, this verdict, i.e. traceability of the impact of +both the quantum and the classical part, is crucial to +infer the architecture quality from common metrics. +Overall, with current approaches we find a mismatch +between the goal of exploring viable architectures and +the process applied. +We therefore propose the application of Sequen- +tial Quantum Enhanced Training (SEQUENT), an +adapted architecture and training procedure for hybrid +quantum transfer learning, where the effect of both +classical and quantum parts are separably assessable. +We see our contributions as follows: +• We provide formal evidence that current quantum +transfer learning architectures might result in an +optimal network configuration (perfect classifica- +tion / regression results) with the least-most quan- +tum impact, i.e., a solution equivalent to a purely +classical one. +• We propose SEQUENT, a two-step procedure of +classical pre-training and quantum fine-tuning us- +ing an adapted architecture to reduce the number +of features classically extracted to the number of +features manageable by the VQC producing the +final classification. +• We show competitive results with a traceable im- +pact of the chosen VQC on the overall perfor- +mance using preliminary benchmark datasets. +2 +BACKGROUND +To delimit SEQUENT, the following section pro- +vides a brief general introduction to the related fields +of quantum computation, quantum machine learning, +deep learning and transfer learning. +2.1 +Quantum Computing +Quantum Computation +works fundamentally dif- +ferent than classical computation, since QC uses +qubits instead of classical bits. Where classical bit +can be in the state 0 or 1, the corresponding state of +a qubit is described in Dirac notation as | 0⟩ and | 1⟩. +However, more importantly, qubits can be in a super- +position, i.e., a linear combination of both: +| ψ⟩ = α | 0⟩+β | 1⟩ +(1) +To alter this state, a set of reversible unitary op- +erations like rotations can be applied sequentially to +individual target qubits or in conjunction with a con- +trol qubit. Upon measurement, the superposition col- +lapses and the qubit takes on either the state | 0⟩ or +| 1⟩ according to a probability. Note that α and β in +(1) are complex numbers where | α |2 and | β |2 give +the probability of measuring the qubit in state | 0⟩ or +| 1⟩ respectively. Note that | α |2 + | β |2= 1, i.e., the +probabilities sum up to 1. (Nielsen and Chuang, 2010) +Quantum algorithms like Grover (Grover, 1996) +or Shor (Shor, 1994) provide a theoretical speedup +compared to classical algorithms. Moreover, in 2019 +quantum supremacy was claimed (Arute et al., 2019), +and the race to find more algorithms providing a quan- +tum advantage is currently underway. However, the +current state of quantum computing is often referred +to as the noisy-intermediate-scale quantum (NISQ) +era (Preskill, 2018), a period when relatively small +and noisy quantum computers are available, however, +still no error-correction to mitigate them, limiting the +execution to small quantum circuits. +Furthermore, +current quantum computers are not yet capable to ex- +ecute algorithms that provide any quantum advantage +in a practically useful setting. +Thus, much research has recently been put into the +investigation of hybrid-classical-quantum algorithms. +That is, algorithms that consist of quantum and clas- +sical parts, each responsible for a distinct task. In this +regard, quantum machine learning has been gaining +in popularity. +Quantum +Machine +Learning +algorithms +have +been proposed in several varieties over the last years +(Farhi et al., 2014; Dong et al., 2008; Biamonte et al., +2017). +Besides quantum kernel methods (Schuld +and Killoran, 2019) variational quantum algorithms +(VQAs) seem to be the most relevant in the current +NISQ-era for various reasons (Cerezo et al., 2021). +VQAs generally are comprised of multiple com- +ponents, but the central part is the structure of the ap- +plied circuit or Ansatz. Furthermore, a VQA Ansatz +is intrinsically parameterized in order to use it as a + +predictive model by optimizing the parameterization +towards a given objective, i.e. to minimize a given +loss. Overall, given a set of data and targets, a param- +eterized circuit and an objective, an approximation +of the generator underlying the data can be learned. +Applying methods like gradient descent, this model +can be trained to predict the label of unseen data +(Cerezo et al., 2021; Mitarai et al., 2018). For the +field of QML, various circuit architectures have been +proposed (Biamonte et al., 2017; Khairy et al., 2020; +Schuld et al., 2020). +For the remainder of this paper, we consider the +following simple φ-parameterized variational quan- +tum circuit (VQC) for η qubits: +VQCφ(z) = meassureσ ◦entangleφδ ◦···◦ +◦entangleφ1 ◦embedη(z) +(2) +with the depth δ, and the output dimension σ given +the input z = (z1,...,zη), where embedη loads the +data-points z into η balanced qubits in superposi- +tion via z-rotations, entangleφ applies controlled +not gates to entangle neighboring qubits followed by +φ-parameterized z rotations, and measureσ applies +the Pauli-Z operator and measures the first σ qubits +(Schuld and Killoran, 2019; Mitarai et al., 2018). +This architecture has also been shown to be di- +rectly applicable to classification tasks, using the +measurement expectation value as a one-hot encoded +prediction of the target (Schuld et al., 2020). +Overall, VQAs have been shown to be applica- +ble to a wide variety of classification tasks (Abo- +hashima et al., 2020) and successfully utilized by +Mari et al. (2020), using the simple architecture de- +fined in (2). Thus, to provide a proof-of-concept for +SEQUENT, we will focus on said architecture for +classification tasks and leave the optimization of em- +beddings (LaRose and Coyle, 2020) and architectures +(Khairy et al., 2020) to future research. +2.2 +Deep Learning +Deep Neural Networks (DNNs) +refer to parame- +terized networks consisting of a set of fully-connected +layers. +A layer comprises a set of distinct neu- +rons, whereas each neuron takes a vector of inputs +x = (x1,x2,...xn), which is multiplied with the cor- +responding weight vector w j = (w j1,w j2,...w jn). A +bias b j is added before being passed into an activa- +tion function ϕ. Therefore, the output of neuron z +at position j takes the following form (Bishop and +Nasrabadi, 2006): +z j = ϕ +� +n +∑ +i=1 +w jixi +bj +� +(3) +Given a target function f(x) : X �→ y, we can de- +fine the approximate +ˆfθ(x) : X �→ ˆy = Lhd→o ◦···◦Ln→h1 +(4) +as a composition of multiple layers L with multiple +neurons z parameterized by θ, d − 1 h-dimensional +hidden layers, and the respective input and target di- +mensions n and o. Using the prediction error J = +(y − ˆfθ(x))2, ˆfθ can be optimized by propagating the +error backwards through the network using the gradi- +ent ∇θJ (Bishop and Nasrabadi, 2006). +Those feed forward models have been shown ca- +pable of approximating arbitrary functions, given a +sufficient amount of data and either a sufficient depth +(i.e. number of hidden layers) or width (i.e. size of +hidden state) (Leshno et al., 1993). +Deep neural networks for image classification +tasks are comprised of two parts: A feature extrac- +tor containing a composite of convolutional layers to +extract a υ-sized vector of features FE : X �→ υ, and +a composite of fully connected layers to classify the +extracted feature vector FC : υ �→ ˆy. Thus, the overall +model is defined as ˆf : X �→ ˆy = FCθ ◦FEθ(x). Those +models have been successfully applied to a wide vari- +ety of real-world classification tasks (He et al., 2016; +Krizhevsky et al., 2012). However, to find a parame- +terization that optimally separates the given dataset, a +large amount of training data is required. +Transfer Learning +aims to solve the problem of in- +sufficient training data by transferring already learned +knowledge (weights, biases) from a task Ts of a source +domain Ds to a related target task Tt of a target do- +main Dt. More specifically, a domain D = X,P(x) +comprises a feature space X and the probability dis- +tribution P(x) where x = (x1,x2,...,xn) ∈ X. The cor- +responding task T is given by T = {y, f(x)} with la- +bel space y and target function f(x) (Zhuang et al., +2021). A deep transfer learning task is defined by +⟨Ds,Ts,Dt,Tt, ˆft(·)⟩, where ˆft(·) is defined according +to Equation 4 (Tan et al., 2018). +Generally, transfer learning is a two-stage process. +Initially, a source model is trained according to a spe- +cific task Ts in the source domain Ds. Consequently, +transfer learning aims to enhance the performance of +the target predictive function ˆft(·) for the target learn- +ing task Tt in target domain Dt by transferring la- +tent knowledge from Ts in Ds, where Ds ̸= Dt and/or +Ts ̸= Tt. Usually, the size of Ds >> Dt (Tan et al., +2018). The knowledge transfer and learning step is +commonly achieved via feature extraction and/or fine- +tuning. + +The feature extraction process freezes the source +model and adds a new classifier to the output of the +pre-trained model. Thereby, the feature maps learned +from Ts in Ds can be repurposed and the newly-added +classifier is trained according to the target task Tt +(Donahue et al., 2014). The fine-tuning process ad- +ditionally unfreezes top layers from the source model +and jointly trains the unfreezed feature representa- +tions from the source model with the added classifier. +By this, the time and space complexity for the tar- +get task Tt can be reduced by transferring and/or fine- +tuning the already learned features of a pre-trained +source model to a target model (Girshick et al., 2014). +3 +RELATED WORK +In the context of machine learning, VQAs are of- +ten applied to the problem of classification (Schuld +et al., 2020; Mitarai et al., 2018; Havl´ıˇcek et al., 2019; +Schuld and Killoran, 2019), although other applica- +tion areas exist. Different techniques, e.g. embed- +ding (Lloyd et al., 2020; LaRose and Coyle, 2020), +or problems, e.g. barren plateaus (McClean et al., +2018), have been widely discussed in the QML liter- +ature. However, we focus on hybrid quantum transfer +learning (Mari et al., 2020) in this paper. +Classical Transfer Learning is widely applied in +present-day machine learning algorithms (Torrey and +Shavlik, 2010; Pan and Yang, 2010; Pratt, 1992) and +can be extended with concepts of the emerging quan- +tum computing technology (Zen et al., 2020). Mari +et al. (2020) propose various hybrid transfer learning +architectures ranging from classical to quantum (CQ), +quantum to classical (QC) and quantum to quantum +(QQ). The authors focus on the former CQ architec- +ture, which which comprises the previously explained +DQC. In the current era of intermediate-scale quan- +tum technology the DQC transfer learning approach +is the most widely investigated and applied one, as +it allows to some extend optimally pre-process high- +dimensional data and afterwards load the most rele- +vant features into a quantum computer. Gokhale et al. +(2020) used this architecture to classify and detect im- +age splicing forgeries, while Acar and Yilmaz (2021) +applied it to detect COVID-19 from CT images. Also, +Mari et al. (2020) assess their approach exemplary on +image classification tasks. Although the results are +quite promising it is not clear from the evaluation, +whether the dressed quantum circuit is advantageous +over a fully classical approach. +4 +DQC QUANTUM IMPACT +We argue that within certain problem instance DQCs +may yield accurate results while not making active +use of any quantum effects in the VQC. This possi- +bility exists especially for easy to solve problem in- +stances, when all purely classical layers are sufficient +to yield accurate results and the quantum layer rep- +resents the identity. +This can be seen by realizing +that the classical pre-processing layer acts as a hid- +den layer with a non-polynomial activation function, +hence being capable of approximating arbitrary con- +tinuous functions depending on the number of hidden +units by the universal approximation theorem (Leshno +et al., 1993). Therefore, the overall DQC architecture +is portrayed in Figure 1. +The central VQC is defined according to sec- +tion 2.1 as introduced above. +Both pre- and post- +processing layers are implemented by fully connected +layers of neurons with a non-linear activation function +according to subsection 2.2. Formally, the DQC for η +qubits can thus be depicted as: +DQC = Lη→σ ◦VQCφ ◦Ln→η +(5) +where Ln→η and Lη→σ are the fully connected clas- +sical dressing layers according to Equation 3, map- +ping from the input size n to the number of qubits η +and from the number of qubits η to the target size σ +respectively, and VQCφ is the actual variational quan- +tum circuit according to Equation 2 with η qubits and +σ = η measured outputs. +Now let us consider a parameterization φ, where +VQCφ(z) = id(z) = z resembles the identity function. +Consequently (5) collapses into the following purely +classical, 2-layer feed-forward network with the hid- +den dimension η: +DQC = Lη→σ ◦id ◦Ln→η = Lη→σ ◦Ln→η +(6) +By the universal function approximation theorem, +this allows DQC to approximate any polynomial func- +tion f : Rn → Ro of degree 1 arbitrarily well, even if +the VQC is not affecting the prediction at all. +� +� +� +� +� +�������� +�������� +�������� +������ +������ +������ +�������� +�������� +�������� +��������� +������������ +������������ +� +Pre- +processing +Post- +processing +������ +Figure 1: Dressed Quantum Circuit Architecture + +Consequently, one has to be careful in the selec- +tion of suitable problem instances, as they must not +be too easy in order to ensure that the VQC is even +needed to yield the desired results. This becomes es- +pecially difficult as current quantum hardware is quite +limited, typically restricting the choice to fairly easy +problem instances. On top of this, no necessity to use +a post-processing layer seems apparent, as it has been +shown in various publications (Schuld et al., 2020; +Schuld and Killoran, 2019) that variational quantum +classifiers, i.e, VQCs can successfully complete clas- +sification tasks without any post-processing. Overall, +whilst conveying a proof-of-concept, that the com- +bination of classical neural networks and variational +quantum circuits in the dressed quantum circuit hy- +brid architecture is able to produce competitive re- +sults, this architecture is neither able to convey the +advantageousness of the chosen quantum circuit nor +exclude the possibility of the classical part just being +able to compensate for quantum in-steadiness. +5 +SEQUENT +To improve the traceability of quantum impact in hy- +brid architectures, we propose Sequential Quantum +Enhanced Training. SEQUENT improves upon the +dressed quantum circuit architecture by introducing +two adaptations to it: First, we omit the classical +post-processing layer and use the variational quan- +tum circuit output directly as the classification result. +Therefore we reduce the measured outputs σ from the +number of qubits η (cf. Figure 1) to the dimension of +the target ˆy (cf. Figure 2). +The direct use of VQCs as a classifier has been +frequently proposed and shown equally applicable as +classical counterparts (Schuld et al., 2020). By this, +the overall quality of the chosen circuit and parame- +terization are directly assessable by the classification +result, thus the final accuracy. Moreover, a parame- +ter setting of universal approximation capabilities (cf. +Equation 6) with the least (identitary) quantum con- +tribution is mathematically precluded by the removal +of the hidden state (compare Equation 5). +Concurrently omitting the pre-processing or com- +pression layer however would increase the number of +at least required qubits to the number of output fea- +tures of the problem domain, or, when applied to im- +age classification, the chosen feature extractor (e.g. +512 for Resnet-18). However, both current quantum +hardware and simulators do not allow for arbitrate +sized circuits, especially maxing out at around 100 +qubits. +������ +� +� +� +� +� +�������� +�������� +�������� +������ +������ +������ +�������� +�������� +�������� +��������� +������������ +������������ +������ +Figure 2: SEQUENT Architecture: Sequential Quantum +Enhanced Training comprised of a classical compression +layer (CCL) parameterized by θ and a variational quantum +circuit (VQC) parameterized by φ with separate phases for +classical (blue) and quantum (green) training for variable +sets of input data X, prediction targets ˆy and VQCs with η +qubits and δ entangling layers. +We therefore secondly propose to maintain the +classical compression layer to provide a map- +ping/compression X �→ η and, in order to fully clas- +sically pre-train the compression layer, add a surro- +gate classical classification layer η �→ ˆy. Replacing +this surrogate classical classification layer with the +chosen variational quantum circuit to be assessed and +freezing the pre-trained weights of the classical com- +pression layer then allows for a second, purely quan- +tum training phase and yield the following sequential +training procedure depicted in Figure 3: +1. Pre-train SEQUENT: ˆf : X �→ η �→ ˆy = CCLθ(x)◦ +CCLθ(z) containing a classical compression layer +and a surrogate classification layer by optimizing +the classical weights θ +2. Freeze the classical weights θ, replace the sur- +rogate classical classification layer by the vari- +ational quantum classification circutit VQCφ(cf. +Equation 2) and optimize the quantum weights φ. +This two-step procedure can be seen as an applica- +tion of transfer learning on its own, transferring from +classical to quantum weights in a hybrid architecture. +Overall, the SEQUENT architecture displayed in +Figure 2 can be formalized as: +SEQUENTθ,φ : X �→ η �→ ˆy = VQCφ(z)◦CCLθ(x) +(7) +CCLθ(x) : X �→ η = Ln→η +(cf. Equation 3) +VQCφ(z) : η �→ ˆy +(cf. Equation 2) +������������������� +������������������� +����������������� +������������������� +����������������� +����������������������������� +��������������������������� +������������������� +��� +��� +Figure 3: +SEQUENT Training Process consisting of +a classical (blue) pre-training phase (1) and a quantum +(green) fine-tuning phase (2). + +To be used for the classification of high- +dimensional data, like images, the input x needs to +be replaced by the intermediate output of an image +recognition model z (cf. subsection 2.2). Combining +both two-step transfer learning procedures, the fol- +lowing three-step procedure is yielded: +1. Classically pre-train a full classification model +(e.g. Resnet (He et al., 2016)) ˆf : X �→ υ �→ ˆy = +FCθ(z) ◦ FEθ(x) to a large generic dataset (com- +pare subsection 2.2) +2. Freeze convolutional feature extraction layers FE +and fine-tune fully-connected layers consisting of +a compression layer and a surrogate classification +layer FE : υ �→ η �→ ˆy = CCLθ(z)◦CCLθ(x). +3. Freeze classical weights and replace surrogate +classification layer with VQC to train the quan- +tum weights φ of the hybrid model: +ˆfθ,φ : X �→ υ �→ η �→ ˆy = VQCφ(z)◦CCLθ(x)◦FE +For a classification task with n classes, at least η ≥ n +qubits are required. Whilst we use the simple Ansatz +introduced in Equation 2 with η = 6 qubits and a +circuit depth of δ = 10 to validate our approach in +the following, any VQC architecture yielding a direct +classification result would be conceivable. +6 +EVALUATION +We evaluate SEQUENT by comparing its perfor- +mance to its predecessor, the DQC, and a purely clas- +sical feed forward neural network. All models were +trained on 2000 datapoints of the moons and spirals +(Lang and Witbrock, 1988) benchmark dataset for +two and four epochs of sequential, hybrid and clas- +sical training respectively. To guarantee comparabil- +ity, we set the size of the hidden state of the classical +model to h = η = 6. The code for all experiments +is available here1. The classification results are vi- +sualized in Figure 4. Looking at the result for the +moons dataset, all compared models are able to de- +pict the shape underlying data. Note, that even the +considerably simpler classical model is perfectly able +to separate the given classes. Hence, these experi- +mental results support the concerns about the impact +of the VQC to the overall DQC’s performance (cf. +section 4). +With a final test accuracy of 95%, the +DQC performs even worse than the purely classical +model reaching 96%. Looking at the SEQUENT re- +sults however, these concerns are eliminated, as the +performance and final accuracy of 97%, besides out- +performing both compared models, can certainly be +1https://github.com/philippaltmann/SEQUENT +Figure 4: Classification Results of SEQUENT, DQC and +Classical Feed Forward Neural Network for moons (left) +and spirals (right) benchmark datasets +denoted to VQC, due to the applied training process +and the used architecture. Similar results show for the +second benchmark dataset of intertwined spirals on +the right side of Figure 4. The overall best accuracy +of 86% however suggests, that further adjustments to +the VQC could be beneficial. This result also depicts +the application of SEQUENT we imagine for bench- +marking and optimizing VQC architectures. +7 +CONCLUSIONS +We proposed Sequential Quantum Enhanced Train- +ing (SEQUENT), a two-step transfer learning proce- +dure applied to training hybrid QML algorithms com- +bined with an adapted hybrid architecture to allow +for tracing both the classical and quantum impact on +the overall performance. +Furthermore, we showed +the need for said adaptions by formally pointing out +weaknesses of the DQC, the current state-of-the-art +approach to this regard. Finally, we showed that SE- +QUENT yields competitive results for two representa- + +SEQUENT +SEQUENT +.97 +.86 +DQC +DQC +.95 +.81 +Classical +Classical +.96 +.79tive benchmark datasets compared to DQCs and clas- +sical neural networks. Thus, we a provided proof- +of-concept for both the proposed reduced architecture +and the adapted transfer learning training procedure. +However, whilst SEQUENT theoretically is appli- +cable to any kind of VQC, we only considered the +simple architecture with fixed angle embeddings and +δ entangling layers as proposed by (Mari et al., 2020). +Furthermore, we only supplied preliminary experi- +mental implications and did not yet test any high di- +mensional real-world applications. Overall, we do not +expect superior results that outperform state-of-the- +art approaches in the first place, as viable circuit ar- +chitectures for quantum machine learning are still an +active and fast-moving field of research. +Thus, both the real world applicability and the de- +velopment of circuit architectures that indeed offer +a benefit over classical ones should undergo further +research attention. To empower real-world applica- +tions, the use of hybrid quantum methods should also +be kept in mind when pre-training large classification +models like Resnet. Also, applying more advanced +techniques to train the pre-processing or compression +layer to take full advantage of the chosen quantum +circuit should be examined. Therefore, auto-encoder +architectures might be applicable to train a more gen- +eralized mapping from the classical input-space to +the quantum-space. Overall, we belief, that applying +the proposed concepts and building upon SEQUENT, +both valuable hybrid applications and beneficial quan- +tum circuit architectures can be found. +ACKNOWLEDGEMENTS +This work is part of the Munich Quantum Valley, +which is supported by the Bavarian state govern- +ment with funds from the Hightech Agenda Bayern +Plus and was partially funded by the German BMWK +Project PlanQK (01MK20005I). +REFERENCES +Abohashima, Z., Elhosen, M., Houssein, E. H., and +Mohamed, W. M. 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A comprehensive sur- +vey on transfer learning. +Proceedings of the IEEE, +109(1):43–76. + diff --git a/09E0T4oBgHgl3EQfuQEC/content/tmp_files/load_file.txt b/09E0T4oBgHgl3EQfuQEC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f7c43943efe46dc0a8c1db84921622ea76a89f1 --- /dev/null +++ b/09E0T4oBgHgl3EQfuQEC/content/tmp_files/load_file.txt @@ -0,0 +1,572 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf,len=571 +page_content='SEQUENT: Towards Traceable Quantum Machine Learning using Sequential Quantum Enhanced Training ∗ Philipp Altmann1, Leo S¨unkel1, Jonas Stein1, Tobias M¨uller2, Christoph Roch1 and Claudia Linnhoff-Popien1 1LMU Munich 2SAP SE, Walldorf, Germany philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='altmann@ifi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='de Keywords: Quantum Machine Learning, Transfer Learning, Supervised Learning, Hybrid Quantum Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Abstract: Applying new computing paradigms like quantum computing to the field of machine learning has recently gained attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, as high-dimensional real-world applications are not yet feasible to be solved us- ing purely quantum hardware, hybrid methods using both classical and quantum machine learning paradigms have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' For instance, transfer learning methods have been shown to be successfully applicable to hybrid image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Nevertheless, beneficial circuit architectures still need to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore, tracing the impact of the chosen circuit architecture and parameterization is crucial for the devel- opment of beneficially applicable hybrid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, current methods include processes where both parts are trained concurrently, therefore not allowing for a strict separability of classical and quantum impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, those architectures might produce models that yield a superior prediction accuracy whilst employing the least possible quantum impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' To tackle this issue, we propose Sequential Quantum Enhanced Training (SE- QUENT) an improved architecture and training process for the traceable application of quantum computing methods to hybrid machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Furthermore, we provide formal evidence for the disadvantage of current methods and preliminary experimental results as a proof-of-concept for the applicability of SEQUENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 1 INTRODUCTION With classical computation evolving towards per- formance saturation, new computing paradigms like quantum computing arise, promising superior perfor- mance in complex problem domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, cur- rent architectures merely reach numbers of 100 quan- tum bits (qubits), prone to noise, and classical com- puters run out of resources simulating similar sized systems (Preskill, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, most real world appli- cations are not yet feasible solely relying on quantum compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Especially in the field of machine learn- ing, where parameter spaces sized upwards of 50 mil- lion are required for tasks like image classification, the resources of current quantum hardware or simula- tors is not yet sufficient for pure quantum approaches (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore, hybrid approaches have been proposed, where the power of both classical and quantum computation are united for improved results (Bergholm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' By this, it is possible to lever- age the advantages of quantum computing for tasks with parameter spaces that cannot be computed solely accepted for publication at ICAART 2023 by quantum computers due to hardware and simula- tion limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Within those hybrid algorithms the quantum part is, analogue to the classical deep neu- ral networks (DNNs), represented by so called vari- ational quantum circuits (VQCs), which are param- eterized and can be trained in a supervised manner using labeled data (Cerezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' For hybrid machine learning, we will from hereon refer to VQCs as quantum parts and to DNNs as classical parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' To solve large-scale real-world tasks, like image classification, the concept of transfer learning has been applied for training such hybrid models (Gir- shick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Pan and Yang, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Given a com- plex model, with high-dimensional input- and param- eter spaces, the term transfer leaning classically refers to the two-step procedures of first pre-training using a large but generic dataset and secondly fine-tuning us- ing a smaller but more specific dataset (Torrey and Shavlik, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Usually, a subset of the model’s weights are frozen for the fine-tuning to compensate for insufficient amounts of fine-tuning data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Applied to hybrid quantum machine learning (QML), the pre-trained model is used as a feature ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='02601v1 [quant-ph] 6 Jan 2023 tractor and the dense classifier is replaced by a hybrid model referred to as dressed quantum circuit (DQC) including classical pre- and post-processing layers, and the central VQC (Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This archi- tecture results in concurrent updates to both classical and quantum weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Even though, this produces up- dates towards overall optimal classification results, it does not allow for tracing the advantageousness of the quantum part of the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, besides pro- viding competitive classification results, such hybrid approaches do not allow for valid judgment whether the chosen quantum circuit benefits the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The only arguable result is that it does not harm the overall performance or that the introduced inaccura- cies may be compensated by the classical layers in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, as we currently are still only exploring VQCs, this verdict, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' traceability of the impact of both the quantum and the classical part, is crucial to infer the architecture quality from common metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, with current approaches we find a mismatch between the goal of exploring viable architectures and the process applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' We therefore propose the application of Sequen- tial Quantum Enhanced Training (SEQUENT), an adapted architecture and training procedure for hybrid quantum transfer learning, where the effect of both classical and quantum parts are separably assessable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' We see our contributions as follows: We provide formal evidence that current quantum transfer learning architectures might result in an optimal network configuration (perfect classifica- tion / regression results) with the least-most quan- tum impact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', a solution equivalent to a purely classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' We propose SEQUENT, a two-step procedure of classical pre-training and quantum fine-tuning us- ing an adapted architecture to reduce the number of features classically extracted to the number of features manageable by the VQC producing the final classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' We show competitive results with a traceable im- pact of the chosen VQC on the overall perfor- mance using preliminary benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 2 BACKGROUND To delimit SEQUENT, the following section pro- vides a brief general introduction to the related fields of quantum computation, quantum machine learning, deep learning and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='1 Quantum Computing Quantum Computation works fundamentally dif- ferent than classical computation, since QC uses qubits instead of classical bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Where classical bit can be in the state 0 or 1, the corresponding state of a qubit is described in Dirac notation as | 0⟩ and | 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, more importantly, qubits can be in a super- position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', a linear combination of both: | ψ⟩ = α | 0⟩+β | 1⟩ (1) To alter this state, a set of reversible unitary op- erations like rotations can be applied sequentially to individual target qubits or in conjunction with a con- trol qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Upon measurement, the superposition col- lapses and the qubit takes on either the state | 0⟩ or | 1⟩ according to a probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Note that α and β in (1) are complex numbers where | α |2 and | β |2 give the probability of measuring the qubit in state | 0⟩ or | 1⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Note that | α |2 + | β |2= 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', the probabilities sum up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (Nielsen and Chuang, 2010) Quantum algorithms like Grover (Grover, 1996) or Shor (Shor, 1994) provide a theoretical speedup compared to classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Moreover, in 2019 quantum supremacy was claimed (Arute et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2019), and the race to find more algorithms providing a quan- tum advantage is currently underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, the current state of quantum computing is often referred to as the noisy-intermediate-scale quantum (NISQ) era (Preskill, 2018), a period when relatively small and noisy quantum computers are available, however, still no error-correction to mitigate them, limiting the execution to small quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Furthermore, current quantum computers are not yet capable to ex- ecute algorithms that provide any quantum advantage in a practically useful setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, much research has recently been put into the investigation of hybrid-classical-quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' That is, algorithms that consist of quantum and clas- sical parts, each responsible for a distinct task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' In this regard, quantum machine learning has been gaining in popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Quantum Machine Learning algorithms have been proposed in several varieties over the last years (Farhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Biamonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Besides quantum kernel methods (Schuld and Killoran, 2019) variational quantum algorithms (VQAs) seem to be the most relevant in the current NISQ-era for various reasons (Cerezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' VQAs generally are comprised of multiple com- ponents, but the central part is the structure of the ap- plied circuit or Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Furthermore, a VQA Ansatz is intrinsically parameterized in order to use it as a predictive model by optimizing the parameterization towards a given objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' to minimize a given loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, given a set of data and targets, a param- eterized circuit and an objective, an approximation of the generator underlying the data can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Applying methods like gradient descent, this model can be trained to predict the label of unseen data (Cerezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Mitarai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' For the field of QML, various circuit architectures have been proposed (Biamonte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Khairy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' For the remainder of this paper, we consider the following simple φ-parameterized variational quan- tum circuit (VQC) for η qubits: VQCφ(z) = meassureσ ◦entangleφδ ◦···◦ entangleφ1 ◦embedη(z) (2) with the depth δ, and the output dimension σ given the input z = (z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=',zη), where embedη loads the data-points z into η balanced qubits in superposi- tion via z-rotations, entangleφ applies controlled not gates to entangle neighboring qubits followed by φ-parameterized z rotations, and measureσ applies the Pauli-Z operator and measures the first σ qubits (Schuld and Killoran, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Mitarai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This architecture has also been shown to be di- rectly applicable to classification tasks, using the measurement expectation value as a one-hot encoded prediction of the target (Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, VQAs have been shown to be applica- ble to a wide variety of classification tasks (Abo- hashima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020) and successfully utilized by Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (2020), using the simple architecture de- fined in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, to provide a proof-of-concept for SEQUENT, we will focus on said architecture for classification tasks and leave the optimization of em- beddings (LaRose and Coyle, 2020) and architectures (Khairy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020) to future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='2 Deep Learning Deep Neural Networks (DNNs) refer to parame- terized networks consisting of a set of fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' A layer comprises a set of distinct neu- rons, whereas each neuron takes a vector of inputs x = (x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='xn), which is multiplied with the cor- responding weight vector w j = (w j1,w j2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='w jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' A bias b j is added before being passed into an activa- tion function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore, the output of neuron z at position j takes the following form (Bishop and Nasrabadi, 2006): z j = ϕ � n ∑ i=1 w jixi +bj � (3) Given a target function f(x) : X �→ y, we can de- fine the approximate ˆfθ(x) : X �→ ˆy = Lhd→o ◦···◦Ln→h1 (4) as a composition of multiple layers L with multiple neurons z parameterized by θ, d − 1 h-dimensional hidden layers, and the respective input and target di- mensions n and o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Using the prediction error J = (y − ˆfθ(x))2, ˆfθ can be optimized by propagating the error backwards through the network using the gradi- ent ∇θJ (Bishop and Nasrabadi, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Those feed forward models have been shown ca- pable of approximating arbitrary functions, given a sufficient amount of data and either a sufficient depth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' number of hidden layers) or width (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' size of hidden state) (Leshno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Deep neural networks for image classification tasks are comprised of two parts: A feature extrac- tor containing a composite of convolutional layers to extract a υ-sized vector of features FE : X �→ υ, and a composite of fully connected layers to classify the extracted feature vector FC : υ �→ ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, the overall model is defined as ˆf : X �→ ˆy = FCθ ◦FEθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Those models have been successfully applied to a wide vari- ety of real-world classification tasks (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, to find a parame- terization that optimally separates the given dataset, a large amount of training data is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Transfer Learning aims to solve the problem of in- sufficient training data by transferring already learned knowledge (weights, biases) from a task Ts of a source domain Ds to a related target task Tt of a target do- main Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' More specifically, a domain D = X,P(x) comprises a feature space X and the probability dis- tribution P(x) where x = (x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=',xn) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The cor- responding task T is given by T = {y, f(x)} with la- bel space y and target function f(x) (Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' A deep transfer learning task is defined by ⟨Ds,Ts,Dt,Tt, ˆft(·)⟩, where ˆft(·) is defined according to Equation 4 (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Generally, transfer learning is a two-stage process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Initially, a source model is trained according to a spe- cific task Ts in the source domain Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Consequently, transfer learning aims to enhance the performance of the target predictive function ˆft(·) for the target learn- ing task Tt in target domain Dt by transferring la- tent knowledge from Ts in Ds, where Ds ̸= Dt and/or Ts ̸= Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Usually, the size of Ds >> Dt (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The knowledge transfer and learning step is commonly achieved via feature extraction and/or fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The feature extraction process freezes the source model and adds a new classifier to the output of the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thereby, the feature maps learned from Ts in Ds can be repurposed and the newly-added classifier is trained according to the target task Tt (Donahue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The fine-tuning process ad- ditionally unfreezes top layers from the source model and jointly trains the unfreezed feature representa- tions from the source model with the added classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' By this, the time and space complexity for the tar- get task Tt can be reduced by transferring and/or fine- tuning the already learned features of a pre-trained source model to a target model (Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 3 RELATED WORK In the context of machine learning, VQAs are of- ten applied to the problem of classification (Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Mitarai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Havl´ıˇcek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Schuld and Killoran, 2019), although other applica- tion areas exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Different techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' embed- ding (Lloyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' LaRose and Coyle, 2020), or problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' barren plateaus (McClean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2018), have been widely discussed in the QML liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, we focus on hybrid quantum transfer learning (Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Classical Transfer Learning is widely applied in present-day machine learning algorithms (Torrey and Shavlik, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Pan and Yang, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Pratt, 1992) and can be extended with concepts of the emerging quan- tum computing technology (Zen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (2020) propose various hybrid transfer learning architectures ranging from classical to quantum (CQ), quantum to classical (QC) and quantum to quantum (QQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The authors focus on the former CQ architec- ture, which which comprises the previously explained DQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' In the current era of intermediate-scale quan- tum technology the DQC transfer learning approach is the most widely investigated and applied one, as it allows to some extend optimally pre-process high- dimensional data and afterwards load the most rele- vant features into a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Gokhale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (2020) used this architecture to classify and detect im- age splicing forgeries, while Acar and Yilmaz (2021) applied it to detect COVID-19 from CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Also, Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (2020) assess their approach exemplary on image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Although the results are quite promising it is not clear from the evaluation, whether the dressed quantum circuit is advantageous over a fully classical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 4 DQC QUANTUM IMPACT We argue that within certain problem instance DQCs may yield accurate results while not making active use of any quantum effects in the VQC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This possi- bility exists especially for easy to solve problem in- stances, when all purely classical layers are sufficient to yield accurate results and the quantum layer rep- resents the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This can be seen by realizing that the classical pre-processing layer acts as a hid- den layer with a non-polynomial activation function, hence being capable of approximating arbitrary con- tinuous functions depending on the number of hidden units by the universal approximation theorem (Leshno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore, the overall DQC architecture is portrayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The central VQC is defined according to sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='1 as introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Both pre- and post- processing layers are implemented by fully connected layers of neurons with a non-linear activation function according to subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Formally, the DQC for η qubits can thus be depicted as: DQC = Lη→σ ◦VQCφ ◦Ln→η (5) where Ln→η and Lη→σ are the fully connected clas- sical dressing layers according to Equation 3, map- ping from the input size n to the number of qubits η and from the number of qubits η to the target size σ respectively, and VQCφ is the actual variational quan- tum circuit according to Equation 2 with η qubits and σ = η measured outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Now let us consider a parameterization φ, where VQCφ(z) = id(z) = z resembles the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Consequently (5) collapses into the following purely classical, 2-layer feed-forward network with the hid- den dimension η: DQC = Lη→σ ◦id ◦Ln→η = Lη→σ ◦Ln→η (6) By the universal function approximation theorem, this allows DQC to approximate any polynomial func- tion f : Rn → Ro of degree 1 arbitrarily well, even if the VQC is not affecting the prediction at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' � � � � � �������� �������� �������� ������ ������ ������ �������� �������� �������� ��������� ������������ ������������ � Pre- processing Post- processing ������ Figure 1: Dressed Quantum Circuit Architecture Consequently, one has to be careful in the selec- tion of suitable problem instances, as they must not be too easy in order to ensure that the VQC is even needed to yield the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This becomes es- pecially difficult as current quantum hardware is quite limited, typically restricting the choice to fairly easy problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' On top of this, no necessity to use a post-processing layer seems apparent, as it has been shown in various publications (Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Schuld and Killoran, 2019) that variational quantum classifiers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='e, VQCs can successfully complete clas- sification tasks without any post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, whilst conveying a proof-of-concept, that the com- bination of classical neural networks and variational quantum circuits in the dressed quantum circuit hy- brid architecture is able to produce competitive re- sults, this architecture is neither able to convey the advantageousness of the chosen quantum circuit nor exclude the possibility of the classical part just being able to compensate for quantum in-steadiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 5 SEQUENT To improve the traceability of quantum impact in hy- brid architectures, we propose Sequential Quantum Enhanced Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' SEQUENT improves upon the dressed quantum circuit architecture by introducing two adaptations to it: First, we omit the classical post-processing layer and use the variational quan- tum circuit output directly as the classification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore we reduce the measured outputs σ from the number of qubits η (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Figure 1) to the dimension of the target ˆy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The direct use of VQCs as a classifier has been frequently proposed and shown equally applicable as classical counterparts (Schuld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' By this, the overall quality of the chosen circuit and parame- terization are directly assessable by the classification result, thus the final accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Moreover, a parame- ter setting of universal approximation capabilities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Equation 6) with the least (identitary) quantum con- tribution is mathematically precluded by the removal of the hidden state (compare Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Concurrently omitting the pre-processing or com- pression layer however would increase the number of at least required qubits to the number of output fea- tures of the problem domain, or, when applied to im- age classification, the chosen feature extractor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 512 for Resnet-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, both current quantum hardware and simulators do not allow for arbitrate sized circuits, especially maxing out at around 100 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' ������ � � � � � �������� �������� �������� ������ ������ ������ �������� �������� �������� ��������� ������������ ������������ ������ Figure 2: SEQUENT Architecture: Sequential Quantum Enhanced Training comprised of a classical compression layer (CCL) parameterized by θ and a variational quantum circuit (VQC) parameterized by φ with separate phases for classical (blue) and quantum (green) training for variable sets of input data X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' prediction targets ˆy and VQCs with η qubits and δ entangling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' We therefore secondly propose to maintain the classical compression layer to provide a map- ping/compression X �→ η and, in order to fully clas- sically pre-train the compression layer, add a surro- gate classical classification layer η �→ ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Replacing this surrogate classical classification layer with the chosen variational quantum circuit to be assessed and freezing the pre-trained weights of the classical com- pression layer then allows for a second, purely quan- tum training phase and yield the following sequential training procedure depicted in Figure 3: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Pre-train SEQUENT: ˆf : X �→ η �→ ˆy = CCLθ(x)◦ CCLθ(z) containing a classical compression layer and a surrogate classification layer by optimizing the classical weights θ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Freeze the classical weights θ, replace the sur- rogate classical classification layer by the vari- ational quantum classification circutit VQCφ(cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Equation 2) and optimize the quantum weights φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This two-step procedure can be seen as an applica- tion of transfer learning on its own, transferring from classical to quantum weights in a hybrid architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, the SEQUENT architecture displayed in Figure 2 can be formalized as: SEQUENTθ,φ : X �→ η �→ ˆy = VQCφ(z)◦CCLθ(x) (7) CCLθ(x) : X �→ η = Ln→η (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Equation 3) VQCφ(z) : η �→ ˆy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Equation 2) ������������������� ������������������� ����������������� ������������������� ����������������� ����������������������������� ��������������������������� ������������������� ��� ��� Figure 3: SEQUENT Training Process consisting of a classical (blue) pre-training phase (1) and a quantum (green) fine-tuning phase (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' To be used for the classification of high- dimensional data, like images, the input x needs to be replaced by the intermediate output of an image recognition model z (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Combining both two-step transfer learning procedures, the fol- lowing three-step procedure is yielded: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Classically pre-train a full classification model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Resnet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2016)) ˆf : X �→ υ �→ ˆy = FCθ(z) ◦ FEθ(x) to a large generic dataset (com- pare subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Freeze convolutional feature extraction layers FE and fine-tune fully-connected layers consisting of a compression layer and a surrogate classification layer FE : υ �→ η �→ ˆy = CCLθ(z)◦CCLθ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Freeze classical weights and replace surrogate classification layer with VQC to train the quan- tum weights φ of the hybrid model: ˆfθ,φ : X �→ υ �→ η �→ ˆy = VQCφ(z)◦CCLθ(x)◦FE For a classification task with n classes, at least η ≥ n qubits are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Whilst we use the simple Ansatz introduced in Equation 2 with η = 6 qubits and a circuit depth of δ = 10 to validate our approach in the following, any VQC architecture yielding a direct classification result would be conceivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 6 EVALUATION We evaluate SEQUENT by comparing its perfor- mance to its predecessor, the DQC, and a purely clas- sical feed forward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' All models were trained on 2000 datapoints of the moons and spirals (Lang and Witbrock, 1988) benchmark dataset for two and four epochs of sequential, hybrid and clas- sical training respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' To guarantee comparabil- ity, we set the size of the hidden state of the classical model to h = η = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The code for all experiments is available here1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The classification results are vi- sualized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Looking at the result for the moons dataset, all compared models are able to de- pict the shape underlying data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Note, that even the considerably simpler classical model is perfectly able to separate the given classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Hence, these experi- mental results support the concerns about the impact of the VQC to the overall DQC’s performance (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' With a final test accuracy of 95%, the DQC performs even worse than the purely classical model reaching 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Looking at the SEQUENT re- sults however, these concerns are eliminated, as the performance and final accuracy of 97%, besides out- performing both compared models, can certainly be 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='com/philippaltmann/SEQUENT Figure 4: Classification Results of SEQUENT, DQC and Classical Feed Forward Neural Network for moons (left) and spirals (right) benchmark datasets denoted to VQC, due to the applied training process and the used architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Similar results show for the second benchmark dataset of intertwined spirals on the right side of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' The overall best accuracy of 86% however suggests, that further adjustments to the VQC could be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' This result also depicts the application of SEQUENT we imagine for bench- marking and optimizing VQC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' 7 CONCLUSIONS We proposed Sequential Quantum Enhanced Train- ing (SEQUENT), a two-step transfer learning proce- dure applied to training hybrid QML algorithms com- bined with an adapted hybrid architecture to allow for tracing both the classical and quantum impact on the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Furthermore, we showed the need for said adaptions by formally pointing out weaknesses of the DQC, the current state-of-the-art approach to this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Finally, we showed that SE- QUENT yields competitive results for two representa- SEQUENT SEQUENT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='97 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='86 DQC DQC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='95 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='81 Classical Classical .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='96 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content='79tive benchmark datasets compared to DQCs and clas- sical neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, we a provided proof- of-concept for both the proposed reduced architecture and the adapted transfer learning training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' However, whilst SEQUENT theoretically is appli- cable to any kind of VQC, we only considered the simple architecture with fixed angle embeddings and δ entangling layers as proposed by (Mari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Furthermore, we only supplied preliminary experi- mental implications and did not yet test any high di- mensional real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, we do not expect superior results that outperform state-of-the- art approaches in the first place, as viable circuit ar- chitectures for quantum machine learning are still an active and fast-moving field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Thus, both the real world applicability and the de- velopment of circuit architectures that indeed offer a benefit over classical ones should undergo further research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' To empower real-world applica- tions, the use of hybrid quantum methods should also be kept in mind when pre-training large classification models like Resnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Also, applying more advanced techniques to train the pre-processing or compression layer to take full advantage of the chosen quantum circuit should be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Therefore, auto-encoder architectures might be applicable to train a more gen- eralized mapping from the classical input-space to the quantum-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Overall, we belief, that applying the proposed concepts and building upon SEQUENT, both valuable hybrid applications and beneficial quan- tum circuit architectures can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is part of the Munich Quantum Valley, which is supported by the Bavarian state govern- ment with funds from the Hightech Agenda Bayern Plus and was partially funded by the German BMWK Project 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=', and He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' A comprehensive sur- vey on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} +page_content=' Proceedings of the IEEE, 109(1):43–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09E0T4oBgHgl3EQfuQEC/content/2301.02601v1.pdf'} diff --git a/09E1T4oBgHgl3EQf5AUq/content/2301.03506v1.pdf b/09E1T4oBgHgl3EQf5AUq/content/2301.03506v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..84c667cbc31ba89636fd71341f571f4acaddd8c2 --- /dev/null +++ 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sha256:a6abf254adc67e7741ce57701e36c7576b5f321d275fc83eeefc733749c3aa50 +size 111432 diff --git a/19FST4oBgHgl3EQfXDjl/content/tmp_files/2301.13783v1.pdf.txt b/19FST4oBgHgl3EQfXDjl/content/tmp_files/2301.13783v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c896c9bd7d33726e422d28117abadf981ef394df --- /dev/null +++ b/19FST4oBgHgl3EQfXDjl/content/tmp_files/2301.13783v1.pdf.txt @@ -0,0 +1,2753 @@ +An analytical approach to Bayesian evidence computation +Juan Garc´ıa-Bellido +Departamento de F´ısica Te´orica C-XI, Universidad Aut´onoma de Madrid, +Cantoblanco, 28049 Madrid, Spain +April 14th, 2005 +Abstract +The Bayesian evidence is a key tool in model selection, allowing a comparison of models with differ- +ent numbers of parameters. Its use in analysis of cosmological models has been limited by difficulties +in calculating it, with current numerical algorithms requiring supercomputers. In this paper we give +exact formulae for the Bayesian evidence in the case of Gaussian likelihoods with arbitrary correlations +and top-hat priors, and approximate formulae for the case of likelihood distributions with leading non- +Gaussianities (skewness and kurtosis). We apply these formulae to cosmological models with and without +isocurvature components, and compare with results we previously obtained using numerical thermody- +namic integration. We find that the results are of lower precision than the thermodynamic integration, +while still being good enough to be useful. +1 +Introduction +Model selection refers to the statistical problem of deciding which model description of observational data +is the best [1, 2]. It differs from parameter estimation, where the choice of a single model (i.e. choice of +parameters to be varied) has already been made and the aim is to find their best-fitting values and ranges. +While there have been widespread applications of parameter estimation techniques, usually likelihood fitting, +to cosmological data, there has so far been quite limited application of model selection statistics [3, 4, 5]. +This is unfortunate, as model selection techniques are necessary to robustly distinguish between models +with different numbers of parameters, and many of the most interesting issues in cosmology concern the +desirability or otherwise of incorporating additional parameters to describe new physical effects. +Within the context of Bayesian inference, model selection should be carried out using the Bayesian +evidence [1, 2], which measures the probability of the model in light of the observational data (i.e. the +average likelihood over the prior distribution). The Bayesian evidence associates a single number with each +model, and the models can then be ranked in order of the evidence, with the ratios of those values interpretted +as the relative probability of the models. This process sets up a desirable tension between model simplicity +and ability to fit the data. +Use of the Bayesian evidence has so far been limited by difficulties in calculating it. +The standard +technique is thermodynamic integration [6, 7], which varies the temperature in a Monte Carlo Markov Chain +(MCMC) approach in order that the distribution is sampled in a way covering both posterior and prior +distributions. However, in recent work [5] we showed that in order to obtain sufficiently-accurate results +in a cosmological context, around 107 likelihood evaluations are required per model. +Such analyses are +CPU-limited by the time needed to generate the predicted spectra to compare with the data, and this +requirement pushes the problem into the supercomputer class (for comparison, parameter estimation runs +typically employ 105 to 106 likelihood evaluations). +In this paper, we propose and exploit a new analytic method to compute the evidence based on an +expansion of the likelihood distribution function. +The method pre-supposes that the covariance of the +posterior distribution has been obtained, for instance via an MCMC parameter estimation run, and in its +1 +arXiv:2301.13783v1 [astro-ph.CO] 31 Jan 2023 + +present form requires that the prior distributions of the parameters are uniform top-hat priors.1 While the +method will not be applicable for general likelihood distributions, we include the leading non-gaussianities +(skewness and kurtosis) in approximating the likelihood shape, with the expectation of obtaining good +results whenever the likelihood distribution is sufficiently simple. Cosmological examples commonly exhibit +likelihood distributions with only a single significant peak. +We apply the method both to toy model examples and to genuine cosmological situations. In particular, +we calculate the evidences for adiabatic and isocurvature models, which we previously computed using +thermodynamic integration in Ref. [5]. We find that the discrepancies between the methods are typically no +worse than 1 in ln(Evidence), meaning that the analytic method is somewhat less accurate than would be +ideal, but is accurate enough to give a useful indication of model preference. +2 +The Bayesian evidence +The posterior probability distribution P(θ, M|D) for the parameters θ of the model M, given the data D, +is related to the likelihood function L(D|θ, M) within a given set of prior distribution functions π(θ, M) for +the parameters of the model, by Bayes’ theorem: +P(θ, M|D) = L(D|θ, M) π(θ, M) +E(D|M) +, +(1) +where E is the Bayesian evidence, i.e. the average likelihood over the priors, +E(D|M) = +� +dθ L(D|θ, M) π(θ, M) , +(2) +where θ is a vector with n-components characterising the n independent parameters. The prior distribution +function π contains all the information about the parameters before observing the data, i.e. our theoretical +prejudices, our physical understanding of the model, and input from previous experiments. +In the case of a large number of parameters (n ≫ 1), the evidence integral cannot be performed straight- +forwardly and must be obtained either numerically or via an analytic approximation. Amongst numerical +methods the most popular is thermodynamic integration [6, 7] but this can be computationally extremely +intensive [5]. The simplest analytical approximation is the Laplace approximation, valid when the distribu- +tion can be approximated by a multivariate Gaussian. This may hold when the quantity and quality of the +data is optimal, but is likely to be valid only in limited cosmological circumstances. +The Bayesian evidence is of interest because it allows a comparison of models amongst an exclusive and +exhaustive set {Mi}i=1...N. We can compute the posterior probability for each hypothesis given the data D +using Bayes theorem: +P(Mi|D) ∝ E(D|Mi) π(Mi) , +(3) +where E(D|Mi) is the evidence of the data under the model Mi, and π(Mi) is the prior probability of the +ith model before we see the data. The ratio of the evidences for the two competing models is called the +Bayes factor [8] +Bij = E(D|Mi) +E(D|Mj) , +(4) +and this is also equal to the ratio of the posterior model probabilities if we assume that we do not favour +any model a priori, so that π(M1) = π(M2) = ... = π(MN) = 1/N. +The Bayes factor Eq. (4) provides a mathematical representation of Occam’s razor, because more complex +models tend to be less predictive, lowering their average likelihood in comparison to simpler, more predictive +models. More complex models can only be favoured if they are able to provide a significantly improved fit to +the data. In simple cases where models give vastly different maximum likelihoods there is no need to employ +model selection techniques, but they are essential for properly discussing cases where the improvement +1An extension to gaussian priors should be feasible, but not one to arbitrary priors. +2 + +of fit is marginal. This latter situation is more or less inevitable whenever the possibility of requiring an +additional parameter arises from new data, unless the new data is of vastly greater power than that preceding +it; cosmological examples include the inclusion of spectral tilt, dark energy density variation, or the case +explored later in this paper of trace isocurvature perturbations. +In this paper we will obtain an analytical formula which approximates the Bayesian evidence by consid- +ering the higher-order cumulants of the distribution in a systematic way. The advantage is that with these +analytical formulae one can compute the evidence for a given model with an arbitrary number of parame- +ters, given the hierarchy of cumulants of the distribution, assumed previously computed for the likelihood +distribution function within the parameter estimation programme. +The evidence needs to be calculated to sufficient precision for robust conclusions to be drawn. +The +standard interpretational scale, due to Jeffreys [1] and summarized in Ref. [5], strengthens its verdict roughly +each time the difference in ln(Evidence) increases by one. The evidence therefore needs to be computed more +accurately than this, with an uncertainty of 0.1 in ln(Evidence) easily sufficient, and a factor two worse than +that acceptable. This accuracy requirement ensures that the relative model probabilities are little changed +by the uncertainty. +The first thing we need is to characterize the distribution function for the model with n parameters. Let +f(x) be this function, and let us assume that it is properly normalized, +� ∞ +−∞ +dnx f(x) = 1 . +(5) +Then, the p-point correlation function is given by +⟨xi1 . . . xip⟩ = +� ∞ +−∞ +dnx xi1 . . . xip f(x) . +(6) +From this distribution function one can always construct the generating functional, φ(u), as the Fourier +transform +φ(u) = +� ∞ +−∞ +dnx ei u·x f(x) . +(7) +This function can be expanded as +φ(u) = exp +� ∞ +� +p=1 +ip +p! Ai1...ip ui1 . . . uip +� +, +(8) +where Ai1...ip are totally symmetric rank-p tensors. For instance, if we restrict ourselves to order 4, we can +write +φ(u) = exp +� +i µiui − 1 +2! Cij uiuj − i +3! Bijk uiujuk + 1 +4! Dijkl uiujukul + · · · + in +n! Ai1...in ui1 . . . uin +� +, +(9) +where µi is the mean value of variable xi; Cij is the covariance matrix; Bijk is the trilinear matrix associated +with the third cumulant or skewness; Dijkl is the rank-4 tensor associated with the fourth cumulant or +kurtosis, and Ai1...in is the rank-n tensor associated with the n-th cumulant. Their expressions in terms of +n-point correlation functions can be obtained from Eq. (7), by realising that +⟨xi1 . . . xin⟩ = (−i)n +∂nφ(u) +∂ui1 . . . ∂uin +���� +u=0 +. +(10) +For instance, the first-order term gives +⟨xi⟩ = (−i) ∂φ(u) +∂ui +���� +u=0 += µi . +(11) +3 + +The second-order correlation function gives +⟨xixj⟩ = (−i)2 ∂2φ(u) +∂ui∂uj +���� +u=0 += Cij + µiµj , +(12) +such that the covariance matrix is obtained, as usual, from +Cij = ⟨xixj⟩ − ⟨xi⟩⟨xj⟩ . +The third-order correlation function gives +⟨xixjxk⟩ = (−i)3 +∂3φ(u) +∂ui∂uj∂uk +���� +u=0 += Bijk + µiCjk + µjCki + µkCij + µiµjµk , +(13) +such that the skewness matrix is obtained from +Bijk = ⟨xixjxk⟩ − ⟨xi⟩⟨xjxk⟩ − ⟨xj⟩⟨xkxi⟩ − ⟨xk⟩⟨xixj⟩ + 2⟨xi⟩⟨xj⟩⟨xk⟩ . +(14) +The fourth-order correlation function gives +⟨xixjxkxl⟩ = (−i)4 +∂4φ(u) +∂ui∂uj∂uk∂ul +���� +u=0 += +Dijkl + CijCkl + CikCjl + CilCjk +(15) ++ +Bijkµl + Bijlµk + Bjklµi + Biklµj ++ +Cijµkµl + Cikµjµl + Cilµjµk ++ +Cjkµiµl + Cjlµiµk + Cklµiµj ++ +µiµjµkµl , +such that the kurtosis matrix is obtained from +Dijkl += +⟨xixjxkxl⟩ − ⟨xixj⟩⟨xkxl⟩ − ⟨xixk⟩⟨xjxl⟩ − ⟨xixl⟩⟨xjxk⟩ +(16) +− +⟨xixjxk⟩⟨xl⟩ − ⟨xixjxl⟩⟨xk⟩ − ⟨xixkxl⟩⟨xj⟩ − ⟨xjxkxl⟩⟨xi⟩ ++ +2 ⟨xixj⟩⟨xk⟩⟨xl⟩ + 2 ⟨xixk⟩⟨xj⟩⟨xl⟩ + 2 ⟨xixl⟩⟨xj⟩⟨xk⟩ + 2 ⟨xjxk⟩⟨xi⟩⟨xl⟩ ++ +2 ⟨xjxl⟩⟨xi⟩⟨xk⟩ + 2 ⟨xkxl⟩⟨xi⟩⟨xj⟩ − 6 ⟨xi⟩⟨xj⟩⟨xk⟩⟨xl⟩ , +and so on, for the higher order cumulants. +3 +The Gaussian approximation +Let us first evaluate the evidence for a multivariate Gaussian distribution, that is, one in which all the +cumulants are zero except the covariance matrix Cij and the means µi. In this case, the generating functional +and the distribution are given by +φ(u) = exp +� +− iµiui − 1 +2 Cij uiuj +� +, +(17) +f(x) = +1 +(2π)n +� ∞ +−∞ +dnu e−i u·x φ(u) +(18) += +1 +(2π)n/2√ +det C +exp +� +− 1 +2C−1 +ij (xi − µi)(xj − µj) +� +, +(19) +4 + +which satisfies +⟨xi⟩ = µi , +⟨xixj⟩ = Cij + µiµj , +⟨xixjxk⟩ = µ(iCjk) + µiµjµk , +. . . +(20) +where the subindices in parenthesis, (ijk), indicate a cyclic sum. Notice that all the n-point correlation +functions can be written in terms of the first two moments of the distribution, and all the higher-order +cumulants vanish. +3.1 +Centred priors +For initial calculations, we assume a top-hat prior and make the unrealistic assumption, to be lifted later, +that it is centered at the mean value: +π(x, a) ≡ +� +(2a)−1 +−a < x − µ < a , +0 +otherwise . +(21) +Since the Fourier transform of a top-hat function is +� ∞ +−∞ +dx eiux π(x, a) = sin au +au +exp[iµu] , +we can write the evidence either way +E(a1, . . . , an) += +� ∞ +−∞ +dnx f(x) +n +� +i=1 +π(xi, ai) = +n +� +i=1 +(2ai)−1 +� a1 +−a1 +dx1· · · +� an +−an +dxn f(˜x) +(22) += +1 +(2π)n +� ∞ +−∞ +dnu φ(u) +n +� +i=1 +sin aiui +aiui +. +(23) +In Eq. (22) we integrate over the displaced coordinate, ˜xi ≡ xi − µi, such that ⟨˜xi⟩ = 0 and ⟨˜xi˜xj⟩ = Cij. +From now on, we ignore the tildes, and assume we have moved to those coordinates. Note that the choice +of prior is not crucial. We could have chosen a Gaussian prior, and the result would not be very different, +except that the window functions, sin z/z, would then be Gaussians. Let us now perform the integration +Eq. (22) in the case of 1, 2 and then n variables. +1 variable. Suppose the covariance is just C = σ2. The evidence is then +E(a) = +1 +2a σ +√ +2π +� a +−a +dx e− x2 +2σ2 = 1 +2π +� ∞ +−∞ +du sin au +au +e− 1 +2 σ2u2 = 1 +2aErf +� +a +σ +√ +2 +� +, +(24) +where Erf[x] is the error function, which asymptotes very quickly to one for x ≥ 2, or a ≥ 3σ. Therefore, +the evidence of a model with centred top-hat prior of width 2a is well approximated by (2a)−1. The wider +is the theoretical prior, the smaller is the evidence, as expected. +2 variables. Suppose we have two correlated variables, x1 and x2, with covariance matrix +C = +� +C11 +C12 +C12 +C22 +� += +� +σ2 +1 +ρσ1σ2 +ρσ1σ2 +σ2 +2 +� +. +(25) +where the cross-correlation ρ is defined by +ρ = +⟨x1x2⟩ +� +⟨x2 +1⟩⟨x2 +2⟩ += ⟨x1x2⟩ +σ1σ2 +, +5 + +with σ1 and σ2 the corresponding quadratic dispersions. In this case, the normalized 2-dimensional distri- +bution function is +f(x) = +1 +2πσ1σ2 +� +1 − ρ2 exp +� +−1 +1 − ρ2 +� x2 +1 +2σ2 +1 +− ρx1x2 +σ1σ2 ++ x2 +2 +2σ2 +2 +�� +, +(26) +which has the property that integrating (“marginalizing”) over one of the two variables, leaves a properly- +normalized Gaussian distribution for the remaining variable, +� ∞ +−∞ +dx2 f(x) = +1 +σ1 +√ +2π e +− +x2 +1 +2σ2 +1 . +(27) +Let us now evaluate the evidence Eq. (22) by integrating first over the prior in x2, +1 +2a2 +� a2 +−a2 +dx2 f(x) = e +− +x2 +1 +2σ2 +1 +σ1 +√ +2π · +1 +4a2 +� +Erf +� a2σ1 + ρσ2 x1 +σ1σ2 +� +2(1 − ρ2) +� ++ Erf +� a2σ1 − ρσ2 x1 +σ1σ2 +� +2(1 − ρ2) +�� +. +(28) +The first term is the result we would have obtained if we had been marginalizing over x2; the second is a +sum of error functions that still depend on x1, and modulates the marginalization. We can use the series +expansion of the error function to second order, +1 +2 +� +Erf[a + x] + Erf[a − x] +� += Erf[a] − 2a x2 +√π e−a2 + O(x4) , +to write Eq. (28) to order x2 +1 as +1 +2a2 +� a2 +−a2 +dx2 f(x) = e +− +x2 +1 +2σ2 +1 +σ1 +√ +2π +� +�� 1 +2a2 +Erf +� +a2 +σ2 +� +2(1 − ρ2) +� +− +ρ2 x2 +1 e +− +a2 +2 +2σ2 +2(1−ρ2) +2σ2 +1σ2(1 − ρ2) +� +2π(1 − ρ2) +� +�� . +(29) +Integrating now over the x1 prior, we finally obtain the evidence +E(a1, a2) += +1 +4a1a2 +� a1 +−a1 +dx1 +� a2 +−a2 +dx2 f(x) += +1 +4a1a2 +Erf +� +a2 +σ2 +� +2(1 − ρ2) +� +Erf +� +a1 +σ1 +√ +2 +� +(30) +− +ρ2 e +− +a2 +2 +2σ2 +2(1−ρ2) +2σ1σ2(1 − ρ2) +� +2π(1 − ρ2) +Erf +� +a1 +σ1 +√ +2 +� +2a1 ++ ρ2 e +− +a2 +2 +2σ2 +2(1−ρ2) − +a2 +1 +2σ2 +1 +4πσ2 +1σ2 +� +1 − ρ2 +. +Note that in the limit of no cross-correlations, ρ → 0, the integral factorizes and we can write an exact +expression for the evidence, +E(a1, a2) += +1 +4a1a2 +1 +2πσ1σ2 +� a1 +−a1 +dx1 +� a2 +−a2 +dx2 e +− +x2 +1 +2σ2 +1 +− +x2 +2 +2σ2 +2 +(31) += +1 +4π2 +� ∞ +−∞ +du1 +� ∞ +−∞ +du2 +sin a1u1 +a1u1 +sin a2u2 +a2u2 +e− 1 +2 σ2 +1u2 +1− 1 +2 σ2 +2u2 +2 +(32) += +1 +4a1a2 +Erf +� +a1 +σ1 +√ +2 +� +Erf +� +a2 +σ2 +√ +2 +� +. +(33) +6 + +It happens, however, that even in the presence of cross-correlations, if the prior is wide (ai ≥ 2σi), then the +terms proportional to exponentials are negligible and the evidence becomes, to very good approximation, +E(a1, a2) = +1 +4a1a2 +Erf +� +a2 +σ2 +� +2(1 − ρ2) +� +Erf +� +a1 +σ1 +√ +2 +� +. +(34) +Moreover, in that case, the error functions are very approximately given by 1. +n variables. Suppose we have n correlated variables, x = (x1, . . . , xn), with covariance matrix +Cn = +� +� +� +� +� +� +� +� +C11 +C12 +. . . +C1n +C12 +C22 +. . . +C2n +... +... +... +... +C1n +C2n +. . . +Cnn +� +� +� +� +� +� +� +� +. +(35) +In that case, the probability distribution function can be expressed as +f(x) = +1 +(2π)n/2√det Cn +exp +� +− 1 +2xT C−1 +n x +� +, +(36) +which has the property that marginalizing over the last variable, xn, we obtain a correlated probability +distribution function for the n − 1 variables, x = (x1, . . . , xn−1), +f(x) = +1 +(2π)(n−1)/2� +det Cn−1 +exp +� +− 1 +2xT C−1 +n−1x +� +, +(37) +where the Cn−1 covariance matrix is given by Eq. (35) without the last column and the last row. +We will now evaluate the evidence Eq. (22) for this multivariate Gaussian, starting with the integration +over the last variable, xn, +1 +2an +� an +−an +dxn f(x) += +1 +(2π)(n−1)/2� +det Cn−1 +exp +� +− 1 +2xT C−1 +n−1x +� +× +� +1 +2an +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ O +� +e− +a2 +n det Cn−1 +2 det Cn +�� +. +(38) +Integrating now over the next variable, xn−1, we find +1 +4anan−1 +� an +−an +dxn +� an−1 +−an−1 +dxn−1 f(x) = +1 +(2π)(n−2)/2� +det Cn−2 +exp +� +− 1 +2 xT C−1 +n−2x +� +× +� +1 +4anan−1 +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� +Erf +� +an +√ +2 +� +det Cn−2 +det Cn−1 +� ++ O +� +e− +a2 +n det Cn−1 +2 det Cn +�� +. +(39) +Continuing the integration over the priors, we end up with the evidence for the n-dimensional distribution, +E(a1, . . . , an) += +1 +�n +p=1 2ap +� a1 +−a1 +· · · +� an +−an +dnx f(x) += +n +� +p=1 +1 +2ap +Erf +� +ap +√ +2 +� +det Cp−1 +det Cp +� ++ O +� +exp +� +− +n +� +p=1 +a2 +p det Cp−1 +2 det Cp +�� +, +(40) +7 + +where the covariance matrices Cp are constructed as above, by eliminating the n−p last rows and columns, un- +til we end up with C0 ≡ 1. Note that the approximation is very good whenever �n +p=1(a2 +p det Cp−1)/(2 det Cp) ≫ +1, which is often the case. Note also that we recover the previous result Eq. (34) for the particular case +n = 2. +In the limit that the cross-correlation between the n variables vanishes, the evidence (40) reduces to the +exact result +E(a1, . . . , an) = +n +� +p=1 +1 +2ap +Erf +� +ap +σp +√ +2 +� +. +(41) +Note that the evidence Eq. (40) reflects correctly the limit in which we eliminate the need for a new variable +xn, by making its prior vanish, +lim +an→0 E(a1, . . . , an) = E(a1, . . . , an−1) +1 +√ +2π +� +det Cn−1 +det Cn +, +(42) +and thus we recover in that limit a properly-normalized distribution, f(x1, . . . , xn) → f(x1, . . . , xn−1), while +the inspection of the likelihood function alone would not have been able to give a reasonable answer. +On the other hand, in the case that our theoretical prejudice cannot assign a concrete prior to a given +variable, we see that the evidence decreases as 1/2a as a increases. Therefore, the Bayesian evidence seems +to be a very good discriminator between theoretical priors, and penalizes including too many parameters, a +la Occam’s razor. +3.2 +Uncentered priors +It is unlikely that the priors will actually be centred on the mean of the distribution, as the priors are not +supposed to know what the data will tell us. We therefore need to generalize the above for uncentred priors. +We continue to assume that the priors are top hats. +We also continue to assume for the moment that the probability distribution is well approximated by +a Gaussian with mean value µ. We will then use displaced variables ˜xi = xi − µi, and write the Gaussian +distribution function as in Eq. (36). The normalized top-hat prior is now uncentered with respect to the +mean value, +π(˜x; a, b) ≡ +� +(a + b)−1 +−a < ˜x < b , +0 +otherwise . +(43) +For a single variable, the result is exact, +E(a; b) = +� ∞ +−∞ +dx f(x) π(x; a, b) = +1 +2a + 2b +� +Erf +� +a +σ +√ +2 +� ++ Erf +� +b +σ +√ +2 +�� +. +(44) +where we are integrating over the displaced variable ˜x, from now on renamed as x. Note that we recover the +result Eq. (24) for the centered prior case in the limit b → a. +For two variables, with distribution function Eq. (26), the uncentered Bayesian evidence is +E(a1, a2; b1, b2) += +1 +(a1 + b1)(a2 + b2) +� b1 +−a1 +dx1 +� b2 +−a2 +dx2 f(x1, x2) +(45) += +1 +(2a1 + 2b1)(2a2 + 2b2) +�� +Erf +� +a1 +σ1 +√ +2 +� ++ Erf +� +b1 +σ1 +√ +2 +�� +(46) +× +� +Erf +� +a2 +σ2 +� +2(1 − ρ2) +� ++ Erf +� +b2 +σ2 +� +2(1 − ρ2) +�� +− +ρ +2π +� +1 − ρ2 +� +e +− +a2 +1 +2σ2 +1 − e +− +b2 +1 +2σ2 +1 +� � +e +− +a2 +2 +2σ2 +2(1−ρ2) + e +− +b2 +2 +2σ2 +2(1−ρ2) +�� +8 + +The evidence for the multiple-variable case Eq. (36) is +E(a, b) = +� ∞ +−∞ +dnx f(x) +n +� +i=1 +π(xi; ai, bi) = +n +� +i=1 +(ai + bi)−1 +� b1 +−a1 +d˜x1· · · +� bn +−an +d˜xn f(˜x) . +(47) +Let us now evaluate it for the multivariate Gaussian Eq. (36), starting with the integration over the last +variable, xn, +1 +an + bn +� bn +−an +dxn f(x) = +1 +(2π)(n−1)/2� +det Cn−1 +exp +� +− 1 +2xT C−1 +n−1x +� +1 +(2an + 2bn) +× +� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +� ++ O +� +e− +a2 +n det Cn−1 +2 det Cn ++ e− +b2 +n det Cn−1 +2 det Cn +�� +(48) +Integrating now over the next variable, xn−1, we find +1 +(an + bn)(an−1 + bn−1) +� bn +−an +dxn +� bn−1 +−an−1 +dxn−1 f(x) = +1 +(2π)(n−2)/2� +det Cn−2 +exp +� +− 1 +2 xT C−1 +n−2x +� +1 +(2an + 2bn)(2an−1 + 2bn−1) +(49) +× +�� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +�� +(50) +× +� +Erf +� +an−1 +√ +2 +� +det Cn−2 +det Cn−1 +� ++ Erf +� +bn−1 +√ +2 +� +det Cn−2 +det Cn−1 +�� +(51) ++ O +� +e− +a2 +n det Cn−1 +2 det Cn ++ e− +b2 +n det Cn−1 +2 det Cn +� +× +� +e +− +a2 +n−1 det Cn−2 +2 det Cn−1 ++ e +− +b2 +n−1 det Cn−2 +2 det Cn−1 +�� +. +Continuing the integration over the priors, we end up with the evidence for the n-dimensional distribution, +E(a, b) += +1 +�n +p=1(ap + bp) +� b1 +−a1 +· · · +� bn +−an +dnx f(x) += +n +� +p=1 +1 +(2ap + 2bp) +� +Erf +� +ap +√ +2 +� +det Cp−1 +det Cp +� ++ Erf +� +bp +√ +2 +� +det Cp−1 +det Cp +�� +(52) ++ O +� n +� +p=1 +� +exp +� +− a2 +p det Cp−1 +2 det Cp +� ++ exp +� +− b2 +p det Cp−1 +2 det Cp +��� +, +where the covariance matrices Cp are constructed as above, by eliminating the n−p last rows and columns, un- +til C0 ≡ 1. Note that the approximation is very good whenever the exponents are large, �n +p=1(a2 +p det Cp−1)/(2 det Cp) ≫ +1, which is often the case. Note also that we recover the expression of the evidence for the centered priors +Eq. (40) in the limit b → a. +Let us now evaluate the evidence for a distribution normalized to the maximum of the likelihood distri- +bution, +f(x) = Lmax exp +� +− 1 +2xT C−1 +n x +� +(53) +9 + +In this case, the evidence is given by Eq. (52), multiplied by a factor Lmax × (2π)n/2√det Cn from the nor- +malization. We can then evaluate the logarithm of the evidence, ignoring the exponentially-small corrections, +as +ln E += +ln Lmax + n +2 ln(2π) + 1 +2 ln det Cn − +n +� +p=1 +ln(2ap + 2bp) ++ +n +� +p=1 +ln +� +Erf +� +ap +√ +2 +� +det Cp−1 +det Cp +� ++ Erf +� +bp +√ +2 +� +det Cp−1 +det Cp +�� +. +(54) +Uncorrelated case. Suppose we have a multivariate Gaussian distribution without correlations between +variables, i.e. Cij = σ2 +i δij is a diagonal matrix; then the evidence reads exactly, +E(a, b) = +1 +�n +p=1(ap + bp) +� b1 +−a1 +· · · +� bn +−an +dnx f(x) = +n +� +p=1 +1 +2(ap + bp) +� +Erf +� +ap +σp +√ +2 +� ++ Erf +� +bp +σp +√ +2 +�� +, +(55) +where σp are the dispersions of each variable ˜xp, and thus the logarithm of the evidence becomes +ln E = ln Lmax + n +2 ln(2π) + +n +� +p=1 +ln σp − +n +� +p=1 +ln(2ap + 2bp) + +n +� +p=1 +ln +� +Erf +� +ap +σp +√ +2 +� ++ Erf +� +bp +σp +√ +2 +�� +(56) +Laplace approximation. The Laplacian approximation to the evidence assumes the distribution is a +correlated Gaussian, and that the priors are large enough so that the whole distribution fits easily inside +them, in which case the error functions are approximately unity and do not contribute to the evidence; from +Eq. (54) we now have +ln E = ln Lmax + n +2 ln(2π) + 1 +2 ln det Cn − +n +� +p=1 +ln ∆θp , +(57) +where ∆θp = ap + bp is the parameter interval associated to the prior. In the next section we will compare +the different approximations. +4 +Non-Gaussian corrections +The advantage of this method is that one can perform a systematic computation of the evidence of a given +model with its own priors, given an arbitrary set of moments of the distribution. Here we will consider the +first two beyond the covariance matrix, i.e. the skewness and the kurtosis terms, see Eq. (9). +4.1 +Skewness +Let us start with the first correction to the Gaussian approximation, the trilinear term Bijk. For this, we +write the generating functional (9) as +φ(u) = exp +� +i µiui − 1 +2! Cij uiuj − i +3! Bijk uiujuk +� +. +(58) +10 + +By performing a change of variable, ui = yi −i C−1 +ik (xk −µk), we can evaluate the Fourier transform integral +and obtain the properly-normalized probability distribution function +f(x) += +1 +(2π)n/2√det Cn +exp +� +− 1 +2xT C−1 +n x +� +× +� +1 − 1 +2Bijk C−1 +ij C−1 +kl xl + 1 +6Bijk C−1 +il C−1 +jmC−1 +kn xlxmxn +� +, +(59) +where xk are the displaced coordinates (xk − µk). This skewed distribution function satisfies +⟨xi⟩ = 0 , +⟨xixj⟩ = Cij , +⟨xixjxk⟩ = Bijk , +⟨xixjxkxl⟩ = 0 , +. . . +(60) +as can be confirmed by direct evaluation. Let us now compute the evidence Eq. (22) for this skewed model. +Since the extra terms in the parenthesis of Eq. (59) are both odd functions of x, when integrating over an +even range like that of the centered top-hat prior Eq. (21), their contribution to the evidence vanish, and +thus the final evidence for the skewed model does not differ from that of the Gaussian model Eq. (40). In +case the prior is off-centered with respect to the mean, e.g. like in Eq. (43), then the contribution of the odd +terms to the evidence would not vanish. Let us evaluate their contribution. +For a single variable (n = 1), the correctly-normalized likelihood function can be written as +f(x) = e−x2/2σ2 +σ +√ +2π +� +1 − B x +2σ4 + B x3 +6σ6 +� +, +satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = B, and the Bayesian integral can be computed exactly as +E(a, b) = +1 +2a + 2b +� +Erf +� +a +σ +√ +2 +� ++ Erf +� +b +σ +√ +2 +�� +− Bσ−3 +6 +√ +2π +�� +1 − a2 +σ2 +� +e− a2 +2σ2 − +� +1 − b2 +σ2 +� +e− b2 +2σ2 +� +1 +a + b . +(61) +Note that for even (centered) priors, with b = a, the evidence reduces to Eq. (24). +For an arbitrary number of variables, the computation is more complicated. Let us start with the n-th +variable and, in order to compute the integral, let us define the auxiliary function +g(λ) += +� bn +−an +dxn xn +exp +� +− λ +2 xT C−1 +n x +� +(2π)n/2√det Cn += +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +× +× +1 +λ +√ +2π +� +exp +� +− λa2 +n +2 +det Cn−1 +det Cn +� +− exp +� +− λb2 +n +2 +det Cn−1 +det Cn +�� +, +(62) +such that, using Erf′[x] = +2 +√π e−x2, +−2g′(λ = 1) = +� bn +−an +dxn xn +(xT C−1 +n x) exp +� +− 1 +2xT C−1 +n x +� +(2π)n/2√det Cn += +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +× +× +1 +√ +2π +�� +2 + a2 +n +det Cn−1 +det Cn +� +exp +� +− a2 +n +2 +det Cn−1 +det Cn +� +− +� +2 + b2 +n +det Cn−1 +det Cn +� +exp +� +− b2 +n +2 +det Cn−1 +det Cn +�� +. +(63) +Therefore, with the use of Eq. (63), the integral of the skewness-corrected distribution function Eq. (59) over +the xn uncentered prior, becomes +� bn +−an +dxn f(x) = +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +� +1 +2 +� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +�� +− 1 +6Bijn C−1 +ij +1 +√ +2π +� +det Cn−1 +det Cn +�� +1 − a2 +n +det Cn−1 +det Cn +� +e− +a2 +n det Cn−1 +2 det Cn +− +� +1 − b2 +n +det Cn−1 +det Cn +� +e− +b2 +n det Cn−1 +2 det Cn +�� +. +(64) +11 + +Let us define two new functions, +Ei(ai, bi) += +1 +2 +� +Erf +� +ai +√ +2 +� +det Ci−1 +det Ci +� ++ Erf +� +bi +√ +2 +� +det Ci−1 +det Ci +�� +, +(65) +Fi(ai, bi) += +1 +6 +√ +2π +� +det Ci−1 +det Ci +�� +1 − a2 +i +det Ci−1 +det Ci +� +e− +a2 +i det Ci−1 +2 det Ci +− +� +1 − b2 +i +det Ci−1 +det Ci +� +e− +b2 +i det Ci−1 +2 det Ci +� +. +Integrating iteratively over xn−1, . . . , x1, we end up with the Bayesian evidence for the third-order-corrected +probability distribution function f(x), +E(a, b) = +n +� +p=1 +Ep(ap, bp) +(ap + bp) +� +1 − +n +� +k=1 +Bijk C−1 +ij +Fk(ak, bk) +Ek(ak, bk) +� +. +(66) +Unless Bijk C−1 +ij +is very large, the correction to the error function is exponentially suppressed, and we do +not expect significant departures from the Gaussian case Eq. (40). Note also that if the prior is symmetric, +it is easy to see that the skewness part of the integral vanishes, Fk(ak, bk) → 0, as can be checked explicitly +by taking bk → ak. +4.2 +Kurtosis +The next correction beyond skewness is the fourth order moment or kurtosis, given by the Dijkl term in +Eq. (9). Let us ignore for the moment the third order skewness and write +φ(u) = exp +� +i µiui − 1 +2! Cij uiuj + 1 +4! Dijkl uiujukul +� +. +(67) +By performing the same change of variables, ui = yi − i C−1 +ik (xk − µk), we can now compute the Fourier +transform and obtain the properly-normalized probability distribution function +f(x) += +1 +(2π)n/2√det Cn +exp +� +− 1 +2xT C−1 +n x +� � +1 + 1 +8Dijkl C−1 +ij C−1 +kl +−1 +4Dijkl C−1 +ij C−1 +kmC−1 +ln xmxn + 1 +24Dijkl C−1 +im C−1 +jn C−1 +kp C−1 +lq xmxnxpxq +� +. +(68) +Performing the integrals, it is easy to see that this distribution satisfies +⟨xixj⟩ = Cij , +⟨xixjxkxl⟩ = Dijkl + CijCkl + CikCjl + CilCjk , +. . . +(69) +Note that in order for the new likelihood distribution (68) to be positive definite, it is required that +DijklC−1 +ij C−1 +kl +< 4, and if we impose that there is only one maximum at the center, then it must sat- +isfy DijklC−1 +ij C−1 +kl < 2. These conditions impose bounds on the maximum possible deviation of the evidence +from a that of a gaussian. +Let us now compute the evidence Eq. (22) for this kurtosis model. The extra terms in the parenthesis of +Eq. (68) are both even functions of x, and we cannot ignore them, even for centered priors. +For a single variable (n = 1), the correctly-normalized likelihood function can be written as +f(x) = e− x2 +2σ2 +σ +√ +2π +� +1 + D +8σ4 − D x2 +4σ6 + D x4 +24σ8 +� +, +satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = 0, ⟨x4⟩ = D + 3σ4, etc. The Bayesian integral can be computed exactly +as +E(a, b) = +1 +2a + 2b +� +Erf +� +a +σ +√ +2 +� ++ Erf +� +b +σ +√ +2 +�� ++ Dσ−4 +8 +√ +2π +� a +σ +� +1 − a2 +3σ2 +� +e− a2 +2σ2 + b +σ +� +1 − b2 +3σ2 +� +e− b2 +2σ2 +� +1 +a + b . +(70) +12 + +For arbitrary number of variables, the computation is again much more complicated. Let us start with +the n-th variable and, in order to compute the first integral, let us define a new auxiliary function +h(λ) += +� bn +−an +dxn +exp +� +− λ +2 xT C−1 +n x +� +(2π)n/2√det Cn += +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +× +× +1 +2 +√ +λ +� +Erf +� +an +√ +λ +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +λ +√ +2 +� +det Cn−1 +det Cn +�� +, +(71) +such that, +−2h′(λ = 1) += +� bn +−an +dxn +(xT C−1 +n x) exp +� +− 1 +2xT C−1 +n x +� +(2π)n/2√det Cn += +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +× +× +� +1 +2 +� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +�� +(72) +− +1 +√ +2π +� +det Cn−1 +det Cn +� +an exp +� +− a2 +n +2 +det Cn−1 +det Cn +� ++ bn exp +� +− b2 +n +2 +det Cn−1 +det Cn +��� +. +4h′′(λ = 1) += +� bn +−an +dxn +(xT C−1 +n x)2 exp +� +− 1 +2xT C−1 +n x +� +(2π)n√det Cn += +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +× +× +� +3 +2 +� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +�� +(73) +− +3 +√ +2π +� +det Cn−1 +det Cn +� +an exp +� +− a2 +n +2 +det Cn−1 +det Cn +� ++ bn exp +� +− b2 +n +2 +det Cn−1 +det Cn +�� +− +a2 +n +√ +2π +�det Cn−1 +det Cn +�3/2 � +an exp +� +− a2 +n +2 +det Cn−1 +det Cn +� ++ bn exp +� +− b2 +n +2 +det Cn−1 +det Cn +��� +. +Therefore, with the use of Eqs. (72) and (73), the integral of the kurtosis-corrected distribution function (68) +over the xn prior, becomes +� bn +−an +dxn f(x) = +exp +� +− 1 +2xT C−1 +n−1x +� +(2π)(n−1)/2� +det Cn−1 +� +1 +2 +� +Erf +� +an +√ +2 +� +det Cn−1 +det Cn +� ++ Erf +� +bn +√ +2 +� +det Cn−1 +det Cn +�� ++ +(74) ++ 1 +8Dijkl C−1 +ij C−1 +kl +1 +√ +2π +� +det Cn−1 +det Cn +� +an +� +1 − a2 +n +3 +det Cn−1 +det Cn +� +e− +a2 +n det Cn−1 +2 det Cn ++ bn +� +1 − b2 +n +3 +det Cn−1 +det Cn +� +e− +b2 +n det Cn−1 +2 det Cn +�� +. +We can now define a new function +Gi(ai, bi) = +1 +8 +√ +2π +� +det Ci−1 +det Ci +� +ai +� +1 − a2 +i +3 +det Ci−1 +det Ci +� +e− +a2 +i det Ci−1 +2 det Ci +− bi +� +1 − b2 +i +3 +det Ci−1 +det Ci +� +e− +b2 +i det Ci−1 +2 det Ci +� +. +(75) +Integrating iteratively over xn−1, . . . , x1, we end up with the Bayesian evidence for the fourth-order-corrected +probability distribution function f(x), +E(a, b) = +n +� +p=1 +Ep(ap, bp) +(ap + bp) +� +1 + Dijkl C−1 +ij C−1 +kl +n +� +m=1 +Gm(am, bm) +Em(am, bm) +� +. +(76) +13 + +so, unless Dijkl C−1 +ij C−1 +kl +is very large, the correction to the error function is exponentially suppressed, and +we do not expect significant departures from the Gaussian case, Eq. (40). +In order to compare models it is customary to compute the logarithm of the evidence. Let us assume that +we are given a likelihood distribution function normalized by the maximum likelihood, and with corrections +up to fourth order, +f(x) = Lmax exp +� +− 1 +2xT C−1 +n x +� � +1 + 1 +8Dijkl C−1 +ij C−1 +kl +�−1� +1 − 1 +2Bijk C−1 +ij C−1 +kl xl + 1 +6Bijk C−1 +il C−1 +jmC−1 +kn xlxmxn ++ 1 +8Dijkl C−1 +ij C−1 +kl − 1 +4Dijkl C−1 +ij C−1 +kmC−1 +ln xmxn + 1 +24Dijkl C−1 +im C−1 +jn C−1 +kp C−1 +lq xmxnxpxq +� +. +(77) +Note that it is normalized so that the maximum corresponds to the mean-centered distribution, i.e. x = 0. +In this case, the evidence of the normalized distribution is given by +E(a, b) = Lmax (2π)n/2� +det Cn +� +1 + 1 +8Dijkl C−1 +ij C−1 +kl +�−1 +× +(78) +n +� +p=1 +Ep(ap, bp) +(ap + bp) +� +1 − +n +� +k=1 +Bijk C−1 +ij +Fk(ak, bk) +Ek(ak, bk) + Dijkl C−1 +ij C−1 +kl +n +� +m=1 +Gm(am, bm) +Em(am, bm) +� +. +We can then evaluate the logarithm of the evidence by +ln E += +ln Lmax + n +2 ln(2π) + 1 +2 ln det Cn − ln +� +1 + 1 +8Dijkl C−1 +ij C−1 +kl +� +− +n +� +p=1 +ln(2ap + 2bp) ++ +n +� +p=1 +ln +� +Erf +� +ap +√ +2 +� +det Cp−1 +det Cp +� ++ Erf +� +bp +√ +2 +� +det Cp−1 +det Cp +�� +(79) ++ ln +� +1 − +n +� +k=1 +Bijk C−1 +ij +Fk(ak, bk) +Ek(ak, bk) + Dijkl C−1 +ij C−1 +kl +n +� +m=1 +Gm(am, bm) +Em(am, bm) +� +. +Note that the condition DijklC−1 +ij C−1 +kl +< 2 constrains the maximum amount that the kurtosis corrections +can contribute to the evidence. +Uncorrelated case. In the case where the likelihood distribution had no correlations among the different +variables, the exact expression for the Bayesian evidence is +ln E = ln Lmax + n +2 ln(2π) + +n +� +p=1 +ln σp − +n +� +p=1 +ln(2ap + 2bp) + +n +� +p=1 +ln +� +Erf +� +ap +σp +√ +2 +� ++ Erf +� +bp +σp +√ +2 +�� +(80) +− ln +� +1 + 1 +8Diijj σ−2 +i +σ−2 +j +� ++ ln +� +1 − +n +� +k=1 +Biik σ−2 +k +Fk(ak, bk) +Ek(ak, bk) + Diijj σ−2 +i +σ−2 +j +n +� +m=1 +Gm(am, bm) +Em(am, bm) +� +, +where σp are the corresponding dispersions of variables xp, and the functions Ei, Fi and Gi are the corre- +sponding limiting functions of Eqs. (65) and (75) for uncorrelated matrices. +5 +Model comparison +Finally we turn to specific applications of the formalism discussed above. Initially we will carry out some +toy model tests of its performance, and then examine real cosmological applications for which we previously +obtained results by thermodynamic integration [5]. +14 + +Figure 1: This figure shows the calculated evidence as a function of the number of likelihood evaluations. +Note that the horizontal axis is logarithmic. The solid line corresponds to the thermodynamic integration. +The dotted line and dot-dashed lines are the analytical methods with and without non-Gaussian corrections +applied. The horizontal dashed line is the number obtained by the direct integration. The upper two panels +correspond to Lg, while the lower two to Lng. The left-hand side panels correspond to wide flat priors of +(−7, 10) on both parameters, while the right-hand side to the narrow priors of (−2, 3) on both parameters. +See text for discussion. +5.1 +A baby-toy model comparison +We begin with a very simple two-dimensional toy model. The purpose of this section is to illustrate the +ineffectiveness of the thermodynamic integration and to give an indication of the performance of the method +we propose here. In addition, the two-dimensional model is simple enough to allow a brute-force direct +numerical integration of evidence allowing us to check the accuracy at the same time. We use the following +two forms of likelihood: +Lg(x, y) += +exp +� +−2x2 − 2(y − 1)2 − xy +2 +� +(81) +Lng(x, y) += +exp +� +−2x2 − 2(y − 1)2 − xy +2 +� ++ exp +� +−2x2 − 2y2 − 3xy +2 +� +(82) +The subscripts g and ng indicate the Gaussian and non-Gaussian cases respectively. +Firstly, we calculate the evidence by the analytical method using Eqs. (56) and (80) and covariance +15 + +matrices inferred from sampling the likelihood using the vanilla Metropolis–Hastings algorithm with fixed +proposal widths. Chains ranging from few to several million samples were used. We also calculate evidence +using thermodynamic algorithm explained in Ref. [5]. Again, we vary algorithm parameters to get evidence +values of varying accuracy. The resulting evidence as a function of number of likelihood evaluations is plotted +in the Figure 1, together with the correct value inferred by direct numerical integration. The number of +likelihood evaluations is crucial as this is the time-limiting step in the cosmological parameter estimation +and model comparison exercises. The results are what could have been anticipated. We note that the size +of the prior does not seem to be of crucial importance. This is comforting, given that the analytical method +requires the knowledge of the true covariance information, while we can only supply a covariance matrix +estimated from the prior-truncated likelihood. We also note that the thermodynamic integration converges +to the correct value in all cases. However, it does so after very many likelihood evaluations; typically about +a million or so even for a two-dimensional problem. The analytical method becomes limited by systematics +already by the ten-thousand samples. +For Gaussian case, there is no systematic by construction, while +the non-gaussian case suffers a systematic of about 0.1 in ln E. The non-Gaussian correction reduces the +error by about a half and thus correctly estimates the uncertainty associated with the purely Gaussian +approximation. In the case of wide priors, the only non-Gaussian correction of an appreciable size is the +ln(1 + DijklC−1 +ij C−1 +kl /8). +5.2 +A toy model comparison +We now proceed by calculating the Bayesian evidence for simple toy models with 5 and 6 parameters, shown +in Table I. The purpose is to compare results with those obtained from thermodynamic integration again, +but this time using a model that bears more resemblance to a typical problem one encounters in cosmology. +Parameter +Mean +Prior Range +Model +x1 +0.022 +[0.0001, 0.044] +toy5,toy6 +x2 +0.12 +[0.001, 0.3] +toy5,toy6 +x3 +1.04 +[0.8, 1.4] +toy5,toy6 +x4 +0.1 +[0.01, 0.3] +toy5,toy6 +x5 +3.1 +[2.6, 3.6] +toy5,toy6 +x6 +0.98 +[0.5, 1.5] +toy6 +Table 1: +The parameters used in the analytical evaluation of the toy model evidences, with 5 and 6 +parameters respectively. The maximum likelihod of the toy models is taken (arbitrarily) to be Lmax = 1. +Beginning with the five-parameter model, we assume first that it has an uncorrelated multivariate Gaus- +sian likelihood distribution. In this case the aim is to test the thermodynamic integration method, which +gives ln Enum +toy5 = −8.65 ± 0.03, while the exact expression gives ln Eana +toy5 = −8.66. Therefore, we conclude +that the thermodynamic integration method is rather good in obtaining the correct evidence of the model. +The Laplace approximation Eq. (57) also fares well for uncorrelated distributions, ln ELap +toy5 = −8.67. +We now consider a likelihood function with a correlated covariance matrix Cij, with the same mean +values and dispersions as the previous case, but with significant correlations. The analytic formula needed, +Eq. (54), is no longer exact,2 and gives ln Eana +toy5c = −7.32. For comparison thermodynamic integration gives +ln Enum +toy5c = −7.28 ± 0.06, again in perfect agreement within errors. In this case the Laplace approximation +fails significantly, ln ELap +toy5c = −6.89, the reason being that the correlations chosen bring the posterior into +significant contact with the edges of the priors. +Let us now return to the uncorrelated case and include a new parameter, x6, as in Table I, and evaluate the +different evidences that appear because of this new parameter, in order to see the sensitivity to systematic +errors in the evaluation of the Bayesian evidence and their effects on model comparison. The numerical +2One could rotate the parameter basis to remove the correlations, but then the priors wouldn’t be top-hats. +16 + +result is ln Enum +toy6 = −10.75 ± 0.03, while the exact analytical expression gives ln Eana +toy6 = −10.74, in perfect +agreement, within errors. The Laplace approximation Eq. (57) again fares well for uncorrelated distributions, +ln ELap +toy6 = −10.74. +When the likelihood function has large correlations, and the priors are not too large, the naive Laplace +approximation, Eq. (57), fares less well than the analytical approximation, Eq. (54). +5.3 +A real model comparison +In this subsection we will make use of the results obtained in Ref. [5], where we evaluated the evidence for +5- and 6-parameter adiabatic models, and for three 10-parameter mixed adiabatic plus isocurvature models. +The prior ranges used are given in Table II. The latter models give a marginally better fit to the data but +require more parameters, which is exactly the situation where model selection techniques are needed to draw +robust conclusions. In Ref. [5] we used thermodynamic integration to compute the evidence and showed that +the isocurvature models ware less favoured than the adiabatic ones, but only at a mild significance level.3 +Beginning with the simplest adiabtic model, which uses the Harrison–Zel’dovich spectrum, we have +used the analytical formulae above, Eq. (54), together with the covariance matrix provided by the cosmoMC +programme [10], and obtained ln Eana +ad += −854.07, while the thermodynamical integration gave ln Enum +ad += +−854.1±0.1 [5]. The agreement is excellent; this is because the distribution function for the adiabatic model +is rather well approximated by a Gaussian, and the priors are rather large, so the formula Eq. (54) is very +close to that obtained in the Laplace approximation, ln ELap +ad += −854.08. +Parameter +Mean +Prior Range +Model +ωb +0.022 +[0.018, 0.032] +AD-HZ,AD-ns,ISO +ωdm +0.12 +[0.04, 0.16] +AD-HZ,AD-ns,ISO +θ +1.04 +[0.98, 1.10] +AD-HZ,AD-ns,ISO +τ +0.17 +[0, 0.5] +AD-HZ,AD-ns,ISO +ln[1010Rrad] +3.1 +[2.6, 4.2] +AD-HZ,AD-ns,ISO +ns +1.0 +[0.8, 1.2] +AD-ns,ISO +niso +1.5 +[0, 3] +ISO +δcor +1.5 +[−0.14, 0.4] +ISO +√α +0 +[−1, 1] +ISO +β +0 +[−1, 1] +ISO +Table 2: +The parameters used in the models; see Ref. [5] for nomenclature and other details. For the +AD-HZ model ns was fixed to 1 and niso, δcor, α and β were fixed to 0. In the AD-ns model, ns also varies. +Every isocurvature model holds the same priors for the whole set of parameters. +However the analytic method fares less well for the adiabatic model with varying ns, with both the +analytic and Laplace methods giving ln EAD−ns = −853.4, while the numerical method gives the smaller +value -854.1, a discrepency of nearly unity. +Turning now to the iscurvature cases, we found an extremely good result for the CDI model, gaining from +Eq. (54) the value ln Eana +cdi = −855.08, while the thermodynamical integration gives ln Enum +cdi += −855.1 ± 0.1. +This is surprising, given the relatively large non-gaussianities for at least three variables: niso, β and δcor, +whose priors are not centered with respect to the mean. +However the NID case shows much less good +agreement, with a discrepency of 0.6. That suggests that the closeness of the CDI comparison is to some +extent a statistical fluke, with the underlying method less accurate. +A summary of the different models can be found in Table 3. +3Recently Trotta [9] used a different technique to analyze a restricted class of isocurvature model featuring just one extra +parameter, and found it highly disfavoured. The different conclusion is primarily due to the very different prior he chose on +the isocurvature amplitude, such that almost all the models under the prior are domintaed by isocurvature modes and in poor +agreement with the data. +17 + +Model +ln Lmax +ln Enum +ln Eana +ln ELap +toy5 +0 +−8.65 ± 0.03 +−8.66 +−8.67 +toy5c +0 +−7.28 ± 0.06 +−7.32 +−6.89 +toy6 +0 +−10.75 ± 0.03 +−10.74 +−10.74 +toy6c +0 +−9.73 ± 0.06 +−9.71 +−9.63 +AD +−840.78 +−854.1 ± 0.1 +−854.1 +−854.1 +AD-ns +−838.50 +−854.1 ± 0.1 +−853.4 +−853.4 +CDI +−838.05 +−855.1 ± 0.2 +−855.1 +−854.5 +NID +−836.60 +−855.1 ± 0.2 +−854.5 +−854.5 +NIV +−842.53 +−855.1 ± 0.3 +−854.9 +−854.9 +Table 3: +The different models, both toy and real, with their maximum likelihoods and evidences. +5.4 +Savage–Dickey method +Another numerical method for evidence calculation is the Savage–Dickey method, first described in Ref. [11] +and recently used in Ref. [9]. This technique allows one to calculate the evidence ratio of two models from a +simple and quick analysis of the Markov chains used for parameter estimation, provided that the models are +nested; i.e., that one of them is included in the parameter space of the other. For instance, the AD model +is nested within the AD-ns model, and the AD and AD-ns models are both nested within the CDI, NID +and NIV ones. In the context of Markov chains, the Savage–Dickey method is essentially a measure of how +much time the sampler spends in the nested model, weighted by the respective volumes of the two models. +When the outer model has extra parameters, this method relies on approximating the nested model as a +model with negligibly narrow priors in directions of extra parameters. We note, however, that when many +extra parameters are present, this method must fail for reasons similar to those why grid-based parameter +estimation approaches fail with models with many parameters. The MCMC parameter estimation simply +does not have high enough dynamic range to probe the two models given the large prior volume ratio. +The AD and AD-ns models differ by one parameter. Using the same AD+ns samples as for the analytic +method (i.e., the samples from which we extracted the covariance matrix), we obtained ln(EAD/EAD+ns) = +0.03. The result from the precise thermodynamical integration, ln(EAD/EAD−ns) = 0 ± 0.1 is in excellent +agreement. The AD-ns and CDI (or NID, NIV) models differ by four parameters. With most simple choices +of parametrization (including in particular the isocurvature and cross-correlation tilts), the AD-ns is not a +point, but a hypersurface within the parameter space of the isocurvature models (i.e. α = 0 and other three +parameters act as dummy, unconstrained, parameters which do not affect the evidence). In these cases, the +evidence ratios given by the Savage–Dickey method do not converge as the priors of the extra parameters +are tightened up around the nested model, although they match thermodynamically-determined values to +within a unit of ln E. +6 +Discussion and Conclusions +We have developed an analytical formalism for computing the Bayesian evidence in the case of an arbitrary +likelihood distribution with a hierarchy of non-Gaussian corrections, and with arbitrary top-hat priors, +centered or uncentered. This analysis can be of great help for the problem of model comparison in the +present context of cosmology, where observational data is still unable to rule out most extensions of the +standard model based on the ΛCDM inflationary paradigm. +As an application of the exact and approximate formulae obtained for the Bayesian evidence of a model +with approximately Gaussian likelihood distributions, we have compared the value predicted analytically +with that computed with a time-consuming algorithm based on the thermodynamical integration approach. +The values obtained analytically agree surprisingly well with those obtained numerically. While one can +estimate the magnitude of the higher order corrections for the analytical formulae, it is very difficult to +18 + +estimate the systematic effects of the numerical approach. Thus, with this analytical method we can test +for systematics in the thermodynamical integration approach. So far, the values obtained agree, so it seems +that the numerical approach is a good tool for estimating the evidence. However, it takes considerable effort +and machine time to do the correct evaluation, and therefore, we propose the use of the analytical estimate, +whose corrections are well under control, in the sense that one can compute the next order corrections and +show that they are small. +Note added: Many years after my work was finished, a book appeared [12] which thoroughly discussed +Bayesian Methods in Cosmology. +References +[1] H. Jeffreys, Theory of Probability, 3rd ed, Oxford University Press (1961). +[2] D. J. C. MacKay, Information theory, inference and learning algorithms, Cambridge University Press +(2003). +[3] A. Jaffe, Astrophys. J. 471, 24 (1996); P. S. Drell, T. J. Loredo, and I. Wasserman I, Astrophys. J. 530, +593 (2000); M. V. John and J. V. Narlikar, Phys. Rev. D 65, 043506 (2002); M. P. Hobson, S. L. Bridle, +and O. Lahav, Mon. Not. Roy. Astr. Soc. 335, 377 (2002); A. Slosar et al., Mon. Not. Roy. Astr. Soc. +341, L29 (2003); T. D. Saini, J. Weller, and S. L. Bridle, Mon. Not. Roy. Astr. Soc. 348, 603 (2004); +A. Niarchou, A. H. Jaffe, and L. Pogosian, Phys. Rev. D 69, 063515 (2004); P. Marshall, N. Rajguru, +and A. Slosar, Phys. Rev. D 73, 067302 (2006). +[4] A. R. Liddle, Mon. Not. Roy. Astr. Soc. 351, L49 (2004). +[5] M. Beltran, J. Garc´ıa-Bellido, J. Lesgourgues, A. R. Liddle and A. Slosar, Phys. Rev. D 71, 063532 +(2005). +[6] J. J. K. ´O’Ruanaidh and W. J. Fitzgerald, Numerical Bayesian Methods Applied to Signal Processing, +Springer–Verlag, New York (1996). +[7] M. P. Hobson and C. McLachlan, Mon. Not. Roy. Astr. Soc. 338, 765 (2003). +[8] R. E. Kass and A. E. Raftery, Journ. Amer. Stat. Assoc. 90, 773 (1995). +[9] R. Trotta, Mon. Not. Roy. Astr. Soc. 378, 72 (2007). +[10] A. Lewis and S. Bridle, Phys. Rev. D66, 103511 (2002). +[11] J. M. Dickey, Ann. Math. Stat 42, 204 (1971). +[12] M. P. Hobson, A. H. Jaffe, A. R. Liddle, P. Mukherjee & D. Parkinson, Bayesian Methods in Cosmology, +Cambridge University Press (2010). +19 + diff --git a/19FST4oBgHgl3EQfXDjl/content/tmp_files/load_file.txt b/19FST4oBgHgl3EQfXDjl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2953940194fa776650ed6abb67d05fc7ea7db5ad --- /dev/null +++ b/19FST4oBgHgl3EQfXDjl/content/tmp_files/load_file.txt @@ -0,0 +1,898 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf,len=897 +page_content='An analytical approach to Bayesian evidence computation Juan Garc´ıa-Bellido Departamento de F´ısica Te´orica C-XI, Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain April 14th, 2005 Abstract The Bayesian evidence is a key tool in model selection, allowing a comparison of models with differ- ent numbers of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Its use in analysis of cosmological models has been limited by difficulties in calculating it, with current numerical algorithms requiring supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this paper we give exact formulae for the Bayesian evidence in the case of Gaussian likelihoods with arbitrary correlations and top-hat priors, and approximate formulae for the case of likelihood distributions with leading non- Gaussianities (skewness and kurtosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We apply these formulae to cosmological models with and without isocurvature components, and compare with results we previously obtained using numerical thermody- namic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We find that the results are of lower precision than the thermodynamic integration, while still being good enough to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 1 Introduction Model selection refers to the statistical problem of deciding which model description of observational data is the best [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' It differs from parameter estimation, where the choice of a single model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' choice of parameters to be varied) has already been made and the aim is to find their best-fitting values and ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' While there have been widespread applications of parameter estimation techniques, usually likelihood fitting, to cosmological data, there has so far been quite limited application of model selection statistics [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This is unfortunate, as model selection techniques are necessary to robustly distinguish between models with different numbers of parameters, and many of the most interesting issues in cosmology concern the desirability or otherwise of incorporating additional parameters to describe new physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Within the context of Bayesian inference, model selection should be carried out using the Bayesian evidence [1, 2], which measures the probability of the model in light of the observational data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the average likelihood over the prior distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Bayesian evidence associates a single number with each model, and the models can then be ranked in order of the evidence, with the ratios of those values interpretted as the relative probability of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This process sets up a desirable tension between model simplicity and ability to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Use of the Bayesian evidence has so far been limited by difficulties in calculating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The standard technique is thermodynamic integration [6, 7], which varies the temperature in a Monte Carlo Markov Chain (MCMC) approach in order that the distribution is sampled in a way covering both posterior and prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' However, in recent work [5] we showed that in order to obtain sufficiently-accurate results in a cosmological context, around 107 likelihood evaluations are required per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Such analyses are CPU-limited by the time needed to generate the predicted spectra to compare with the data, and this requirement pushes the problem into the supercomputer class (for comparison, parameter estimation runs typically employ 105 to 106 likelihood evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this paper, we propose and exploit a new analytic method to compute the evidence based on an expansion of the likelihood distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The method pre-supposes that the covariance of the posterior distribution has been obtained, for instance via an MCMC parameter estimation run, and in its 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='13783v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='CO] 31 Jan 2023 present form requires that the prior distributions of the parameters are uniform top-hat priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 While the method will not be applicable for general likelihood distributions, we include the leading non-gaussianities (skewness and kurtosis) in approximating the likelihood shape, with the expectation of obtaining good results whenever the likelihood distribution is sufficiently simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cosmological examples commonly exhibit likelihood distributions with only a single significant peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We apply the method both to toy model examples and to genuine cosmological situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In particular, we calculate the evidences for adiabatic and isocurvature models, which we previously computed using thermodynamic integration in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We find that the discrepancies between the methods are typically no worse than 1 in ln(Evidence), meaning that the analytic method is somewhat less accurate than would be ideal, but is accurate enough to give a useful indication of model preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 2 The Bayesian evidence The posterior probability distribution P(θ, M|D) for the parameters θ of the model M, given the data D, is related to the likelihood function L(D|θ, M) within a given set of prior distribution functions π(θ, M) for the parameters of the model, by Bayes’ theorem: P(θ, M|D) = L(D|θ, M) π(θ, M) E(D|M) , (1) where E is the Bayesian evidence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the average likelihood over the priors, E(D|M) = � dθ L(D|θ, M) π(θ, M) , (2) where θ is a vector with n-components characterising the n independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The prior distribution function π contains all the information about the parameters before observing the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' our theoretical prejudices, our physical understanding of the model, and input from previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the case of a large number of parameters (n ≫ 1), the evidence integral cannot be performed straight- forwardly and must be obtained either numerically or via an analytic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Amongst numerical methods the most popular is thermodynamic integration [6, 7] but this can be computationally extremely intensive [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The simplest analytical approximation is the Laplace approximation, valid when the distribu- tion can be approximated by a multivariate Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This may hold when the quantity and quality of the data is optimal, but is likely to be valid only in limited cosmological circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Bayesian evidence is of interest because it allows a comparison of models amongst an exclusive and exhaustive set {Mi}i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We can compute the posterior probability for each hypothesis given the data D using Bayes theorem: P(Mi|D) ∝ E(D|Mi) π(Mi) , (3) where E(D|Mi) is the evidence of the data under the model Mi, and π(Mi) is the prior probability of the ith model before we see the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The ratio of the evidences for the two competing models is called the Bayes factor [8] Bij = E(D|Mi) E(D|Mj) , (4) and this is also equal to the ratio of the posterior model probabilities if we assume that we do not favour any model a priori, so that π(M1) = π(M2) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' = π(MN) = 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Bayes factor Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (4) provides a mathematical representation of Occam’s razor, because more complex models tend to be less predictive, lowering their average likelihood in comparison to simpler, more predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' More complex models can only be favoured if they are able to provide a significantly improved fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In simple cases where models give vastly different maximum likelihoods there is no need to employ model selection techniques, but they are essential for properly discussing cases where the improvement 1An extension to gaussian priors should be feasible, but not one to arbitrary priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 2 of fit is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This latter situation is more or less inevitable whenever the possibility of requiring an additional parameter arises from new data, unless the new data is of vastly greater power than that preceding it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' cosmological examples include the inclusion of spectral tilt, dark energy density variation, or the case explored later in this paper of trace isocurvature perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this paper we will obtain an analytical formula which approximates the Bayesian evidence by consid- ering the higher-order cumulants of the distribution in a systematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The advantage is that with these analytical formulae one can compute the evidence for a given model with an arbitrary number of parame- ters, given the hierarchy of cumulants of the distribution, assumed previously computed for the likelihood distribution function within the parameter estimation programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The evidence needs to be calculated to sufficient precision for robust conclusions to be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The standard interpretational scale, due to Jeffreys [1] and summarized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5], strengthens its verdict roughly each time the difference in ln(Evidence) increases by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The evidence therefore needs to be computed more accurately than this, with an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 in ln(Evidence) easily sufficient, and a factor two worse than that acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This accuracy requirement ensures that the relative model probabilities are little changed by the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The first thing we need is to characterize the distribution function for the model with n parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let f(x) be this function, and let us assume that it is properly normalized, � ∞ −∞ dnx f(x) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (5) Then, the p-point correlation function is given by ⟨xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' xip⟩ = � ∞ −∞ dnx xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' xip f(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (6) From this distribution function one can always construct the generating functional, φ(u), as the Fourier transform φ(u) = � ∞ −∞ dnx ei u·x f(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (7) This function can be expanded as φ(u) = exp � ∞ � p=1 ip p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Ai1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='ip ui1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' uip � , (8) where Ai1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='ip are totally symmetric rank-p tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For instance, if we restrict ourselves to order 4, we can write φ(u) = exp � i µiui − 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cij uiuj − i 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Bijk uiujuk + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Dijkl uiujukul + · · · + in n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Ai1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='in ui1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' uin � , (9) where µi is the mean value of variable xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cij is the covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Bijk is the trilinear matrix associated with the third cumulant or skewness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Dijkl is the rank-4 tensor associated with the fourth cumulant or kurtosis, and Ai1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='in is the rank-n tensor associated with the n-th cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Their expressions in terms of n-point correlation functions can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (7), by realising that ⟨xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' xin⟩ = (−i)n ∂nφ(u) ∂ui1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' ∂uin ���� u=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (10) For instance, the first-order term gives ⟨xi⟩ = (−i) ∂φ(u) ∂ui ���� u=0 = µi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (11) 3 The second-order correlation function gives ⟨xixj⟩ = (−i)2 ∂2φ(u) ∂ui∂uj ���� u=0 = Cij + µiµj , (12) such that the covariance matrix is obtained, as usual, from Cij = ⟨xixj⟩ − ⟨xi⟩⟨xj⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The third-order correlation function gives ⟨xixjxk⟩ = (−i)3 ∂3φ(u) ∂ui∂uj∂uk ���� u=0 = Bijk + µiCjk + µjCki + µkCij + µiµjµk , (13) such that the skewness matrix is obtained from Bijk = ⟨xixjxk⟩ − ⟨xi⟩⟨xjxk⟩ − ⟨xj⟩⟨xkxi⟩ − ⟨xk⟩⟨xixj⟩ + 2⟨xi⟩⟨xj⟩⟨xk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (14) The fourth-order correlation function gives ⟨xixjxkxl⟩ = (−i)4 ∂4φ(u) ∂ui∂uj∂uk∂ul ���� u=0 = Dijkl + CijCkl + CikCjl + CilCjk (15) + Bijkµl + Bijlµk + Bjklµi + Biklµj + Cijµkµl + Cikµjµl + Cilµjµk + Cjkµiµl + Cjlµiµk + Cklµiµj + µiµjµkµl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' such that the kurtosis matrix is obtained from Dijkl = ⟨xixjxkxl⟩ − ⟨xixj⟩⟨xkxl⟩ − ⟨xixk⟩⟨xjxl⟩ − ⟨xixl⟩⟨xjxk⟩ (16) − ⟨xixjxk⟩⟨xl⟩ − ⟨xixjxl⟩⟨xk⟩ − ⟨xixkxl⟩⟨xj⟩ − ⟨xjxkxl⟩⟨xi⟩ + 2 ⟨xixj⟩⟨xk⟩⟨xl⟩ + 2 ⟨xixk⟩⟨xj⟩⟨xl⟩ + 2 ⟨xixl⟩⟨xj⟩⟨xk⟩ + 2 ⟨xjxk⟩⟨xi⟩⟨xl⟩ + 2 ⟨xjxl⟩⟨xi⟩⟨xk⟩ + 2 ⟨xkxl⟩⟨xi⟩⟨xj⟩ − 6 ⟨xi⟩⟨xj⟩⟨xk⟩⟨xl⟩ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' and so on,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' for the higher order cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 3 The Gaussian approximation Let us first evaluate the evidence for a multivariate Gaussian distribution, that is, one in which all the cumulants are zero except the covariance matrix Cij and the means µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this case, the generating functional and the distribution are given by φ(u) = exp � − iµiui − 1 2 Cij uiuj � , (17) f(x) = 1 (2π)n � ∞ −∞ dnu e−i u·x φ(u) (18) = 1 (2π)n/2√ det C exp � − 1 2C−1 ij (xi − µi)(xj − µj) � , (19) 4 which satisfies ⟨xi⟩ = µi , ⟨xixj⟩ = Cij + µiµj , ⟨xixjxk⟩ = µ(iCjk) + µiµjµk , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (20) where the subindices in parenthesis, (ijk), indicate a cyclic sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Notice that all the n-point correlation functions can be written in terms of the first two moments of the distribution, and all the higher-order cumulants vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 Centred priors For initial calculations, we assume a top-hat prior and make the unrealistic assumption, to be lifted later, that it is centered at the mean value: π(x, a) ≡ � (2a)−1 −a < x − µ < a , 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (21) Since the Fourier transform of a top-hat function is � ∞ −∞ dx eiux π(x, a) = sin au au exp[iµu] , we can write the evidence either way E(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , an) = � ∞ −∞ dnx f(x) n � i=1 π(xi, ai) = n � i=1 (2ai)−1 � a1 −a1 dx1· · · � an −an dxn f(˜x) (22) = 1 (2π)n � ∞ −∞ dnu φ(u) n � i=1 sin aiui aiui .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (23) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) we integrate over the displaced coordinate, ˜xi ≡ xi − µi, such that ⟨˜xi⟩ = 0 and ⟨˜xi˜xj⟩ = Cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' From now on, we ignore the tildes, and assume we have moved to those coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that the choice of prior is not crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We could have chosen a Gaussian prior, and the result would not be very different, except that the window functions, sin z/z, would then be Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us now perform the integration Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) in the case of 1, 2 and then n variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 1 variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Suppose the covariance is just C = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The evidence is then E(a) = 1 2a σ √ 2π � a −a dx e− x2 2σ2 = 1 2π � ∞ −∞ du sin au au e− 1 2 σ2u2 = 1 2aErf � a σ √ 2 � , (24) where Erf[x] is the error function, which asymptotes very quickly to one for x ≥ 2, or a ≥ 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Therefore, the evidence of a model with centred top-hat prior of width 2a is well approximated by (2a)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The wider is the theoretical prior, the smaller is the evidence, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Suppose we have two correlated variables, x1 and x2, with covariance matrix C = � C11 C12 C12 C22 � = � σ2 1 ρσ1σ2 ρσ1σ2 σ2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (25) where the cross-correlation ρ is defined by ρ = ⟨x1x2⟩ � ⟨x2 1⟩⟨x2 2⟩ = ⟨x1x2⟩ σ1σ2 , 5 with σ1 and σ2 the corresponding quadratic dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this case, the normalized 2-dimensional distri- bution function is f(x) = 1 2πσ1σ2 � 1 − ρ2 exp � −1 1 − ρ2 � x2 1 2σ2 1 − ρx1x2 σ1σ2 + x2 2 2σ2 2 �� , (26) which has the property that integrating (“marginalizing”) over one of the two variables, leaves a properly- normalized Gaussian distribution for the remaining variable, � ∞ −∞ dx2 f(x) = 1 σ1 √ 2π e − x2 1 2σ2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (27) Let us now evaluate the evidence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) by integrating first over the prior in x2, 1 2a2 � a2 −a2 dx2 f(x) = e − x2 1 2σ2 1 σ1 √ 2π · 1 4a2 � Erf � a2σ1 + ρσ2 x1 σ1σ2 � 2(1 − ρ2) � + Erf � a2σ1 − ρσ2 x1 σ1σ2 � 2(1 − ρ2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (28) The first term is the result we would have obtained if we had been marginalizing over x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the second is a sum of error functions that still depend on x1, and modulates the marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We can use the series expansion of the error function to second order, 1 2 � Erf[a + x] + Erf[a − x] � = Erf[a] − 2a x2 √π e−a2 + O(x4) , to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (28) to order x2 1 as 1 2a2 � a2 −a2 dx2 f(x) = e − x2 1 2σ2 1 σ1 √ 2π � �� 1 2a2 Erf � a2 σ2 � 2(1 − ρ2) � − ρ2 x2 1 e − a2 2 2σ2 2(1−ρ2) 2σ2 1σ2(1 − ρ2) � 2π(1 − ρ2) � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (29) Integrating now over the x1 prior, we finally obtain the evidence E(a1, a2) = 1 4a1a2 � a1 −a1 dx1 � a2 −a2 dx2 f(x) = 1 4a1a2 Erf � a2 σ2 � 2(1 − ρ2) � Erf � a1 σ1 √ 2 � (30) − ρ2 e − a2 2 2σ2 2(1−ρ2) 2σ1σ2(1 − ρ2) � 2π(1 − ρ2) Erf � a1 σ1 √ 2 � 2a1 + ρ2 e − a2 2 2σ2 2(1−ρ2) − a2 1 2σ2 1 4πσ2 1σ2 � 1 − ρ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that in the limit of no cross-correlations, ρ → 0, the integral factorizes and we can write an exact expression for the evidence, E(a1, a2) = 1 4a1a2 1 2πσ1σ2 � a1 −a1 dx1 � a2 −a2 dx2 e − x2 1 2σ2 1 − x2 2 2σ2 2 (31) = 1 4π2 � ∞ −∞ du1 � ∞ −∞ du2 sin a1u1 a1u1 sin a2u2 a2u2 e− 1 2 σ2 1u2 1− 1 2 σ2 2u2 2 (32) = 1 4a1a2 Erf � a1 σ1 √ 2 � Erf � a2 σ2 √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (33) 6 It happens, however, that even in the presence of cross-correlations, if the prior is wide (ai ≥ 2σi), then the terms proportional to exponentials are negligible and the evidence becomes, to very good approximation, E(a1, a2) = 1 4a1a2 Erf � a2 σ2 � 2(1 − ρ2) � Erf � a1 σ1 √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (34) Moreover, in that case, the error functions are very approximately given by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' n variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Suppose we have n correlated variables, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , xn), with covariance matrix Cn = � � � � � � � � C11 C12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' C1n C12 C22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' C2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' C1n C2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cnn � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (35) In that case, the probability distribution function can be expressed as f(x) = 1 (2π)n/2√det Cn exp � − 1 2xT C−1 n x � , (36) which has the property that marginalizing over the last variable, xn, we obtain a correlated probability distribution function for the n − 1 variables, x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , xn−1), f(x) = 1 (2π)(n−1)/2� det Cn−1 exp � − 1 2xT C−1 n−1x � , (37) where the Cn−1 covariance matrix is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (35) without the last column and the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We will now evaluate the evidence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) for this multivariate Gaussian, starting with the integration over the last variable, xn, 1 2an � an −an dxn f(x) = 1 (2π)(n−1)/2� det Cn−1 exp � − 1 2xT C−1 n−1x � × � 1 2an Erf � an √ 2 � det Cn−1 det Cn � + O � e− a2 n det Cn−1 2 det Cn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (38) Integrating now over the next variable, xn−1, we find 1 4anan−1 � an −an dxn � an−1 −an−1 dxn−1 f(x) = 1 (2π)(n−2)/2� det Cn−2 exp � − 1 2 xT C−1 n−2x � × � 1 4anan−1 Erf � an √ 2 � det Cn−1 det Cn � Erf � an √ 2 � det Cn−2 det Cn−1 � + O � e− a2 n det Cn−1 2 det Cn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (39) Continuing the integration over the priors, we end up with the evidence for the n-dimensional distribution, E(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , an) = 1 �n p=1 2ap � a1 −a1 · · � an −an dnx f(x) = n � p=1 1 2ap Erf � ap √ 2 � det Cp−1 det Cp � + O � exp � − n � p=1 a2 p det Cp−1 2 det Cp �� , (40) 7 where the covariance matrices Cp are constructed as above, by eliminating the n−p last rows and columns, un- til we end up with C0 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that the approximation is very good whenever �n p=1(a2 p det Cp−1)/(2 det Cp) ≫ 1, which is often the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note also that we recover the previous result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (34) for the particular case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the limit that the cross-correlation between the n variables vanishes, the evidence (40) reduces to the exact result E(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , an) = n � p=1 1 2ap Erf � ap σp √ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (41) Note that the evidence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (40) reflects correctly the limit in which we eliminate the need for a new variable xn, by making its prior vanish, lim an→0 E(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , an) = E(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , an−1) 1 √ 2π � det Cn−1 det Cn , (42) and thus we recover in that limit a properly-normalized distribution, f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , xn) → f(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , xn−1), while the inspection of the likelihood function alone would not have been able to give a reasonable answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' On the other hand, in the case that our theoretical prejudice cannot assign a concrete prior to a given variable, we see that the evidence decreases as 1/2a as a increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Therefore, the Bayesian evidence seems to be a very good discriminator between theoretical priors, and penalizes including too many parameters, a la Occam’s razor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 Uncentered priors It is unlikely that the priors will actually be centred on the mean of the distribution, as the priors are not supposed to know what the data will tell us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We therefore need to generalize the above for uncentred priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We continue to assume that the priors are top hats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We also continue to assume for the moment that the probability distribution is well approximated by a Gaussian with mean value µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We will then use displaced variables ˜xi = xi − µi, and write the Gaussian distribution function as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The normalized top-hat prior is now uncentered with respect to the mean value, π(˜x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' a, b) ≡ � (a + b)−1 −a < ˜x < b , 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (43) For a single variable, the result is exact, E(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' b) = � ∞ −∞ dx f(x) π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' a, b) = 1 2a + 2b � Erf � a σ √ 2 � + Erf � b σ √ 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (44) where we are integrating over the displaced variable ˜x, from now on renamed as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that we recover the result Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (24) for the centered prior case in the limit b → a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For two variables, with distribution function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (26), the uncentered Bayesian evidence is E(a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' b1, b2) = 1 (a1 + b1)(a2 + b2) � b1 −a1 dx1 � b2 −a2 dx2 f(x1, x2) (45) = 1 (2a1 + 2b1)(2a2 + 2b2) �� Erf � a1 σ1 √ 2 � + Erf � b1 σ1 √ 2 �� (46) × � Erf � a2 σ2 � 2(1 − ρ2) � + Erf � b2 σ2 � 2(1 − ρ2) �� − ρ 2π � 1 − ρ2 � e − a2 1 2σ2 1 − e − b2 1 2σ2 1 � � e − a2 2 2σ2 2(1−ρ2) + e − b2 2 2σ2 2(1−ρ2) �� 8 The evidence for the multiple-variable case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (36) is E(a, b) = � ∞ −∞ dnx f(x) n � i=1 π(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' ai, bi) = n � i=1 (ai + bi)−1 � b1 −a1 d˜x1· · · � bn −an d˜xn f(˜x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (47) Let us now evaluate it for the multivariate Gaussian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (36),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' starting with the integration over the last variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 1 an + bn � bn −an dxn f(x) = 1 (2π)(n−1)/2� det Cn−1 exp � − 1 2xT C−1 n−1x � 1 (2an + 2bn) × � Erf � an √ 2 � det Cn−1 det Cn � + Erf � bn √ 2 � det Cn−1 det Cn � + O � e− a2 n det Cn−1 2 det Cn + e− b2 n det Cn−1 2 det Cn �� (48) Integrating now over the next variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' xn−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' we find ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(an + bn)(an−1 + bn−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� bn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='−an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='dxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� bn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='−an−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='dxn−1 f(x) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(2π)(n−2)/2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 xT C−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n−2x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(2an + 2bn)(2an−1 + 2bn−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(49) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='Erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ Erf ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ Erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='bn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(51) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ e− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n−1 det Cn−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n−1 det Cn−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Continuing the integration over the priors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' we end up with the evidence for the n-dimensional distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' E(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' b) = 1 �n p=1(ap + bp) � b1 −a1 · · � bn −an dnx f(x) = n � p=1 1 (2ap + 2bp) � Erf � ap √ 2 � det Cp−1 det Cp � + Erf � bp √ 2 � det Cp−1 det Cp �� (52) + O � n � p=1 � exp � − a2 p det Cp−1 2 det Cp � + exp � − b2 p det Cp−1 2 det Cp ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' where the covariance matrices Cp are constructed as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' by eliminating the n−p last rows and columns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' un- til C0 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that the approximation is very good whenever the exponents are large, �n p=1(a2 p det Cp−1)/(2 det Cp) ≫ 1, which is often the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note also that we recover the expression of the evidence for the centered priors Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (40) in the limit b → a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us now evaluate the evidence for a distribution normalized to the maximum of the likelihood distri- bution, f(x) = Lmax exp � − 1 2xT C−1 n x � (53) 9 In this case, the evidence is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (52), multiplied by a factor Lmax × (2π)n/2√det Cn from the nor- malization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We can then evaluate the logarithm of the evidence, ignoring the exponentially-small corrections, as ln E = ln Lmax + n 2 ln(2π) + 1 2 ln det Cn − n � p=1 ln(2ap + 2bp) + n � p=1 ln � Erf � ap √ 2 � det Cp−1 det Cp � + Erf � bp √ 2 � det Cp−1 det Cp �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54) Uncorrelated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Suppose we have a multivariate Gaussian distribution without correlations between variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cij = σ2 i δij is a diagonal matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' then the evidence reads exactly, E(a, b) = 1 �n p=1(ap + bp) � b1 −a1 · · � bn −an dnx f(x) = n � p=1 1 2(ap + bp) � Erf � ap σp √ 2 � + Erf � bp σp √ 2 �� , (55) where σp are the dispersions of each variable ˜xp, and thus the logarithm of the evidence becomes ln E = ln Lmax + n 2 ln(2π) + n � p=1 ln σp − n � p=1 ln(2ap + 2bp) + n � p=1 ln � Erf � ap σp √ 2 � + Erf � bp σp √ 2 �� (56) Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Laplacian approximation to the evidence assumes the distribution is a correlated Gaussian, and that the priors are large enough so that the whole distribution fits easily inside them, in which case the error functions are approximately unity and do not contribute to the evidence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54) we now have ln E = ln Lmax + n 2 ln(2π) + 1 2 ln det Cn − n � p=1 ln ∆θp , (57) where ∆θp = ap + bp is the parameter interval associated to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the next section we will compare the different approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 4 Non-Gaussian corrections The advantage of this method is that one can perform a systematic computation of the evidence of a given model with its own priors, given an arbitrary set of moments of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Here we will consider the first two beyond the covariance matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the skewness and the kurtosis terms, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 Skewness Let us start with the first correction to the Gaussian approximation, the trilinear term Bijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For this, we write the generating functional (9) as φ(u) = exp � i µiui − 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cij uiuj − i 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Bijk uiujuk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (58) 10 By performing a change of variable, ui = yi −i C−1 ik (xk −µk), we can evaluate the Fourier transform integral and obtain the properly-normalized probability distribution function f(x) = 1 (2π)n/2√det Cn exp � − 1 2xT C−1 n x � × � 1 − 1 2Bijk C−1 ij C−1 kl xl + 1 6Bijk C−1 il C−1 jmC−1 kn xlxmxn � , (59) where xk are the displaced coordinates (xk − µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This skewed distribution function satisfies ⟨xi⟩ = 0 , ⟨xixj⟩ = Cij , ⟨xixjxk⟩ = Bijk , ⟨xixjxkxl⟩ = 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (60) as can be confirmed by direct evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us now compute the evidence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) for this skewed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Since the extra terms in the parenthesis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (59) are both odd functions of x, when integrating over an even range like that of the centered top-hat prior Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (21), their contribution to the evidence vanish, and thus the final evidence for the skewed model does not differ from that of the Gaussian model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In case the prior is off-centered with respect to the mean, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' like in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (43), then the contribution of the odd terms to the evidence would not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us evaluate their contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For a single variable (n = 1), the correctly-normalized likelihood function can be written as f(x) = e−x2/2σ2 σ √ 2π � 1 − B x 2σ4 + B x3 6σ6 � , satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = B, and the Bayesian integral can be computed exactly as E(a, b) = 1 2a + 2b � Erf � a σ √ 2 � + Erf � b σ √ 2 �� − Bσ−3 6 √ 2π �� 1 − a2 σ2 � e− a2 2σ2 − � 1 − b2 σ2 � e− b2 2σ2 � 1 a + b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (61) Note that for even (centered) priors, with b = a, the evidence reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For an arbitrary number of variables, the computation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us start with the n-th variable and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' in order to compute the integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' let us define the auxiliary function g(λ) = � bn −an dxn xn exp � − λ 2 xT C−1 n x � (2π)n/2√det Cn = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 × × 1 λ √ 2π � exp � − λa2 n 2 det Cn−1 det Cn � − exp � − λb2 n 2 det Cn−1 det Cn �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (62) such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' using Erf′[x] = 2 √π e−x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' −2g′(λ = 1) = � bn −an dxn xn (xT C−1 n x) exp � − 1 2xT C−1 n x � (2π)n/2√det Cn = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 × × 1 √ 2π �� 2 + a2 n det Cn−1 det Cn � exp � − a2 n 2 det Cn−1 det Cn � − � 2 + b2 n det Cn−1 det Cn � exp � − b2 n 2 det Cn−1 det Cn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (63) Therefore, with the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (63), the integral of the skewness-corrected distribution function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (59) over the xn uncentered prior, becomes � bn −an dxn f(x) = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 � 1 2 � Erf � an √ 2 � det Cn−1 det Cn � + Erf � bn √ 2 � det Cn−1 det Cn �� − 1 6Bijn C−1 ij 1 √ 2π � det Cn−1 det Cn �� 1 − a2 n det Cn−1 det Cn � e− a2 n det Cn−1 2 det Cn − � 1 − b2 n det Cn−1 det Cn � e− b2 n det Cn−1 2 det Cn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (64) 11 Let us define two new functions, Ei(ai, bi) = 1 2 � Erf � ai √ 2 � det Ci−1 det Ci � + Erf � bi √ 2 � det Ci−1 det Ci �� , (65) Fi(ai, bi) = 1 6 √ 2π � det Ci−1 det Ci �� 1 − a2 i det Ci−1 det Ci � e− a2 i det Ci−1 2 det Ci − � 1 − b2 i det Ci−1 det Ci � e− b2 i det Ci−1 2 det Ci � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Integrating iteratively over xn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , x1, we end up with the Bayesian evidence for the third-order-corrected probability distribution function f(x), E(a, b) = n � p=1 Ep(ap, bp) (ap + bp) � 1 − n � k=1 Bijk C−1 ij Fk(ak, bk) Ek(ak, bk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (66) Unless Bijk C−1 ij is very large, the correction to the error function is exponentially suppressed, and we do not expect significant departures from the Gaussian case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note also that if the prior is symmetric, it is easy to see that the skewness part of the integral vanishes, Fk(ak, bk) → 0, as can be checked explicitly by taking bk → ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 Kurtosis The next correction beyond skewness is the fourth order moment or kurtosis, given by the Dijkl term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us ignore for the moment the third order skewness and write φ(u) = exp � i µiui − 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Cij uiuj + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Dijkl uiujukul � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (67) By performing the same change of variables, ui = yi − i C−1 ik (xk − µk), we can now compute the Fourier transform and obtain the properly-normalized probability distribution function f(x) = 1 (2π)n/2√det Cn exp � − 1 2xT C−1 n x � � 1 + 1 8Dijkl C−1 ij C−1 kl −1 4Dijkl C−1 ij C−1 kmC−1 ln xmxn + 1 24Dijkl C−1 im C−1 jn C−1 kp C−1 lq xmxnxpxq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (68) Performing the integrals, it is easy to see that this distribution satisfies ⟨xixj⟩ = Cij , ⟨xixjxkxl⟩ = Dijkl + CijCkl + CikCjl + CilCjk , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (69) Note that in order for the new likelihood distribution (68) to be positive definite, it is required that DijklC−1 ij C−1 kl < 4, and if we impose that there is only one maximum at the center, then it must sat- isfy DijklC−1 ij C−1 kl < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' These conditions impose bounds on the maximum possible deviation of the evidence from a that of a gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us now compute the evidence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (22) for this kurtosis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The extra terms in the parenthesis of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (68) are both even functions of x, and we cannot ignore them, even for centered priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For a single variable (n = 1), the correctly-normalized likelihood function can be written as f(x) = e− x2 2σ2 σ √ 2π � 1 + D 8σ4 − D x2 4σ6 + D x4 24σ8 � , satisfying ⟨x⟩ = 0, ⟨x2⟩ = σ2, ⟨x3⟩ = 0, ⟨x4⟩ = D + 3σ4, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Bayesian integral can be computed exactly as E(a, b) = 1 2a + 2b � Erf � a σ √ 2 � + Erf � b σ √ 2 �� + Dσ−4 8 √ 2π � a σ � 1 − a2 3σ2 � e− a2 2σ2 + b σ � 1 − b2 3σ2 � e− b2 2σ2 � 1 a + b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (70) 12 For arbitrary number of variables, the computation is again much more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us start with the n-th variable and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' in order to compute the first integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' let us define a new auxiliary function h(λ) = � bn −an dxn exp � − λ 2 xT C−1 n x � (2π)n/2√det Cn = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 × × 1 2 √ λ � Erf � an √ λ √ 2 � det Cn−1 det Cn � + Erf � bn √ λ √ 2 � det Cn−1 det Cn �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (71) such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' −2h′(λ = 1) = � bn −an dxn (xT C−1 n x) exp � − 1 2xT C−1 n x � (2π)n/2√det Cn = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 × × � 1 2 � Erf � an √ 2 � det Cn−1 det Cn � + Erf � bn √ 2 � det Cn−1 det Cn �� (72) − 1 √ 2π � det Cn−1 det Cn � an exp � − a2 n 2 det Cn−1 det Cn � + bn exp � − b2 n 2 det Cn−1 det Cn ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4h′′(λ = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� bn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='−an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='dxn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(xT C−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n x)2 exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2xT C−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(2π)n√det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2xT C−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n−1x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='(2π)(n−1)/2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='Erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='+ bn exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='− b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='det Cn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Therefore, with the use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (72) and (73),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the integral of the kurtosis-corrected distribution function (68) over the xn prior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' becomes � bn −an dxn f(x) = exp � − 1 2xT C−1 n−1x � (2π)(n−1)/2� det Cn−1 � 1 2 � Erf � an √ 2 � det Cn−1 det Cn � + Erf � bn √ 2 � det Cn−1 det Cn �� + (74) + 1 8Dijkl C−1 ij C−1 kl 1 √ 2π � det Cn−1 det Cn � an � 1 − a2 n 3 det Cn−1 det Cn � e− a2 n det Cn−1 2 det Cn + bn � 1 − b2 n 3 det Cn−1 det Cn � e− b2 n det Cn−1 2 det Cn �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We can now define a new function Gi(ai, bi) = 1 8 √ 2π � det Ci−1 det Ci � ai � 1 − a2 i 3 det Ci−1 det Ci � e− a2 i det Ci−1 2 det Ci − bi � 1 − b2 i 3 det Ci−1 det Ci � e− b2 i det Ci−1 2 det Ci � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (75) Integrating iteratively over xn−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' , x1, we end up with the Bayesian evidence for the fourth-order-corrected probability distribution function f(x), E(a, b) = n � p=1 Ep(ap, bp) (ap + bp) � 1 + Dijkl C−1 ij C−1 kl n � m=1 Gm(am, bm) Em(am, bm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (76) 13 so, unless Dijkl C−1 ij C−1 kl is very large, the correction to the error function is exponentially suppressed, and we do not expect significant departures from the Gaussian case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In order to compare models it is customary to compute the logarithm of the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us assume that we are given a likelihood distribution function normalized by the maximum likelihood, and with corrections up to fourth order, f(x) = Lmax exp � − 1 2xT C−1 n x � � 1 + 1 8Dijkl C−1 ij C−1 kl �−1� 1 − 1 2Bijk C−1 ij C−1 kl xl + 1 6Bijk C−1 il C−1 jmC−1 kn xlxmxn + 1 8Dijkl C−1 ij C−1 kl − 1 4Dijkl C−1 ij C−1 kmC−1 ln xmxn + 1 24Dijkl C−1 im C−1 jn C−1 kp C−1 lq xmxnxpxq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (77) Note that it is normalized so that the maximum corresponds to the mean-centered distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this case, the evidence of the normalized distribution is given by E(a, b) = Lmax (2π)n/2� det Cn � 1 + 1 8Dijkl C−1 ij C−1 kl �−1 × (78) n � p=1 Ep(ap, bp) (ap + bp) � 1 − n � k=1 Bijk C−1 ij Fk(ak, bk) Ek(ak, bk) + Dijkl C−1 ij C−1 kl n � m=1 Gm(am, bm) Em(am, bm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We can then evaluate the logarithm of the evidence by ln E = ln Lmax + n 2 ln(2π) + 1 2 ln det Cn − ln � 1 + 1 8Dijkl C−1 ij C−1 kl � − n � p=1 ln(2ap + 2bp) + n � p=1 ln � Erf � ap √ 2 � det Cp−1 det Cp � + Erf � bp √ 2 � det Cp−1 det Cp �� (79) + ln � 1 − n � k=1 Bijk C−1 ij Fk(ak, bk) Ek(ak, bk) + Dijkl C−1 ij C−1 kl n � m=1 Gm(am, bm) Em(am, bm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that the condition DijklC−1 ij C−1 kl < 2 constrains the maximum amount that the kurtosis corrections can contribute to the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Uncorrelated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the case where the likelihood distribution had no correlations among the different variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' the exact expression for the Bayesian evidence is ln E = ln Lmax + n 2 ln(2π) + n � p=1 ln σp − n � p=1 ln(2ap + 2bp) + n � p=1 ln � Erf � ap σp √ 2 � + Erf � bp σp √ 2 �� (80) − ln � 1 + 1 8Diijj σ−2 i σ−2 j � + ln � 1 − n � k=1 Biik σ−2 k Fk(ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' bk) Ek(ak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' bk) + Diijj σ−2 i σ−2 j n � m=1 Gm(am,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' bm) Em(am,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' bm) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' where σp are the corresponding dispersions of variables xp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' and the functions Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Fi and Gi are the corre- sponding limiting functions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (65) and (75) for uncorrelated matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 5 Model comparison Finally we turn to specific applications of the formalism discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Initially we will carry out some toy model tests of its performance, and then examine real cosmological applications for which we previously obtained results by thermodynamic integration [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 14 Figure 1: This figure shows the calculated evidence as a function of the number of likelihood evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note that the horizontal axis is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The solid line corresponds to the thermodynamic integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The dotted line and dot-dashed lines are the analytical methods with and without non-Gaussian corrections applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The horizontal dashed line is the number obtained by the direct integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The upper two panels correspond to Lg, while the lower two to Lng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The left-hand side panels correspond to wide flat priors of (−7, 10) on both parameters, while the right-hand side to the narrow priors of (−2, 3) on both parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' See text for discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 A baby-toy model comparison We begin with a very simple two-dimensional toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The purpose of this section is to illustrate the ineffectiveness of the thermodynamic integration and to give an indication of the performance of the method we propose here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In addition, the two-dimensional model is simple enough to allow a brute-force direct numerical integration of evidence allowing us to check the accuracy at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We use the following two forms of likelihood: Lg(x, y) = exp � −2x2 − 2(y − 1)2 − xy 2 � (81) Lng(x, y) = exp � −2x2 − 2(y − 1)2 − xy 2 � + exp � −2x2 − 2y2 − 3xy 2 � (82) The subscripts g and ng indicate the Gaussian and non-Gaussian cases respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Firstly, we calculate the evidence by the analytical method using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (56) and (80) and covariance 15 matrices inferred from sampling the likelihood using the vanilla Metropolis–Hastings algorithm with fixed proposal widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Chains ranging from few to several million samples were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We also calculate evidence using thermodynamic algorithm explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Again, we vary algorithm parameters to get evidence values of varying accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The resulting evidence as a function of number of likelihood evaluations is plotted in the Figure 1, together with the correct value inferred by direct numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The number of likelihood evaluations is crucial as this is the time-limiting step in the cosmological parameter estimation and model comparison exercises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The results are what could have been anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We note that the size of the prior does not seem to be of crucial importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This is comforting, given that the analytical method requires the knowledge of the true covariance information, while we can only supply a covariance matrix estimated from the prior-truncated likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We also note that the thermodynamic integration converges to the correct value in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' However, it does so after very many likelihood evaluations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' typically about a million or so even for a two-dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The analytical method becomes limited by systematics already by the ten-thousand samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For Gaussian case, there is no systematic by construction, while the non-gaussian case suffers a systematic of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 in ln E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The non-Gaussian correction reduces the error by about a half and thus correctly estimates the uncertainty associated with the purely Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the case of wide priors, the only non-Gaussian correction of an appreciable size is the ln(1 + DijklC−1 ij C−1 kl /8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 A toy model comparison We now proceed by calculating the Bayesian evidence for simple toy models with 5 and 6 parameters, shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The purpose is to compare results with those obtained from thermodynamic integration again, but this time using a model that bears more resemblance to a typical problem one encounters in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Parameter Mean Prior Range Model x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='022 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='044] toy5,toy6 x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='12 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3] toy5,toy6 x3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='04 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4] toy5,toy6 x4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3] toy5,toy6 x5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='6] toy5,toy6 x6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='98 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5] toy6 Table 1: The parameters used in the analytical evaluation of the toy model evidences, with 5 and 6 parameters respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The maximum likelihod of the toy models is taken (arbitrarily) to be Lmax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Beginning with the five-parameter model, we assume first that it has an uncorrelated multivariate Gaus- sian likelihood distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this case the aim is to test the thermodynamic integration method, which gives ln Enum toy5 = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='03, while the exact expression gives ln Eana toy5 = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Therefore, we conclude that the thermodynamic integration method is rather good in obtaining the correct evidence of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Laplace approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (57) also fares well for uncorrelated distributions, ln ELap toy5 = −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We now consider a likelihood function with a correlated covariance matrix Cij, with the same mean values and dispersions as the previous case, but with significant correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The analytic formula needed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54), is no longer exact,2 and gives ln Eana toy5c = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For comparison thermodynamic integration gives ln Enum toy5c = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='06, again in perfect agreement within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In this case the Laplace approximation fails significantly, ln ELap toy5c = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='89, the reason being that the correlations chosen bring the posterior into significant contact with the edges of the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Let us now return to the uncorrelated case and include a new parameter, x6, as in Table I, and evaluate the different evidences that appear because of this new parameter, in order to see the sensitivity to systematic errors in the evaluation of the Bayesian evidence and their effects on model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The numerical 2One could rotate the parameter basis to remove the correlations, but then the priors wouldn’t be top-hats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 16 result is ln Enum toy6 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='03, while the exact analytical expression gives ln Eana toy6 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='74, in perfect agreement, within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The Laplace approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (57) again fares well for uncorrelated distributions, ln ELap toy6 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' When the likelihood function has large correlations, and the priors are not too large, the naive Laplace approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (57), fares less well than the analytical approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3 A real model comparison In this subsection we will make use of the results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5], where we evaluated the evidence for 5- and 6-parameter adiabatic models, and for three 10-parameter mixed adiabatic plus isocurvature models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The prior ranges used are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The latter models give a marginally better fit to the data but require more parameters, which is exactly the situation where model selection techniques are needed to draw robust conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5] we used thermodynamic integration to compute the evidence and showed that the isocurvature models ware less favoured than the adiabatic ones, but only at a mild significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3 Beginning with the simplest adiabtic model, which uses the Harrison–Zel’dovich spectrum, we have used the analytical formulae above, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54), together with the covariance matrix provided by the cosmoMC programme [10], and obtained ln Eana ad = −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='07, while the thermodynamical integration gave ln Enum ad = −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The agreement is excellent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' this is because the distribution function for the adiabatic model is rather well approximated by a Gaussian, and the priors are rather large, so the formula Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54) is very close to that obtained in the Laplace approximation, ln ELap ad = −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Parameter Mean Prior Range Model ωb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='022 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='018, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='032] AD-HZ,AD-ns,ISO ωdm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='12 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='16] AD-HZ,AD-ns,ISO θ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='04 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='98, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='10] AD-HZ,AD-ns,ISO τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='17 [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5] AD-HZ,AD-ns,ISO ln[1010Rrad] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2] AD-HZ,AD-ns,ISO ns 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='0 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2] AD-ns,ISO niso 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5 [0, 3] ISO δcor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5 [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='14, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4] ISO √α 0 [−1, 1] ISO β 0 [−1, 1] ISO Table 2: The parameters used in the models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [5] for nomenclature and other details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For the AD-HZ model ns was fixed to 1 and niso, δcor, α and β were fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the AD-ns model, ns also varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Every isocurvature model holds the same priors for the whole set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' However the analytic method fares less well for the adiabatic model with varying ns, with both the analytic and Laplace methods giving ln EAD−ns = −853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4, while the numerical method gives the smaller value -854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1, a discrepency of nearly unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Turning now to the iscurvature cases, we found an extremely good result for the CDI model, gaining from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' (54) the value ln Eana cdi = −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='08, while the thermodynamical integration gives ln Enum cdi = −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This is surprising, given the relatively large non-gaussianities for at least three variables: niso, β and δcor, whose priors are not centered with respect to the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' However the NID case shows much less good agreement, with a discrepency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' That suggests that the closeness of the CDI comparison is to some extent a statistical fluke, with the underlying method less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' A summary of the different models can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 3Recently Trotta [9] used a different technique to analyze a restricted class of isocurvature model featuring just one extra parameter, and found it highly disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The different conclusion is primarily due to the very different prior he chose on the isocurvature amplitude, such that almost all the models under the prior are domintaed by isocurvature modes and in poor agreement with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 17 Model ln Lmax ln Enum ln Eana ln ELap toy5 0 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='03 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='66 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='67 toy5c 0 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='06 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='32 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='89 toy6 0 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='03 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='74 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='74 toy6c 0 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='06 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='71 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='63 AD −840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='78 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 AD-ns −838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='50 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 −853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4 −853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4 CDI −838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='05 −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5 NID −836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='60 −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='2 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='5 NIV −842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='53 −855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='3 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='9 −854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='9 Table 3: The different models, both toy and real, with their maximum likelihoods and evidences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='4 Savage–Dickey method Another numerical method for evidence calculation is the Savage–Dickey method, first described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [11] and recently used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This technique allows one to calculate the evidence ratio of two models from a simple and quick analysis of the Markov chains used for parameter estimation, provided that the models are nested;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=', that one of them is included in the parameter space of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' For instance, the AD model is nested within the AD-ns model, and the AD and AD-ns models are both nested within the CDI, NID and NIV ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In the context of Markov chains, the Savage–Dickey method is essentially a measure of how much time the sampler spends in the nested model, weighted by the respective volumes of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' When the outer model has extra parameters, this method relies on approximating the nested model as a model with negligibly narrow priors in directions of extra parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' We note, however, that when many extra parameters are present, this method must fail for reasons similar to those why grid-based parameter estimation approaches fail with models with many parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The MCMC parameter estimation simply does not have high enough dynamic range to probe the two models given the large prior volume ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The AD and AD-ns models differ by one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Using the same AD+ns samples as for the analytic method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=', the samples from which we extracted the covariance matrix), we obtained ln(EAD/EAD+ns) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The result from the precise thermodynamical integration, ln(EAD/EAD−ns) = 0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='1 is in excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The AD-ns and CDI (or NID, NIV) models differ by four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' With most simple choices of parametrization (including in particular the isocurvature and cross-correlation tilts), the AD-ns is not a point, but a hypersurface within the parameter space of the isocurvature models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' α = 0 and other three parameters act as dummy, unconstrained, parameters which do not affect the evidence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' In these cases, the evidence ratios given by the Savage–Dickey method do not converge as the priors of the extra parameters are tightened up around the nested model, although they match thermodynamically-determined values to within a unit of ln E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 6 Discussion and Conclusions We have developed an analytical formalism for computing the Bayesian evidence in the case of an arbitrary likelihood distribution with a hierarchy of non-Gaussian corrections, and with arbitrary top-hat priors, centered or uncentered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' This analysis can be of great help for the problem of model comparison in the present context of cosmology, where observational data is still unable to rule out most extensions of the standard model based on the ΛCDM inflationary paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' As an application of the exact and approximate formulae obtained for the Bayesian evidence of a model with approximately Gaussian likelihood distributions, we have compared the value predicted analytically with that computed with a time-consuming algorithm based on the thermodynamical integration approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' The values obtained analytically agree surprisingly well with those obtained numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' While one can estimate the magnitude of the higher order corrections for the analytical formulae, it is very difficult to 18 estimate the systematic effects of the numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Thus, with this analytical method we can test for systematics in the thermodynamical integration approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' So far, the values obtained agree, so it seems that the numerical approach is a good tool for estimating the evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' However, it takes considerable effort and machine time to do the correct evaluation, and therefore, we propose the use of the analytical estimate, whose corrections are well under control, in the sense that one can compute the next order corrections and show that they are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Note added: Many years after my work was finished, a book appeared [12] which thoroughly discussed Bayesian Methods in Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Jeffreys, Theory of Probability, 3rd ed, Oxford University Press (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Astr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 378, 72 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Lewis and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Bridle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' D66, 103511 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Dickey, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Stat 42, 204 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} 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+page_content=' Mukherjee & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' Parkinson, Bayesian Methods in Cosmology, Cambridge University Press (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19FST4oBgHgl3EQfXDjl/content/2301.13783v1.pdf'} diff --git a/39FQT4oBgHgl3EQfHTWW/content/2301.13248v1.pdf b/39FQT4oBgHgl3EQfHTWW/content/2301.13248v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..20465a28f94df4cc22466d235c5e1524df8ef401 --- /dev/null +++ b/39FQT4oBgHgl3EQfHTWW/content/2301.13248v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:840399aba2314dcf819a1604819d5f1428df4bde1e543f892f6ea7bf56d988f3 +size 1649301 diff --git a/4dAzT4oBgHgl3EQfffz3/content/tmp_files/2301.01455v1.pdf.txt b/4dAzT4oBgHgl3EQfffz3/content/tmp_files/2301.01455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cb8154f0ab1ecc593d1baf4271e2a7bbf2ebd56 --- /dev/null +++ b/4dAzT4oBgHgl3EQfffz3/content/tmp_files/2301.01455v1.pdf.txt @@ -0,0 +1,440 @@ +Engineering sub-Poisson light in a simple mirror and beam +splitter system +Sun-Hyun Youn∗ +Department of Physics, Chonnam National University, Gwangju 500-757, Korea +Abstract +Vacuum fluctuation, which is the intrinsic nature of an electric field can be measured via homo- +dyne detection. Moreover, electric field intensity fluctuation are also related to vacuum fluctuations. +Squeezed vacuum and sub-Poisson light can be obtained by controlling the vacuum fluctuation us- +ing noble nonlinear interaction. Based on the squeezed vacuum by inserting a mirror on the unused +part of the beam splitter was proposed in 1994, we present the mode matching method for the +vacuum and light fields. Light intensity fluctuations also can be reduced by inserting a mirror on +the unused part of the beam splitter. To obtain sub-Poisson light as a function of the distance +between the mirror and detector, a detector with a thinner active layer than the wavelength is +required. +PACS numbers: 03.67.-a,03.70.+k, 03.65.Yz +Keywords: Quantum optics, Squeezed State, Vacuum fluctuation, Sub-Poisson, Beam splitter and Mirror +∗ E-mail: sunyoun@jnu.ac.kr, fax: +82-62-530-3369 +1 +arXiv:2301.01455v1 [quant-ph] 4 Jan 2023 + +I. +INTRODUCTION +When a single photon is in a particular mode, according to the particle nature of light, +photons will be sequentially found in that mode. The probability of finding a photon is +proportional to the absolute square of the wave function related to the electromagnetic +wave. Vacuum fluctuations are related to the spatial characteristics of the electromagnetic +wave. The spontaneous decay caused by the vacuum can be suppressed in cavities [1]. The- +oretical and experimental studies have beem conducted on methods to change the vacuum +fluctuations near mirrors[2–4]. +In this study, in contrast to previous studies on the vacuum noise characteristics of +light using a homodyne detector, we calculate the intensity fluctuations when photons are +directly measured using photon counter. The obtained results are similar to those obtained +in previous studies, but herein we predict the results considering mode matching in the +experiment. +In section II, the fluctuation of light that can be measured using a detector is calculated +with a mirror placed on one side of the beam splitter. In section III, an experimental device +is proposed for perfect mode matching, and in the last section, the practical limits of the +vacuum fluctuation near the mirror are discussed. +II. +VACUUM FLUCTUATION NEAR A MIRROR. +An electric field can be written as +ˆEL = ˆEcl + ˆEQ, +(1) +where +ˆEcl = i +� +ℏω +2ϵ0V (αei(ωt−k0z) − α∗ei(ωt−k0z))⃗x, +ˆEQ = i +� +k +� +ℏωk +2ϵ0V (ˆbke−i(ωkt−kz) − ˆb† +kei(ωkt−kz))⃗x. +(2) +Here, k0 and ω are the wave number and angular frequency of the laser, respectively, ℏ and +ϵ0 have usual meanings, and V is the normalization volume[5]. Considering the laser mode +2 + +FIG. 1: Vacuum mode relations in the beam splitter with a mirror. BS: Beam splitter, M: mirror +in Fig. 1, the modes aout +1 +and aout +2 +can be written as +aout +1 += +√ +Tb + +√ +Rc, +aout +2 += − +√ +Rb + +√ +Tc, +(3) +where the modes c and cout can be written as +c = +� +Tmd − +� +Rmcout, +cout = +√ +Ra1 + +√ +Ta2. +Then the electric field in fluctuating vacuum modes at a1 is +ˆE(+) +vac,1 = +� +k +i +� +ℏωk +4ϵ0V { +√ +Tˆb† +kei(ωkt−kZ1) + µˆa† +1,kei(ωkt+kz1) +−R +� +Rmˆa† +1,kei(ωkt−kz1) − +√ +RT +� +Rmˆa† +2,kei(ωkt−kz1) + +� +RTm ˆd† +kei(ωkt−kZM)} (4) +where Rm(Tm) is the reflectance(transmittance) of the mirror and R(T) is the reflectance +(transmittance) of the beam splitter, z1(Z1) is the distance from the mirror (laser) to the +detector. ZM is related to the vacuum source behind the mirror and it can be any number. +We add the factor +1 +√ +2 for the normalization of the vacuum fluctuation. The vacuum mode +(ˆa† +1ei(ωt−kz1)) at the detector is the reflected vacuum mode (ˆa† +1ei(ωt+kz1)) at the mirror. If two +modes are perfectly matched the µ in Eq. 4 is 1 and the two counterpropagating modes +yield the standing wave mode[2, 3]. If µ = 0, the fluctuation value from Eq. 7 becomes |α|2T +2 , +3 + +b +BS +α2 +C +d +ino +↑ +M +ino +a1 +ait is the square of the constant dc current T|α|2 +2 . In other words, if we directly measure the +fluctuation of the laser intensity, the fluctuation is dependent on the distance (z1) between +the mirror and the detector. +Even in photo counting experiments, the photon number +fluctuation is related to the vacuum fluctuation, therefor, the photon number fluctuation is +also depend on the distance z1. +If we used the photodetetion theory [6] with instantaneous response of the photodetector +[7], +ˆI1 = { +√ +T ˆE(+) +cl ++ ˆE(+) +vac,1} × { +√ +T ˆE(−) +cl ++ ˆE(−) +vac,1}, +(5) +where we normalize the photocurrent. If the electric field of the local oscillator is considerably +greater than the vacuum field, the terms containig α have physical significance. When the +constant dc current T|α|2 +2 +is neglected, Eq. 5 yields +ˆIo +1(z1, Z1) = |α| +√ +2[ +√ +Teiφ{(µe−ik(Z1+z1) − e−ik(Z1−z1)R +� +Rm)ˆa1 − e−ik(Z1−z1)ˆa2} ++ +√ +Te−iφ{(µeik(Z1+z1) − eik(Z1−z1)R +� +Rm)ˆa† +1 − e−k(Z1−z1)ˆa† +2} ++ eiφTˆb + e−iφTˆb† + eiφeik(ZM−Z1)� +TRTm ˆd + e−iφe−ik(ZM−z1)� +TRTm ˆd†], +(6) +We then evaluate the square of the photocurrent to determine the fluctuation. After +squaring Eq. 6, we find the photocurrent fluctuation as follows: +⟨(ˆIo +1)2⟩ = |α|2T +2 +{1 + µ2 − 2µR +� +Rm cos(2kz1)} +(7) +If µ = 0, the fluctuation value from Eq. 7 becomes |α|2T +2 , which is the square of the con- +stant dc current +√ +T|α| +√ +2 . In other words, if we directly measure the laser intensity fluctuation, +the fluctuation is dependent on the distance (z1) between the mirror and detector. Even +in the photo counting experiment, the photon number fluctuation is related to the vacuum +fluctuation; therefore, the photon number fluctuation is also dependent on the distance z1. +If we consider practical limits such as finite linewidth and finite absorption length, Eq. 7 +will change as follows[2, 8]. +⟨(ˆIo +1)2⟩P = |α|2T +2 +{1 + µ2 − 2µR +� +Rme−z2 +1∆k2 +× κ[cos(2k0z1 + φ0) − e−κD cos(2k0(z1 + D) + φ0)] +� +4k2 +0 + κ2 +}, +(8) +4 + +where ∆k is the line width of the local oscillator beam with Gaussian line width distribution +functions. κ is the absorption coefficient, D is the detector active length, and φ0 = arctan 2k +κ . +We assumed that the probability that a photon is converted into an electron hole pair at +distance η from the surface of the detector’s active region is κe−κη[9]. +The two coefficients √Rm and µ depend on the mode matching condition. Even when +we used the total mirror, if the mode from the mirror is not perfectly matched with the +mode from the laser, the effective reflectance √Rm can not be 1. Furthermore, the mode +a1 to the mirror is reflected by the mirror and then meets at the detector. At the detector, +if two counter-propagating modes are not exactly matched, the coefficient µ cannot be 1. +To evaluate this mode matching condition, we assume that the amplitude envelope of the +electromagnetic wave in the transverse plane is given by a Gaussian function. +Considering the Gaussian modes [10] +E(ρ, z) = E0 +w0 +w(z) exp[− +ρ2 +w(z)2] exp[−ikz − ik +ρ2 +2R(z) + iζ(z)] +(9) +, where w0 is the radius of the beam waist and +w(z) = w0 +� +1 + ( z +z0 +)2 +R(z) = z(1 + (z0 +z )2) +ζ(z) = tan−1 z +z0 +(10) +and z0 is defined as follows: +z0 = π +λw2 +0. +(11) +First, we assume that the laser and vacuu modes have the same beam waist w0 at the +detector. Then the laser and vacuum modes are perfectly matched; thus, √Rm = 1. On +the other hand, the vacuum Ev(0) starting from the detector propagates to the mirror and +reflects at the mirror. The returned vacuum Ev(2z1) is not the same Ev(0). The coefficient +µ can be calculated as follow: +µ = +| < Ev(0)Ev(2z1)∗ > | +� +< Ev(0)2 >< Ev(2z1)2 > += +(1 + 4z2 +1 +z2 +0 ) +1 +4 +(1 + 5z2 +1 +z2 +0 + 4 z4 +1 +z4 +0 ) +1 +4 +(12) +5 + +FIG. 2: +Mode matching value µ as a function of w0 and z1. +In Fig. 2, µ is plotted as a function of z1 and w0, where z1 is the distance between the +mirror and detector We assume that the detector and mirror are large enough that all the +waves are detected and reflected. If the distance between the mirror and detector and the +size of the beam waist are small enough, the coefficient µ remains near 1. +If we consider the case where the vacuum field has waist at the mirror, the coefficient µ +automatically becomes 1 due to the symmetry, but the vacuum field Ev(z1) at the detector +does not matche the laser field EL(0). We assumed that the laser field has beam waist w0 at +the detector, and the vacuum field has a beam waist wm at the mirror. Then the effective +reflectance √Rm becomes +� +Rm = +| < Ev(z1)EL(0)∗ > | +� +< Ev(z1)2 >< EL(0)2 > += +√ +2 +� +wm +w0 (1 + z2 +1 +z2m) +1 +4 +({(1 + w2m +w2 +0 )2 + z2 +1 +z2 +0 }{1 + z2 +1 +z2m}) +1 +4 +, +(13) +where zm = π +λw2 +m. +In Fig. 3, √Rm is plotted as a function of z1 and wm, where z1 is the distance between +the mirror and detector. We set w0 to 100λ. Additionally, we also assume that the detector +and mirror are large enough that all the waves are detected and reflected. The coefficient +√Rm can be 1 only when the distance between the mirror and detector is small and the size +of the beam waist is sufficiently small. +6 + +μ +1 +0.5 +3 +4 +5 +log( +0m +3 +入 +log() +7 +2FIG. 3: Mode matching value √Rm as a function of w0 and z1, with w0 equal to 100λ +The mode matching condition is crucial for detecting the modulation effect of the vacuum +fluctuation near the mirror, as denoted by Eq. 8. With the usual setup, we can not satisfy +the conditions µ = 1 and √Rm = 1. In the next section, we suggest a noble experimental +setup that satisfies two mode-matching conditions. +III. +SET UP FOR MODE MATCHING +For a laser that has a Gaussian transverse mode, we have to establish a vacuum mode that +also has a Gaussian transverse mode. Fig. 4 displays the setup for perfect mode matching +between the laser light mode and a vacuum mode. +The laser used in the experiment passes through lens L1 and is divided into two by +the beam splitter (BS1). The laser is a Gaussian beam and it proceeds according to the +Gaussian approximation. The light passing through BS1 and traveling to mirror M2 reaches +the partial mirror B and yields a beam waist on the L3 side surface of B. Similarly, the +light reflecting from the mirror M1 passes through the partial reflector A and yields a beam +waist on the L2 side surface of A. +The light passing through A and B passes through the L2 and L3 of the same focal length, +respectively, and yields another beam waist on the detector surface. The transmittance of +light passing through A from M1 is almost 0, and the reflectance of light stemming from the +L2 side is almost 1. In this way, if the mode is perfectly matched using the light passing +through B and A, an experimental setup can be established wherein one side of the beam +splitter BS2 is a mirror (A). +7 + +/Rm +1 +0.5 +3 +4 +5 +3 +1og +wm +log() +入 +7 +2Using this method, the degree of mode matching can be increased compared to that +when the experiment is performed by simply placing a plane mirror on one side of the beam +splitter. Additionally the experimental constraints caused by the mode matching can be +overcome. The experimental setup in Fig. 4 enables the measurement of how the vacuum +fluctuations of the light passing through the beam splitter change when a mirror is placed +on one side of the beam splitter. +FIG. 4: Mode matching setup +IV. +CONCLUSION AND DISCUSSION. +The quantum nature of photons is highly dependent on their vacuum fluctuations. Vac- +uum fluctuations can be directly measured via homodyne detection. The fluctuation of one +quadrature of the vacuum can be less than that of the usual vacuum, e.g., squeezed vacuum. +Light intensity fluctuations are also dependent on vacuum fluctuations. Sub-Poisson light +can be generated by controlling the vacuum fluctuations based on the nonlinear interaction +of light and matter. In this study, we proposed the modulation of vacuum fluctuations by +inserting a mirror on the unused part of the beam splitter in a homodyne measuring system. +Furthermore, we calculated the effect of the line width of the laser and the thickness of the +detector layer. The line width can be practically reduced to modulate vacuum fluctuations, +but the decrease of the thickness of the detector to modulate vacuum fluctuations is chal- +lenging. We calculated the effect of mode matching between the vacuum and light fields and +8 + +BS1 +M1 +A +L1 +L2 +DD1 +M2 +BS2 +B +L3 +D2showed that the degree of mode matching obtained by adding a simple mirror in the unused +beam splitter may not be sufficient to modulate the vacuum fluctuations. We present the +perfect mode matching method for the vacuum and light fields. Then, the light intensity +fluctuations can be reduced by inserting a beam splitter and a mirror. We still require a +detector with an active layer thinner than the wavelength to obtain a sub-Poisson light as a +function of the distance between the mirror and detector. We expect that our simple method +of reducing vacuum fluctuations will play a great role in quantum information science. +[1] W. Jhe, A Anderson, E. A. Hinds, D. Meschede, L. Moi, and S. Haroche, Phys. Rev. Lett. +58, 666 (1987) +[2] S. H. Youn, J. H. Lee, J. S. Chang, Opt. and Quant. Elec. 27, 355 (1995) +[3] S. H. Youn, J. H. Lee, J. S. Chang, International Workshop on Squeezed States and Uncer- +tainty Relations, N95-13921 (1994) +[4] S. A. Wadood, J. T, Schultz, A. N. Vamivakas, and C.R. Stroud Jr, J. of Mod. Opt. 66, 1116 +(2019) +[5] A. Yariv, Quantum Electronics 3rd ed., John Wiley & Sons. Inc, (1989) +[6] P. D. Drummond, Phys. Rev. A 35, 4253 (1987). +[7] B. Yurke, Phys. Rev. A 32, 311 (1985) +[8] A. E. Siegman, Laser (Oxford University Press, Oxford, 1986 ) +[9] S. M. Sze, +Semiconductor Devices Physics and Technology (AT&T Bell Lab. Murray Hill, +New Jersey, 1985) +[10] B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics ( Wiley, Nw York, 1991) +9 + diff --git a/4dAzT4oBgHgl3EQfffz3/content/tmp_files/load_file.txt b/4dAzT4oBgHgl3EQfffz3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c06c1458bc4349c10a3bdc00a9242340290ad28 --- /dev/null +++ b/4dAzT4oBgHgl3EQfffz3/content/tmp_files/load_file.txt @@ -0,0 +1,179 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf,len=178 +page_content='Engineering sub-Poisson light in a simple mirror and beam splitter system Sun-Hyun Youn∗ Department of Physics, Chonnam National University, Gwangju 500-757, Korea Abstract Vacuum fluctuation, which is the intrinsic nature of an electric field can be measured via homo- dyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Moreover, electric field intensity fluctuation are also related to vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Squeezed vacuum and sub-Poisson light can be obtained by controlling the vacuum fluctuation us- ing noble nonlinear interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Based on the squeezed vacuum by inserting a mirror on the unused part of the beam splitter was proposed in 1994, we present the mode matching method for the vacuum and light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Light intensity fluctuations also can be reduced by inserting a mirror on the unused part of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' To obtain sub-Poisson light as a function of the distance between the mirror and detector, a detector with a thinner active layer than the wavelength is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' PACS numbers: 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='-a,03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='+k, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='Yz Keywords: Quantum optics, Squeezed State, Vacuum fluctuation, Sub-Poisson, Beam splitter and Mirror ∗ E-mail: sunyoun@jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='kr, fax: +82-62-530-3369 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='01455v1 [quant-ph] 4 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' INTRODUCTION When a single photon is in a particular mode, according to the particle nature of light, photons will be sequentially found in that mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The probability of finding a photon is proportional to the absolute square of the wave function related to the electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Vacuum fluctuations are related to the spatial characteristics of the electromagnetic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The spontaneous decay caused by the vacuum can be suppressed in cavities [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The- oretical and experimental studies have beem conducted on methods to change the vacuum fluctuations near mirrors[2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In this study, in contrast to previous studies on the vacuum noise characteristics of light using a homodyne detector, we calculate the intensity fluctuations when photons are directly measured using photon counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The obtained results are similar to those obtained in previous studies, but herein we predict the results considering mode matching in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In section II, the fluctuation of light that can be measured using a detector is calculated with a mirror placed on one side of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In section III, an experimental device is proposed for perfect mode matching, and in the last section, the practical limits of the vacuum fluctuation near the mirror are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' VACUUM FLUCTUATION NEAR A MIRROR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' An electric field can be written as ˆEL = ˆEcl + ˆEQ, (1) where ˆEcl = i � ℏω 2ϵ0V (αei(ωt−k0z) − α∗ei(ωt−k0z))⃗x, ˆEQ = i � k � ℏωk 2ϵ0V (ˆbke−i(ωkt−kz) − ˆb† kei(ωkt−kz))⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' (2) Here, k0 and ω are the wave number and angular frequency of the laser, respectively, ℏ and ϵ0 have usual meanings, and V is the normalization volume[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Considering the laser mode 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 1: Vacuum mode relations in the beam splitter with a mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' BS: Beam splitter, M: mirror in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 1, the modes aout 1 and aout 2 can be written as aout 1 = √ Tb + √ Rc, aout 2 = − √ Rb + √ Tc, (3) where the modes c and cout can be written as c = � Tmd − � Rmcout, cout = √ Ra1 + √ Ta2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Then the electric field in fluctuating vacuum modes at a1 is ˆE(+) vac,1 = � k i � ℏωk 4ϵ0V { √ Tˆb† kei(ωkt−kZ1) + µˆa† 1,kei(ωkt+kz1) −R � Rmˆa† 1,kei(ωkt−kz1) − √ RT � Rmˆa† 2,kei(ωkt−kz1) + � RTm ˆd† kei(ωkt−kZM)} (4) where Rm(Tm) is the reflectance(transmittance) of the mirror and R(T) is the reflectance (transmittance) of the beam splitter, z1(Z1) is the distance from the mirror (laser) to the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' ZM is related to the vacuum source behind the mirror and it can be any number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We add the factor 1 √ 2 for the normalization of the vacuum fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The vacuum mode (ˆa† 1ei(ωt−kz1)) at the detector is the reflected vacuum mode (ˆa† 1ei(ωt+kz1)) at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If two modes are perfectly matched the µ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 4 is 1 and the two counterpropagating modes yield the standing wave mode[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If µ = 0, the fluctuation value from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 7 becomes |α|2T 2 , 3 b BS α2 C d ino ↑ M ino a1 ait is the square of the constant dc current T|α|2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In other words, if we directly measure the fluctuation of the laser intensity, the fluctuation is dependent on the distance (z1) between the mirror and the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Even in photo counting experiments, the photon number fluctuation is related to the vacuum fluctuation, therefor, the photon number fluctuation is also depend on the distance z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If we used the photodetetion theory [6] with instantaneous response of the photodetector [7], ˆI1 = { √ T ˆE(+) cl + ˆE(+) vac,1} × { √ T ˆE(−) cl + ˆE(−) vac,1}, (5) where we normalize the photocurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If the electric field of the local oscillator is considerably greater than the vacuum field, the terms containig α have physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' When the constant dc current T|α|2 2 is neglected, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 5 yields ˆIo 1(z1, Z1) = |α| √ 2[ √ Teiφ{(µe−ik(Z1+z1) − e−ik(Z1−z1)R � Rm)ˆa1 − e−ik(Z1−z1)ˆa2} + √ Te−iφ{(µeik(Z1+z1) − eik(Z1−z1)R � Rm)ˆa† 1 − e−k(Z1−z1)ˆa† 2} + eiφTˆb + e−iφTˆb† + eiφeik(ZM−Z1)� TRTm ˆd + e−iφe−ik(ZM−z1)� TRTm ˆd†], (6) We then evaluate the square of the photocurrent to determine the fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' After squaring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 6, we find the photocurrent fluctuation as follows: ⟨(ˆIo 1)2⟩ = |α|2T 2 {1 + µ2 − 2µR � Rm cos(2kz1)} (7) If µ = 0, the fluctuation value from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 7 becomes |α|2T 2 , which is the square of the con- stant dc current √ T|α| √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In other words, if we directly measure the laser intensity fluctuation, the fluctuation is dependent on the distance (z1) between the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Even in the photo counting experiment, the photon number fluctuation is related to the vacuum fluctuation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' therefore, the photon number fluctuation is also dependent on the distance z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If we consider practical limits such as finite linewidth and finite absorption length, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 7 will change as follows[2, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' ⟨(ˆIo 1)2⟩P = |α|2T 2 {1 + µ2 − 2µR � Rme−z2 1∆k2 × κ[cos(2k0z1 + φ0) − e−κD cos(2k0(z1 + D) + φ0)] � 4k2 0 + κ2 }, (8) 4 where ∆k is the line width of the local oscillator beam with Gaussian line width distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' κ is the absorption coefficient, D is the detector active length, and φ0 = arctan 2k κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We assumed that the probability that a photon is converted into an electron hole pair at distance η from the surface of the detector’s active region is κe−κη[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The two coefficients √Rm and µ depend on the mode matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Even when we used the total mirror, if the mode from the mirror is not perfectly matched with the mode from the laser, the effective reflectance √Rm can not be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Furthermore, the mode a1 to the mirror is reflected by the mirror and then meets at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' At the detector, if two counter-propagating modes are not exactly matched, the coefficient µ cannot be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' To evaluate this mode matching condition, we assume that the amplitude envelope of the electromagnetic wave in the transverse plane is given by a Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Considering the Gaussian modes [10] E(ρ, z) = E0 w0 w(z) exp[− ρ2 w(z)2] exp[−ikz − ik ρ2 2R(z) + iζ(z)] (9) , where w0 is the radius of the beam waist and w(z) = w0 � 1 + ( z z0 )2 R(z) = z(1 + (z0 z )2) ζ(z) = tan−1 z z0 (10) and z0 is defined as follows: z0 = π λw2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' (11) First, we assume that the laser and vacuu modes have the same beam waist w0 at the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Then the laser and vacuum modes are perfectly matched;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' thus, √Rm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' On the other hand, the vacuum Ev(0) starting from the detector propagates to the mirror and reflects at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The returned vacuum Ev(2z1) is not the same Ev(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The coefficient µ can be calculated as follow: µ = | < Ev(0)Ev(2z1)∗ > | � < Ev(0)2 >< Ev(2z1)2 > = (1 + 4z2 1 z2 0 ) 1 4 (1 + 5z2 1 z2 0 + 4 z4 1 z4 0 ) 1 4 (12) 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 2: Mode matching value µ as a function of w0 and z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 2, µ is plotted as a function of z1 and w0, where z1 is the distance between the mirror and detector We assume that the detector and mirror are large enough that all the waves are detected and reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If the distance between the mirror and detector and the size of the beam waist are small enough, the coefficient µ remains near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' If we consider the case where the vacuum field has waist at the mirror, the coefficient µ automatically becomes 1 due to the symmetry, but the vacuum field Ev(z1) at the detector does not matche the laser field EL(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We assumed that the laser field has beam waist w0 at the detector, and the vacuum field has a beam waist wm at the mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Then the effective reflectance √Rm becomes � Rm = | < Ev(z1)EL(0)∗ > | � < Ev(z1)2 >< EL(0)2 > = √ 2 � wm w0 (1 + z2 1 z2m) 1 4 ({(1 + w2m w2 0 )2 + z2 1 z2 0 }{1 + z2 1 z2m}) 1 4 , (13) where zm = π λw2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 3, √Rm is plotted as a function of z1 and wm, where z1 is the distance between the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We set w0 to 100λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Additionally, we also assume that the detector and mirror are large enough that all the waves are detected and reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The coefficient √Rm can be 1 only when the distance between the mirror and detector is small and the size of the beam waist is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 6 μ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='5 3 4 5 log( 0m 3 入 log() 7 2FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 3: Mode matching value √Rm as a function of w0 and z1, with w0 equal to 100λ The mode matching condition is crucial for detecting the modulation effect of the vacuum fluctuation near the mirror, as denoted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' With the usual setup, we can not satisfy the conditions µ = 1 and √Rm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In the next section, we suggest a noble experimental setup that satisfies two mode-matching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' SET UP FOR MODE MATCHING For a laser that has a Gaussian transverse mode, we have to establish a vacuum mode that also has a Gaussian transverse mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 4 displays the setup for perfect mode matching between the laser light mode and a vacuum mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The laser used in the experiment passes through lens L1 and is divided into two by the beam splitter (BS1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The laser is a Gaussian beam and it proceeds according to the Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The light passing through BS1 and traveling to mirror M2 reaches the partial mirror B and yields a beam waist on the L3 side surface of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Similarly, the light reflecting from the mirror M1 passes through the partial reflector A and yields a beam waist on the L2 side surface of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The light passing through A and B passes through the L2 and L3 of the same focal length, respectively, and yields another beam waist on the detector surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The transmittance of light passing through A from M1 is almost 0, and the reflectance of light stemming from the L2 side is almost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In this way, if the mode is perfectly matched using the light passing through B and A, an experimental setup can be established wherein one side of the beam splitter BS2 is a mirror (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 7 /Rm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='5 3 4 5 3 1og wm log() 入 7 2Using this method, the degree of mode matching can be increased compared to that when the experiment is performed by simply placing a plane mirror on one side of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Additionally the experimental constraints caused by the mode matching can be overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The experimental setup in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 4 enables the measurement of how the vacuum fluctuations of the light passing through the beam splitter change when a mirror is placed on one side of the beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' 4: Mode matching setup IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' CONCLUSION AND DISCUSSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The quantum nature of photons is highly dependent on their vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Vac- uum fluctuations can be directly measured via homodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The fluctuation of one quadrature of the vacuum can be less than that of the usual vacuum, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=', squeezed vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Light intensity fluctuations are also dependent on vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Sub-Poisson light can be generated by controlling the vacuum fluctuations based on the nonlinear interaction of light and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' In this study, we proposed the modulation of vacuum fluctuations by inserting a mirror on the unused part of the beam splitter in a homodyne measuring system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Furthermore, we calculated the effect of the line width of the laser and the thickness of the detector layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' The line width can be practically reduced to modulate vacuum fluctuations, but the decrease of the thickness of the detector to modulate vacuum fluctuations is chal- lenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We calculated the effect of mode matching between the vacuum and light fields and 8 BS1 M1 A L1 L2 DD1 M2 BS2 B L3 D2showed that the degree of mode matching obtained by adding a simple mirror in the unused beam splitter may not be sufficient to modulate the vacuum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We present the perfect mode matching method for the vacuum and light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Then, the light intensity fluctuations can be reduced by inserting a beam splitter and a mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We still require a detector with an active layer thinner than the wavelength to obtain a sub-Poisson light as a function of the distance between the mirror and detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' We expect that our simple method of reducing vacuum fluctuations will play a great role in quantum information science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Jhe, A Anderson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Hinds, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Inc, (1989) [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Drummond, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' A 35, 4253 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Yurke, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' A 32, 311 (1985) [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Siegman, Laser (Oxford University Press, Oxford, 1986 ) [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Sze, Semiconductor Devices Physics and Technology (AT&T Bell Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Murray Hill, New Jersey, 1985) [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Saleh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} +page_content=' Teich, Fundamentals of Photonics ( Wiley, Nw York, 1991) 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf'} diff --git a/59FAT4oBgHgl3EQfnB39/content/2301.08627v1.pdf b/59FAT4oBgHgl3EQfnB39/content/2301.08627v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..20d90be34cd2b5b88f6d185a7622a2627faea71f --- /dev/null +++ b/59FAT4oBgHgl3EQfnB39/content/2301.08627v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:aeff6a7369ac0c1d85eef2be35c6202c6b95330852fdafd658be2ee8729d9faa +size 160979 diff --git a/59FAT4oBgHgl3EQfnB39/vector_store/index.faiss b/59FAT4oBgHgl3EQfnB39/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..61929f3c7f6845b29cf897d7ce886b92ceffd0da --- /dev/null +++ b/59FAT4oBgHgl3EQfnB39/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dafd1b99039cf71614020e3030ea8a3b2d078e881ef8cdb3264cda8ac4587cfa +size 1900589 diff --git 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systems increases to unprecedented levels, two problems +occur. First, the spatial stationarity assumption along the antenna elements is no longer valid. Second, +the large array size results in an unacceptably high power consumption if high-resolution analog-to- +digital converters are used. To address these two challenges, we consider a Bussgang linear minimum +mean square error (BLMMSE)-based channel estimator for large scale massive MIMO systems with +one-bit quantizers and a spatially non-stationary channel. Whereas other works usually assume that the +channel covariance is known at the base station, we consider a plug-in BLMMSE estimator that uses an +estimate of the channel covariance and rigorously analyze the distortion produced by using an estimated, +rather than the true, covariance. To cope with the spatial non-stationarity, we introduce dithering into the +quantized signals and provide a theoretical error analysis. In addition, we propose an angular domain +fitting procedure which is based on solving an instance of non-negative least squares. For the multi-user +data transmission phase, we further propose a BLMMSE-based receiver to handle one-bit quantized data +signals. Our numerical results show that the performance of the proposed BLMMSE channel estimator +is very close to the oracle-aided scheme with ideal knowledge of the channel covariance matrix. The +BLMMSE receiver outperforms the conventional maximum-ratio-combining and zero-forcing receivers +in terms of the resulting ergodic sum rate. +1Communications and Information Theory Group (CommIT), Technische Universit¨at Berlin, 10587 Berlin, Germany (e-mail: +{tianyu.yang, caire}@tu-berlin.de). +2Ludwig-Maximilians-Universit¨at Munich, 80333 Munich, Germany (e-mail: maly@math.lmu.de). +3Munich Center for Machine Learning (MCML). +4Utrecht University, 3584 CD Utrecht, Netherlands (e-mail: s.dirksen@uu.nl). +This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which +this version may no longer be accessible. +arXiv:2301.04641v1 [cs.IT] 11 Jan 2023 + +2 +Index Terms +Extra-Large Scale Massive MIMO, Spatially Non-Stationary, One-Bit Quantization, Dithering, Buss- +gang Linear MMSE (BLMMSE). +I. INTRODUCTION +Massive multiple-input-multiple-output (MIMO) has been vastly researched and considered as +an essential technology in 5G wireless communication systems within sub-6 GHz bands [1–3]. +Benefiting from the large number (tens to hundreds) of antennas at the base station (BS) array, +dozens of users can be served in the same time-frequency slots. This results in higher spectrum +and energy efficiency due to the spatial multiplexing and high array gain [1, 4]. Theory shows that +by increasing the array dimension, i.e., the number of antenna elements, it is possible to achieve +higher data rates and to mitigate the impacts of inter-cell interference and thermal noise [5]. +Nevertheless, as the array dimension increases, two new challenges occur. First, some inherent +properties of the channel environment change compared to small-scale MIMO, so that the basic +assumptions of massive MIMO design are no longer valid for large arrays. Specifically, most +existing massive MIMO works are based on the assumption of a spatially stationary channel, +where all antenna elements observe the same far-field propagation from the channel scatters [6–9]. +However, in the very large antenna array regime, spatial non-stationarity has been experimentally +observed [10]. Two main reasons for this non-stationarity are further discussed in [11–13]. First, +with large arrays, the distance between the BS array and some scattering clusters may be smaller +than the Rayleigh distance. As a consequence, user signals impinging onto the BS array cannot +be assumed as far-field propagation and have a spherical wavefront instead of a plane wavefront. +Second, due to the physical large size of the array, some of the scattering clusters may only be +visible to a part of the array. Furthermore, a new deployment of large arrays that are usually +integrated into large structures, e.g., along the walls of buildings [14], was considered as an +extension of massive MIMO and referred to as extra-large scale massive MIMO (XL-MIMO) in +[15]. It is also pointed out in [15] that due to the large dimension (tens of meters) of XL-MIMO, +spatially non-wide sense stationary (non-WSS) characteristics appear along the array. +Aside from the spatially non-WSS property of large antenna arrays, a second major concern is +the hardware cost and power consumption of high-resolution analog-to-digital converters (ADCs). +Commercial high-resolution ADCs (12 to 16 bits) are expensive and their power consumption + +3 +grows exponentially in terms of the number of quantization bits [16]. This problem is even more +severe for wideband systems, where the power consumption of high-resolution ADCs increases +linearly with the signal bandwidth due to required higher sampling rates [17]. To alleviate the +issue of high power consumption, low-resolution ADCs (e.g., 1-3 bits) are utilized for massive +MIMO systems [18–20] and it is shown in [18, 19] that the capacity loss due to the coarse +quantization is approximately equal to only π/2 at low signal-to-noise ratios (SNRs). In massive +MIMO systems the SNR per antenna element may be relatively low, while still achieving an +overall large spectral efficiency over data stream due to the large number of antennas per user +(data stream), such that both spatial multiplexing gain and array gain are achieved. +A. Contributions +Accurate estimation of channel state information (CSI) at the BS is a key factor to achieve +the potential benefits of massive MIMO systems. Taking the spatially non-WSS property into +account, an adaptive grouping sparse Bayesian learning scheme was proposed for uplink channel +estimation in [21]. A model-driven deep learning-based channel reconstruction scheme was +proposed in [22]. On the other hand, many recent works have investigated channel estimators +with one-bit quantized signals in massive MIMO systems, see e.g., [23, 24] and references +therein. Very recently, in [25] a covariance recovery scheme for one-bit sampled non-stationary +signals with time-varying sampling thresholds was proposed, where a modified arcsine law was +further generalized to fit the non-stationary case. However, the study in [25] is not within the +massive MIMO regime. To the best of our knowledge, no work has addressed the problem of +channel estimation with both low-resolution quantization and spatially non-WSS channels in +massive MIMO systems. To fill this gap, in this paper we adopt the Bussgang linear minimum +mean square estimation (BLMMSE) method that was initially proposed in [23], and propose +a BLMMSE-based “plug-in” channel estimator for one-bit Massive MIMO systems with the +spatially non-WSS property.1 Our main contributions are summarized as follows. +• We adopt a BLMMSE channel estimator to deal with the one-bit quantized signal. However, +in contrast to [23] that assumes exact knowledge of the channel covariance at the BS, we +1By “plug-in” we mean that the BLMMSE requires the knowledge of the channel covariance, which is typically assumed to +be known. However, in our setting, also the channel covariance needs to be estimated from low-resolution quantized samples. +The plug-in estimator consists of using the estimated covariance “as if” it were the true one, in the BLMMSE. + +4 +propose a “plug-in” version that instead uses an estimate of the channel covariance. Our first +contribution is a theoretical analysis of the distortion caused by the use of this estimate: we +estimate the mean squared distance between the BLMMSE estimate based on an estimate of +the channel covariance versus the BLMMSE estimate based on the true channel covariance +(see Lemma 1). +• Our second contribution is a method to estimate the channel covariance matrix based on +one-bit samples. We introduce dithering into the one-bit ADCs to cope with the non-Toeplitz +structure of the channel covariance matrix (resulting from the spatially non-WSS channel). +We propose a covariance estimator based on dithered quantized samples and derive bounds +on the estimation error in terms of the maximum norm and the Frobenius norm (Theorem 1). +By combining this result with the aforementioned bound on the MSE achieved by the +BLMMSE with given estimated covariance, we derive a bound on the expected MSE of the +channel estimator in terms of the number of samples used to estimate the channel covariance +(Theorem 2). +• We empirically further enhance the proposed channel covariance estimator by exploiting +the angle domain of the spatially non-WSS channel. Using dictionary functions in the angle +domain, we formulate the channel covariance estimation as a non-negative least-squares +problem (NNLS), which can be efficiently solved by a standard numerical NNLS solver, +e.g., [26], even for very large problem dimensions. +• We design a linear receiver for the uplink (UL) data transmission phase, based on an estimate +of the channel matrix obtained in the training phase, to achieve better rate detection and thus +improve the ergodic sum rate of multi-user severing. Contrary to the conventional maximum- +ratio-combining (MRC) and zero-forcing (ZF) receivers that do not take the quantization into +account, the proposed receiver considers the Bussgang decomposition of one-bit quantized +data signals and uses BLMMSE-based estimation with knowledge of only the estimated +channel covariance matrix. +• Our numerical results show that the proposed BLMMSE channel estimator, which uses the +proposed channel covariance estimator based on dithered quantized samples, is superior +to benchmark methods and achieves a performance very close to the performance of an +oracle-aided scheme using the true channel covariance matrix. The proposed BLMMSE- +based receiver also significantly outperforms MRC and ZF receivers as expected due to its + +5 +specific consideration of quantized signals. +B. Organization +The rest of this paper is organized as follows. In Section II, we introduce the channel model +with the spatially non-stationary property. Section III is devoted to the analysis of the BLMMSE +channel estimator and our results on channel covariance estimation from one-bit quantized +samples. In Section IV, we propose the BLMMSE receiver for the data transmission phase +to obtain a higher sum rate. The numerical results are then provided in Section V. Finally, in +Section VI we conclude our work and provide a discussion of possible future research directions. +C. Notation +For any N ∈ N we write [N] = {1, 2, . . . , N}. We use lower-case, bold lower-case, and +bold upper-case letters to denote scalars, column vectors, and matrices, respectively. The trace, +transpose and Hermitian transpose are respectively denoted by tr(·), (·)T and (·)H. E[·] returns the +mathematical expectation. diag(A) gives a diagonal matrix with diagonal of A, while diag(a) +denotes the diagonal matrix with diagonal equal to a. We denote the M × M identity matrix by +IM. The i-th element of a vector a is denoted by [a]i, while the i-th row and column of a matrix +A are respectively denoted by [A]i,· and [A]·,i. An all-zero matrix is denoted by 0. ∥a∥2 denotes +the Euclidean norm of a vector a. ∥A∥F, ∥A∥, and ∥A∥∞ denote the Frobenius, operator, and +maximum norms of a matrix A. We use ⟨A, B⟩F := tr(AHB) to denote the Frobenius inner +product. We furthermore use a ≲ b to abbreviate a ≤ Cb, for some absolute constant C > 0. +II. SYSTEM MODEL WITH SPATIALLY NON-STATIONARY CHANNEL +Consider a BS equipped with M antennas in a uniform linear array (ULA). We assume that +the channel scattering clusters consist of common and local clusters, where common clusters +are visible to all antennas while local clusters are only visible to a sub-array. An illustration of +the considered scattering geometry is shown in Fig. 1. The channel vector resulting from the +contribution of the common clusters at the n-th time slot is given by +hc +n = +Lc +� +i=1 +ρc +i(n)a(θc +i), +(1) +where Lc denotes the total number of multipaths in the common clusters, ρc +i(n) ∼ CN(0, γc +i ) +is the i-th complex channel gain of the common clusters with its power γc +i , θc +i is the i-th + +6 +User +Local Cluster +BS +ULA +Local Cluster +Common +Cluster +Fig. 1: Illustration of the studied large-scale Massive MIMO system in ULA with spatially non- +WSS channel, where local clusters are only visible to a part of antenna elements while the +common clusters are visible to the whole array. +angle of arrival (AoA), and where a(θ) ∈ CM×1 is the steering vector, whose m-th entry is +[a(θ)]m += ejπ(m−1) sin(θ) by assuming that the antenna spacing is equal to half of the carrier +wavelength, for all m ∈ M, where M = [M] is the antenna index set of all M antennas. +Assume that there are L local clusters and that each local cluster is visible to a consecutive sub- +array. The i-th local cluster is thus visible to M l +i antennas with index set Ml +i, where M l +i = |Ml +i|. +The channel vector resulting from the contribution of the paths in the i-th local cluster at the +n-th time slot is given by +hl +n,i = +Ll +j +� +j=1 +ρl +ij(n)Sia(θl +ij), +(2) +where Ll +i denotes the total number of multipaths in the i-th local cluster, ρl +ij(n) ∼ CN(0, γl +ij) is +the j-th complex channel gain of the i-th local cluster with its power γl +ij, θl +ij is the AoA, and +where Si ∈ CM×M is the diagonal selection matrix indicating the visible sub-array of the i-th +local cluster, whose diagonal is defined as +[Si]m,m = +� +� +� +� +� +1, +m ∈ Ml +i +0, +m ∈ M \ Ml +i. +(3) +We further assume that the channel gains of different paths in all common and local clusters at + +7 +each time slot n, {ρc +i(n)}Lc +i=1 and +� +ρl +ij(n) +�Ll +i +j=1, ∀i ∈ [L], are uncorrelated2. Note that we implicitly +assume that the channel geometry and visibility of all clusters do not change over the channel +geometry coherent time Tc, which is a much longer time period than the channel coherent time +(see [30] and references therein). Concretely, the Angular Power Spectrum (APS) {γc +i }Lc +i=1 and +� +γl +ij +�Ll +i +j=1, ∀i ∈ [L] along with the AoAs {θc +i}Lc +i=1 and +� +θl +ij +�Ll +j +j=1 , ∀i ∈ [L] as well as the selection +matrices {Si}L +i=1 are constant over Tc. Under these assumptions, the total channel vector at n-th +time slot hn and the corresponding total channel covariance matrix Ch are given by +hn = hc +n + +L +� +i=1 +hl +n,i, +(4) +Ch = E[hnhH +n] = Chc + +L +� +i=1 +Chl +i +(5) += +Lc +� +i=1 +γc +i a(θc +i)a(θc +i)H + +L +� +i=1 +Ll +j +� +j=1 +γl +ijSia(θl +ij)a(θl +ij)HSH +i +(6) += Acdiag(γc)(Ac)H + +L +� +i=1 +SiAl +idiag(γl +i)(Al +i)HSH +i , +(7) +where Ac := [a(θc +1), . . . , a(θc +Lc)], γc := [γc +1, . . . , γc +Lc]T and Al +i := [a(θl +i,1), . . . , a(θl +i,Ll +i)], γl +i := +[γl +i,1, . . . , γl +i,Ll +i]T, ∀i ∈ [L]. +The total channel power gain of all common clusters and all local clusters are given by +P c = +Lc +� +i=1 +γc +i +and +P l = +L +� +i=1 +P l +i = +L +� +i=1 +Ll +j +� +j=1 +γl +ij, +(8) +where we assume that P c and P l are normalized such that max(diag(Ch)) = 1. To help +readability, Table I summarizes the model notation. +III. CHANNEL ESTIMATION WITH ONE-BIT SAMPLES +For a generic user under a normalized pilot, the BS receives at the n-th time slot the signal +yn = hn + nn, +(9) +where hn ∼ CN(0, Ch) is the M × 1 channel vector and nn ∼ CN(0, N0IM) is additive white +Gaussian noise (AWGN) with noise power N0. The SNR is thus defined by 1/N0 due to the +2Note that this is a standard assumption specified in, e.g., the channel models of 3GPP standard TR 38.901 [27] and TR +25.996 [28]. This assumption is also implicitly included in the documentation of the well-known channel simulator QuaDRiGa +[29]. + +8 +L +Number of local clusters +Lc, Ll +i +Number of multipaths of common and local clusters +ρc +i, ρl +ij +Complex channel gain of common and local clusters +γc +i, γl +ij +APS of common and local clusters +Si +Diagonal selection matrices of local clusters +θc +i, θl +ij +AoAs of common and local clusters +Ac, Al +i +Matrices of steering vectors of common and local clusters +TABLE I: Summary of the used notations +assumption that max(diag(Ch)) = 1. After one-bit ADC the quantized signal becomes +rn = Q(yn), +(10) +where Q(·) is a suitable one-bit quantizer that is applied separately to the real and imaginary +part. One popular instance for such a quantizer is the complex-sign operator [31] +rnd +n = csign(yn) = 1 +√ +2 +� +sign(Re(yn)) + j sign(Im(yn)) +� +, +(11) +which quantizes the entries of Re(yn) and Im(yn) independently, i.e., the sign-function +sign: R → {−1, 1} +sign(x) = +� +� +� +� +� +1 +x ≥ 0 +−1 +x < 0 +(12) +acts componentwise (memoryless scalar quantization). We use the superscript “nd” (“non-dithered”) +in (11) since the sign-function is applied directly to the samples without dithering. Note that +this type of one-bit quantization looses any scaling information. +A. Bussgang LMMSE channel estimator +We consider channel estimation for a generic time slot. Thus, we ignore the time slot index +n for simplicity. In order to estimate the channel vector h from a quantized sample r, we first +transfer the nonlinear quantizer operation to a statistically equivalent linear formulation via the +well-known Bussgang decomposition [32], which yields +r = Q(y) = Ay + q +(13) += Ah + An + q +(14) += Ah + �n, +(15) +where the linear operator A is called the Bussgang gain, q is a mean-zero random vector that +is uncorrelated with y, and �n := An+q is the total noise. To enforce q to be uncorrelated with + +9 +y, the Bussgang gain A is chosen to minimize the power of the equivalent quantization noise +[33] such that +A = E +� +ryH� +E +� +yyH�−1 = CryC−1 +y , +(16) +where Cry = E +� +ryH� +denotes the covariance between the quantized signal r and the received +signal y. The so-called BLMMSE estimator [23] of the channel vector h given the quantized +signal r is then expressed as +�hBLM = ChrC−1 +r r. +(17) +Note that this is not the optimal MMSE estimator since q is not Gaussian noise. We however +know that the vector q is uncorrelated with the vector y and one can prove that q is also +uncorrelated with the channel vector h, see [23, App. A] for the proof of E[hqH] = 0. Thus, h +is uncorrelated with the total noise �n and consequently we obtain from (15) that +Chr = ChAH. +(18) +Similarly as in [23], A and Cr can be easily computed as follows. For the one-bit quantizer in +(11) and Gaussian inputs, Cry is given as [34], [35, Ch.12] +Cry = +� +2 +πdiag(Cy)− 1 +2Cy +(19) +and combining (19) and (16) we obtain +A = +� +2 +πdiag(Cy)− 1 +2. +(20) +Furthermore, Cr can be obtained using the map Parcsine(·) by the arcsine law [34, 36] as +Cr = Parcsine(Cy) += 2 +π +� +arcsin +� +diag(Cy)− 1 +2Re(Cy)diag(Cy)− 1 +2 +� ++ j arcsin +� +diag(Cy)− 1 +2Im(Cy)diag(Cy)− 1 +2 +� � +. +(21) +With the BLMMSE estimator in hand, (17) has a closed form that only depends on Cy = +Ch + N0IM. Whereas the noise power N0 is normally assumed to be known at the BS3, the +channel covariance matrix Ch still needs to be estimated from samples to finally apply the +BLMMSE estimator (17). Consider an “plug-in estimator” �Cy of Cy, we define the estimated +3This can be achieved via, e.g., low rate control channel. + +10 +channel vector as +�h = �Chr �C−1 +r r, +(22) +where �Chr and �Cr are the estimators of Chr and Cr obtained by replacing Cy by its estimator +�Cy, i.e., +�Chr = �Ch �AH, +�Ch = �Cy−N0IM, +�A = +� +2 +πdiag(�Cy)− 1 +2, +�Cr = Parcsine(�Cy). (23) +The following lemma controls the estimation error in (17) if an estimator �Cy of Cy is used. +Lemma 1: There are absolute constants c1, c2, C > 0 such that the following holds. Let +θ ∈ (0, 1) be fixed. Assume that +���� +� +diag(Cy)− 1 +2Cydiag(Cy)− 1 +2 +� +i,j +���� ≤ 1 − θ, +for all i ̸= j +(24) +and +min +i∈[M] |[Cy]i,i| ≥ θ, +λmin(Cr) ≥ θ, +(25) +where λmin(·) gives the minimal eigenvalue of the matrix. Consider εF > 0, ε∞ > 0 such that +∥�Cy − Cy∥F < εF, +∥�Cy − Cy∥∞ < ε∞ +and assume that +ε∞ ≤ c1 min +� +εF +∥Cy∥F +, +θ3 +∥Cy∥∞ +, θ, 1 +� +and εF ≤ c2 min +� +θ4, +θ6∥Ch∥F +max{1, ∥Ch∥} ∥Cy∥ +� +. +(26) +Then, +E +�����h − �hBLM��� +2 +2 +� +≤ Cθ−6 max{1, ∥Ch∥} ∥Ch∥FεF, +(27) +where the expectation is taken with respect to r. +Proof: See Appendix A. +Remark 1: Let us briefly comment on the assumptions in Lemma 1. As the construction +of the BLMMSE involves the inverses of diag(Cy) and Cr, it is to be expected that in the +situation that these matrices are near-singular, a small error in the estimation of the covariance +can lead to a large difference between �h and �hBLM. This expected behaviour is quantified in +Lemma 1 using the parameter θ. The lower bound on λmin(Cr) is an implicit condition on +Cy. To give a more explicit condition, let us write offdiag(Cr) for the off-diagonal part. Using +that ∥ arcsin(B)∥ ≤ π +2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary material]), we can make the +potentially crude estimate +λmin(Cr) ≥ λmin(diag(Cr)) − ∥ offdiag(Cr)∥ ≥ 1 − ∥ offdiag(Cy)∥, +(28) + +11 +so that it is sufficient if +∥ offdiag(Cy)∥ ≤ 1 − θ. +(29) +Note that the latter condition also implies (24). Finally, let us comment on the condition linking +ε∞ and εF in (26). In the application that follows, we will see that the ℓ∞-error achieved by the +estimator �Cy is a factor M smaller than the achieved Frobenius norm error. As a consequence, +the relation between ε∞ and εF will be satisfied. +♦ +B. Channel covariance estimation from quantized samples +In this part, we present an approach to estimate the covariance matrix Cy from a finite number +of samples so that we can use the estimate �Cy to apply the plug-in BLMMSE channel estimator in +(22). Assume that the BS collects N unquantized i.i.d. samples {yn}N +n=1 for covariance estimation +and applies coarse quantization in the ADCs. In the case of a spatially WSS channel, the diagonal +of Ch is constant and the (non-dithered) one-bit samples rnd +n defined in (11) can be used. Defining +the sample covariance of the quantized samples +�Cnd +r = 1 +N +N +� +n=1 +rnd +n +� +rnd +n +�H , +(30) +the true covariance matrix Cy can then be estimated via the arcsin-law [31, 36, 37] +�Cnd +y = sin +�π +2 Re +� +�Cnd +r +�� ++ j sin +�π +2 Im +� +�Cnd +r +�� +. +(31) +Due to the spatially non-WSS property in our model, however, it is seen from the formulation in +(6) that the channel covariance may have a non-constant diagonal and non-Toeplitz structure. In +such a scenario, the estimator in (31) will perform poorly since it enforces a constant diagonal. +To overcome this limitation of the quantizer csign, we will introduce random dithering [38– +40]. The beneficial effect of dithering in memoryless one-bit quantization was recently rigorously +analyzed in the context of one-bit compressed sensing, see, e.g., [41–46]. We will adapt a +covariance estimator from [37] that uses two-bit dithered quantized samples. Specifically, we +assume that the real and imaginary parts of each entry are quantized independently with two +independent dithers, so that we are given the (dithered) four-bit samples +� +Re(rd +n), Im(rd +n), Re(�rd +n), Im(�rd +n) +� +:= +(32) +� +sign +� +Re(yn) + τ Re +n +� +, sign +� +Im(yn) + τ Im +n +� +, sign +� +Re(yn) + �τ Re +n +� +, sign +� +Im(yn) + �τ Im +n +�� +, + +12 +RF +S/H +1-bit ADC +DSP +S/H +Switch +S/H +1-bit ADC +S/H +Switch +DG +Fig. 2: Illustration of the implementation of dithered one-bit quantizer in the m-th antenna chain. +where the real dithering vectors τ Re +n , τ Im +n , �τ Re +n , �τ Im +n +∈ RM, for n ∈ [N], are independent and +uniformly distributed in [−λ, λ]M and λ > 0 is a tuning parameter. An example of an im- +plementation of such a dithered quantization design is illustrated in Fig. 2. The real part and +imaginary part of the received signal after the radio frequency (RF) circuits are sampled and +stored separately in two sample-and-hold (S/H) circuits. Then, a switch is used to extract in turn +the signals from two S/H circuits and forward them to the one-bit ADC. Meanwhile, a dithering +signal generated by the dithering generator (DG) is added into the one-bit ADC and the analog +signal is dithered quantized. For instance, if the switches connect the a points, the DG generates +random dithering signals τ Re and τ Im. Oppositely, if the b points are connected, �τ Re and �τ Im +are generated from DG. The quantized signals of all antenna chains will be processed in the +digital signal processor (DSP). Note that we use S/H circuits to avoid using two one-bit ADCs +for each real or imaginary part signal. Also, this circuit can be directly used for non-dithered +one-bit quantization by fixing the connection of switches and turn off the DG. +Given N dithered quantized samples from (32), we can estimate Cy via +�Cd +y = 1 +2 +�Cd + 1 +2 +� +�Cd�H +, +(33) +where +�Cd = λ2 +N +N +� +n=1 +rd +n +��rd +n +�H +(34) +is an asymmetric version of the sample covariance matrix scaled with λ2. We can now quantify +the approximation of Cy by �Cd +y for all random vectors y ∈ CM with S-subgaussian coordinates. +Definition 1: We say that a random vector y ∈ CM with covariance matrix Cy has S- + +13 +subgaussian coordinates if, for all p ≥ 2 and j ∈ [M], +max +�� +E +� +|[Re(y)]j|p�� 1 +p, +� +E +� +|[Im(y)]j|p�� 1 +p� +≤ S√p∥Cy∥ +1 +2∞. +(35) +Note that if y ∈ CM is complex Gaussian with mean zero, then both Re(y) and Im(y) are +mean-zero real Gaussian vectors with covariance matrix 1 +2Re(Cy). Hence, y has S-subgausian +coordinates for some absolute constant S. The following estimates, which complement operator +norm error bounds derived in [37], are tailored to be used in Lemma 1. +Theorem 1: Let y ∈ CM be a mean-zero random vector vector with covariance matrix +E +� +yyH� += Cy and S-subgaussian coordinates. Let y1, ..., yN +i.i.d. +∼ y. Then there exists a constant +c > 0 which only depends on S such that if λ2 ≳ log(N)∥Cy∥∞, the covariance estimator �Cd +y +fulfills, for any t ≥ 0, with probability at least 1 − 8e−cNt +����Cd +y − Cy +��� +∞ ≲ λ2 +� +log(M) + t +N +. +(36) +and +����Cd +y − Cy +��� +F ≲ λ2 +� +M 2(log(M) + t) +N +. +(37) +Proof: See Appendix B. +By combining Theorem 1 with Lemma 1, we can derive a bound on the expected estimation +error of the channel vector in terms of the number of samples N used to estimate Cy. +Theorem 2: There exist constants c1, . . . , c4 > 0 depending only on S such that the following +holds. Let y ∈ CM be a zero-mean random vector with covariance matrix Cy and S-subgaussian +coordinates. Let y1, ..., yN +i.i.d. +∼ y. Suppose that Cy, Cr, and θ ∈ (0, 1) satisfy (24) and (25). +Further suppose that λ2 ≥ c1 log(N)∥Cy∥∞ and +N ≥ c2λ4M 2� +θ−6 +θ−12∥Ch∥−2 +F +max{1, ∥Ch∥2}∥Cy∥2� +max{1, ∥Cy∥2 +∞} +� +log(M)+t +� +. (38) +Then, for any t ≥ 0, with probability at least 1 − 8e−c3Nt +E +�����h − �hBLM��� +2 +2 +� +≤ c4λ2θ−6M max{1, ∥Cy∥∞} max{1, ∥Ch∥} ∥Ch∥F +� +log(M) + t +N +. +(39) +Proof: See Appendix C. +Remark 2: Theorem 2 implies that the parameter λ of the uniform distribution of the dithering +vectors must be carefully tuned. Developing a corresponding method is thus desirable. We defer +this open point to future work. +♦ + +14 +C. APS-based channel covariance estimation +Let us now revisit the problem of estimating the channel covariance based on an estimator +�Cy of Cy. Previously we used the basic estimator for the channel covariance given by +�Ch = +� +�Cy − N0IM +� +, +(40) +which may not necessarily be a positive semi-definite matrix. To heuristically improve the +performance of this basic estimator, we further exploit the angle domain and apply a commonly +considered APS-based covariance fitting to estimate the APS and subsequently enhance the +estimate of the channel covariance, see e.g., [9, 47–49]. Specifically, assuming that the visible +antennas of all scattering clusters are known at the BS, i.e., the BS has the exact knowledge of +the selection matrices {Si}L +i=1. Using G Dirac delta functions that are equally spaced in the angle +domain with AoAs {θi}G +i=1 as dictionary, the channel covariance matrix can be approximated as +�Ch(�γ) = �Adiag(�γL+1)�AH + +L +� +i=1 +Si �Adiag(�γi)�AHSH +i , +(41) +where �A := [a(θ1), . . . , a(θG)] and we define �γ ∈ R �G ++ := [�γT +1 , . . . , �γT +L+1]T as the non-negative +coefficients to be estimated, where �G = (L + 1)G. Then, we estimate the coefficients by fitting +the parametric channel covariance �Ch(�γ) to the basic estimator �Ch in terms of the Frobenius +norm. We denote the estimated coefficients by +�γ⋆ = arg min +�γ∈R � +G ++ +����Ch(�γ) − �Ch +��� +2 +F . +(42) +By defining b := vec(�Ci +h) and B := +� +Bl +1, . . . , Bl +L, Bc� +, where +Bc := +� +vec(a(θ1)a(θ1)H), . . . , vec(a(θG)a(θG)H) +� +, +(43) +Bl +i := +� +vec(Sia(θ1)a(θ1)HSH +i ), . . . , vec(Sia(θG)a(θG)HSH +i ) +� +, +∀i ∈ [L], +(44) +we see that �γ⋆ can be obtained by solving the nonnegative least squares (NNLS) problem +min +�γ∈R � +G ++ +∥B�γ − b∥2 +2 . +(45) +This problem can be efficiently solved using a variety of convex optimization techniques (see, +e.g., [50, 51]). In our simulations, we use the novel MATLAB NNLS solver [26] which is much +faster and stabler than the built-in MATLAB function lsqnonneg, especially under large problem +dimensions as considered here. With the estimated ASF �γ⋆ in hand, we obtain �C⋆ +h := �Ch(�γ⋆) +as the final estimator of the channel covariance. + +15 +IV. DATA TRANSMISSION RATE +In the UL data transmission phase, we assume K users simultaneously transmit their data. +The received signal at the BS before quantization is given by +yD = Hs + nD, +(46) +where H ∈ CM×K is the channel matrix of K users, s ∼ CN(0, IK) is the vector of the data +signals of K users and nD ∼ CN(0, N0IM) is additive white Gaussian noise. Note that we use +the subscript ‘D’ to indicate the signal during data transmission. After the one-bit non-dithered +quantization, the quantized signal is given by +rD = Q(yD) = ADHs + ADnD + qD, +(47) +where AD = +� +2 +π(diag(HHH+N0IM))− 1 +2 is the Bussgang gain calculated similarly as in (16) and +(20). By treating interference as noise, we apply a linear receiver WH to separate the quantized +signal into K streams as +�s = WHrD = WHADHs + WH(ADnD + qD). +(48) +The data signal of the k-th user is then decoded by the k-th element of �s as +�sk = wH +k ADhksk + wH +k +K +� +i̸=k +ADhisi + wH +k (ADnD + qD), +(49) +where wk and hk are the k-th columns of the W and H, respectively. The covariance matrix of +the statistically equivalent quantizer noise qD is given by +CqD = CrD − ADCyDAH +D, +(50) +where CyD = HHH+N0IM and the covariance matrix CrD of rD can be obtained via the arcsine +law as CrD = Parcsine(CyD). Note that the quantizer noise qD is non-Gaussian. Considering the +worst case by treating qD as Gaussian distributed with the same covariance matrix CqD, we can +obtain a lower bound of the optimistic ergodic sum rate4 of K users [53], given as +Rsum = +K +� +k=1 +E +� +log2 +� +1 + +|wH +k ADhk|2 +�K +i̸=k |wH +k ADhi|2 + N0∥wH +k AD∥2 +2 + wH +k CqDwk +�� +. +(51) +Now, we consider the design of the linear receiver W. Using the proposed plug-in channel +estimator, the channel matrix H is estimated as �H. Then, the conventional MRC and ZF receivers +4Note that the “optimistic ergodic sum rate” is an upper bound assuming Gaussian signaling since it assumes that the useful +signal coefficient and the interference variance are perfectly known [52]. + +16 +are given as +WH +MRC = �HH, +(52) +WH +ZF = +� +�HH �H +�−1 �HH. +(53) +Note that the conventional MRC and ZF receivers might not perform so well due to the quantized +signal. Therefore, we propose a BLMMSE receiver that takes directly into account the quantized +signal by considering its Bussgang decomposition, which is expected to yield better performance. +Specifically, the BLMMSE receiver is given as +WH +BLM = CsrDC−1 +rD , +(54) +where +CsrD = E +� +srH +D +� += HHAH +D, +CrD = E +� +rDrH +D +� += Parcsine(CyD). +(55) +In practice, using the estimated channel matrix �H, the BLMMSE receiver WH +BLM is given as +WH +BLM = �HH �AH +D +� +Parcsine +� +�CyD +��−1 +, +(56) +where �AD = +� +2 +π(diag( �H �HH + N0IM))− 1 +2 and �CyD = �H �HH + N0IM. +V. SIMULATION RESULTS +In our simulation, we take M = 256 antennas at the BS in ULA. The channel consists of Lc = +3 multipaths in common clusters and L = 2 local clusters, where each local cluster is composed +of three multipaths, i.e., Ll +i = 3, i = 1, 2. The first local cluster is visible to the first quarter +of antennas and the second local cluster is visible to the last quarter of antennas, i.e., Ml +1 = +{1, 2, . . . , M +4 }, Ml +2 = { 3M +4 +1, 3M +4 +2, . . . , M}. The AoAs of the common clusters and the first +and second local clusters are uniformly and randomly generated from [−60, 60], [−60, 0] and +[0, 60] degrees, respectively, i.e., θc +i ∼ U(−60, 60), ∀i ∈ [Lc], θl +1,j ∼ U(−60, 0), ∀j ∈ [Ll +1], θl +2,j ∼ +U(0, 60), ∀j ∈ [Ll +2]. The APS of all multipaths are randomly generated with the constraints P c = +0.3, P l +1 = 0.7, and P l +2 = 0.5. Note that this setting satisfies the assumption of max(diag(Ch)) = +1. The SNR is set to 10dB (equivalently, N0 = 0.1). We consider three different estimators �Cy +for Cy: the estimator (31), based on non-dithered one-bit quantized samples, the estimator (33), +based on dithered one-bit quantized samples, and finally, as a reference, the sample covariance +matrix of the unquantized samples. In the first two cases with quantized samples, we use the +estimator to produce the APS-based estimator �C⋆ +h of the channel covariance Ch, as is detailed +in Section III-C. All results presented below are averaged over 10 random channel geometry +realizations, each with 20 groups of N i.i.d. random channel realizations. + +17 +A. Channel covariance estimation +Given an estimator �C⋆ +h of the channel covariance matrix, we first evaluate it in terms of the +normalized Frobenius norm error, which is given by +ENF = E +� +�� +���Ch − �C⋆ +h +��� +2 +F +∥Ch∥2 +F +� +�� . +(57) +The numerical results under different values of λ are shown in Fig. 3a. It is seen that the choice +of λ significantly influences the results of dithered quantization. A larger number of samples N +can result in a more robust covariance estimation in terms of tuning λ. Moreover, the results +under various numbers of samples N are shown in Fig. 3b, where 3 different choices of λ for +dithered quantization are included. It is observed that the Frobenius norm errors of the dithered +estimators are much smaller than the ones of the non-dithered estimators over a large range of +number of samples N. It is also seen that by finding a proper λ, the estimation performance +can be much improved. Furthermore, in the regime of a small number of samples (e.g., when +N < 100 in our case), the results for the estimator based on dithered quantized samples are even +better than the sample covariance of the unquantized samples. This shows that our algorithm +has a benefit in practical cases with limited number of samples. +B. Channel vector estimation via BLMMSE +Next, we numerically evaluate the BLMMSE-based channel vector estimator in terms of the +normalized MSE, which is given by +ENMSE = +E +����h − �h +��� +2 +2 +� +tr(Ch) +. +(58) +Given an estimated channel covariance, we calculate the NMSE with 100 i.i.d. random channel +realizations. The averaged results under various λ and various N are depicted in Fig. 4a and +Fig. 4b, respectively, where the lower bound is given by the BLMMSE estimator obtained using +the true channel covariance. It is observed again that the choice of λ significantly influences the +estimation performance of the dithered case. By applying a proper λ (e.g., λ = 1 in Fig. 4b) +the channel estimate can be considerably improved compared to the non-dithered case for a +large range of number of samples. Moreover, it is seen from Figs. 3a and 4a that the trend of +tuning λ in BLMMSE based channel estimation is different from the trend in channel covariance +estimation. In the range of 1 ≤ λ ≤ 2, the results of BLMMSE-based channel estimation are no + +18 +longer as robust as the results in covariance estimation even under large N. The optimal choices +of λ for the two estimation problems are also different, e.g., λ⋆ ≈ 1.5 in Fig. 3a and λ⋆ ≈ 1.2 +in Fig. 4a under N = 500. +C. Ergodic sum rate evaluation +Finally, we evaluate the proposed scheme in terms of the ergodic sum rate given in (51). We +test with K = 4 users and assume that the channel geometry of all users follows the setting +described at the beginning of this section. For each channel estimation we test with MRC, ZF, and +BLMMSE receivers given in (52), (53), and (56), respectively. Similarly as in the previous part, +beside the non-dithered and dithered schemes we also provide results based on the true channel +covariance matrix. However, unlike for the channel MSE criterion used in the previous part, the +use of the true covariance is not guaranteed to yield a better sum rate, as a channel estimator +with smaller MSE does not necessarily yield a higher sum rate. Furthermore, we provide results +based on the true channel vectors as upper bounds. +We first present the resulting sum rates under various λ with N = 50 samples for covariance +estimation and BLMMSE channel estimation in Fig. 5a. It is firstly observed that the BLMMSE +receiver performs much better than the ZF and MRC receivers. This is expected since the +BLMMSE receiver takes the quantization into account whereas the conventional ZF and MRC +receivers do not. Next, we observe that the results based on the true covariance do not always +provide the largest sum rate as previously explained. Specifically, among the results with ZF +receivers (the dashed lines) the non-dithered one is the best. Since the ZF and MRC receivers are +designed to deal with non-quantized received signals and perform much worse than the BLMMSE +receiver, we now focus on the results of the BLMMSE receiver, which is also individually +depicted in Fig. 5b with one more case of N = 500 samples. It is noticed again that a larger +number of samples N makes the results with dithering in terms of the sum rate not only better +but also more robust against λ. From Fig. 5b it is seen that the highest sum rate obtained by +the proposed scheme with dithering (under N = 500 and λ ≈ 0.6) is very close to the result +based on the true covariance matrix. This shows the advantage of the proposed scheme for both +channel estimation and multi-user receivers under one-bit quantization. +Finally, we focus on the influence of the number of samples N. The sum rates under various +N of MRC, ZF, and BLMMSE are depicted in Fig. 6a and of only BLMMSE with 3 different + +19 +0.5 +1 +1.5 +2 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +ENF +Unquantized, N = 50 +Unquantized, N = 500 +Non-Dithered, N = 50 +Non-Dithered, N = 500 +Dithered, N = 50 +Dithered, N = 500 +(a) ENF v.s. λ +101 +102 +103 +N +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +ENF +Unquantized +Non-Dithered +Dithered, = 0.66667 +Dithered, = 1 +Dithered, = 1.5 +(b) ENF v.s. N +Fig. 3: Normalized Frobenius-norm error of channel covariance under various λ in (a) and i.i.d. +samples N in (b). +choices of λ for the dithered case are depicted in Fig. 6b. In Fig. 6a we see a similar behavior as +before: the BLMMSE receiver produces better sum rates than the ZF and MRC receivers over a +large range of N. It is seen from Fig. 6b that under the best λ⋆ ≈ 0.6 the proposed scheme with +dithering produces sum rates comparable to the results based on the true channel covariance when +N ≥ 100. It is additionally observed from Fig. 6b that as the number of samples N increases +the difference between the results obtained with different λ is decreasing. This indicates again +that under larger N the results with dithering are more robust to variations in λ. +VI. CONCLUSION AND DISCUSSION +In this work, we proposed a plug-in channel estimator for massive MIMO systems with +spatially non-stationary channels and one-bit quantizers. We analyzed the quantized signal via +the Bussgang decomposition and analyzed the distortion produced by using an estimated, rather +than the true, channel covariance in the construction of the BLMMSE estimator of the channel. +To obtain an estimate of the covariance of the spatially non-stationary channel, we introduced +a channel covariance estimator based on dithered quantized samples and theoretically analyzed +its performance. We further enhanced this estimator using an APS-based NNLS solution. Our +numerical results showed large performance gains of the proposed scheme with dithering in +terms of both channel vector and covariance estimation. Finally, we proposed a BLMMSE- +based receiver tailored to one-bit quantized data signals for the multi-user data transmission + +20 +0.5 +1 +1.5 +2 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +ENMSE +Non-Dithered, N = 50 +Non-Dithered, N = 500 +Dithered, N = 50 +Dithered, N = 500 +True Covariance +(a) EMMSE v.s. λ +101 +102 +103 +N +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +ENMSE +Non-Dithered +Dithered, = 0.66667 +Dithered, = 1 +Dithered, = 1.5 +True Covariance +(b) EMMSE v.s. N +Fig. 4: Normalized MSE of channel vectors via BLMMSE under various λ in (a) and i.i.d. +samples N in (b). +0.5 +1 +1.5 +2 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Rsum +Non-Dithered, N = 50, MRC +Non-Dithered, N = 50, ZF +Non-Dithered, N = 50, BLMMSE +Dithered, N = 50, MRC +Dithered, N = 50, ZF +Dithered, N = 50, BLMMSE +True Covariance, MRC +True Covariance, ZF +True Covariance, BLMMSE +True Channel, MRC +True Channel, ZF +True Channel, BLMMSE +(a) Rsum v.s. λ via MRC, ZF, BLMMSE receivers +0.5 +1 +1.5 +2 +12.6 +12.7 +12.8 +12.9 +13 +13.1 +13.2 +13.3 +13.4 +13.5 +Rsum +Non-Dithered, N = 50, BLMMSE +Non-Dithered, N = 500, BLMMSE +Dithered, N = 50, BLMMSE +Dithered, N = 500, BLMMSE +True Covariance, BLMMSE +(b) Rsum v.s. λ via BLMMSE receiver +Fig. 5: Ergodic sum rate of K = 4 users under various λ via MRC, ZF and BLMMSE receivers +in (a) and enlarged view of results via BLMMSE receiver in (b). +phase and showed in numerical experiments that it outperforms the conventional MRC and ZF +receivers in terms of the resulting ergodic sum rate. +There are two important aspects of our work that can be improved. First, we observed in +the numerical experiments that the hyperparameter λ of the dithering generation influences the +channel estimation significantly. Even though this influence was observed to diminish as the +sample size increases, it is still of significant interest to develop a data-driven method to optimally + +21 +101 +102 +103 +N +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Rsum +Non-Dithered, MRC +Non-Dithered, ZF +Non-Dithered, BLMMSE +Dithered, = 0.66667, MRC +Dithered, = 0.66667, ZF +Dithered, = 0.66667, BLMMSE +True Covariance, MRC +True Covariance, ZF +True Covariance, BLMMSE +True Channel, MRC +True Channel, ZF +True Channel, BLMMSE +(a) Rsum v.s. N via MRC, ZF, BLMMSE receivers +101 +102 +103 +N +12.4 +12.6 +12.8 +13 +13.2 +13.4 +13.6 +Rsum +Non-Dithered, BLMMSE +Dithered, = 0.66667, BLMMSE +Dithered, = 1, BLMMSE +Dithered, = 1.5, BLMMSE +True Covariance, BLMMSE +(b) Rsum v.s. N via BLMMSE receiver +Fig. 6: Ergodic sum rate of K = 4 users under various number of i.i.d. samples N via MRC, +ZF and BLMMSE receivers in (a) and enlarged view of results via BLMMSE receiver in (b). +tune λ. Second, in the proposed APS-based channel covariance estimation scheme, the visibility +of local clusters is assumed to be known at the BS. In practice, however, the visibility of local +clusters is usually not easy to estimate. It is therefore desirable to develop a scheme without +the assumption of known visibility of local clusters. We will investigate these two questions in +future work. +APPENDIX A +PROOF OF LEMMA 1 +Recall that +�hBLM = ChrC−1 +r r = ChAHC−1 +r r = (Cy − N0I)AHC−1 +r r +(59) +and +�h = +� +�Cy − N0I +� +�AH �C−1 +r r, +(60) +where �A and �Cr are defined like A and Cr with Cy being replaced by �Cy. Let us abbreviate +�hBLM = Mr +and +�h = � +Mr. +(61) +Consider α, β, γ > 0 such that mini∈[M] |[Cy]i,i| ≥ α, λmin(Cr) ≥ γ, and +���� +� +diag(Cy)− 1 +2Cydiag(Cy)− 1 +2 +� +i,j +���� ≤ 1 − β, +for all i ̸= j. +(62) +In particular, θ ≤ min{α, β, γ}. We start by writing +E +�����h − �hBLM��� +2 +2 +� += E +� +tr +� +�hBLM � +�hBLM�H +− �hBLM�hH − �h +� +�hBLM�H ++ �h�hH +�� +(63) + +22 += tr +� +MCr +� +M − � +M +�H ++ � +MCr +� +� +M − M +�H� +(64) += +� +MCr + � +MCr, +� +M − � +M +�� +F +(65) += 2 +� +MCr, +� +M − � +M +�� +F − +�� +� +M − M +� +Cr, +� +M − � +M +�� +F +(66) +≤ 2∥MCr∥F ∥M − � +M∥F + ∥Cr∥ ∥M − � +M∥2 +F +(67) +Observe that ∥A∥ ≤ +1 +√α by assumption such that +∥MCr∥F = ∥ChAH∥F ≤ ∥Ch∥F ∥AH∥ ≤ +1 +√α∥Ch∥F. +(68) +Moreover, using that ∥ arcsin(B)∥ ≤ π +2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary Material, Eq. +(4)]), we find +∥Cr∥ ≤ +���diag(Cy)− 1 +2Re(Cy)diag(Cy)− 1 +2 +��� + +���diag(Cy)− 1 +2Im(Cy)diag(Cy)− 1 +2 +��� +(69) +≤ 2 +���diag(Cy)− 1 +2 +��� ∥Cy∥ +���diag(Cy)− 1 +2 +��� ≤ 2 +α ∥Cy∥ . +(70) +We conclude that +E +�����h − �hBLM��� +2 +2 +� +≲ α− 1 +2∥Ch∥F∥M − � +M∥F + α−1 ∥Cy∥ ∥M − � +M∥2 +F. +(71) +We will now show that +���M − � +M +��� +F ≤ κ ≤ α +1 +2 ∥Ch∥F +∥Cy∥ , +(72) +so that we obtain +κ +√α∥Ch∥F as a final estimate. +We start by estimating +���M − � +M +��� +F = +���(Cy − N0I)AHC−1 +r +− +� +�Cy − N0I +� +�AH �C−1 +r +��� +F +(73) +≤ +���Cy − �Cy +��� +F ∥A∥ +��C−1 +r +�� + +����Cy − N0I +��� +F +���A − �A +��� +��C−1 +r +�� +(74) ++ ∥�Cy − N0I∥∥�A∥∥C−1 +r +− �C−1 +r ∥F. +(75) +The first term is clearly bounded by γ−1α− 1 +2εF. To estimate the second term, we note that +����Cy − N0I +��� +F ≤ ∥Ch∥F + +����Cy − Cy +��� +F ≤ ∥Ch∥F + εF and +����Cy − N0I +��� ≤ ∥Ch∥ + εF. +(76) +Furthermore, we use that +Z−1 +1 +− Z−1 +2 += Z−1 +1 (Z2 − Z1)Z−1 +2 +(77) +for any invertible Z1, Z2 of the same dimensions. This yields +����A − A +��� = 2 +π +���diag(�Cy)− 1 +2 +� +diag(Cy) +1 +2 − diag(�Cy) +1 +2 +� +diag(Cy)− 1 +2 +��� +(78) + +23 +≤ 2 +π +���diag(�Cy)− 1 +2 +��� +���diag(Cy) +1 +2 − diag(�Cy) +1 +2 +��� +���diag(Cy)− 1 +2 +��� +(79) +By assumption, we have +���diag(Cy)− 1 +2 +��� = +� +1 +mini[Cy]i,i +≤ +� +1 +α. +(80) +Moreover, since +����Cy − Cy +��� +∞ ≤ α +2 by (26), we find +min +i [�Cy]i,i ≥ min +i [Cy]i,i − +����Cy − Cy +��� +∞ ≥ α +2 +(81) +and so +���diag(�Cy)− 1 +2 +��� ≤ +� +2 +α. +(82) +Note that this also implies that ∥�A∥ ≲ +1 +√α. Using that |√x − √y| ≤ |x−y| +√c +if x ≥ c > 0, y ≥ 0, +we find +���diag(Cy) +1 +2 − diag(�Cy) +1 +2 +��� ≤ +� +1 +α +���Cy − �Cy +��� +∞ = +� +1 +αε∞, +(83) +and hence +���A − �A +��� ≤ 4 +πα− 3 +2ε∞. +(84) +Let us finally estimate the last term on the right hand side of (73). Write cij = [Cy]i,j, ˆcij = +[�Cy]i,j and observe that +����� +ˆcij +� +ˆciiˆcjj +− +cij +√ciicjj +����� ≤ +����� +ˆcij − cij +� +ˆciiˆcjj +����� + |cij| 1 +√ˆcii +����� +1 +� +ˆcjj +− +1 +√cjj +����� + |cij| +1 +� +ˆcjj +���� +1 +√ˆcii +− +1 +√cii +���� (85) +≲ 1 +α +����Cy − Cy +��� +∞ + ∥Cy∥∞ +1 +α2 +����Cy − Cy +��� +∞ +(86) +≲ ∥Cy∥∞ +1 +α2 +����Cy − Cy +��� +∞ ≤ β +2 +(87) +as +ε∞ ≲ β +α2 +∥Cy∥∞ +. +(88) +By (62), this implies that +���� +� +diag(�Cy)− 1 +2 �Cydiag(�Cy)− 1 +2 +� +i,j +���� ≤ 1 − β +2 , +for all i ̸= j. +(89) +Clearly, for any +| arcsin(x) − arcsin(y)| ≤ Lβ|x − y|, +(90) +for all x, y ∈ (−1 + β +2, 1 − β +2) where +Lβ = +sup +0≤z<1− β +2 +� +1 +1 − z2 = +� +1 +1 − (1 − β +2)2 ≤ +� 2 +β . +(91) + +24 +Together with (62) and (89) this yields +∥Cr − �Cr∥F ≲ β−1/2 ���diag(�Cy)− 1 +2Re(�Cy)diag(�Cy)− 1 +2 − diag(Cy)− 1 +2Re(Cy)diag(Cy)− 1 +2 +��� +F ++ β−1/2 ���diag(�Cy)− 1 +2Im(�Cy)diag(�Cy)− 1 +2 − diag(Cy)− 1 +2Im(Cy)diag(Cy)− 1 +2 +��� +F . +(92) +Now observe that +���diag(�Cy)− 1 +2Re(�Cy)diag(�Cy)− 1 +2 − diag(Cy)− 1 +2Re(Cy)diag(Cy)− 1 +2 +��� +F +≤ +���diag(�Cy)− 1 +2 − diag(Cy)− 1 +2 +��� ∥�Cy∥F +���diag(�Cy)− 1 +2 +��� ++ +���diag(Cy)− 1 +2 +��� ∥�Cy − Cy∥F +���diag(�Cy)− 1 +2 +��� ++ +���diag(Cy)− 1 +2 +��� ∥Cy∥F +���diag(�Cy)− 1 +2 − diag(Cy)− 1 +2 +��� +(93) +≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF +(94) +and analogously, +���diag(�Cy)− 1 +2Im(�Cy)diag(�Cy)− 1 +2 − diag(Cy)− 1 +2Im(Cy)diag(Cy)− 1 +2 +��� +F +≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF. +(95) +Hence, +���Cr − �Cr +��� +F ≲ β− 1 +2α−2∥Cy∥Fε∞ + β− 1 +2(α−2ε∞ + α−1)εF. +(96) +By our assumptions on ε∞ and εF, the right hand side is bounded by γ/2 and hence the +assumption ∥C−1 +r ∥ ≤ γ−1 implies that ∥�C−1 +r ∥ ≤ 2γ−1. Using now again (77) we finally arrive +at +���C−1 +r +− �C−1 +r +��� +F ≲ β− 1 +2γ−2 � +α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF +� +. +(97) +Combining all our estimates in (73), we find +���M − � +M +��� +F ≲ γ−1α− 1 +2εF + γ−1(∥Ch∥F + εF)α− 3 +2ε∞ ++ (∥Ch∥ + εF)α− 1 +2β− 1 +2γ−2� +α−2∥Cy∥Fε∞ + +� +α−2ε∞ + α−1� +εF +� +. +(98) +Since +ε∞ ≤ min +� +εF +∥Cy∥F +, 1 +� +, +(99) +we can estimate the right hand side by +κ := c α− 5 +2β− 1 +2γ−2 max{1, ∥Ch∥}εF, +(100) + +25 +for an absolute constant c > 0. Clearly, +κ ≤ α− 1 +2 ∥Ch∥F +∥Cy∥ +(101) +by our assumption on εF, which completes the proof. +APPENDIX B +PROOF OF THEOREM 1 +In the proof of Theorem 1 we will use the following lemmas. The first one bounds the bias +of (34) in terms of λ. +Lemma 2: Let S > 0. There exist constants c1, c2 > 0 depending only on S such that the fol- +lowing holds. Let y ∈ CM be a mean-zero random vector with covariance matrix E +� +yyH� += Cy +and S-subgaussian coordinates. Let λ > 0 and let rRe = sign(Re(y)+τ Re), rIm = sign(Im(y)+ +τ Im), �rRe = sign(Re(y)+�τ Re), and �rIm = sign(Im(y)+�τ Im), where τ Re, τ Im, �τ Re, �τ Im are inde- +pendent and uniformly distributed in [−λ, λ]M and independent of y. Abbreviate r = rRe +jrIm +and �r = �rRe + j�rIm. Then, +��λ2E +� +r�rH� +− Cy +�� +∞ ≤ c1(λ2 + ∥Cy∥∞)e +−c2λ2 +∥Cy∥∞ . +(102) +Proof: The proof of this lemma is a straightforward extension of [37, Lemma 17] to the +complex domain. We include it for the convenience of the reader. First note that +��λ2E +� +r�rH� +− Cy +�� +∞ += +��λ2E +� +(rRe + jrIm)(�rRe + j�rIm)H� +− E +� +(Re(y) + jIm(y))(Re(y) + jIm(y))H��� +∞ +≤ +��λ2E +� +rRe(�rRe)T� +− E +� +Re(y)Re(y)T��� +∞ + +��λ2E +� +rRe(�rIm)T� +− E +� +Re(y)Im(y)T��� +∞ ++ +��λ2E +� +rIm(�rRe)T� +− E +� +Im(y)Re(y)T��� +∞ + +��λ2E +� +rIm(�rIm)T� +− E +� +Im(y)Im(y)T��� +∞ +(103) +Since y has S-subgaussian coordinates, we get from (35) that ∥[Re(y)]i∥ψ2, ∥[Im(y)]i∥ψ2 ≤ +S∥Cy∥ +1 +2∞, for any i ∈ [M], where ∥ · ∥ψ2 denotes the subgaussian norm. Applying [37, Lemma +17] for U = [Re(y)]i and V = [Re(y)]j yields +���λ2E +� +sign +� +[Re(y)]i + [τ Re]i +� +· sign +� +[Re(y)]j + [�τ Re]j +�� +− E +� +[Re(y)]i[Re(y)]j +���� +≲ (λ2 + S2∥Cy∥∞)e +−c +λ2 +S2∥Cy∥∞ . +(104) +Since this holds for any choice of i, j ∈ [M], the first term on the right-hand side of (103) +satisfies the claimed bound. The three other terms can be treated in the same way such that our +claim follows. + +26 +The second lemma is a simple concentration inequality that applies to dithered samples of real +distributions. +Lemma 3: There exist absolute constants c1, c2 > 0 such that the following holds. Let y, �y ∈ +RM be random vectors. Let y1, ..., yN +i.i.d. +∼ y, let �y1, ..., �yN +i.i.d. +∼ �y, and let τ 1, . . . , τ N, �τ 1, . . . , �τ N +be independent and uniformly distributed in [−λ, λ], for λ > 0. Define rk = sign(yk + τ k) and +�rk = sign(�yk + �τ k). If N ≥ c1 log(M), then +Pr +������ +λ2 +N +N +� +k=1 +rk�rT +k − E +� +rk�rT +k +� +����� +∞ +≥ +� +λ4 +� +c1 +log(M) +N ++ t +�� +≤ 2e−c2Nt. +(105) +In particular, the claim holds if y = �y and yi = �yi, for all i ∈ [N]. +Proof: Write Rk +i,j = [rk]i[�rk]j for i, j ∈ [M]. Since |Rk +i,j − E[Rk +i,j]| ≤ 2 for all i, j, k, the +bound is trivial for t ≥ 4. Moreover, by Bernstein’s inequality for bounded random variables +(see, e.g., [54, Theorem 2.8.4]), we find for any u ≤ 8λ2 +Pr +� +1 +N +����� +N +� +k=1 +λ2 � +Rk +i,j − E[Rk +i,j] +� +����� ≥ u +� +≤ 2e +−c min +� +N2u2 +σ2 +i,j +, Nu +2λ2 +� +(106) +≤ 2e +−cN min +� +u2 +λ4 , u +λ2 +� +≤ 2e−c2N u2 +λ4 , +(107) +as +σ2 +i,j := +N +� +k=1 +λ4E +�� +Rk +i,j − E[Rk +i,j] +�2� += +N +� +k=1 +λ4 � +E +� +(Rk +i,j)2� +− +� +E[Rk +i,j] +�2� +≤ λ4N. +(108) +Hence, for any given t < 4 we can set u = +� +λ4 +� +c1 +log(M) +N ++ t +� +and note that u ≤ 8λ2 as +N ≥ c1 log(M). By applying the union bound over all M 2 entries we obtain the result. +Proof of Theorem 1: By the triangle inequality, +����Cd +y − Cy +��� +∞ ≤ +����Cd +y − E +� +�Cd +y +���� +∞ + +���E +� +�Cd +y +� +− Cy +��� +∞ +(109) +Write r = rRe + jrIm and �r = �rRe + j�rIm, where rRe = sign(Re(y) + τ Re), rIm = sign(Im(y) + +τ Im), �rRe = sign(Re(y) + �τ Re), and �rIm = sign(Im(y) + �τ Im). By Lemma 2, +���E +� +�Cd +y +� +− Cy +��� +∞ = +��λ2E +� +r�rH� +− Cy +�� +∞ ≲ +� +λ2 + ∥Cy∥∞ +�2 e +−c2λ2 +∥Cy∥∞ ≲ λ2 +√ +N +, +(110) +where we have used that λ2 ≳ log(N)∥Cy∥∞. To estimate the first term in (109), observe that +����Cd +y − E +� +�Cd +y +���� +∞ = +����Cd − E +� +�Cd���� +∞ +(111) += +����� +� +λ2 +N +N +� +k=1 +(rRe +k + jrIm +k )(�rRe +k + j�rIm +k )H +� +− E +� +λ2 +N +N +� +k=1 +(rRe +k + jrIm +k )(�rRe +k + j�rIm +k )H +� ����� +∞ + +27 +≤ +����� +� +λ2 +N +N +� +k=1 +rRe +k (�rRe +k )T +� +− E +� +λ2 +N +N +� +k=1 +rRe +k (�rRe +k )T +������ +∞ ++ +����� +� +λ2 +N +N +� +k=1 +rRe +k (�rIm +k )T +� +− E +� +λ2 +N +N +� +k=1 +rRe +k (�rIm +k )T +������ +∞ ++ +����� +� +λ2 +N +N +� +k=1 +rIm +k (�rRe +k )T +� +− E +� +λ2 +N +N +� +k=1 +rIm +k (�rRe +k )T +������ +∞ ++ +����� +� +λ2 +N +N +� +k=1 +rIm +k (�rIm +k )T +� +− E +� +λ2 +N +N +� +k=1 +rIm +k (�rIm +k )T +������ +∞ +(112) +Using Lemma 3 for each of the four terms and applying a union bound, we get +Pr +�����Cd +y − E +� +�Cd +y +���� +∞ ≳ λ2 +� +log(M) + t +N +� +≤ 8e−cNt, +(113) +and thus the first statement of Theorem 1. 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Vershynin, High-dimensional probability: An introduction with applications in data science. +Cambridge +University Press, 2018, vol. 47. + diff --git a/5tE3T4oBgHgl3EQfpQqt/content/tmp_files/load_file.txt b/5tE3T4oBgHgl3EQfpQqt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f5da4e464a49880e642c6468f8d13172c86e9e1 --- /dev/null +++ b/5tE3T4oBgHgl3EQfpQqt/content/tmp_files/load_file.txt @@ -0,0 +1,1129 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf,len=1128 +page_content='1 Plug-in Channel Estimation with Dithered Quantized Signals in Spatially Non-Stationary Massive MIMO Systems Tianyu Yang1, Johannes Maly2,3, Sjoerd Dirksen4, and Giuseppe Caire1 Abstract As the array dimension of massive MIMO systems increases to unprecedented levels, two problems occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' First, the spatial stationarity assumption along the antenna elements is no longer valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Second, the large array size results in an unacceptably high power consumption if high-resolution analog-to- digital converters are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To address these two challenges, we consider a Bussgang linear minimum mean square error (BLMMSE)-based channel estimator for large scale massive MIMO systems with one-bit quantizers and a spatially non-stationary channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Whereas other works usually assume that the channel covariance is known at the base station, we consider a plug-in BLMMSE estimator that uses an estimate of the channel covariance and rigorously analyze the distortion produced by using an estimated, rather than the true, covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To cope with the spatial non-stationarity, we introduce dithering into the quantized signals and provide a theoretical error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In addition, we propose an angular domain fitting procedure which is based on solving an instance of non-negative least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' For the multi-user data transmission phase, we further propose a BLMMSE-based receiver to handle one-bit quantized data signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Our numerical results show that the performance of the proposed BLMMSE channel estimator is very close to the oracle-aided scheme with ideal knowledge of the channel covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The BLMMSE receiver outperforms the conventional maximum-ratio-combining and zero-forcing receivers in terms of the resulting ergodic sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 1Communications and Information Theory Group (CommIT), Technische Universit¨at Berlin, 10587 Berlin, Germany (e-mail: {tianyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='yang, caire}@tu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 2Ludwig-Maximilians-Universit¨at Munich, 80333 Munich, Germany (e-mail: maly@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3Munich Center for Machine Learning (MCML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4Utrecht University, 3584 CD Utrecht, Netherlands (e-mail: s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='dirksen@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='nl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='04641v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='IT] 11 Jan 2023 2 Index Terms Extra-Large Scale Massive MIMO, Spatially Non-Stationary, One-Bit Quantization, Dithering, Buss- gang Linear MMSE (BLMMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' INTRODUCTION Massive multiple-input-multiple-output (MIMO) has been vastly researched and considered as an essential technology in 5G wireless communication systems within sub-6 GHz bands [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Benefiting from the large number (tens to hundreds) of antennas at the base station (BS) array, dozens of users can be served in the same time-frequency slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This results in higher spectrum and energy efficiency due to the spatial multiplexing and high array gain [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Theory shows that by increasing the array dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', the number of antenna elements, it is possible to achieve higher data rates and to mitigate the impacts of inter-cell interference and thermal noise [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Nevertheless, as the array dimension increases, two new challenges occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' First, some inherent properties of the channel environment change compared to small-scale MIMO, so that the basic assumptions of massive MIMO design are no longer valid for large arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Specifically, most existing massive MIMO works are based on the assumption of a spatially stationary channel, where all antenna elements observe the same far-field propagation from the channel scatters [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' However, in the very large antenna array regime, spatial non-stationarity has been experimentally observed [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Two main reasons for this non-stationarity are further discussed in [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' First, with large arrays, the distance between the BS array and some scattering clusters may be smaller than the Rayleigh distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' As a consequence, user signals impinging onto the BS array cannot be assumed as far-field propagation and have a spherical wavefront instead of a plane wavefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Second, due to the physical large size of the array, some of the scattering clusters may only be visible to a part of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Furthermore, a new deployment of large arrays that are usually integrated into large structures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', along the walls of buildings [14], was considered as an extension of massive MIMO and referred to as extra-large scale massive MIMO (XL-MIMO) in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is also pointed out in [15] that due to the large dimension (tens of meters) of XL-MIMO, spatially non-wide sense stationary (non-WSS) characteristics appear along the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Aside from the spatially non-WSS property of large antenna arrays, a second major concern is the hardware cost and power consumption of high-resolution analog-to-digital converters (ADCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Commercial high-resolution ADCs (12 to 16 bits) are expensive and their power consumption 3 grows exponentially in terms of the number of quantization bits [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This problem is even more severe for wideband systems, where the power consumption of high-resolution ADCs increases linearly with the signal bandwidth due to required higher sampling rates [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To alleviate the issue of high power consumption, low-resolution ADCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', 1-3 bits) are utilized for massive MIMO systems [18–20] and it is shown in [18, 19] that the capacity loss due to the coarse quantization is approximately equal to only π/2 at low signal-to-noise ratios (SNRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In massive MIMO systems the SNR per antenna element may be relatively low, while still achieving an overall large spectral efficiency over data stream due to the large number of antennas per user (data stream), such that both spatial multiplexing gain and array gain are achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Contributions Accurate estimation of channel state information (CSI) at the BS is a key factor to achieve the potential benefits of massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Taking the spatially non-WSS property into account, an adaptive grouping sparse Bayesian learning scheme was proposed for uplink channel estimation in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' A model-driven deep learning-based channel reconstruction scheme was proposed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' On the other hand, many recent works have investigated channel estimators with one-bit quantized signals in massive MIMO systems, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [23, 24] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Very recently, in [25] a covariance recovery scheme for one-bit sampled non-stationary signals with time-varying sampling thresholds was proposed, where a modified arcsine law was further generalized to fit the non-stationary case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' However, the study in [25] is not within the massive MIMO regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To the best of our knowledge, no work has addressed the problem of channel estimation with both low-resolution quantization and spatially non-WSS channels in massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To fill this gap, in this paper we adopt the Bussgang linear minimum mean square estimation (BLMMSE) method that was initially proposed in [23], and propose a BLMMSE-based “plug-in” channel estimator for one-bit Massive MIMO systems with the spatially non-WSS property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 Our main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We adopt a BLMMSE channel estimator to deal with the one-bit quantized signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' However, in contrast to [23] that assumes exact knowledge of the channel covariance at the BS, we 1By “plug-in” we mean that the BLMMSE requires the knowledge of the channel covariance, which is typically assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' However, in our setting, also the channel covariance needs to be estimated from low-resolution quantized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The plug-in estimator consists of using the estimated covariance “as if” it were the true one, in the BLMMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4 propose a “plug-in” version that instead uses an estimate of the channel covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Our first contribution is a theoretical analysis of the distortion caused by the use of this estimate: we estimate the mean squared distance between the BLMMSE estimate based on an estimate of the channel covariance versus the BLMMSE estimate based on the true channel covariance (see Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Our second contribution is a method to estimate the channel covariance matrix based on one-bit samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We introduce dithering into the one-bit ADCs to cope with the non-Toeplitz structure of the channel covariance matrix (resulting from the spatially non-WSS channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We propose a covariance estimator based on dithered quantized samples and derive bounds on the estimation error in terms of the maximum norm and the Frobenius norm (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By combining this result with the aforementioned bound on the MSE achieved by the BLMMSE with given estimated covariance, we derive a bound on the expected MSE of the channel estimator in terms of the number of samples used to estimate the channel covariance (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We empirically further enhance the proposed channel covariance estimator by exploiting the angle domain of the spatially non-WSS channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using dictionary functions in the angle domain, we formulate the channel covariance estimation as a non-negative least-squares problem (NNLS), which can be efficiently solved by a standard numerical NNLS solver, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [26], even for very large problem dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We design a linear receiver for the uplink (UL) data transmission phase, based on an estimate of the channel matrix obtained in the training phase, to achieve better rate detection and thus improve the ergodic sum rate of multi-user severing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Contrary to the conventional maximum- ratio-combining (MRC) and zero-forcing (ZF) receivers that do not take the quantization into account, the proposed receiver considers the Bussgang decomposition of one-bit quantized data signals and uses BLMMSE-based estimation with knowledge of only the estimated channel covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Our numerical results show that the proposed BLMMSE channel estimator, which uses the proposed channel covariance estimator based on dithered quantized samples, is superior to benchmark methods and achieves a performance very close to the performance of an oracle-aided scheme using the true channel covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The proposed BLMMSE- based receiver also significantly outperforms MRC and ZF receivers as expected due to its 5 specific consideration of quantized signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Organization The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In Section II, we introduce the channel model with the spatially non-stationary property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Section III is devoted to the analysis of the BLMMSE channel estimator and our results on channel covariance estimation from one-bit quantized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In Section IV, we propose the BLMMSE receiver for the data transmission phase to obtain a higher sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The numerical results are then provided in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Finally, in Section VI we conclude our work and provide a discussion of possible future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Notation For any N ∈ N we write [N] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We use lower-case, bold lower-case, and bold upper-case letters to denote scalars, column vectors, and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The trace, transpose and Hermitian transpose are respectively denoted by tr(·), (·)T and (·)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' E[·] returns the mathematical expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' diag(A) gives a diagonal matrix with diagonal of A, while diag(a) denotes the diagonal matrix with diagonal equal to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We denote the M × M identity matrix by IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The i-th element of a vector a is denoted by [a]i, while the i-th row and column of a matrix A are respectively denoted by [A]i,· and [A]·,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' An all-zero matrix is denoted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∥a∥2 denotes the Euclidean norm of a vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∥A∥F, ∥A∥, and ∥A∥∞ denote the Frobenius, operator, and maximum norms of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We use ⟨A, B⟩F := tr(AHB) to denote the Frobenius inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We furthermore use a ≲ b to abbreviate a ≤ Cb, for some absolute constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' SYSTEM MODEL WITH SPATIALLY NON-STATIONARY CHANNEL Consider a BS equipped with M antennas in a uniform linear array (ULA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We assume that the channel scattering clusters consist of common and local clusters, where common clusters are visible to all antennas while local clusters are only visible to a sub-array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' An illustration of the considered scattering geometry is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The channel vector resulting from the contribution of the common clusters at the n-th time slot is given by hc n = Lc � i=1 ρc i(n)a(θc i), (1) where Lc denotes the total number of multipaths in the common clusters, ρc i(n) ∼ CN(0, γc i ) is the i-th complex channel gain of the common clusters with its power γc i , θc i is the i-th 6 User Local Cluster BS ULA Local Cluster Common Cluster Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 1: Illustration of the studied large-scale Massive MIMO system in ULA with spatially non- WSS channel, where local clusters are only visible to a part of antenna elements while the common clusters are visible to the whole array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' angle of arrival (AoA), and where a(θ) ∈ CM×1 is the steering vector, whose m-th entry is [a(θ)]m = ejπ(m−1) sin(θ) by assuming that the antenna spacing is equal to half of the carrier wavelength, for all m ∈ M, where M = [M] is the antenna index set of all M antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Assume that there are L local clusters and that each local cluster is visible to a consecutive sub- array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The i-th local cluster is thus visible to M l i antennas with index set Ml i, where M l i = |Ml i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The channel vector resulting from the contribution of the paths in the i-th local cluster at the n-th time slot is given by hl n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i = Ll j � j=1 ρl ij(n)Sia(θl ij),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (2) where Ll i denotes the total number of multipaths in the i-th local cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ρl ij(n) ∼ CN(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' γl ij) is the j-th complex channel gain of the i-th local cluster with its power γl ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' θl ij is the AoA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' and where Si ∈ CM×M is the diagonal selection matrix indicating the visible sub-array of the i-th local cluster,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' whose diagonal is defined as [Si]m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='m = � � � � � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' m ∈ Ml i 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' m ∈ M \\ Ml i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (3) We further assume that the channel gains of different paths in all common and local clusters at 7 each time slot n, {ρc i(n)}Lc i=1 and � ρl ij(n) �Ll i j=1, ∀i ∈ [L], are uncorrelated2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that we implicitly assume that the channel geometry and visibility of all clusters do not change over the channel geometry coherent time Tc, which is a much longer time period than the channel coherent time (see [30] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Concretely, the Angular Power Spectrum (APS) {γc i }Lc i=1 and � γl ij �Ll i j=1, ∀i ∈ [L] along with the AoAs {θc i}Lc i=1 and � θl ij �Ll j j=1 , ∀i ∈ [L] as well as the selection matrices {Si}L i=1 are constant over Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Under these assumptions, the total channel vector at n-th time slot hn and the corresponding total channel covariance matrix Ch are given by hn = hc n + L � i=1 hl n,i, (4) Ch = E[hnhH n] = Chc + L � i=1 Chl i (5) = Lc � i=1 γc i a(θc i)a(θc i)H + L � i=1 Ll j � j=1 γl ijSia(θl ij)a(θl ij)HSH i (6) = Acdiag(γc)(Ac)H + L � i=1 SiAl idiag(γl i)(Al i)HSH i , (7) where Ac := [a(θc 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , a(θc Lc)], γc := [γc 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , γc Lc]T and Al i := [a(θl i,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , a(θl i,Ll i)], γl i := [γl i,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , γl i,Ll i]T, ∀i ∈ [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The total channel power gain of all common clusters and all local clusters are given by P c = Lc � i=1 γc i and P l = L � i=1 P l i = L � i=1 Ll j � j=1 γl ij, (8) where we assume that P c and P l are normalized such that max(diag(Ch)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To help readability, Table I summarizes the model notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' CHANNEL ESTIMATION WITH ONE-BIT SAMPLES For a generic user under a normalized pilot, the BS receives at the n-th time slot the signal yn = hn + nn, (9) where hn ∼ CN(0, Ch) is the M × 1 channel vector and nn ∼ CN(0, N0IM) is additive white Gaussian noise (AWGN) with noise power N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The SNR is thus defined by 1/N0 due to the 2Note that this is a standard assumption specified in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', the channel models of 3GPP standard TR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='901 [27] and TR 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='996 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This assumption is also implicitly included in the documentation of the well-known channel simulator QuaDRiGa [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 8 L Number of local clusters Lc, Ll i Number of multipaths of common and local clusters ρc i, ρl ij Complex channel gain of common and local clusters γc i, γl ij APS of common and local clusters Si Diagonal selection matrices of local clusters θc i, θl ij AoAs of common and local clusters Ac, Al i Matrices of steering vectors of common and local clusters TABLE I: Summary of the used notations assumption that max(diag(Ch)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' After one-bit ADC the quantized signal becomes rn = Q(yn), (10) where Q(·) is a suitable one-bit quantizer that is applied separately to the real and imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' One popular instance for such a quantizer is the complex-sign operator [31] rnd n = csign(yn) = 1 √ 2 � sign(Re(yn)) + j sign(Im(yn)) � , (11) which quantizes the entries of Re(yn) and Im(yn) independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', the sign-function sign: R → {−1, 1} sign(x) = � � � � � 1 x ≥ 0 −1 x < 0 (12) acts componentwise (memoryless scalar quantization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We use the superscript “nd” (“non-dithered”) in (11) since the sign-function is applied directly to the samples without dithering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that this type of one-bit quantization looses any scaling information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Bussgang LMMSE channel estimator We consider channel estimation for a generic time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Thus, we ignore the time slot index n for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In order to estimate the channel vector h from a quantized sample r, we first transfer the nonlinear quantizer operation to a statistically equivalent linear formulation via the well-known Bussgang decomposition [32], which yields r = Q(y) = Ay + q (13) = Ah + An + q (14) = Ah + �n, (15) where the linear operator A is called the Bussgang gain, q is a mean-zero random vector that is uncorrelated with y, and �n := An+q is the total noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To enforce q to be uncorrelated with 9 y, the Bussgang gain A is chosen to minimize the power of the equivalent quantization noise [33] such that A = E � ryH� E � yyH�−1 = CryC−1 y , (16) where Cry = E � ryH� denotes the covariance between the quantized signal r and the received signal y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The so-called BLMMSE estimator [23] of the channel vector h given the quantized signal r is then expressed as �hBLM = ChrC−1 r r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (17) Note that this is not the optimal MMSE estimator since q is not Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We however know that the vector q is uncorrelated with the vector y and one can prove that q is also uncorrelated with the channel vector h, see [23, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' A] for the proof of E[hqH] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Thus, h is uncorrelated with the total noise �n and consequently we obtain from (15) that Chr = ChAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (18) Similarly as in [23], A and Cr can be easily computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' For the one-bit quantizer in (11) and Gaussian inputs, Cry is given as [34], [35, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='12] Cry = � 2 πdiag(Cy)− 1 2Cy (19) and combining (19) and (16) we obtain A = � 2 πdiag(Cy)− 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (20) Furthermore, Cr can be obtained using the map Parcsine(·) by the arcsine law [34, 36] as Cr = Parcsine(Cy) = 2 π � arcsin � diag(Cy)− 1 2Re(Cy)diag(Cy)− 1 2 � + j arcsin � diag(Cy)− 1 2Im(Cy)diag(Cy)− 1 2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (21) With the BLMMSE estimator in hand, (17) has a closed form that only depends on Cy = Ch + N0IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Whereas the noise power N0 is normally assumed to be known at the BS3, the channel covariance matrix Ch still needs to be estimated from samples to finally apply the BLMMSE estimator (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Consider an “plug-in estimator” �Cy of Cy, we define the estimated 3This can be achieved via, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', low rate control channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 10 channel vector as �h = �Chr �C−1 r r, (22) where �Chr and �Cr are the estimators of Chr and Cr obtained by replacing Cy by its estimator �Cy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', �Chr = �Ch �AH, �Ch = �Cy−N0IM, �A = � 2 πdiag(�Cy)− 1 2, �Cr = Parcsine(�Cy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (23) The following lemma controls the estimation error in (17) if an estimator �Cy of Cy is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Lemma 1: There are absolute constants c1, c2, C > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let θ ∈ (0, 1) be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Assume that ���� � diag(Cy)− 1 2Cydiag(Cy)− 1 2 � i,j ���� ≤ 1 − θ, for all i ̸= j (24) and min i∈[M] |[Cy]i,i| ≥ θ, λmin(Cr) ≥ θ, (25) where λmin(·) gives the minimal eigenvalue of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Consider εF > 0, ε∞ > 0 such that ∥�Cy − Cy∥F < εF, ∥�Cy − Cy∥∞ < ε∞ and assume that ε∞ ≤ c1 min � εF ∥Cy∥F , θ3 ∥Cy∥∞ , θ, 1 � and εF ≤ c2 min � θ4, θ6∥Ch∥F max{1, ∥Ch∥} ∥Cy∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (26) Then, E �����h − �hBLM��� 2 2 � ≤ Cθ−6 max{1, ∥Ch∥} ∥Ch∥FεF, (27) where the expectation is taken with respect to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Remark 1: Let us briefly comment on the assumptions in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' As the construction of the BLMMSE involves the inverses of diag(Cy) and Cr, it is to be expected that in the situation that these matrices are near-singular, a small error in the estimation of the covariance can lead to a large difference between �h and �hBLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This expected behaviour is quantified in Lemma 1 using the parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The lower bound on λmin(Cr) is an implicit condition on Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To give a more explicit condition, let us write offdiag(Cr) for the off-diagonal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using that ∥ arcsin(B)∥ ≤ π 2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary material]), we can make the potentially crude estimate λmin(Cr) ≥ λmin(diag(Cr)) − ∥ offdiag(Cr)∥ ≥ 1 − ∥ offdiag(Cy)∥, (28) 11 so that it is sufficient if ∥ offdiag(Cy)∥ ≤ 1 − θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (29) Note that the latter condition also implies (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Finally, let us comment on the condition linking ε∞ and εF in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In the application that follows, we will see that the ℓ∞-error achieved by the estimator �Cy is a factor M smaller than the achieved Frobenius norm error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' As a consequence, the relation between ε∞ and εF will be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ♦ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Channel covariance estimation from quantized samples In this part, we present an approach to estimate the covariance matrix Cy from a finite number of samples so that we can use the estimate �Cy to apply the plug-in BLMMSE channel estimator in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Assume that the BS collects N unquantized i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' samples {yn}N n=1 for covariance estimation and applies coarse quantization in the ADCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In the case of a spatially WSS channel, the diagonal of Ch is constant and the (non-dithered) one-bit samples rnd n defined in (11) can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Defining the sample covariance of the quantized samples �Cnd r = 1 N N � n=1 rnd n � rnd n �H , (30) the true covariance matrix Cy can then be estimated via the arcsin-law [31, 36, 37] �Cnd y = sin �π 2 Re � �Cnd r �� + j sin �π 2 Im � �Cnd r �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (31) Due to the spatially non-WSS property in our model, however, it is seen from the formulation in (6) that the channel covariance may have a non-constant diagonal and non-Toeplitz structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In such a scenario, the estimator in (31) will perform poorly since it enforces a constant diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To overcome this limitation of the quantizer csign, we will introduce random dithering [38– 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The beneficial effect of dithering in memoryless one-bit quantization was recently rigorously analyzed in the context of one-bit compressed sensing, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [41–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We will adapt a covariance estimator from [37] that uses two-bit dithered quantized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Specifically, we assume that the real and imaginary parts of each entry are quantized independently with two independent dithers, so that we are given the (dithered) four-bit samples � Re(rd n), Im(rd n), Re(�rd n), Im(�rd n) � := (32) � sign � Re(yn) + τ Re n � , sign � Im(yn) + τ Im n � , sign � Re(yn) + �τ Re n � , sign � Im(yn) + �τ Im n �� , 12 RF S/H 1-bit ADC DSP S/H Switch S/H 1-bit ADC S/H Switch DG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 2: Illustration of the implementation of dithered one-bit quantizer in the m-th antenna chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' where the real dithering vectors τ Re n , τ Im n , �τ Re n , �τ Im n ∈ RM, for n ∈ [N], are independent and uniformly distributed in [−λ, λ]M and λ > 0 is a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' An example of an im- plementation of such a dithered quantization design is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The real part and imaginary part of the received signal after the radio frequency (RF) circuits are sampled and stored separately in two sample-and-hold (S/H) circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Then, a switch is used to extract in turn the signals from two S/H circuits and forward them to the one-bit ADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Meanwhile, a dithering signal generated by the dithering generator (DG) is added into the one-bit ADC and the analog signal is dithered quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' For instance, if the switches connect the a points, the DG generates random dithering signals τ Re and τ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Oppositely, if the b points are connected, �τ Re and �τ Im are generated from DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The quantized signals of all antenna chains will be processed in the digital signal processor (DSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that we use S/H circuits to avoid using two one-bit ADCs for each real or imaginary part signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Also, this circuit can be directly used for non-dithered one-bit quantization by fixing the connection of switches and turn off the DG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Given N dithered quantized samples from (32), we can estimate Cy via �Cd y = 1 2 �Cd + 1 2 � �Cd�H , (33) where �Cd = λ2 N N � n=1 rd n ��rd n �H (34) is an asymmetric version of the sample covariance matrix scaled with λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We can now quantify the approximation of Cy by �Cd y for all random vectors y ∈ CM with S-subgaussian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Definition 1: We say that a random vector y ∈ CM with covariance matrix Cy has S- 13 subgaussian coordinates if, for all p ≥ 2 and j ∈ [M], max �� E � |[Re(y)]j|p�� 1 p, � E � |[Im(y)]j|p�� 1 p� ≤ S√p∥Cy∥ 1 2∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (35) Note that if y ∈ CM is complex Gaussian with mean zero, then both Re(y) and Im(y) are mean-zero real Gaussian vectors with covariance matrix 1 2Re(Cy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Hence, y has S-subgausian coordinates for some absolute constant S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The following estimates, which complement operator norm error bounds derived in [37], are tailored to be used in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Theorem 1: Let y ∈ CM be a mean-zero random vector vector with covariance matrix E � yyH� = Cy and S-subgaussian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', yN i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Then there exists a constant c > 0 which only depends on S such that if λ2 ≳ log(N)∥Cy∥∞, the covariance estimator �Cd y fulfills, for any t ≥ 0, with probability at least 1 − 8e−cNt ����Cd y − Cy ��� ∞ ≲ λ2 � log(M) + t N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (36) and ����Cd y − Cy ��� F ≲ λ2 � M 2(log(M) + t) N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (37) Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By combining Theorem 1 with Lemma 1, we can derive a bound on the expected estimation error of the channel vector in terms of the number of samples N used to estimate Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Theorem 2: There exist constants c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , c4 > 0 depending only on S such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y ∈ CM be a zero-mean random vector with covariance matrix Cy and S-subgaussian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', yN i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Suppose that Cy, Cr, and θ ∈ (0, 1) satisfy (24) and (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Further suppose that λ2 ≥ c1 log(N)∥Cy∥∞ and N ≥ c2λ4M 2� θ−6 +θ−12∥Ch∥−2 F max{1, ∥Ch∥2}∥Cy∥2� max{1, ∥Cy∥2 ∞} � log(M)+t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (38) Then, for any t ≥ 0, with probability at least 1 − 8e−c3Nt E �����h − �hBLM��� 2 2 � ≤ c4λ2θ−6M max{1, ∥Cy∥∞} max{1, ∥Ch∥} ∥Ch∥F � log(M) + t N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (39) Proof: See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Remark 2: Theorem 2 implies that the parameter λ of the uniform distribution of the dithering vectors must be carefully tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Developing a corresponding method is thus desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We defer this open point to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ♦ 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' APS-based channel covariance estimation Let us now revisit the problem of estimating the channel covariance based on an estimator �Cy of Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Previously we used the basic estimator for the channel covariance given by �Ch = � �Cy − N0IM � , (40) which may not necessarily be a positive semi-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To heuristically improve the performance of this basic estimator, we further exploit the angle domain and apply a commonly considered APS-based covariance fitting to estimate the APS and subsequently enhance the estimate of the channel covariance, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [9, 47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Specifically, assuming that the visible antennas of all scattering clusters are known at the BS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', the BS has the exact knowledge of the selection matrices {Si}L i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using G Dirac delta functions that are equally spaced in the angle domain with AoAs {θi}G i=1 as dictionary, the channel covariance matrix can be approximated as �Ch(�γ) = �Adiag(�γL+1)�AH + L � i=1 Si �Adiag(�γi)�AHSH i , (41) where �A := [a(θ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , a(θG)] and we define �γ ∈ R �G + := [�γT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , �γT L+1]T as the non-negative coefficients to be estimated, where �G = (L + 1)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Then, we estimate the coefficients by fitting the parametric channel covariance �Ch(�γ) to the basic estimator �Ch in terms of the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We denote the estimated coefficients by �γ⋆ = arg min �γ∈R � G + ����Ch(�γ) − �Ch ��� 2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (42) By defining b := vec(�Ci h) and B := � Bl 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , Bl L, Bc� , where Bc := � vec(a(θ1)a(θ1)H), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , vec(a(θG)a(θG)H) � , (43) Bl i := � vec(Sia(θ1)a(θ1)HSH i ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , vec(Sia(θG)a(θG)HSH i ) � , ∀i ∈ [L], (44) we see that �γ⋆ can be obtained by solving the nonnegative least squares (NNLS) problem min �γ∈R � G + ∥B�γ − b∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (45) This problem can be efficiently solved using a variety of convex optimization techniques (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [50, 51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In our simulations, we use the novel MATLAB NNLS solver [26] which is much faster and stabler than the built-in MATLAB function lsqnonneg, especially under large problem dimensions as considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' With the estimated ASF �γ⋆ in hand, we obtain �C⋆ h := �Ch(�γ⋆) as the final estimator of the channel covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 15 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' DATA TRANSMISSION RATE In the UL data transmission phase, we assume K users simultaneously transmit their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The received signal at the BS before quantization is given by yD = Hs + nD, (46) where H ∈ CM×K is the channel matrix of K users, s ∼ CN(0, IK) is the vector of the data signals of K users and nD ∼ CN(0, N0IM) is additive white Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that we use the subscript ‘D’ to indicate the signal during data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' After the one-bit non-dithered quantization, the quantized signal is given by rD = Q(yD) = ADHs + ADnD + qD, (47) where AD = � 2 π(diag(HHH+N0IM))− 1 2 is the Bussgang gain calculated similarly as in (16) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By treating interference as noise, we apply a linear receiver WH to separate the quantized signal into K streams as �s = WHrD = WHADHs + WH(ADnD + qD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (48) The data signal of the k-th user is then decoded by the k-th element of �s as �sk = wH k ADhksk + wH k K � i̸=k ADhisi + wH k (ADnD + qD), (49) where wk and hk are the k-th columns of the W and H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The covariance matrix of the statistically equivalent quantizer noise qD is given by CqD = CrD − ADCyDAH D, (50) where CyD = HHH+N0IM and the covariance matrix CrD of rD can be obtained via the arcsine law as CrD = Parcsine(CyD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that the quantizer noise qD is non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Considering the worst case by treating qD as Gaussian distributed with the same covariance matrix CqD, we can obtain a lower bound of the optimistic ergodic sum rate4 of K users [53], given as Rsum = K � k=1 E � log2 � 1 + |wH k ADhk|2 �K i̸=k |wH k ADhi|2 + N0∥wH k AD∥2 2 + wH k CqDwk �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (51) Now, we consider the design of the linear receiver W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using the proposed plug-in channel estimator, the channel matrix H is estimated as �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Then, the conventional MRC and ZF receivers 4Note that the “optimistic ergodic sum rate” is an upper bound assuming Gaussian signaling since it assumes that the useful signal coefficient and the interference variance are perfectly known [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 16 are given as WH MRC = �HH, (52) WH ZF = � �HH �H �−1 �HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (53) Note that the conventional MRC and ZF receivers might not perform so well due to the quantized signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Therefore, we propose a BLMMSE receiver that takes directly into account the quantized signal by considering its Bussgang decomposition, which is expected to yield better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Specifically, the BLMMSE receiver is given as WH BLM = CsrDC−1 rD , (54) where CsrD = E � srH D � = HHAH D, CrD = E � rDrH D � = Parcsine(CyD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (55) In practice, using the estimated channel matrix �H, the BLMMSE receiver WH BLM is given as WH BLM = �HH �AH D � Parcsine � �CyD ��−1 , (56) where �AD = � 2 π(diag( �H �HH + N0IM))− 1 2 and �CyD = �H �HH + N0IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' SIMULATION RESULTS In our simulation, we take M = 256 antennas at the BS in ULA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The channel consists of Lc = 3 multipaths in common clusters and L = 2 local clusters, where each local cluster is composed of three multipaths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', Ll i = 3, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The first local cluster is visible to the first quarter of antennas and the second local cluster is visible to the last quarter of antennas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', Ml 1 = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , M 4 }, Ml 2 = { 3M 4 +1, 3M 4 +2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The AoAs of the common clusters and the first and second local clusters are uniformly and randomly generated from [−60, 60], [−60, 0] and [0, 60] degrees, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', θc i ∼ U(−60, 60), ∀i ∈ [Lc], θl 1,j ∼ U(−60, 0), ∀j ∈ [Ll 1], θl 2,j ∼ U(0, 60), ∀j ∈ [Ll 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The APS of all multipaths are randomly generated with the constraints P c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3, P l 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='7, and P l 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Note that this setting satisfies the assumption of max(diag(Ch)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The SNR is set to 10dB (equivalently, N0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We consider three different estimators �Cy for Cy: the estimator (31), based on non-dithered one-bit quantized samples, the estimator (33), based on dithered one-bit quantized samples, and finally, as a reference, the sample covariance matrix of the unquantized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In the first two cases with quantized samples, we use the estimator to produce the APS-based estimator �C⋆ h of the channel covariance Ch, as is detailed in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' All results presented below are averaged over 10 random channel geometry realizations, each with 20 groups of N i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' random channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Channel covariance estimation Given an estimator �C⋆ h of the channel covariance matrix, we first evaluate it in terms of the normalized Frobenius norm error, which is given by ENF = E � �� ���Ch − �C⋆ h ��� 2 F ∥Ch∥2 F � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (57) The numerical results under different values of λ are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is seen that the choice of λ significantly influences the results of dithered quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' A larger number of samples N can result in a more robust covariance estimation in terms of tuning λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Moreover, the results under various numbers of samples N are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3b, where 3 different choices of λ for dithered quantization are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is observed that the Frobenius norm errors of the dithered estimators are much smaller than the ones of the non-dithered estimators over a large range of number of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is also seen that by finding a proper λ, the estimation performance can be much improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Furthermore, in the regime of a small number of samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', when N < 100 in our case), the results for the estimator based on dithered quantized samples are even better than the sample covariance of the unquantized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This shows that our algorithm has a benefit in practical cases with limited number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Channel vector estimation via BLMMSE Next, we numerically evaluate the BLMMSE-based channel vector estimator in terms of the normalized MSE, which is given by ENMSE = E ����h − �h ��� 2 2 � tr(Ch) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (58) Given an estimated channel covariance, we calculate the NMSE with 100 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' random channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The averaged results under various λ and various N are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4b, respectively, where the lower bound is given by the BLMMSE estimator obtained using the true channel covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is observed again that the choice of λ significantly influences the estimation performance of the dithered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By applying a proper λ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', λ = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4b) the channel estimate can be considerably improved compared to the non-dithered case for a large range of number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Moreover, it is seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3a and 4a that the trend of tuning λ in BLMMSE based channel estimation is different from the trend in channel covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In the range of 1 ≤ λ ≤ 2, the results of BLMMSE-based channel estimation are no 18 longer as robust as the results in covariance estimation even under large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The optimal choices of λ for the two estimation problems are also different, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', λ⋆ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3a and λ⋆ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4a under N = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Ergodic sum rate evaluation Finally, we evaluate the proposed scheme in terms of the ergodic sum rate given in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We test with K = 4 users and assume that the channel geometry of all users follows the setting described at the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' For each channel estimation we test with MRC, ZF, and BLMMSE receivers given in (52), (53), and (56), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Similarly as in the previous part, beside the non-dithered and dithered schemes we also provide results based on the true channel covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' However, unlike for the channel MSE criterion used in the previous part, the use of the true covariance is not guaranteed to yield a better sum rate, as a channel estimator with smaller MSE does not necessarily yield a higher sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Furthermore, we provide results based on the true channel vectors as upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We first present the resulting sum rates under various λ with N = 50 samples for covariance estimation and BLMMSE channel estimation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is firstly observed that the BLMMSE receiver performs much better than the ZF and MRC receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This is expected since the BLMMSE receiver takes the quantization into account whereas the conventional ZF and MRC receivers do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Next, we observe that the results based on the true covariance do not always provide the largest sum rate as previously explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Specifically, among the results with ZF receivers (the dashed lines) the non-dithered one is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Since the ZF and MRC receivers are designed to deal with non-quantized received signals and perform much worse than the BLMMSE receiver, we now focus on the results of the BLMMSE receiver, which is also individually depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 5b with one more case of N = 500 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is noticed again that a larger number of samples N makes the results with dithering in terms of the sum rate not only better but also more robust against λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 5b it is seen that the highest sum rate obtained by the proposed scheme with dithering (under N = 500 and λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6) is very close to the result based on the true covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This shows the advantage of the proposed scheme for both channel estimation and multi-user receivers under one-bit quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Finally, we focus on the influence of the number of samples N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The sum rates under various N of MRC, ZF, and BLMMSE are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6a and of only BLMMSE with 3 different 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 ENF Unquantized, N = 50 Unquantized, N = 500 Non-Dithered, N = 50 Non-Dithered, N = 500 Dithered, N = 50 Dithered, N = 500 (a) ENF v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' λ 101 102 103 N 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='9 ENF Unquantized Non-Dithered Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667 Dithered, = 1 Dithered, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 (b) ENF v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' N Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 3: Normalized Frobenius-norm error of channel covariance under various λ in (a) and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' samples N in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' choices of λ for the dithered case are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6a we see a similar behavior as before: the BLMMSE receiver produces better sum rates than the ZF and MRC receivers over a large range of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6b that under the best λ⋆ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6 the proposed scheme with dithering produces sum rates comparable to the results based on the true channel covariance when N ≥ 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is additionally observed from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6b that as the number of samples N increases the difference between the results obtained with different λ is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This indicates again that under larger N the results with dithering are more robust to variations in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' CONCLUSION AND DISCUSSION In this work, we proposed a plug-in channel estimator for massive MIMO systems with spatially non-stationary channels and one-bit quantizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We analyzed the quantized signal via the Bussgang decomposition and analyzed the distortion produced by using an estimated, rather than the true, channel covariance in the construction of the BLMMSE estimator of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To obtain an estimate of the covariance of the spatially non-stationary channel, we introduced a channel covariance estimator based on dithered quantized samples and theoretically analyzed its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We further enhanced this estimator using an APS-based NNLS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Our numerical results showed large performance gains of the proposed scheme with dithering in terms of both channel vector and covariance estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Finally, we proposed a BLMMSE- based receiver tailored to one-bit quantized data signals for the multi-user data transmission 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3 ENMSE Non-Dithered, N = 50 Non-Dithered, N = 500 Dithered, N = 50 Dithered, N = 500 True Covariance (a) EMMSE v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' λ 101 102 103 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='35 ENMSE Non-Dithered Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667 Dithered, = 1 Dithered, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 True Covariance (b) EMMSE v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' N Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 4: Normalized MSE of channel vectors via BLMMSE under various λ in (a) and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' samples N in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 2 6 7 8 9 10 11 12 13 14 15 Rsum Non-Dithered, N = 50, MRC Non-Dithered, N = 50, ZF Non-Dithered, N = 50, BLMMSE Dithered, N = 50, MRC Dithered, N = 50, ZF Dithered, N = 50, BLMMSE True Covariance, MRC True Covariance, ZF True Covariance, BLMMSE True Channel, MRC True Channel, ZF True Channel, BLMMSE (a) Rsum v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' λ via MRC, ZF, BLMMSE receivers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='9 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5 Rsum Non-Dithered, N = 50, BLMMSE Non-Dithered, N = 500, BLMMSE Dithered, N = 50, BLMMSE Dithered, N = 500, BLMMSE True Covariance, BLMMSE (b) Rsum v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' λ via BLMMSE receiver Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 5: Ergodic sum rate of K = 4 users under various λ via MRC, ZF and BLMMSE receivers in (a) and enlarged view of results via BLMMSE receiver in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' phase and showed in numerical experiments that it outperforms the conventional MRC and ZF receivers in terms of the resulting ergodic sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' There are two important aspects of our work that can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' First, we observed in the numerical experiments that the hyperparameter λ of the dithering generation influences the channel estimation significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Even though this influence was observed to diminish as the sample size increases, it is still of significant interest to develop a data-driven method to optimally 21 101 102 103 N 6 7 8 9 10 11 12 13 14 15 Rsum Non-Dithered, MRC Non-Dithered, ZF Non-Dithered, BLMMSE Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667, MRC Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667, ZF Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667, BLMMSE True Covariance, MRC True Covariance, ZF True Covariance, BLMMSE True Channel, MRC True Channel, ZF True Channel, BLMMSE (a) Rsum v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' N via MRC, ZF, BLMMSE receivers 101 102 103 N 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='8 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='6 Rsum Non-Dithered, BLMMSE Dithered, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='66667, BLMMSE Dithered, = 1, BLMMSE Dithered, = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='5, BLMMSE True Covariance, BLMMSE (b) Rsum v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' N via BLMMSE receiver Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 6: Ergodic sum rate of K = 4 users under various number of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' samples N via MRC, ZF and BLMMSE receivers in (a) and enlarged view of results via BLMMSE receiver in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' tune λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Second, in the proposed APS-based channel covariance estimation scheme, the visibility of local clusters is assumed to be known at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' In practice, however, the visibility of local clusters is usually not easy to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It is therefore desirable to develop a scheme without the assumption of known visibility of local clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We will investigate these two questions in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 1 Recall that �hBLM = ChrC−1 r r = ChAHC−1 r r = (Cy − N0I)AHC−1 r r (59) and �h = � �Cy − N0I � �AH �C−1 r r, (60) where �A and �Cr are defined like A and Cr with Cy being replaced by �Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let us abbreviate �hBLM = Mr and �h = � Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (61) Consider α, β, γ > 0 such that mini∈[M] |[Cy]i,i| ≥ α, λmin(Cr) ≥ γ, and ���� � diag(Cy)− 1 2Cydiag(Cy)− 1 2 � i,j ���� ≤ 1 − β, for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (62) In particular, θ ≤ min{α, β, γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We start by writing E �����h − �hBLM��� 2 2 � = E � tr � �hBLM � �hBLM�H − �hBLM�hH − �h � �hBLM�H + �h�hH �� (63) 22 = tr � MCr � M − � M �H + � MCr � � M − M �H� (64) = � MCr + � MCr, � M − � M �� F (65) = 2 � MCr, � M − � M �� F − �� � M − M � Cr, � M − � M �� F (66) ≤ 2∥MCr∥F ∥M − � M∥F + ∥Cr∥ ∥M − � M∥2 F (67) Observe that ∥A∥ ≤ 1 √α by assumption such that ∥MCr∥F = ∥ChAH∥F ≤ ∥Ch∥F ∥AH∥ ≤ 1 √α∥Ch∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (68) Moreover, using that ∥ arcsin(B)∥ ≤ π 2∥B∥ if ∥B∥∞ ≤ 1 (see [37, Supplementary Material, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (4)]), we find ∥Cr∥ ≤ ���diag(Cy)− 1 2Re(Cy)diag(Cy)− 1 2 ��� + ���diag(Cy)− 1 2Im(Cy)diag(Cy)− 1 2 ��� (69) ≤ 2 ���diag(Cy)− 1 2 ��� ∥Cy∥ ���diag(Cy)− 1 2 ��� ≤ 2 α ∥Cy∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (70) We conclude that E �����h − �hBLM��� 2 2 � ≲ α− 1 2∥Ch∥F∥M − � M∥F + α−1 ∥Cy∥ ∥M − � M∥2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (71) We will now show that ���M − � M ��� F ≤ κ ≤ α 1 2 ∥Ch∥F ∥Cy∥ , (72) so that we obtain κ √α∥Ch∥F as a final estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We start by estimating ���M − � M ��� F = ���(Cy − N0I)AHC−1 r − � �Cy − N0I � �AH �C−1 r ��� F (73) ≤ ���Cy − �Cy ��� F ∥A∥ ��C−1 r �� + ����Cy − N0I ��� F ���A − �A ��� ��C−1 r �� (74) + ∥�Cy − N0I∥∥�A∥∥C−1 r − �C−1 r ∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (75) The first term is clearly bounded by γ−1α− 1 2εF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To estimate the second term, we note that ����Cy − N0I ��� F ≤ ∥Ch∥F + ����Cy − Cy ��� F ≤ ∥Ch∥F + εF and ����Cy − N0I ��� ≤ ∥Ch∥ + εF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (76) Furthermore, we use that Z−1 1 − Z−1 2 = Z−1 1 (Z2 − Z1)Z−1 2 (77) for any invertible Z1, Z2 of the same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This yields ����A − A ��� = 2 π ���diag(�Cy)− 1 2 � diag(Cy) 1 2 − diag(�Cy) 1 2 � diag(Cy)− 1 2 ��� (78) 23 ≤ 2 π ���diag(�Cy)− 1 2 ��� ���diag(Cy) 1 2 − diag(�Cy) 1 2 ��� ���diag(Cy)− 1 2 ��� (79) By assumption, we have ���diag(Cy)− 1 2 ��� = � 1 mini[Cy]i,i ≤ � 1 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (80) Moreover, since ����Cy − Cy ��� ∞ ≤ α 2 by (26), we find min i [�Cy]i,i ≥ min i [Cy]i,i − ����Cy − Cy ��� ∞ ≥ α 2 (81) and so ���diag(�Cy)− 1 2 ��� ≤ � 2 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (82) Note that this also implies that ∥�A∥ ≲ 1 √α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using that |√x − √y| ≤ |x−y| √c if x ≥ c > 0, y ≥ 0, we find ���diag(Cy) 1 2 − diag(�Cy) 1 2 ��� ≤ � 1 α ���Cy − �Cy ��� ∞ = � 1 αε∞, (83) and hence ���A − �A ��� ≤ 4 πα− 3 2ε∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (84) Let us finally estimate the last term on the right hand side of (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Write cij = [Cy]i,j, ˆcij = [�Cy]i,j and observe that ����� ˆcij � ˆciiˆcjj − cij √ciicjj ����� ≤ ����� ˆcij − cij � ˆciiˆcjj ����� + |cij| 1 √ˆcii ����� 1 � ˆcjj − 1 √cjj ����� + |cij| 1 � ˆcjj ���� 1 √ˆcii − 1 √cii ���� (85) ≲ 1 α ����Cy − Cy ��� ∞ + ∥Cy∥∞ 1 α2 ����Cy − Cy ��� ∞ (86) ≲ ∥Cy∥∞ 1 α2 ����Cy − Cy ��� ∞ ≤ β 2 (87) as ε∞ ≲ β α2 ∥Cy∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (88) By (62), this implies that ���� � diag(�Cy)− 1 2 �Cydiag(�Cy)− 1 2 � i,j ���� ≤ 1 − β 2 , for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (89) Clearly, for any | arcsin(x) − arcsin(y)| ≤ Lβ|x − y|, (90) for all x, y ∈ (−1 + β 2, 1 − β 2) where Lβ = sup 0≤z<1− β 2 � 1 1 − z2 = � 1 1 − (1 − β 2)2 ≤ � 2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (91) 24 Together with (62) and (89) this yields ∥Cr − �Cr∥F ≲ β−1/2 ���diag(�Cy)− 1 2Re(�Cy)diag(�Cy)− 1 2 − diag(Cy)− 1 2Re(Cy)diag(Cy)− 1 2 ��� F + β−1/2 ���diag(�Cy)− 1 2Im(�Cy)diag(�Cy)− 1 2 − diag(Cy)− 1 2Im(Cy)diag(Cy)− 1 2 ��� F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (92) Now observe that ���diag(�Cy)− 1 2Re(�Cy)diag(�Cy)− 1 2 − diag(Cy)− 1 2Re(Cy)diag(Cy)− 1 2 ��� F ≤ ���diag(�Cy)− 1 2 − diag(Cy)− 1 2 ��� ∥�Cy∥F ���diag(�Cy)− 1 2 ��� + ���diag(Cy)− 1 2 ��� ∥�Cy − Cy∥F ���diag(�Cy)− 1 2 ��� + ���diag(Cy)− 1 2 ��� ∥Cy∥F ���diag(�Cy)− 1 2 − diag(Cy)− 1 2 ��� (93) ≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF (94) and analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ���diag(�Cy)− 1 2Im(�Cy)diag(�Cy)− 1 2 − diag(Cy)− 1 2Im(Cy)diag(Cy)− 1 2 ��� F ≲ α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (95) Hence, ���Cr − �Cr ��� F ≲ β− 1 2α−2∥Cy∥Fε∞ + β− 1 2(α−2ε∞ + α−1)εF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (96) By our assumptions on ε∞ and εF, the right hand side is bounded by γ/2 and hence the assumption ∥C−1 r ∥ ≤ γ−1 implies that ∥�C−1 r ∥ ≤ 2γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Using now again (77) we finally arrive at ���C−1 r − �C−1 r ��� F ≲ β− 1 2γ−2 � α−2∥Cy∥Fε∞ + (α−2ε∞ + α−1)εF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (97) Combining all our estimates in (73), we find ���M − � M ��� F ≲ γ−1α− 1 2εF + γ−1(∥Ch∥F + εF)α− 3 2ε∞ + (∥Ch∥ + εF)α− 1 2β− 1 2γ−2� α−2∥Cy∥Fε∞ + � α−2ε∞ + α−1� εF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (98) Since ε∞ ≤ min � εF ∥Cy∥F , 1 � , (99) we can estimate the right hand side by κ := c α− 5 2β− 1 2γ−2 max{1, ∥Ch∥}εF, (100) 25 for an absolute constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Clearly, κ ≤ α− 1 2 ∥Ch∥F ∥Cy∥ (101) by our assumption on εF, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 1 In the proof of Theorem 1 we will use the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The first one bounds the bias of (34) in terms of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Lemma 2: Let S > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' There exist constants c1, c2 > 0 depending only on S such that the fol- lowing holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y ∈ CM be a mean-zero random vector with covariance matrix E � yyH� = Cy and S-subgaussian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let λ > 0 and let rRe = sign(Re(y)+τ Re), rIm = sign(Im(y)+ τ Im), �rRe = sign(Re(y)+�τ Re), and �rIm = sign(Im(y)+�τ Im), where τ Re, τ Im, �τ Re, �τ Im are inde- pendent and uniformly distributed in [−λ, λ]M and independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Abbreviate r = rRe +jrIm and �r = �rRe + j�rIm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Then, ��λ2E � r�rH� − Cy �� ∞ ≤ c1(λ2 + ∥Cy∥∞)e −c2λ2 ∥Cy∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (102) Proof: The proof of this lemma is a straightforward extension of [37, Lemma 17] to the complex domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' We include it for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' First note that ��λ2E � r�rH� − Cy �� ∞ = ��λ2E � (rRe + jrIm)(�rRe + j�rIm)H� − E � (Re(y) + jIm(y))(Re(y) + jIm(y))H��� ∞ ≤ ��λ2E � rRe(�rRe)T� − E � Re(y)Re(y)T��� ∞ + ��λ2E � rRe(�rIm)T� − E � Re(y)Im(y)T��� ∞ + ��λ2E � rIm(�rRe)T� − E � Im(y)Re(y)T��� ∞ + ��λ2E � rIm(�rIm)T� − E � Im(y)Im(y)T��� ∞ (103) Since y has S-subgaussian coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' we get from (35) that ∥[Re(y)]i∥ψ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∥[Im(y)]i∥ψ2 ≤ S∥Cy∥ 1 2∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' for any i ∈ [M],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' where ∥ · ∥ψ2 denotes the subgaussian norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Applying [37, Lemma 17] for U = [Re(y)]i and V = [Re(y)]j yields ���λ2E � sign � [Re(y)]i + [τ Re]i � sign � [Re(y)]j + [�τ Re]j �� − E � [Re(y)]i[Re(y)]j ���� ≲ (λ2 + S2∥Cy∥∞)e −c λ2 S2∥Cy∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (104) Since this holds for any choice of i, j ∈ [M], the first term on the right-hand side of (103) satisfies the claimed bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The three other terms can be treated in the same way such that our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' 26 The second lemma is a simple concentration inequality that applies to dithered samples of real distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Lemma 3: There exist absolute constants c1, c2 > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y, �y ∈ RM be random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', yN i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∼ y, let �y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', �yN i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' ∼ �y, and let τ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , τ N, �τ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' , �τ N be independent and uniformly distributed in [−λ, λ], for λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Define rk = sign(yk + τ k) and �rk = sign(�yk + �τ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' If N ≥ c1 log(M), then Pr ������ λ2 N N � k=1 rk�rT k − E � rk�rT k � ����� ∞ ≥ � λ4 � c1 log(M) N + t �� ≤ 2e−c2Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (105) In particular, the claim holds if y = �y and yi = �yi, for all i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Proof: Write Rk i,j = [rk]i[�rk]j for i, j ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Since |Rk i,j − E[Rk i,j]| ≤ 2 for all i, j, k, the bound is trivial for t ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Moreover, by Bernstein’s inequality for bounded random variables (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=', [54, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='4]), we find for any u ≤ 8λ2 Pr � 1 N ����� N � k=1 λ2 � Rk i,j − E[Rk i,j] � ����� ≥ u � ≤ 2e −c min � N2u2 σ2 i,j , Nu 2λ2 � (106) ≤ 2e −cN min � u2 λ4 , u λ2 � ≤ 2e−c2N u2 λ4 , (107) as σ2 i,j := N � k=1 λ4E �� Rk i,j − E[Rk i,j] �2� = N � k=1 λ4 � E � (Rk i,j)2� − � E[Rk i,j] �2� ≤ λ4N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (108) Hence, for any given t < 4 we can set u = � λ4 � c1 log(M) N + t � and note that u ≤ 8λ2 as N ≥ c1 log(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By applying the union bound over all M 2 entries we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' Proof of Theorem 1: By the triangle inequality, ����Cd y − Cy ��� ∞ ≤ ����Cd y − E � �Cd y ���� ∞ + ���E � �Cd y � − Cy ��� ∞ (109) Write r = rRe + jrIm and �r = �rRe + j�rIm, where rRe = sign(Re(y) + τ Re), rIm = sign(Im(y) + τ Im), �rRe = sign(Re(y) + �τ Re), and �rIm = sign(Im(y) + �τ Im).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' By Lemma 2, ���E � �Cd y � − Cy ��� ∞ = ��λ2E � r�rH� − Cy �� ∞ ≲ � λ2 + ∥Cy∥∞ �2 e −c2λ2 ∥Cy∥∞ ≲ λ2 √ N , (110) where we have used that λ2 ≳ log(N)∥Cy∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' To estimate the first term in (109),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' observe that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='����Cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='y − E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='�Cd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='∞ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='����Cd − E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='�Cd���� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='rIm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k (�rRe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k )T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='rIm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k (�rRe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k )T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='����� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k (�rIm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k )T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='− E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='λ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='rIm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k (�rIm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='k )T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='(112) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content='Using Lemma 3 for each of the four terms and applying a union bound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' we get Pr �����Cd y − E � �Cd y ���� ∞ ≳ λ2 � log(M) + t N � ≤ 8e−cNt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (113) and thus the first statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' The second statement follows trivially using ����Cd y − Cy ��� F ≤ M ����Cd y − Cy ��� ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' (114) APPENDIX C PROOF OF THEOREM 2 By Theorem 1, we can apply Lemma 1 with ε∞ ∼ λ2 � log(M) + t N , εF ∼ M max{1, ∥Cy∥∞}λ2 � log(M) + t N Note that we do not pick the “minimal setting” for εF suggested by Theorem 1: the additional factor max{1, ∥Cy∥∞} ensures that ε∞ ≲ εF/∥Cy∥F holds (as required in (26)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' It remains to note that all other conditions on ε∞ and εF in Lemma 1 under the stated assumption on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE3T4oBgHgl3EQfpQqt/content/2301.04641v1.pdf'} +page_content=' This completes the proof.' metadata={'source': 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b/6tAyT4oBgHgl3EQfcvc9/content/tmp_files/2301.00288v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..211977f55851b90532270d02fbe6d3fbb1a01a1e --- /dev/null +++ b/6tAyT4oBgHgl3EQfcvc9/content/tmp_files/2301.00288v1.pdf.txt @@ -0,0 +1,3087 @@ +arXiv:2301.00288v1 [math.AP] 31 Dec 2022 +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR +NON-MONOTONIC SHEAR FLOWS +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +Abstract. We give a proof of linear inviscid damping and vorticity depletion for non-monotonic +shear flows with one critical point in a bounded periodic channel. In particular, we obtain +quantitative depletion rates for the vorticity function without any symmetry assumptions. +Dedicated to Carlos Kenig, on the occasion of his 70th birthday. +Key Words: Inviscid damping, vorticity depletion, non-monotonic shear flows. +Mathematics Subject Classification: 35B40, 35Q31, 35P25 +Contents +1. +Introduction +1 +2. +Spectral property and representation formula +5 +3. +Bounds on the Green’s function and modified Green’s function +7 +4. +The limiting absorption principle +13 +5. +Bounds on ψι +k,ǫ: the non-degenerate case +18 +6. +Bounds on ψι +k,ǫ: the degenerate case +20 +7. +Proof of Theorem 1.2 +27 +References +30 +1. Introduction +The study of stability problems in mathematical analysis of fluid dynamics has a long and +distinguished history, dating back to the work of Kelvin [18], Orr [25] and Rayleigh [26] among +many others, and continuing to the present day. Hydrodynamical stability problems can be +considered in both two and three dimensions. In this paper we work with two dimensional +inviscid flows. +For the Euler equations, there are significant recent progresses on the asymptotic stability +of monotonic shear flows and vortices, assuming spectral stability, see for example [9, 30, 34, +35, 14, 16, 3, 17, 22, 28] for linear results. +The main mechanism of stabilization is the so +called “inviscid damping”, which refers to the transfer of energy of vorticity to higher and +higher frequencies leading to decay of the stream and velocity functions, as t → ∞. Extending +the linearized stability analysis for inviscid fluid equations to the full nonlinear setting is a +challenging problem, and the only available results are on spectrally stable monotonic shear +The first author was supported in part by NSF grant DMS-2007008. The second author is partially supported +by a UC Davis startup grant. The third author was supported in part by NSF grant DMS-1945179. +1 + +2 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +flows [2, 23, 10, 12], and on point vortices [11]. We refer also to the recent review article [13] +for a more in-depth discussion of recent developments of both linear and nonlinear inviscid +damping. +Many physically important shear flows are not monotonic, such as Poiseuille flow and Kol- +mogorov flows. For such flows on the linear inviscid level, there is an additional significant +physical phenomenon called “vorticity depletion” which refers to the asymptotic vanishing of +vorticity as t → ∞ near the critical point where the derivative of the shear flow is zero, first +predicted in Bouchet and Morita [5], and proved rigorously in Wei-Zhang-Zhao [31]. A similar +phenomenon was proved in Bedrossian-Coti Zelati-Vicol [3] for the case of vortices. See also +[17] by the first and third author for a refined description of the dynamics in Gevrey spaces as +a step towards proving nonlinear vortex symmetrization. +In [31] by Wei-Zhang-Zhao, sharp linear inviscid damping estimates and quantitative deple- +tion estimates were obtained for an important class of “symmetric shear flows” in a periodic +channel (see also [32] by Wei-Zhang-Zhao for a similar result for Kolmogorov flow). When +no symmetry is assumed, only qualitative bounds are available. +Heuristically the general +case should be similar to the symmetric one, since the main vorticity depletion mechanism +is completely local and asymptotically all shear flows approach symmetric ones at the (non- +degenerate) critical points. However there are significant difficulties in using the approach of +[31] to extend the quantitative depletion bounds of [31] to the general case, as the argument +in [31] relies heavily on decomposition of functions into odd and even parts, which are specific +to symmetric shear flows. +In this paper we prove linear inviscid damping estimates and quantitative vorticity depletion +estimates for a class of stable non-monotonic shear flows with one non-degenerate critical +point. The main new features of our results are that we do not need symmetry condition on +the background shear flow, and that our formulation on quantitative depletion for vorticity +function seem to be new even for general symmetric shear flows (see however Wei-Zhang-Zhao +[32] which contains a sharp depletion rate at the critical points for Kolmogorov flow), see +Theorem 1.2 below for the precise statements. +We begin with the description of our main +equations and theorem. +1.1. Main equations. Consider the two dimensional Euler equation linearized around a shear +flow (b(y), 0), in the periodic channel (x, y, t) ∈ T × [0, 1] × [0, ∞): +∂tω + b(y)∂xω − b′′(y)uy = 0, +div u = 0 +and +ω = −∂yux + ∂xuy, +(1.1) +with the natural non-penetration boundary condition uy|y=0,1 = 0. +For the linearized flow, +� +T×[0, 1] +ux(x, y, t) dxdy and +� +T×[0, 1] +ω(x, y, t) dxdy are conserved quan- +tities. In this paper, we will assume that +� +T×[0,1] +ux +0(x, y) dxdy = +� +T×[0,1] +ω0 dxdy = 0. +These assumptions can be dropped by adjusting b(y) with a linear shear flow C0y + C1. Then +one can see from the divergence free condition on u that there exists a stream function ψ(t, x, y) +with ψ(t, x, 0) = ψ(t, x, 1) ≡ 0, such that +ux = −∂yψ, uy = ∂xψ. +(1.2) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS +3 +The stream function ψ can be solved through +∆ψ = ω, +ψ|y=0,1 = 0. +(1.3) +We summarize our equations as follows + + + +∂tω + b(y)∂xω − b′′(y)∂xψ = 0, +∆ψ(t, x, y) = ω(t, x, y), +ψ(t, x, 0) = ψ(t, x, 1) = 0, +(ux, uy) = (−∂yψ, ∂xψ), +(1.4) +for t ≥ 0, (x, y) ∈ T × [0, 1]. +Our goal is to understand the long time behavior of ω(t) as t → ∞, with Sobolev regular +initial vorticity ω0. +1.2. The main results. We describe more precisely the main assumptions and our main +conclusion. The main conditions we shall assume on the shear flow b(y) ∈ C4([0, 1]) are as +follows. +Assumption 1.1. We assume that the background flow b(y) ∈ C4([0, 1]) satisfies the following +conditions. +(1) +S := {y ∈ [0, 1] : b′(y) = 0} = {y∗} ⊂ (0, 1). +(1.5) +In addition, b′′(y∗) ̸= 0. +(2) For k ∈ Z\{0}, the linearized operator Lk : L2(0, 1) → L2(0, 1) defined as +Lkg(y) := b(y)g(y) + b′′(y) +� 1 +0 +Gk(y, z)g(z) dz +(1.6) +has no discrete eigenvalues nor generalized embedded eigenvalues. In the above Gk is +the Green’s function for k2 − +d2 +dy2 on the interval (0, 1) with zero Dirichlet boundary +condition. +We refer to section 2 below for the definition and more discussion about generalized embed- +ded eigenvalues. +Our main result is the following theorem. +Theorem 1.2. Assume that ω(t, ·) ∈ C([0, ∞), H4(T×[0, 1])) with the associated stream func- +tion ψ(t, ·) is the unique solution to (1.4), with initial data ω0 ∈ H4(T × [0, 1]) satisfying for +all y ∈ [0, 1], +� +T +ω0(x, y) dx = 0. +(1.7) +Then we have the following bounds. +(i) Inviscid damping estimates: +∥ψ(t, ·)∥L2(T×[0,1]) ≲ +1 +⟨t⟩2 ∥ω0∥H4(T×[0,1]), +(1.8) +∥ux(t, ·)∥L2(T×[0,1]) ≲ 1 +⟨t⟩∥ω0∥H4(T×[0,1]), +∥uy(t, ·)∥L2(T×[0,1]) ≲ +1 +⟨t⟩2 ∥ω0∥H4(T×[0,1]). +(1.9) +(ii) Vorticity depletion estimates: there exists a decomposition +ω(t, x, y) := ωloc(t, x, y) + ωnloc(t, x, y), +(1.10) + +4 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +where for (x, y, t) ∈ T × [0, 1] × [0, ∞), +|ωloc(t, x, y)| ≲ |y − y∗|7/4∥ω0∥H4(T×[0,1]), +|ωnloc(t, x, y)| ≲ +1 +⟨t⟩7/8 ∥ω0∥H4(T×[0,1]). +(1.11) +1.3. Remarks and main ideas of proof. We have the following remarks on Theorem 1.2. +Firstly, in the above theorem we have not tracked the minimal regularity required for the +bounds (1.8), (1.9) and (1.11) to hold, and a more careful argument can probably significantly +reduce the number of derivatives needed on the initial data ω0. Secondly, we note also that +the argument here can be applied to non-monotonic shear flows with multiple non-degenerate +points, although the presentation will be more complicated. +Thirdly, a more sophisticated +analysis may yield a sharper rate of vorticity depletion with rate +|ωloc(t, x, y)| ≲ |y − y∗|2−, +|ωnloc(t, x, y)| ≲ ⟨t⟩−1+. +It is not clear to us though if one can reach the optimal rates of |y − y∗|2 and ⟨t⟩−1. +We briefly explain the main ideas of the proof. +By a standard spectral representation formula, see (2.7), it suffices to study the spectral +density functions and the associated Rayleigh equation (2.8). There are two main cases to +consider. When the spectral parameter λ is not close to the critical value b(y∗), the situation +is similar to monotonic shear flows and can be treated as in [14]. The main new case is when +the spectral parameter λ is close to the critical value b(y∗). In this case, the Rayleigh equation +(2.8) is very singular, and the potential term +b′′(y) +b(y)−λ+iǫ has a quadratic singularity roughly of +the form +2 +(y−y∗)2+(λ−b(y∗))+iǫ for y close to y∗. +The key observation here, as in [17], is that the potential term +b′′(y) +b(y)−λ+iǫ is critically singular +and has real part with a favorable sign for 1 ≫ |y − y∗| ≫ |λ − b(y∗)|1/2, which needs to be +incorporated as part of the main term. We therefore define a modified Green’s function for +the main term, see (3.12)-(3.13), which has strong vanishing conditions near y = y∗, leading +ultimately to vorticity depletion. After extracting the main terms in the Rayleigh equation +(2.8), the rest of the terms can be treated as compact perturbations, and can be bounded using +a limiting absorption principle, see Lemma 4.4, thanks to the spectral assumption 1.1. +The limiting absorption principle provides preliminary bounds on the spectral density func- +tions ψι +k,ǫ(y, λ) with ι ∈ {±}. To obtain the desired quantitative decay rates, we take up to +two derivatives in λ of the spectral density functions, and again use the limiting absorption +principle to estimate the resulting derivatives, after extracting the main singular terms. The +procedure is more or less straightforward but the calculations are quite lengthy. We refer to +[14] also for similar calculations in a simpler setting. Lastly, we note that there are important +cancellations between ψ+ +k,ǫ(y, λ) and ψ− +k,ǫ(y, λ) in the limit ǫ → 0+, which is the reason why we +need two versions of the limiting absorption principle, see Lemma 4.4, with different weighted +spaces. +1.4. Notations. We summarize here some notations that are specific for this paper for the +reader’s conveniences. +For positive numbers α, β, we set α ∧ β := min{α, β}. +We denote +for d > 0, Σd := {b(y) : +y ∈ [y∗ − d, y∗ + d]}, Sd := [y∗ − d, y∗ + d]. +We also denote +Σ := {b(y) : y ∈ [0, 1]} and I := [0, 1]. For k ∈ Z\{0}, we define for f ∈ H1(I) the norm +∥f∥H1 +k(I) := ∥f∥L2(I) + |k|−1∥f ′∥L2(I). + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS +5 +2. Spectral property and representation formula +Taking Fourier transform in x in the equation (1.4) for ω, we obtain that +∂tωk + ikb(y)ωk − ikb′′(y)ψk = 0, +(2.1) +for k ∈ Z, t ≥ 0, y ∈ [0, 1]. In the above, ωk and ψk are the k-th Fourier coefficients of ω, ψ in +x respectively. For each k ∈ Z\{0}, recall from (1.6) that for any g ∈ L2(0, 1), +Lkg(y) = b(y)g(y) + b′′(y) +� 1 +0 +Gk(y, z)g(z)dz, +(2.2) +where Gk is the Green’s function for the operator k2− d2 +dy2 on (0, 1) with zero Dirichlet boundary +condition. Then (2.1) can be reformulated abstractly as +∂tωk + ikLkωk = 0. +(2.3) +In contrast to the spectral property of the linearized operator around monotonic shear flows, +the spectral property of Lk is less understood, especially on the generation of discrete eigen- +values and embedded eigenvalues. From general spectral theory, we know that the spectrum +of Lk consists of the continuous spectrum +Σ := +� +b(y) : y ∈ [0, 1] +� +, +(2.4) +together with some discrete eigenvalues with nonzero imaginary part which can only accumulate +at the set of continuous spectrum Σ. Unlike the case of monotonic shear flows where the discrete +eigenvalues can accumulate only at inflection points of the background shear flow, there appears +no simple characterization of the possible accumulation points for non-monotonic shear flows. +Recall that λ ∈ Σ is called an embedded eigenvalue if there exists a nontrivial g ∈ L2(0, 1), +such that +Lkg = λg. +(2.5) +For non-monotonic shear flows, this definition is too restrictive, as accumulation points of +discrete eigenvalues may no longer be embedded eigenvalues. To capture the discrete eigen- +values, we recall the following definition of “generalized embedded eigenvalues”, which can be +found already in [31], adapted to our setting. +Definition 2.1. We call λ ∈ Σ a generalized embedded eigenvalue, if one of the following +conditions is satisfied. +• λ is an embedded eigenvalue. +• λ ̸= b(y∗) and there exists a nontrivial ψ ∈ H1 +0(0, 1) : (0, 1) → C such that in the sense +of distributions on (0, 1), +(k2 − ∂2 +y)ψ(y) + P.V.b′′(y)ψ(y) +b(y) − λ + iπ +� +z∈[0,1], b(z)=λ +b′′(z)ψ(z) +|b′(z)| +δ(y − z) = 0. +(2.6) +We remark that our assumption that the critical point y∗ of b(y) being non-degenerate +implies that the sum in (2.6) is finite, and that the spectral assumption 1.1 is satisfied if b′′ > 0 +on [0, 1]. + +6 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +Proposition 2.2. Suppose that k ∈ Z\{0} and ωk +0 ∈ L2([0, 1]). Then the stream function +ψk(t, y) for k ∈ Z\{0}, y ∈ [0, 1], t ≥ 0 has the representation +ψk(t, y) = − 1 +2πi lim +ǫ→0+ +� +Σ +e−ikλt � +ψ− +k,ǫ(y, λ) − ψ+ +k,ǫ(y, λ) +� +dλ, +(2.7) +where ψι +k,ǫ(y, λ) for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ, k ∈ Z\{0}, and sufficiently small ǫ ∈ +[−1/4, 1/4]\{0}, are the solutions to +−k2ψι +k,ǫ(y, λ) + d2 +dy2 ψι +k,ǫ(y, λ) − +b′′(y) +b(y) − λ + iιǫψι +k,ǫ(y, λ) = +−ωk +0(y) +b(y) − λ + iιǫ, +(2.8) +with zero Dirichlet boundary condition. +Proof. By standard theory of spectral projection, from (2.3), we obtain that for y ∈ [0, 1], +ωk(t, y) = +1 +2πi lim +ǫ→0+ +� +Σ +eiλt �� +(λ + kLk − iǫ)−1 − (λ + kLk + iǫ)−1� +ωk +0 +� +(y) dλ. +(2.9) +We then obtain for y ∈ [0, 1], +ψk(t, y) = − 1 +2πi lim +ǫ→0+ +� +Σ +e−ikλt +� 1 +0 +Gk(y, z) +× +�� +(−λ + Lk − iǫ)−1 − (−λ + Lk + iǫ)−1� +ωk +0 +� +(z) dzdλ += − 1 +2πi lim +ǫ→0+ +� +Σ +e−ikλt � +ψ− +k,ǫ(y, λ) − ψ+ +k,ǫ(y, λ) +� +dλ. +(2.10) +In the above, for y ∈ [0, 1] and λ ∈ Σ, +ψ+ +k,ǫ(y, λ) := +� 1 +0 +Gk(y, z) +� +(−λ + Lk + iǫ)−1ωk +0 +� +(z) dz, +ψ− +k,ǫ(y, λ) := +� 1 +0 +Gk(y, z) +� +(−λ + Lk − iǫ)−1ωk +0 +� +(z) dz. +(2.11) +Therefore for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ, +� +k2 − d2 +dy2 +� +ψι +k,ǫ(y, y0) = (−λ + Lk + iιǫ)−1ωk +0(y), +(2.12) +which implies +ωk +0(y) =(−λ + Lk + iιǫ) +� +k2 − d2 +dy2 +� +ψι +k,ǫ(y, λ) +=(b(y) − λ + iιǫ) +� +k2 − d2 +dy2 +� +ψι +k,ǫ(y, λ) + b′′(y)ψι +k,ǫ(y, λ). +(2.13) +It follows from (2.13) that ψ+ +k,ǫ(y, λ), ψ− +k,ǫ(y, λ) satisfy (2.8). The proposition is now proved. +□ +Remark 2.3. The existence of ψι +k,ǫ for sufficiently small ǫ ̸= 0 follows from our spectral +assumptions, which imply the solvability of (2.8) for sufficiently small ǫ ̸= 0, see also (4.9). + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS +7 +3. Bounds on the Green’s function and modified Green’s function +3.1. Elementary properties of the standard Green’s function. For integers k ∈ Z\{0}, +recall that the Green’s function Gk(y, z) solves +− d2 +dy2 Gk(y, z) + k2Gk(y, z) = δ(y − z), +(3.1) +with Dirichlet boundary conditions Gk(0, z) = Gk(1, z) = 0, z ∈ (0, 1). Gk has the explicit +formula +Gk(y, z) = +1 +k sinh k +� +sinh(k(1 − z)) sinh(ky) +if y ≤ z, +sinh(kz) sinh(k(1 − y)) +if y ≥ z, +(3.2) +and the symmetry +Gk(y, z) = Gk(z, y), +for k ∈ Z\{0}, y, z ∈ [0, 1]. +(3.3) +We note the following bounds for Gk +sup +y∈[0,1],|A|≤10 +� +|k|2��Gk(y, z)(log |z − A|)m�� +L1(z∈[0,1]) + |k| +��∂y,zGk(y, z)(log |z − A|)m�� +L1(z∈[0,1]) +� ++ +sup +y∈[0,1],α∈{0,1} +� +|k|3/2−α ��∂α +y,zGk(y, z) +�� +L2(z∈[0,1]) +� +≲ | log ⟨k⟩|m, +for m ∈ {0, 1, 2, 3}. +(3.4) +Define +Fk(y, z) = +1 +sinh k +� +−k cosh (k(1 − z)) cosh (ky), +0 ≤ y ≤ z ≤ 1; +−k cosh (kz) cosh (k(1 − y)), +1 ≥ y > z ≥ 0. +(3.5) +We note that +∂y∂zGk(y, z) = ∂z∂yGk(y, z) = δ(y − z) + Fk(y, z), +for y, z ∈ [0, 1]. +(3.6) +By direct computation, we see Fk satisfies the bounds +sup +y∈[0,1],|A|≤10 +���Fk(y, z)(log |z − A|)m�� +L1(z∈[0,1]) + |k|−1��∂y,zFk(y, z)(log |z − A|)m�� +L1(z∈[0,1]) +� ++ +sup +y∈[0,1],α∈{0,1} +� +|k|−1/2−α ��∂α +y,zFk(y, z) +�� +L2(z∈[0,1]) +� +≲ | log ⟨k⟩|m, +for m ∈ {0, 1, 2, 3}. +(3.7) +The bounds (3.4) and (3.7) can be proved by explicit calculations and are useful in the proof +of Lemma 4.1 below. +3.2. Bounds on the modified Green’s function. It follows from Assumption 1.1 that there +exists a δ0 ∈ (0, 1/8) such that +inf{|y∗|, |y∗ − 1|} > 10δ0 +and +sup +y∈(y∗−4δ0,y∗+4δ0) +|b′′′(y)|δ0 < |b′′(y∗)|/10. +(3.8) +Define the set +Σδ0 := {b(y) : y ∈ [y∗ − δ0, y∗ + δ0]}, +(3.9) + +8 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +and fix a standard smooth cutoff function ϕ ∈ C∞ +c (−2, 2) satisfying ϕ ≡ 1 on [−3/2, 3/2]. For +simplicity of notations, we denote +I := (0, 1). +(3.10) +To simplify notations we define also for d ∈ (0, 1/10), +Sd := [y∗ − d, y∗ + d]. +(3.11) +For applications below, we also need to study the “modified Green’s function” Gk(y, z; λ+iǫ) +for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0}, which satisfies for y, z ∈ (0, 1), +(k2−∂2 +y)Gk(y, z; λ+iǫ)+ +b′′(y) +b(y) − λ + iǫ +� +ϕ +�y − y∗ +δ0 +� +−ϕ +�y − y∗ +δ(λ) +�� +Gk(y, z; λ+iǫ) = δ(y−z), (3.12) +with the boundary condition +Gk(y, z; λ + iǫ)|y∈{0,1} = 0. +(3.13) +In the above, we have used the notation that +δ(λ) := 8 +� +|λ − b(y∗)|/b′′(y∗). +(3.14) +Define the weight ̺(y; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} as +̺(y; λ + iǫ) :=|λ − b(y∗)|1/2 + |ǫ|1/2 + |y − y∗|. +(3.15) +The crucial bounds we need for the modified Green’s function Gk(y, z; λ + iǫ) is the following. +Lemma 3.1. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} be defined as +in (3.12). Then we have the identity for y, z ∈ [0, 1], +Gk(y, z; λ + iǫ) = Gk(z, y; λ + iǫ), +(3.16) +and the following statements hold. +(i) We have the bounds +sup +y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|} +|Gk(y, z; λ + iǫ)| ≲ min{̺(z; λ + iǫ), 1/|k|}, +sup +y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|} +|∂yGk(y, z; λ + iǫ)| ≲ 1; +(3.17) +(ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≳ 1/|k|, we have +the bounds with α ∈ {0, 1} +|∂α +y Gk(y1, z; λ + iǫ)| +≲ +� +|k| + ̺−1(y1; λ + iǫ) +�α +e−|k||y1−y2| +� +|k| +� +[y2−1/|k|,y2+1/|k|]∩I +|Gk(y, z; λ + iǫ)|2 dy +�1/2 +. +(3.18) +(iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≪ 1/|k|, we have +the bounds with α ∈ {0, 1} +|∂α +y Gk(y1, z; λ + iǫ)| ≲ +� +|k| + ̺−1(y1; λ + iǫ) +�α +min +�̺2(y1; λ + iǫ) +̺2(y2; λ + iǫ), ̺(y2; λ + iǫ) +̺(y1; λ + iǫ) +� +M, +(3.19) +where +M := +� +1 +̺(y2; λ + iǫ) +� +[y2−̺(y2;λ+iǫ),y2+̺(y2;λ+iǫ)]∩I +|Gk(y, z; λ + iǫ)|2 dy +�1/2 +. +(3.20) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS +9 +Proof. The proof is based on energy estimates and “entanglement inequalities”, as in [15]. See +also the earlier work [33] where this type of inequality was used. We divide the proof into +several steps. +Step 1: +the proof of (3.17). +We first establish the bounds (3.17). +For simplicity of +notation, we suppress the dependence on z, λ + iǫ and set for y ∈ [0, 1], +h(y) := Gk(y, z; λ + iǫ), +V (y) := +b′′(y) +b(y) − λ + iǫ +� +ϕ +�y − y∗ +δ0 +� +− ϕ +�y − y∗ +δ +�� +. +(3.21) +Multiplying h to (3.12) and integrating over [0, 1], we obtain that +� 1 +0 +|∂yh(y)|2 + |k|2|h(y)|2 dy + +� 1 +0 +b′′(y) +b(y) − λ + iǫ +� +ϕ +�y − y∗ +δ0 +� +− ϕ +�y − y∗ +δ +�� +|h(y)|2 dy = h(z). +(3.22) +Note that for y ∈ [0, 1], ℜV (y) ≥ 0, and in addition, for y ∈ Sδ0 and +|y − y∗| > C0 +� +|λ − b(y∗)|1/2 + |ǫ|1/2� +with sufficiently large C0 ≫ 1, +1 + ℜV (y) ≳ +1 +̺2(y; λ + iǫ). +(3.23) +It follows from (3.22) that +� 1 +0 +|∂yh(y)|2 + |k|2|h(y)|2 dy + +� +y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2) +1 +� +̺(y; λ + iǫ) +�2 |h(y)|2 dy +≲ |h(z)|. +(3.24) +Using the Sobolev type inequality +∥h∥L∞(J) ≲ ∥h∥L2(J∗)|J|−1/2 + ∥∂yh∥L2(J)|J|1/2, +(3.25) +for any interval J, J∗ with J∗ ⊆ J and |J∗| ≳ |J|, and choosing the interval J ⊂ I as an interval +containing z with length of the size C1 min{1/|k|, ̺(z; λ + iǫ)}, we obtain from (3.24) that +� 1 +0 +|∂yh(y)|2 + |k|2|h(y)|2 dy + +� +y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2) +1 +� +̺(y; λ + iǫ) +�2 |h(y)|2 dy +≲ min{1/|k|, ̺(z; λ + iǫ)}. +(3.26) +The desired bound (3.17) follows from (3.26), (3.25), and equation (3.12). +Step 2: the proof of (3.18). Denote +M1 := +� +|k| +� +[y2−1/|k|,y2+1/|k|]∩I +|Gk(y, z; λ + iǫ)|2 dy +�1/2 +. +(3.27) +For the sake of concreteness, we assume that y1 > z (so y2 ∈ [z, y1]). We shall also assume +that y1 − y2 ≫ 1/|k| as the other case is analogous but easier. For ϕ ∈ C1 +p([y2, 1]), the space of +piecewise C1 functions, with ϕ(y2) = 0, we multiply ϕ2h to equation (3.12) and integrate over + +10 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +[y2, 1] to obtain that +� 1 +y2 +|∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0. +(3.28) +Taking the real part of (3.28) and using Cauchy-Schwarz inequality, we get that +� 1 +y2 +� +|∂yϕ(y)|2 − |k|2|ϕ(y)|2� +|h(y)|2 dy ≥ 0. +(3.29) +We now choose ϕ more specifically as follows. We require that +ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + 1/|k|], ϕ(y2 + 1/|k|) = 1, +ϕ′(y) = |k|ϕ(y) for y ∈ [y2 + 1/|k|, y1 − 1/|k|], ϕ′(y) = 0 for y ∈ [y1 − 1/|k|, 1]. +(3.30) +It follows from (3.29)-(3.30) that +� 1 +y1−1/|k| +|k|2ϕ2(y)|h(y)|2 dy ≲ |k|M2 +1 , +ϕ(y) ≈ e|k||y1−y2| for y ∈ [y1 − 1/|k|, y1 + 1/|k|] ∩ I. +(3.31) +The desired bounds (3.18) follow from (3.31) and equation (3.12). +Step 3: the the proof of (3.19). For the sake of concreteness, we assume that y1 > z (and +so y2 ∈ [z, y1]). We shall also assume that y1 − y2 ≫ ̺(y2; λ + iǫ) and that y2 > y∗ + δ + |ǫ|1/2 +as the other cases are analogous. +For ϕ ∈ C1 +p([y2, 1]) with ϕ(y2) = 0, we multiply ϕ2h to equation (3.12) and integrate over +[y2, 1] to obtain that +� 1 +y2 +|∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0. +(3.32) +Write for y ∈ [y2, 1] +h(y) = (y − y∗)1/2h∗(y). +(3.33) +Simple calculations show that +� 1 +y2 +(y − y∗)|∂yh∗(y)|2ϕ2(y) + 2(y − y∗)∂yϕ(y)ϕ(y)∂yh∗(y)h∗(y) + +1 +4(y − y∗)|h∗(y)|2ϕ2(y) ++ |k|2|h(y)|2ϕ2(y) + (y − y∗)V (y)ϕ2(y)|h∗(y)|2 dy = 0. +(3.34) +Therefore +� 1 +y2 +� +1 +4(y − y∗) + (y − y∗)ℜVy∗(y) +� +ϕ2(y)|h∗(y)|2 dy ≤ +� 1 +y2 +(y − y∗)(∂yϕ)2(y)|h∗(y)|2 dy, (3.35) +which implies that +� 1 +y2 +1 +y − y∗ +�� +(y − y∗)∂yϕ +�2(y) − +� +1/4 + (y − y∗)2ℜV (y) +� +ϕ2(y) +� +|h∗(y)|2 dy ≥ 0. +(3.36) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 11 +We notice the pointwise bounds for y ∈ [y2, 1], +1/4 + (y − y∗)2ℜV (y) ≥ max +� +0, 9/4 − C2 +̺2(y2; λ + iǫ) +(y − y∗)2 +− C2|y − y∗| +� +. +(3.37) +Now we choose ϕ ∈ C1 +p([y2, 1]) more precisely as follows. We require that +ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + ̺(y2; λ + iǫ)], ϕ(y2 + ̺(y2; λ + iǫ)) = 1, +(y − y∗)ϕ′(y) = +� +1/4 + (y − y∗)2ℜV (y) +�1/2ϕ(y) +for y ∈ [y2 + ̺(y2; λ + iǫ), y1 − ̺(y1; λ + iǫ)], and ϕ′(y) = 0 for y ∈ [y1 − ̺(y1; λ + iǫ), 1]. +(3.38) +It follows from (3.36)-(3.38) that +� y1 +y1−̺(y1;λ+iǫ) +1 +̺(y1; λ + iǫ)ϕ2(y)|h∗(y)|2 dy ≲ M2/̺(y2; λ + iǫ), +ϕ(y) ≈ +(y1 − y∗)3/2 +̺3/2(y2; λ + iǫ) for y ∈ [y1 − ̺(y1; λ + iǫ), y1]. +(3.39) +The desired bounds (3.19) follow from the change of variable (3.33), the bound (3.36), (3.39) +and equation (3.12). +□ +As a corollary of Lemma 3.1, we have the following additional bounds on the modified +Green’s function. +Lemma 3.2. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0, k ∈ Z\{0} and ǫ ∈ [−1/8, 1/8]\{0} +be defined as in (3.12). Recall the definition (3.14) for δ = δ(λ) > 0. Define +h := 10(δ + |ǫ|1/2), +(3.40) +and also for y, z ∈ [0, 1], +Hk(y, z; λ + iǫ) := +� +∂z + ϕ +�y − y∗ +h +� +∂y +� +Gk(y, z; λ + iǫ). +(3.41) +Then the following statements hold for z ∈ S4δ. +(i) We have the bounds +sup +y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|} +|Hk(y, z; λ + iǫ)| ≲ 1, +sup +y∈[0,1], |y−z|≤min{̺(z;λ+iǫ),1/|k|} +|∂yHk(y, z; λ + iǫ)| ≲ 1/ min{̺(z; λ + iǫ), 1/|k|}; +(3.42) +(ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≳ 1/|k|, we have +the bounds with α ∈ {0, 1} +� +min{̺(y1; λ + iǫ), 1/|k|} +�α|∂α +y Hk(y1, z; λ + iǫ)| +≲ +e−|k||y1−y2| +min{̺(z; λ + iǫ), 1/|k|} +� +|k| +� +[y2−1/|k|,y2+1/|k|]∩I +|Gk(y, z; λ + iǫ)|2 dy +�1/2 +. +(3.43) + +12 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +(iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2; λ + iǫ) ≪ 1/|k|, we have +the bounds with α ∈ {0, 1} +� +min{̺(y1; λ + iǫ), 1/|k|} +�α|∂α +y Hk(y1, z; λ + iǫ)| +≲ +1 +min{̺(z; λ + iǫ), 1/|k|} min +�̺2(y1; λ + iǫ) +̺2(y2; λ + iǫ), ̺(y2; λ + iǫ) +̺(y1; λ + iǫ) +� +M, +(3.44) +where +M := +� +1 +̺(y2; λ + iǫ) +� +[y2−̺(y2;λ+iǫ),y2+̺(y2;λ+iǫ)]∩I +|Gk(y, z; λ + iǫ)|2 dy +�1/2 +. +(3.45) +Proof. Denote with a slight abuse of notation for y ∈ [0, 1], +ϕ†(y) := ϕ +�y − y∗ +h +� +, +V (y) := +b′′(y) +b(y) − λ + iǫ +� +ϕ +�y − y∗ +δ0 +� +− ϕ +�y − y∗ +δ(λ) +�� +. +(3.46) +Then Hk,j(y, z; λ + iǫ) satisfies for y ∈ [0, 1], z ∈ S4δ, +(k2 − ∂2 +y)Hk(y, z; λ + iǫ) + V (y)Hk(y, z; λ + iǫ) += −∂2 +yϕ†(y)∂yGk(y, z; λ + iǫ) − ∂yV (y)ϕ†(y)Gk(y, z; λ + iǫ) − 2∂yϕ†(y)∂2 +yGk(y, z; λ + iǫ). +(3.47) +The desired bounds then follow from equation (3.47), Lemma 3.1 and standard elliptic regu- +larity theory. +□ +The bounds in Lemma 3.1 and Lemma 3.2 are quite sharp, since we can exploit the decay +coming from both k2 and +b′′(y) +b(y)−λ+iǫ +� +ϕ +� y−y∗ +δ0 +� +− ϕ +� y−y∗ +δ(λ) +�� +. It is however somewhat complicated +to formulate a concrete bound that is easy to use. Instead, the following simple bounds are +more often used. +Corollary 3.3. Let Gk(y, z; λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\{0} be defined +as in (3.12). Then we have the following bounds. +(i) For y, z ∈ [0, 1], we have the bounds with α ∈ {0, 1} +� +|k| + ̺−1(y; λ + iǫ) +�−α +|∂α +y Gk(y, z; λ + iǫ)| +≲ +1 +|k| + ̺−1(z; λ + iǫ) min +� +e−|k||y−z|, ̺2(y; λ + iǫ) +̺2(z; λ + iǫ), ̺(z; λ + iǫ) +̺(y; λ + iǫ) +� +. +(3.48) +(iii) For y ∈ [0, 1], z ∈ S4δ, we have the bounds with α ∈ {0, 1, 2} +� +|k| + ̺−1(y; λ + iǫ) +�−α +|∂α +y Hk(y, z; λ + iǫ)| ≲ min +� +e−|k||y−z|, ̺2(y; λ + iǫ) +̺2(z; λ + iǫ), ̺(z; λ + iǫ) +̺(y; λ + iǫ) +� +. +(3.49) +Proof. The desired bounds (3.48)-(3.49) follow directly from Lemma 3.1 and Lemma 3.2, by +choosing, if necessary, another point y′ between y and z such that ̺(y′; λ + iǫ) ≈ 1/|k|, and +applying (3.48)-(3.49) on intervals [min{z, y′}, max{z, y′}] and [min{y′, y}, max{y′, y}] succes- +sively. +□ + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 13 +4. The limiting absorption principle +In this section we study the solvability of the main Rayleigh equations (2.8). It turns out +that the situation is very different for the spectral range λ ∈ Σ\Σδ0/2 (the non-degenerate case) +and λ ∈ Σδ0 (the degenerate case). We first consider the non-degenerate case. +4.1. The non-degenerate case. Fix ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σ\Σδ0/2, k ∈ Z\{0}. Define for +each g ∈ L2(0, 1) the operator +Tk,λ,ǫg(y) := +� 1 +0 +Gk(y, z) +b′′(z)g(z) +b(z) − λ + iǫdz. +(4.1) +For applications below, we fix a smooth cutoff function Φ ∈ C∞ +0 (y∗ − δ0/3, y∗ + δ0/3) with +Φ ≡ 1 on [y∗ − δ0/4, y∗ + δ0/4]. To obtain the optimal dependence on the frequency variable +k, we define +∥g∥H1 +k(I) := ∥g∥L2(I) + |k|−1∥g′∥L2(I). +(4.2) +Lemma 4.1. For ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, the operator Tk,λ,ǫ satisfies the +bound +∥Tk,λ,ǫg∥H1 +k(I) ≲ |k|−1/3∥g∥H1 +k(I), +for all g ∈ H1 +k(I). +(4.3) +In addition, we have the more precise regularity structure +����∂yTk,λ,ǫg(y) + b′′(y)(1 − Φ(y))g(y) +b′(y) +log (b(y) − λ + iǫ) +���� +W 1,1(R) +≲ ⟨k⟩4/3∥g∥H1 +k(I). +(4.4) +Proof. We can decompose for y ∈ [0, 1], +Tk,λ,ǫg(y) := T 1 +k,λ,ǫg(y) + T 2 +k,λ,ǫg(y), +(4.5) +where +T 1 +k,λ,ǫg(y) := +� 1 +0 +Gk(y, z)Φ(z)b′′(z)g(z) +b(z) − λ − iǫ dz, +T 2 +k,λ,ǫg(y) := +� 1 +0 +Gk(y, z)(1 − Φ(z))b′′(z)g(z) +b(z) − λ + iǫ +dz. +(4.6) +It follows from the definition of Φ that T 1 +k,λ,ǫg(y) satisfies the bound +∥T 1 +k,λ,ǫg(y)∥H1 +k(I) ≲ |k|−1/3∥g∥H1 +k(I), +∥∂yT 1 +k,λ,ǫg(y)∥W 1,1(I) ≲ ⟨k⟩4/3∥g∥H1 +k(I). +(4.7) +To bound T 2 +k,λ,ǫg(y), we follow the approach in [14]. Using integration by parts, we obtain that +T 2 +k,λ,ǫg(y) = +� 1 +0 +Gk(y, z)(1 − Φ(z))b′′(z)g(z) +b′(z) +∂z log(b(z) − λ + iǫ) dz += − +� 1 +0 +∂zGk(y, z)(1 − Φ(z))b′′(z)g(z) +b′(z) +log(b(z) − λ + iǫ) dz +− +� 1 +0 +Gk(y, z)∂z +�(1 − Φ(z))b′′(z)g(z) +b′(z) +� +log(b(z) − λ + iǫ) dz. +(4.8) +The desired bounds follow from the bound (3.4), the formula (3.6) and (3.7). +□ +We now prove the limiting absorption principle, using the assumption that there is no discrete +or generalized embedded eigenvalues. + +14 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +Lemma 4.2. There exist ǫ0, κ > 0, such that the following statement holds. +For all λ ∈ +Σ\Σδ0/2, k ∈ Z\{0}, 0 < |ǫ| < ǫ0 and any g ∈ H1 +k(I), we have the bound +∥g + Tk,λ,ǫg∥H1 +k(I) ≥ κ∥g∥H1 +k(I). +(4.9) +Proof. We prove (4.9) by contradiction. Assume that there exist for j ≥ 1, a sequence of num- +bers kj ∈ Z\{0}, λj ∈ Σ\Σδ0/2, ǫj ∈ R\{0} → 0 and functions gj ∈ H1 +kj(I) with ∥gj∥H1 +kj (I) = 1, +satisfying kj → k∗ ∈ (Z\{0}) ∪ {±∞}, λj → λ∗ ∈ Σ\Σδ0 as j → ∞, such that +��gj + Tkj,λj,ǫjgj +�� +H1 +kj (I) → 0, +as j → ∞. +(4.10) +The bounds (4.3) and (4.10) imply that |kj| ≲ 1. Thus k∗ ∈ Z\{0}. Using ∥gj∥H1 +kj (I) = 1, the +bounds (4.4) and the compact embedding W 1,1(I) → L2(I), we conclude that by passing to a +subsequence, Tkj,λj,ǫjgj converges in H1(I). In view of (4.10) we can assume that gj → g in +H1(I), where ∥g∥H1 +k∗ = 1. +Using formula (4.1), we obtain from (4.10) that for y ∈ I, +g(y) + lim +j→∞ +� 1 +0 +Gk∗(y, z) +b′′(z)g(z) +b(z) − λ + iǫj +dz = 0. +(4.11) +Applying k2 +∗ − d2 +dy2 to (4.11), we get that for y ∈ I, +k2 +∗g(y) − g′′(y) + lim +j→∞ +(b(y) − λ∗)b′′(y)g(y) +(b(y) − λ∗)2 + ǫ2 +j ++ iπ +� +z∈[0,1],b(z)=λ +b′′(z)g(z) +|b′(z)| +δ(y − z) = 0, +(4.12) +in the sense of distributions for y ∈ (0, 1), which contradicts our spectral assumption that λ∗ +is not a generalized embedded eigenvalue for Lk. The lemma is then proved. +□ +4.2. The degenerate case λ ∈ Σδ0. Recall the definition (3.14) for δ = δ(λ). For λ ∈ Σδ0, k ∈ +Z\{0}, y ∈ I and ǫ ∈ [−1/8, 1/8]\{0}, we denote +dk(λ, ǫ) := +� +|λ − b(y∗)|1/2 + |ǫ|1/2� +∧ 1 +|k|, +̺k(y; λ + iǫ) := ̺(y; λ + iǫ) ∧ 1 +|k|. +(4.13) +Define the weight and the associated weighted Sobolev spaces XN,̺k and XL,̺k as +∥g∥XN,̺k (I) := +� +α∈{0,1} +(δ + |ǫ|1/2)−1/2��� +� +dk(λ, ǫ) +�(−7/4+α)∂α +y g +��� +L2(S3(δ+|ǫ|1/2)) ++ +� +α∈{0,1} +∥̺−7/4+α +k +(·; λ + iǫ)∂α +y g∥L∞(I\S3(δ+|ǫ|1/2)), +(4.14) +and +∥g∥XL,̺k (I) := +� +α∈{0,1} +(δ + |ǫ|1/2)−1/2��dα +k(λ, ǫ)∂α +y g +�� +L2(S3(δ+|ǫ|1/2)) ++ +� +α∈{0,1} +��dk(λ, ǫ)−1̺α+1 +k +(·; λ + iǫ)∂α +y g +�� +L∞(I\S3(δ+|ǫ|1/2)), +(4.15) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 15 +Fix ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σδ0, k ∈ Z\{0}. Recall the definition (3.14) for δ = δ(λ) > 0. +Define for each g ∈ L2(0, 1) the operator +T ∗ +k (λ + iǫ)g(y) := +� 1 +0 +Gk(y, z; λ + iǫ) +� +1 − ϕ +�y − y∗ +δ0 +� ++ ϕ +�y − y∗ +δ +�� +b′′(z)g(z) +b(z) − λ + iǫdz. +(4.16) +Then we have the following bounds for T ∗ +k (λ + iǫ). +Lemma 4.3. For ǫ ∈ [−1/4, 1/4]\{0}, λ ∈ Σδ0, k ∈ Z\{0}, the operator T ∗ +k (λ + iǫ) satisfies the +bound for X ∈ {XN,̺k(I), XL,̺k(I)} +∥T ∗ +k (λ + iǫ)g∥X ≲ (1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2))−1/4∥g∥X, +for all g ∈ H1 +k(I). +(4.17) +Proof. We provide the detailed proof only for the case X = XN,̺k(I) as the other case is +analogous. Since k, λ, ǫ are fixed, for simplicity of notations, we suppress the dependence on +k, λ, ǫ to write T ∗ as T ∗ +k (λ + iǫ), and decompose for y ∈ I, +T ∗g(y) := T ∗ +1 g(y) + T ∗ +2 g(y), +(4.18) +where +T ∗ +1 g(y) := +� 1 +0 +Gk(y, z; λ + iǫ) +� +1 − ϕ +�z − y∗ +δ0 +�� +b′′(z)g(z) +b(z) − λ + iǫdz, +T ∗ +2 g(y) := +� 1 +0 +Gk(y, z; λ + iǫ)ϕ +�z − y∗ +δ +� +b′′(z)g(z) +b(z) − λ + iǫdz. +(4.19) +It follows from the bounds on modified Green’s function Gk(y, z; λ + iǫ), see Lemma 3.1, that +��T ∗ +1 g +�� +XN,̺k(I) ≲ |k|−1/2��g +�� +XN,̺k (I). +(4.20) +To prove (4.17), it suffices to prove +∥T ∗ +2 g∥XN,̺k (I) ≲ +� +1 + |k|(δ + |ǫ|1/2) +�−1/4∥g∥XN,̺k (I). +(4.21) +We assume momentarily that |ǫ| ≲ |λ − b(y∗)| and explain how to remove this assumption +at the end of the proof. We decompose further for y ∈ I, +T ∗ +2 g(y) = +� 1 +0 +Gk(y, z; λ + iǫ)ϕ +�z − y∗ +δ′ +� +ϕ +�z − y∗ +δ +� +b′′(z)g(z) +b(z) − λ + iǫdz ++ +� 1 +0 +Gk(y, z; λ + iǫ) +� +1 − ϕ +�z − y∗ +δ′ +�� +ϕ +�z − y∗ +δ +� +b′′(z)g(z) +b(z) − λ + iǫdz +:= T ∗ +2,Rg(y) + T ∗ +2,Sg(y), +(4.22) +where we have chosen δ′ = δ/C3 with a large constant C3 so that |b(y) − λ| ≈ |λ − b(y∗)| for +|y − y∗| < δ′. +It suffices to prove for ⋄ ∈ {R, S} +∥T ∗ +2,⋄g∥XN,̺k (I) ≲ +� +1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) +�−1/4∥g∥XN,̺k (I). +(4.23) +Step 1. We first prove (4.23) with ⋄ = R. +Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2. In this case for |z − y∗| ≲ δ and |y − y∗| ≲ 1 we have +the bound +��Gk(y, z; λ + iǫ) +�� ≲ +δ2 + |ǫ| +|y − y∗| + δ + |ǫ|1/2 , +��∂yGk(y, z; λ + iǫ) +�� ≲ +δ2 + |ǫ| +(|y − y∗| + δ + |ǫ|1/2)2 . (4.24) + +16 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +It follows from the bound (4.24) that +∥T ∗ +2,Rg∥XN,̺k (I) ≲ +� +1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) +�−1/4∥g∥XN,̺k (I) +(4.25) +Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2. In this case, we have for |z − y∗| ≲ δ and |y − y∗| ≲ 1 +that +��Gk(y, z; λ + iǫ) +�� + |k|−1��∂yGk(y, z; λ + iǫ) +�� ≲ |k|−1e−|k||y−z|. +(4.26) +The desired bound +∥T ∗ +2,Rg∥XN,̺k (I) ≲ +� +1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) +�−1/4∥g∥XN,̺k (I) +(4.27) +follows from (4.26). +Step 2. We now turn to the proof of (4.23) with ⋄ = S and still consider two cases. +Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2. Denoting for y ∈ I, +ϕ∗�y − y∗ +δ +� +:= +� +1 − ϕ +�z − y∗ +δ′ +�� +ϕ +�z − y∗ +δ +� +, +(4.28) +we can rewrite +T ∗ +2,Sg(y) = +� 1 +0 +Gk(y, z; λ + iǫ)ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +∂z log b(z) − λ + iǫ +δ2 += − +� 1 +0 +∂z +� +Gk(y, z; λ + iǫ)ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +� +log b(z) − λ + iǫ +δ2 +dz. +(4.29) +As a consequence of (4.29), we also have +∂y +� +T ∗ +2,Sg(y) +� += ∂y +� 1 +0 +Gk(y, z; λ + iǫ)ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +∂z log b(z) − λ + iǫ +δ2 +dz += − +� 1 +0 +� +∂y(∂z + ∂y)Gk(y, z; λ, ǫ)ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +� +log b(z) − λ + iǫ +δ2 +dz ++ +� 1 +0 +� +∂2 +yGk(y, z; λ + iǫ)ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +� +log b(z) − λ + iǫ +δ2 +dz +− +� 1 +0 +∂yGk(y, z; λ + iǫ)∂z +� +ϕ∗�z − y∗ +δ +�b′′(z)g(z) +b′(z) +� +log b(z) − λ + iǫ +δ2 +dz. +(4.30) +Note that on the support of ϕ∗(z−y∗ +δ +), we have +|b′(z)| ≈ δ, +̺(z; λ + iǫ) ≈ δ. +(4.31) +The desired bound (4.23) for ⋄ = S follows from (4.29)-(4.30), Lemma 3.1 and 3.2, and we +have, in addition, +(δ + |ǫ|1/2)−1/2 +����∂y +� +∂yT ∗ +2,Sg(y) + ϕ∗�y − y∗ +δ +�b′′(y)g(y) +b′(y) +log b(y) − λ + iǫ +δ2 +����� +L2(S3(δ+|ǫ|1/2),j) +≲ δ−1/4� +1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) +�−1/4 +∥g∥XN,̺k (I). +(4.32) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 17 +Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2. This case is analogous to Case I, using Lemma 3.1 +and Lemma 3.2. +Finally we turn to the assumption that |ǫ|1/2 ≲ δ. +Suppose |ǫ|1/2 ≫ δ, then the factor +1 +b(z)−λ+iǫ is not truly singular, and the desired bounds (4.21) follow directly from the bounds +on the modified Green’s function Gk(y, z; λ + iǫ) from Lemma 3.1 and Lemma 3.2. Indeed, we +have the stronger bound +∥T ∗ +2 g∥XN,̺k (I) ≲ +δ +� +|ǫ| +∥g∥XN,̺k (I), +(4.33) +which will be useful below. +□ +The following limiting absorption principle plays an essential role in establishing the vorticity +depletion phenomenon. +Lemma 4.4. There exist positive numbers ǫ0, κ such that the following statement holds. +For ǫ ∈ [−ǫ0, ǫ0]\{0}, λ ∈ Σδ0, k ∈ Z\{0}, and X ∈ {XN,̺k(I), XL,̺k(I)}, +∥(I + T ∗ +k (λ + iǫ))g∥XN,̺k (I) ≥ κ∥g∥XN,̺k (I), +for all g ∈ H1 +k(I). +(4.34) +Proof. We only consider the case X = XN,̺k(I) as the other case is analogous. We prove (4.34) +by a contradiction argument. Assume (4.34) does not hold for any ǫ0 > 0. Then there exist +for ℓ ∈ Z ∩ [1, ∞), +λℓ → λ∗ ∈ Σδ0, ǫℓ ̸= 0 with ǫℓ → 0, kℓ → k∗ ∈ (Z\{0}) ∪ {±∞}, +(4.35) +and functions gℓ satisfying +∥gℓ∥XN,̺kℓ (I) = 1 +(4.36) +such that +��(I + T ∗ +kℓ(λℓ + iǫℓ))gℓ +�� +XN,̺kℓ (I) → 0. +(4.37) +We can assume that λ∗ = b(y∗), otherwise the proof follows from the argument in the non- +degenerate case. We consider several cases. +Case I: lim supℓ→∞ ∥gℓ∥H1(I\Sδ0) > 0. By the bound (4.20), we can assume that k∗ ∈ Z\{0}. +By the bounds (4.36) and (4.37), we can assume (passing to a subsequence if necessary) that +gℓ → g, in H1 +loc(I\{y∗}) as ℓ → ∞, +g(0) = g(1) = 0. +(4.38) +Then it follows from (4.36) and (4.37) that g satisfies +|g(y)| ≲ |y − y∗|7/4, +(4.39) +and for y ∈ (0, 1), +(k2 +∗ − ∂2 +y)g(y) + +b′′(y) +b(y) − b(y∗)g(y) = 0, +(4.40) +which imply that b(y∗) is an embedded eigenvalue for Lk, a contradiction to the spectral +assumption. +Case II: lim supℓ→∞ ∥gℓ∥H1(I\Sδ0) = 0. By the bound (4.17) we can assume that |kℓ|(δℓ + +|ǫℓ|1/2) ≲ 1. In this case, using (4.37), we obtain that (passing to a subsequence if necessary) +��(|λℓ − b(y∗)| + |ǫ|)−9/8gℓ +�� +L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2]) ++ +��(|λℓ − b(y∗)| + |ǫ|)−5/8∂ygℓ +�� +L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2]) ≥ σ > 0, +(4.41) + +18 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +where we recall from (3.14) that +δℓ ≈ |λℓ − b(y∗)|1/2. +(4.42) +We divide into several subcases. +Subcase II.1: |ǫℓ|1/2 ≈ δℓ for a subsequence. +Define the change of variables for ℓ ≥ 1, y ∈ I, +y − y∗ = δℓY, +gℓ(y) := (|λℓ − b(y∗)| + |ǫℓ|)−7/8Hℓ(Y ). +(4.43) +It follows from (4.32) that we can extract a nontrivial limit H ∈ H1(R) of Hℓ satisfying for +Y ∈ R, +(β2 − ∂2 +Y )H(Y ) + +b′′(y∗) +b′′(y∗)Y 2/2 + γ + iαH(Y ) = 0, +(4.44) +where β ∈ R, α, γ ∈ R\{0}. This is impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is +spectrally stable. +Subcase II.2: |ǫℓ|1/2 = o(δℓ) for a subsequence. +Passing to a subsequence and using rescaling +as in (4.43) we can extract a nontrivial limit H ∈ H1(R), such that +(β2 − ∂2 +Y )H(Y ) + lim +ǫ→0 +b′′(y∗) +b′′(y∗)Y 2/2 + γ + iǫH(Y ) = 0. +(4.45) +This is again impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is spectrally stable. +Subcase II.3: δℓ = o(|ǫℓ|1/2) for a subsequence. +This case is not possible thanks to the +bound (4.33). The lemma is now proved. +□ +5. Bounds on ψι +k,ǫ: the non-degenerate case +In this section we obtain bounds on ψι +k,ǫ(y, λ) in the non-degenerate case, i.e. when λ ∈ +Σ\Σδ0/2. Since the arguments are analogous to those in [14], we will be brief in the proofs, and +provide only comments on the main ideas involved. +We begin with the following preliminary bounds. +Lemma 5.1. For λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the bounds +∥ψι +k,ǫ(·, λ)∥H1 +k(I) ≲ |k|−1/2∥ω0k∥H1 +k(I). +(5.1) +Proof. The desired bounds (5.1) follow directly from the Rayleigh equation (2.8) and Lemma +4.2, once we use the Green’s function Gk to invert k2 − ∂2 +y and formulate (2.8) as an integral +equation. +□ +To obtain control on ∂λψι +k,ǫ(·, λ) for λ ∈ Σ\Σδ0/2, we take derivative in (2.8), and obtain +that +(k2 − ∂2 +y)∂λψι +k,ǫ(y, λ) + +b′′(y)∂λψι +k,ǫ(y, λ) +b(y) − λ + iιǫ += +ωk +0(y) +(b(y) − λ + iιǫ)2 − +b′′(y)ψι +k,ǫ(z, λ) +(b(y) − λ + iιǫ)2 , +(5.2) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 19 +for y ∈ I with zero boundary value at y ∈ {0, 1}. Reformulating (5.2) as an integral equation, +we obtain that +∂λψι +k,ǫ(y, λ) + +� 1 +0 +Gk(y, z) +b′′(z)∂λψι +k,ǫ(z, λ) +b(z) − λ + iιǫ +dz += +� 1 +0 +Gk(y, z) +ωk +0(z) +(b(z) − λ + iιǫ)2 dz − +� 1 +0 +Gk(y, z) +b′′(z)ψι +k,ǫ(z, λ) +(b(z) − λ + iιǫ)2 dz. +(5.3) +Recall the definition of the smooth cutoff function Φ below (4.1). We have the following bounds +for ∂λψι +k,ǫ(y, λ) when λ ∈ Σ\Σδ0. +Lemma 5.2. For λ ∈ Σ\Σδ0/2, k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, ∂λψι +k,ǫ(y, λ) satisfies the +following decomposition +∂λψι +k,ǫ(y, λ) = +�b′(y0)ωk +0(y) +|b′(y)|2 +− +b′′(y)ψι +k,ǫ(y, λ) +|b′(y)|2 +� +(1 − Φ(y)) log (b(y) − λ + iιǫ) ++ +� +σ=0,1 +ωk +0(σ)Ψι +k,σ,ǫ(y, λ) log (b(σ) − λ + iιǫ) + Rι +σ,k,y0,ǫ(y). +(5.4) +In the above for σ ∈ {0, 1}, ι ∈ {±}, 0 < ǫ < ǫ0, and λ ∈ Σ\Σδ0/2, +��Rι +σ,k,y0,ǫ +�� +H1 +k(I) ≲ |k|1/2∥ω0k∥H2 +k(I), +��Ψι +k,σ,ǫ(·, λ) +�� +H1 +k(I) ≲ |k|−1/2. +(5.5) +Proof. The basic idea is to expand the right hand side of (5.3) using integration by parts, and +apply Lemma 4.2 after removing the most singular parts. Indeed, denoting schematically, +U := +� 1 +0 +Gk(y, z) +ωk +0(z) +(b(z) − λ + iιǫ)2 dz − +� 1 +0 +Gk(y, z) +b′′(z)ψι +k,ιǫ(z, λ) +(b(z) − λ + iιǫ)2 dz, +(5.6) +we note that ∂λψι +k,ǫ(y, λ) − U satisfies the equation (recalling (4.1) for the definition of Tk,λ,ιǫ), +(I + Tk,λ,ιǫ) +� +∂λψι +k,ǫ(y, λ) +� += −Tk,λ,ιǫU. +(5.7) +The term Tk,λ,ιǫU ∈ H1 +k(I) (noting however that for the boundary terms we need to track the +singular coefficient log (b(σ) − λ + iιǫ), σ ∈ {0, 1}), and we can apply Lemma 4.2 to (5.7) in +order to obtain the desired conclusions. We refer to [14] for the detailed proof. +□ +To obtain bounds on ∂2 +λψι +k,ǫ(y, λ) for λ ∈ Σ\Σδ0/2, we take two derivatives in (2.8) and obtain +that +(k2 − ∂2 +y)∂2 +λψι +k,ǫ(y, λ) + +b′′(y)∂2 +λψι +k,ǫ(y, λ) +b(y) − λ + iιǫ += 2 +ωk +0(y) +(b(y) − λ + iιǫ)3 − 2 +b′′(y)ψι +k,ǫ(z, λ) +(b(y) − λ + iιǫ)3 + +b′′(y)∂λψι +k,ǫ(z, λ) +(b(y) − λ + iιǫ)2 , +(5.8) + +20 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +for y ∈ I with zero boundary value at y ∈ {0, 1}. We can reformulate (5.8) in the integral form +for y ∈ I, as +∂2 +λψι +k,ǫ(y, λ) + +� 1 +0 +Gk(y, z) +b′′(z)∂2 +λψι +k,ǫ(z, λ) +b(z) − λ + iιǫ +dz += +� 1 +0 +Gk(y, z) +� +2 +ωk +0(z) +(b(z) − λ + iιǫ)3 − 2 +b′′(z)ψι +k,ǫ(z, λ) +(b(z) − λ + iιǫ)3 + +b′′(z)∂λψι +k,ǫ(z, λ) +(b(z) − λ + iιǫ)2 +� +dz. +(5.9) +We have the following bounds on ∂2 +λψι +k,ǫ(y, λ) for λ ∈ Σ\Σδ0/2. +Lemma 5.3. For k ∈ Z\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the following bound +����∂2 +λψι +k,ǫ(y, λ) − +ωk +0(1)Φ1ι +k,ǫ(y, λ) +b(1) − λ + iιǫ − +ωk +0(0)Φ0ι +k,ǫ(y, λ) +b(0) − λ + iιǫ − +b′′(y)ψι +k,ǫ(y, λ) − ωk +0(y) +|b′(y)|2(b(y) − λ + iιǫ) +���� +L2(y∈I,λ∈Σ\Σδ0/2) +≲ |k|3/2∥ω0k∥H3 +k(I) +(5.10) +In the above the functions Φσι +k,ǫ, σ ∈ {0, 1} satisfy the equation for y ∈ I +(I + Tk,λ,ιǫ)Φ1ι +k,ǫ = +sinh (ky) +|b′(1)|2 sinh k, +(I + Tk,λ,ιǫ)Φ0ι +k,ǫ = sinh (k(1 − y)) +|b′(0)|2 sinh k . +(5.11) +Proof. The main idea of the proof is to expand the right side of (5.9) and apply Lemma 4.2 +after removing the most singular terms. Indeed, denoting schematically, +U∗ := +� 1 +0 +Gk(y, z) +� +2 +ωk +0(z) +(b(z) − λ + iιǫ)3 − 2 +b′′(z)ψι +k,ιǫ(z, λ) +(b(z) − λ + iιǫ)3 + +b′′(z)∂λψι +k,ιǫ(z, λ) +(b(z) − λ + iιǫ)2 +� +dz, +(5.12) +we have +(I + Tk,λ,ιǫ) +� +∂2 +λψι +k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗� += +� +Tk,λ,ιǫ +�2U∗. +(5.13) +We note that ∂2 +λψι +k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗ ∈ H1 +k(I) (however we again need to track the +singularities in λ in the boundary terms, involving log(b(σ) − λ + iιǫ) and 1/(b(σ) − λ + iιǫ) +for σ ∈ {0, 1}), and we can apply Lemma (4.2) in order to obtain the desired conclusions. We +refer to [14] for the detailed proof. +□ +6. Bounds on ψι +k,ǫ: the degenerate case +In this section we use the limiting absorption principle to study the Rayleigh equation (2.8) +for λ ∈ Σδ0. More precisely, write for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0, (recall the +definition of ǫ0 from Lemma 4.4) +ψι +k,ǫ(y, λ) = φι +k,ǫ(y, λ) + Ψ(y) +1 +b′′(y)ω0k(y), +(6.1) +where Ψ ∈ C∞ +c (S3δ0) and Ψ ≡ 1 on S2δ0. Recall that Sd = [y∗ −d, y∗ +d] for d > 0 from (3.11). +Then φι +k,ǫ(y, λ) satisfies for y ∈ I, +(k2 − ∂2 +y)φι +k,ǫ(y, λ) + +b′′(y) +b(y) − λ + iιǫφι +k,ǫ(y, λ) = gι +k,ǫ(y, λ), +(6.2) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 21 +where for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0 +gι +k,ǫ(y, λ) := +1 − Ψ(y) +b(y) − λ + iιǫω0k(y) − (k2 − ∂2 +y) +� Ψ(y) +b′′(y)ω0k(y) +� +. +(6.3) +Our main results are bounds for the functions φι +k,ǫ(y, λ). We begin with the following pre- +liminary bounds. +Lemma 6.1. Assume that k ∈ Z\{0}, λ ∈ Σδ0 and let φι +k,ǫ(y, λ) with ι ∈ {±}, ǫ ∈ (0, ǫ0) be as +defined in (6.1)-(6.2). Recall from (3.14) and (4.13) that +δ := δ(λ) = 8 +� +|λ − b(y∗)|/|b′′(y∗)|, +dk = dk(λ, ǫ) := +� +|λ − b(y∗)|1/2 + |ǫ|1/2� +∧ 1 +|k|. +(6.4) +We have the bounds for k ∈ Z\{0}, ǫ ∈ (0, ǫ0), ι ∈ {±}, λ ∈ Σδ0, +� +α∈{0,1} +��d−7/4+α +k +∂α +y φι +k,ǫ(y, λ) +�� +L2� +[y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)] +�(δ + |ǫ|1/2)−1/2 ++ +� +α∈{0,1} +��(|y − y∗| ∧ dk)−7/4+α∂α +y φι +k,ǫ(y, λ) +�� +L∞� +[0,1]\[y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)] +� +≲ |k|5/2��ω0k +�� +H3 +k(I). +(6.5) +Define for y ∈ [0, 1], k ∈ Z\{0}, λ ∈ Σδ0\{b(y∗)}, +ψk(y, λ) := lim +ǫ→0+ +� +ψ+ +k,ǫ(y, λ) − ψ− +k,ǫ(y, λ) +� += lim +ǫ→0+ +� +φ+ +k,ǫ(y, λ) − φ− +k,ǫ(y, λ) +� +. +(6.6) +Then we have the bounds for λ ∈ Σδ0\{b(y∗)}, +� +α∈{0,1} +��(δ ∧ |k|−1)−7/4+α∂α +y ψk(y, λ) +�� +L2([y∗−3δ,y∗+3δ])δ−1/2 ++ +� +α∈{0,1} +��(δ ∧ |k|−1)−11/4(|y − y∗| ∧ 1 +|k|)1+α∂α +y ψk(y, λ) +�� +L∞([0,1]\[y∗−3δ,y∗+3δ])) +≲ |k|5/2��ω0k +�� +H3 +k(I). +(6.7) +Proof. It follows from (6.3) and our assumptions on the initial data ω0k that we have the bound +for k ∈ Z\{0}, ι ∈ {±}, 0 < ǫ < ǫ0 and λ ∈ Σδ0, +��gι +k,ǫ(y, λ) +�� +C2 +k(I) ≲ |k|1/2∥ω0k∥H3 +k(I). +(6.8) +We can reformulate equation (6.2) in the integral form as (recall the definition of T ∗(λ + iǫ) +from (4.16)) +φι +k,ǫ(y, λ) + T ∗ +k (λ + iιǫ)φι +k,ǫ(y, λ) = +� 1 +0 +Gk(y, z; λ + iιǫ)gι +k,ǫ(z, λ)dz, +(6.9) +for y ∈ I. By Lemma 4.4, we obtain the bound +��φι +k,ǫ(·, λ) +�� +XN,̺k(I) ≲ +��� +� 1 +0 +Gk(y, z; λ + iιǫ)gι +k,ǫ(z, λ)dz +��� +XN,̺k +≲ |k|5/2∥ω0k∥H3 +k(I), +(6.10) +which, by the definition of the space XN,̺k, see (4.14), implies the desired bounds (6.5). + +22 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +For applications below on isolating the singularity at λ = b(y), we fix ϕδ(y) ∈ C∞ +c (S2δ) as +ϕδ(y) := ϕ(y +δ ) +� +1 − ϕ( y +δ′ ) +� +, +(6.11) +for y ∈ I, with δ′ := δ/M and an M ≫ 1 sufficiently large such that |b(y) − λ| ≈ |λ − b(y∗)| for +|y − y∗| < δ/M. +To prove (6.7), we note from (6.2) that φ+ +k,ǫ(y, λ) − φ− +k,ǫ(y, λ) satisfies the equation for y ∈ I. +(k2 − ∂2 +y) +� +φ+ +k,ǫ(y, λ) − φ− +k,ǫ(y, λ) +� ++ +b′′(y) +b(y) − λ + iǫ +� +φ+ +k,ǫ(y, λ) − φ− +k,ǫ(y, λ) +� += g+ +k,ǫ(y, λ) − g− +k,ǫ(y, λ) − +� +b′′(y) +b(y) − λ + iǫ − +b′′(y) +b(y) − λ − iǫ +� +φ− +k,ǫ(y, λ). +(6.12) +Denote for λ ∈ Σδ0\{b(y∗)}, ǫ ∈ (0, ǫ0) and y ∈ I the function hk,ǫ(y, λ) as the solution to +(k2 − ∂2 +y)hk,ǫ(y, λ) + +b′′(y) +b(y) − λ + iǫhk,ǫ(y, λ) = ϕδ(y) +� +b′′(y) +b(y) − λ − iǫ − +b′′(y) +b(y) − λ + iǫ +� +φ− +k,ǫ(y, λ), +(6.13) +with zero Dirichlet boundary condition. Then it is clear that for λ ∈ Σδ0\{b(y∗)}, y ∈ I, +ψk(y, λ) = lim +ǫ→0+ hk,ǫ(y, λ). +(6.14) +We can reformulate (6.13) as the following integral equation for λ ∈ Σδ0\{b(y∗)}, y ∈ I, +hk,ǫ(y, λ) + T ∗ +k (λ + iǫ)hk,ǫ(y, λ) += − +� 1 +0 +Gk(y, z; λ + iǫ)ϕδ(z) +� +b′′(z) +b(z) − λ + iǫ − +b′′(z) +b(z) − λ − iǫ +� +φ− +k,ǫ(z, λ) dz. +(6.15) +It follows from the bound (6.5) that for |ǫ| ≲ (δ ∧ 1 +|k|)4, +���� +� 1 +0 +Gk(y, z; λ + iǫ)ϕδ(z) +� +b′′(z) +b(z) − λ + iǫ − +b′′(z) +b(z) − λ − iǫ +� +φ− +k,ǫ(z, λ) dz +���� +XL,̺k +≲ (δ ∧ 1 +|k|)7/4. +(6.16) +The desired bound (6.7) then follows from Lemma 4.4 with X = XL,̺k. +□ +To obtain higher order regularity bounds (in λ) of φι +k,ǫ(·, λ), we take the derivative ∂λ in +(6.2). It follows that ∂λφι +k,ǫ(y, λ) satisfies for y ∈ I, +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iιǫ +� +∂λφι +k,ǫ(y, λ) = − +b′′(y) +(b(y) − λ + iιǫ)2 φι +k,ǫ(y, λ) + ∂λgι +k,ǫ(y, λ), +(6.17) +with zero Dirichlet boundary condition. +Recall the definition of ϕδ from (6.11). We have the following bounds on ∂λφι +k,ǫ(y, λ). +Lemma 6.2. Assume that k ∈ Z\{0}, λ ∈ Σδ0\{b(y∗)}. +Let ψι +k,ǫ(y, λ) and φι +k,ǫ(y, λ) with +ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} be as defined in (2.8) and (6.1) respectively. Recall from +(3.14) that +δ := δ(λ) = 8 +� +|λ − b(y∗)|/b′′(y∗). +(6.18) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 23 +Denote for y ∈ [0, 1], ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, +Λι +1,ǫ(y, λ) := φι +k,ǫ(y, λ)ϕδ(y) b′′(y) +(b′(y))2 log b(y) − λ + iιǫ +δ2 +, +Λ1(y, λ) := ψk(y, λ)ϕδ(y) b′′(y) +(b′(y))2 log b(y) − λ +δ2 +. +(6.19) +We have the bounds for 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, ι ∈ {±}, and λ ∈ Σδ0 that +� +α∈{0,1} +���(δ ∧ |k|−1)1/4+α∂α +y +� +∂λφι +k,ǫ(y, λ) − Λι +1,ǫ(y, λ) +���� +L2([y∗−3δ,y∗+3δ])δ−1/2 ++ +� +α∈{0,1} +���(δ ∧ |k|−1)2(|y − y∗| ∧ 1 +|k|)−7/4+α∂α +y ∂λφι +k,ǫ(y, λ) +��� +L∞([0,1]\[y∗−3δ,y∗+3δ])) +≲ |k|5/2��ω0k +�� +H3 +k(I). +(6.20) +In addition, we have the bounds for λ ∈ Σδ0\{b(y∗)} and k ∈ Z\{0}, +� +α∈{0,1} +��(δ ∧ |k|−1)1/4+α∂α +y +� +∂λψk(y, λ) − Λ1(y, λ) +���� +L2([y∗−3δ,y∗+3δ])δ−1/2 ++ +� +α∈{0,1} +��(δ ∧ |k|−1)−3/4(|y − y∗| ∧ 1 +|k|)1+α∂α +y ∂λψk(y, λ) +�� +L∞([0,1]\[y∗−3δ,y∗+3δ])) +≲ |k|5/2��ω0k +�� +H3 +k(I). +(6.21) +Proof. Define for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I, +∂λφι +k,ǫ(y, λ) := φι +k,ǫ(y, λ; 1) + +� 1 +0 +Gk(y, z; λ + iιǫ) +� +−b′′(z) +(b(z) − λ + iιǫ)2 φι +k,ǫ(z, λ) + ∂λgι +k,ǫ(z, λ) +� +dz. +(6.22) +It follows from (6.17) that φι +k,ǫ(y, λ; 1) satisfies for y ∈ I, +φι +k,ǫ(y, λ; 1) + T ∗ +k (λ + iιǫ)φι +k,ǫ(y, λ; 1) += −T ∗ +k (λ + iιǫ) +� 1 +0 +Gk(y, z; λ + iιǫ) +� +− +b′′(z) +(b(z) − λ + iιǫ)2 φι +k,ǫ(z, λ) + ∂λgι +k,ǫ(z, λ) +� +dz. +(6.23) +Denote for k ∈ Z\{0}, ι ∈ {±}, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, z ∈ I, +hι +k,ǫ(z, λ; 1) := +b′′(z) +(b(z) − λ + iιǫ)2 ϕδ(z)φι +k,ǫ(z, λ), +hι +k,ǫ(z, λ; 2) := +b′′(z) +(b(z) − λ + iιǫ)2 (1 − ϕδ(z))φι +k,ǫ(z, λ), +hι +k,ǫ(z, λ; 3) := ∂λgι +k,ǫ(z, λ). +(6.24) +It follows from the bound (6.5) and Lemma 3.1 that for j ∈ {2, 3} +��T ∗ +k (λ + iιǫ) +� 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz +�� +XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2∥ω0k∥H3 +k(I). +(6.25) + +24 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +Using integration by parts argument similar to (4.29)-(4.30), we have also +����T ∗ +k (λ + iιǫ) +� 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; 1) dz +���� +XN,̺k +≲ (δ ∧ |k|−1)−2|k|5/2��ω0k +�� +H3 +k(I). +(6.26) +It follows from (6.25)-(6.26) and Lemma 4.4 that for λ\{b(y∗)}, +��φι +k,ǫ(y, λ; 1) +�� +XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2��ω0k +�� +H3 +k(I). +(6.27) +The desired bound (6.20) follows, as a consequence of (6.27) and (6.22). +Using (6.17), we get that for y ∈ I, +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iǫ +�� +∂λφ+ +k,ǫ(y, λ) − ∂λφ− +k,ǫ(y, λ) +� += − +� +b′′(y) +(b(y) − λ + iǫ)2 φ+ +k,ǫ(y, λ) − +b′′(y) +(b(y) − λ − iǫ)2 φ− +k,ǫ(y, λ) +� ++ +� +∂λg+ +k,ǫ(y, λ) − ∂λg− +k,ǫ(y, λ) +� +− +� +b′′(y) +b(y) − λ + iǫ − +b′′(y) +b(y) − λ − iǫ +� +∂λφ− +k,ǫ(y, λ), +(6.28) +with zero Dirichlet boundary condition. +Denoting for λ ∈ Σδ0\{b(y∗)} and y ∈ I, Dφk,ǫ(y, λ) as the solution to +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iιǫ +� +Dφk,ǫ(y, λ) += −ϕδ(y) +� +b′′(y) +(b(y) − λ + iǫ)2 φ+ +k,ǫ(y, λ) − +b′′(y) +(b(y) − λ − iǫ)2 φ− +k,ǫ(y, λ) +� +− ϕδ(y) +� +b′′(y) +b(y) − λ + iιǫ − +b′′(y) +b(y) − λ − iιǫ +� +∂λφ− +k,ǫ(y, λ), +(6.29) +for y ∈ I with zero Dirichlet boundary condition. +We notice the identity that for y ∈ I, λ ∈ Σδ0\{b(y∗)}, +∂λψk(y, λ) = lim +ǫ→0+ Dφk,ǫ(y, λ). +(6.30) +We can reformulate (6.29) as the integral equation for y ∈ I, +Dφk,ǫ(y, λ) + T ∗ +k (λ + iǫ)Dφk,ǫ(y, λ) += − +� 1 +0 +Gk(y, z; λ + iǫ)ϕδ(z) +� +b′′(z) +(b(z) − λ + iǫ)2 φ+ +k,ǫ(z, λ) − +b′′(z) +(b(z) − λ − iǫ)2 φ− +k,ǫ(z, λ) +� +dz +− +� 1 +0 +Gk(y, z; λ + iǫ)ϕδ(z) +� +b′′(z) +b(z) − λ + iǫ − +b′′(z) +b(z) − λ − iιǫ +� +∂λφ− +k,ǫ(z, λ) dz +:= Rk,ǫ(y, λ). +(6.31) +We can write for y ∈ I, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, +Dφk,ǫ(y, λ) := Rk,ǫ(y, λ) + Dφk,ǫ(y, λ; 1). +(6.32) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 25 +Then Dφk,ǫ(y, λ; 1) satisfies for y ∈ I, λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, the +equation +Dφk,ǫ(y, λ; 1) + T ∗ +k (λ + iǫ)Dφk,ǫ(y, λ; 1) = −T ∗ +k (λ + iǫ)Rk,ǫ(y, λ). +(6.33) +The desired bounds (6.37) follow from (6.31)-(6.33), and Lemma 3.2 with X = XL,̺k. +□ +Lastly we turn to the highest order derivative ∂2 +λψι +k,ǫ(y, λ) that we need to control. To study +∂2 +λψι +k,ǫ(y, λ), we take the derivative ∂λ in (6.17) and obtain that +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iιǫ +� +∂2 +λφι +k,ǫ(·, λ) = − +2b′′(y) +(b(y) − λ + iιǫ)2 ∂λφι +k,ǫ(·, λ) +− +2b′′(y) +(b(y) − λ + iιǫ)3 φι +k,ǫ(y, λ) + ∂2 +λgι +k,ǫ(y, λ). +(6.34) +Lemma 6.3. Assume that k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)} and let φι +k,ǫ(y, λ) with ι ∈ {±}, 0 < ǫ < +min{|λ − b(y∗)|, ǫ0} be as defined in (6.2). Recall that +δ := δ(λ) = 8 +� +|λ − b(y∗)|/b′′(y∗). +(6.35) +Denoting for y ∈ [0, 1], λ ∈ Λδ0\{b(y∗)}, +Λ2(y, λ) := − ψk(y, λ)ϕδ(y) b′′(y) +(b′(y))2 lim +ǫ→0+ +1 +b(y) − λ + iǫ +− ϕδ(y) b′′(y) +(b′(y))2 lim +ǫ→0+ +� +1 +b(y) − λ + iǫ − +1 +b(y) − λ − iǫ +� +φ− +k,ǫ(y, λ), +(6.36) +then we have the bounds for λ ∈ Λδ0\{b(y∗)}, +� +α∈{0,1} +���(δ ∧ |k|−1)9/4� +∂2 +λψk(y, λ) − Λ2(y, λ) +���� +L2([y∗−3δ,y∗+3δ])δ−1/2 ++ +� +α∈{0,1} +���(δ ∧ |k|−1)5/4(|y − y∗| ∧ 1 +|k|)∂2 +λψk(y, λ) +��� +L∞([0,1]\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k +�� +H3 +k(I). +(6.37) +Proof. Denote for k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and y ∈ I, +hι +k,ǫ(z, λ; 4) := − +2b′′(z) +(b(z) − λ − iιǫ)2 ϕδ(z)∂λφι +k,ǫ(z, λ), +hι +k,ǫ(z, λ; 5) = − +2b′′(z) +(b(z) − λ − iιǫ)3 ϕδ(z)φι +k,ǫ(z, λ) +hι +k,ǫ(z, λ; 6) := − +b′′(z) +(b(z) − λ − iιǫ)2 (1 − ϕδ(z))∂λφι +k,ǫ(z, λ), +hι +k,ǫ(z, λ; 7) = − +2b′′(z) +(b(z) − λ − iιǫ)3 (1 − ϕδ(z))φι +k,ǫ(z, λ), +hι +k,ǫ(z, λ; 8) := ∂2 +λgι +k,ǫ(z, λ). +(6.38) + +26 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +Define for k ∈ Z\{0}, λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and z ∈ I, +∂2 +λφι +k,ǫ(y, λ) := φι +k,ǫ(y, λ; 2) + +8 +� +j=4 +� 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz +− +8 +� +j=4 +T ∗ +k (λ + iιǫ) +� 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz +(6.39) +It follows from (6.34) that φι +k,ǫ(y, λ; 2) satisfies for y ∈ I, +φι +k,ǫ(y, λ; 2) + T ∗ +k (λ + iιǫ)φι +k,ǫ(y, λ; 2) = +8 +� +j=4 +� +T ∗ +k (λ + iιǫ) +�2 � 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz. +(6.40) +It follows from Lemma 6.2 and Lemma 3.1 that for j ∈ {6, 7, 8} +��� +� +T ∗ +k (λ + iǫ) +�2 +� 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz +��� +XN,̺k +≲ (δ ∧ |k|−1)−4|k|5/2∥ω0k∥H3 +k(I). (6.41) +Using integration by parts argument similar to (4.29)-(4.30), we have also for j ∈ {4, 5}, +���� +� +T ∗ +k (λ + iǫ) +�2 � 1 +0 +Gk(y, z; λ + iιǫ)hι +k,ǫ(z, λ; j) dz +���� +XN,̺k +≲ (δ ∧ |k|−1)−4|k|5/2��ω0k +�� +H3 +k(I). +(6.42) +It follows from (6.38)-(6.42) and Lemma 4.4 that for λ ∈ Λδ0\{b(y∗)}, ι ∈ {±}, 0 < ǫ < +min{|λ − b(y∗)|, ǫ0}, +��φι +k,ǫ(y, λ; 2) +�� +XN,̺k ≲ (δ ∧ |k|−1)−4|k|5/2��ω0k +�� +H1 +k(I). +(6.43) +Using (6.34), we get that for y ∈ I, +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iǫ +�� +∂2 +λφ+ +k,ǫ(y, λ) − ∂2 +λφ− +k,ǫ(y, λ) +� += +8 +� +j=4 +� +h+ +k,ǫ(y, λ; j) − h− +k,ǫ(y, λ; j) +� +. +(6.44) +Denoting D2φk,ǫ(y, λ), h ∈ I, λ ∈ Λδ0\{b(y∗)}, as the solution to +� +k2 − ∂2 +y + +b′′(y) +b(y) − λ + iιǫ +� +D2φk,ǫ(y, λ) = +5 +� +j=4 +� +h+ +k,ǫ(y, λ; j) − h− +k,ǫ(y, λ; j) +� +, +(6.45) +for y ∈ I with zero Dirichlet boundary condition. +We note the identity that for y ∈ I, λ ∈ Σδ0\{b(y∗)}, +∂2 +λψk(y, λ) = lim +ǫ→0+ D2φk,ǫ(y, λ). +(6.46) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 27 +We can reformulate (6.45) as the integral equation for y ∈ I +D2φk,ǫ(y, λ) + T ∗ +k (λ + iǫ)D2φk,ǫ(y, λ) += +� 1 +0 +Gk(y, z; λ + iǫ)ϕδ(z) +5 +� +j=4 +� +h+ +k,ǫ(z, λ; j) − h− +k,ǫ(z, λ; j) +� +dz := R∗ +k,ǫ(y, λ). +(6.47) +We can write for λ ∈ Σδ0\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I, +D2φk,ǫ(y, λ) := D2φk,ǫ(y, λ; 2) + R∗ +k,ǫ(y, λ) − T ∗ +k (λ + iǫ)R∗ +k,ǫ(y, λ). +(6.48) +Then D2φk,ǫ(y, λ; 2) satisfies for y ∈ I, λ ∈ Σδ0\{b(y∗)}, +D2φk,ǫ(y, λ; 2) + T ∗ +k (λ + iǫ)D2φk,ǫ(y, λ; 2) = +� +T ∗ +k (λ + iǫ) +�2R∗ +k,ǫ(y, λ). +(6.49) +The desired bounds (6.37) follow from (6.47)-(6.49), and Lemma 3.2 with X = XL,̺k, using +also the bound +��� +T ∗ +k (λ + iǫ) +�2R∗ +k,ǫ(·, λ) +�� +XL,̺k ≲ +� +δ ∧ 1 +|k| +�−4 +|k|5/2∥ω0k∥H3 +k(I). +(6.50) +□ +7. Proof of Theorem 1.2 +In this section, we prove Theorem 1.2. We can assume that t ≥ 1. We first give the proof of +(1.8)-(1.9). Using the representation formula (2.7), we have +ψk(t, y) = +1 +2πi lim +ǫ→0+ +� +Σ +e−ikλt� +ψ+ +k,ǫ(y, λ) − ψ− +k,ǫ(y, λ) +� +dλ += − +1 +2πik2t2 lim +ǫ→0+ +� +Σ +e−ikλt� +∂2 +λψ+ +k,ǫ(y, λ) − ∂2 +λψ− +k,ǫ(y, λ) +� +dλ. +(7.1) +Fix Φ∗ ∈ C∞ +0 (Σδ0) with Φ∗ ≡ 1 on Σ2δ0/3. We can decompose for t ≥ 1, y ∈ [0, 1], +ψk(t, y) := ψ1 +k(t, y) + ψ2 +k(t, y), +(7.2) +where +ψ1 +k(t, y) := − +1 +2πik2t2 lim +ǫ→0+ +� +Σ +e−ikλt(1 − Φ∗(λ)) +� +∂2 +λψι +k,ǫ(y, λ) − ∂2 +λψ− +k,ǫ(y, λ) +� +dλ, +ψ2 +k(t, y) := − +1 +2πik2t2 lim +ǫ→0+ +� +Σ +e−ikλtΦ∗(λ) +� +∂2 +λψι +k,ǫ(y, λ) − ∂2 +λψ− +k,ǫ(y, λ) +� +dλ. +(7.3) +For (1.8), it suffices to prove that for σ ∈ {1, 2}, k ∈ Z\{0} and t ≥ 1, +��ψσ +k(t, ·) +�� +L2([0,1]) ≲ |k|3 +t2 ∥ω0k∥H3 +k([0,1]). +(7.4) +The case σ = 1 in (7.4) corresponding to the non-degenerate case is analogous to the case of +monotonic shear flows, see [14], and follow from Lemma 5.1-Lemma 5.3. We focus on the main +new case σ = 2 in (7.4). Denote for k ∈ Z\{0}, +Mk := |k|5/2∥ω0k∥H3 +k([0,1]). +(7.5) +Our main tools are Lemmas 6.1, Lemma 6.2 and Lemma 6.3, which imply the following bounds +for y ∈ [0, 1], λ ∈ Σδ0. + +28 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +• If |λ − b(y∗)|1/2 < |y − y∗|/20, then +|ψk(y, λ)| ≲ +� +min +� +|λ − b(y∗)|1/2, |k|−1��11/4(|y − y∗|−1 + |k|)Mk, +|∂2 +λψk(y, λ) ≲ +� +min +� +|λ − b(y∗)|1/2, |k|−1��−5/4(|y − y∗|−1 + |k|)Mk; +(7.6) +• If |y − y∗|/20 < |λ − b(y∗)|1/2 < 20|y − y∗|, then +|ψk(y, λ)| ≲ +� +min +� +|λ − b(y∗)|1/2, |k|−1��5/4|λ − b(y∗)|1/4Mk, +|ψk(y, λ) − ψk(y, b(y))| ≲ |λ − b(y)|1/2|λ − b(y∗)|3/8Mk, +��∂2 +λψk(·, λ) − Λ2(·, λ) +�� +L2(|y−y∗|≈|λ−b(y∗)|1/2) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk; +(7.7) +• If |λ − b(y∗)|1/2 > 20|y − y∗|, then +|ψk(y, λ)| ≲ |λ − b(y∗)|1/4� +min +� +|λ − b(y∗)|1/2, |k|−1��5/4Mk, +��∂2 +λψk(·, λ) − Λ2(·, λ) +�� +L2(|y−y∗|<|λ−b(y∗)|1/2/20) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk. +(7.8) +It follows from (7.6)-(7.8) that for y ∈ [0, 1], t ≥ 1, +��� +� +R +e−ikλtΦ∗(λ)Λ2(y, λ)dλ +��� ≲ |y − y∗|−1/4 max +� +1, |k|1/2|y − y∗|1/2� +Mk, +(7.9) +and, by considering the cases |λ − b(y∗)| ≪ |y − y∗|2, |λ − b(y∗)| ≈ |y − y∗|2 and |λ − b(y∗)| ≫ +|y − y∗|2, also that for y ∈ [0, 1], t ≥ 1, +��� +� +R +e−ikλtΦ∗(λ) +� +∂2 +λψk(y, λ) − Λ2(y, λ) +� +dλ +��� +L2([0,1]) ≲ |k|9/4Mk. +(7.10) +The desired bound (7.4) for σ = 2 follows from (7.9)-(7.10). +The proof of (1.9) is similar to the proof of (1.8), using Lemma 6.1 and Lemma 6.2. +We now turn to the proof of the depletion bounds (1.11). Assume that k ∈ Z\{0}. Applying +−k2 + ∂2 +y to ψk(t, y) in (2.7), and using (2.8), we get that for y ∈ [0, 1], t ≥ 1, +ωk(t, y) = ω∗ +k(t, y) + ω∗∗ +k (t, y), +(7.11) +where +ω∗ +k(t, y) +:= +1 +2πi lim +ǫ→0+ +� +Σ +e−ikλt(1 − Φ∗(y)) +�b′′(y)ψ+ +k,ǫ(y, λ) − ω0k(y) +b(y) − λ + iǫ +− +b′′(y)ψ− +k,ǫ(y, λ) − ω0k(y) +b(y) − λ − iǫ +� +dλ, +ω∗∗ +k (t, y) := +1 +2πi lim +ǫ→0+ +� +Σ +e−ikλtΦ∗(y) +�b′′(y)ψ+ +k,ǫ(y, λ) − ω0k(y) +b(y) − λ + iǫ +− +b′′(y)ψ− +k,ǫ(y, λ) − ω0k(y) +b(y) − λ − iǫ +� +dλ. +(7.12) +We have the bound for t ≥ 1, +��ω∗ +k(t, y) +�� +L∞([0,1]) ≲ |k|2Mk. +(7.13) + +LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 29 +For |y − y∗| < δ0/10, t ≥ 1, since (b(y) − λ + iιǫ) with ι ∈ {±} is not singular in this case, we +have in addition by integration by parts that +|ω∗ +k(t, y)| ≲ |k|2 1 +t Mk. +(7.14) +We now turn to ω∗∗ +k (t, y). Using (6.1), we can write for y ∈ [0, 1], t ≥ 1, +2πi ω∗∗ +k (t, y) += lim +ǫ→0+ +� +R +e−ikλtΦ∗(λ) +�φ+ +k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y) +b(y) − λ + iǫ +− +φ− +k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y) +b(y) − λ − iǫ +� +dλ += lim +ǫ→0+ +� +R +e−ikλtΦ∗(λ) +� +φ+ +k,ǫ(y, λ) +b(y) − λ + iǫ − +φ− +k,ǫ(y, λ) +b(y) − λ − iǫ +� +dλ + Wk(t, y), +(7.15) +where Wk(t, y) satisfies the bound for t ≥ 1, +∥Wk(t, ·)∥L∞([0,1]) ≲ t−1Mk, +(7.16) +which follows from simple integration by parts argument. We decompose for y ∈ [0, 1]\{y∗}, +ω∗∗ +k (t, y) − Wk(t, y) +2πi += +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(λ) +� +ψk(y, λ) +b(y) − λ + iǫ +� +dλ ++ +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(λ)φ− +k,ǫ(y, λ) +� +1 +b(y) − λ + iǫ − +1 +b(y) − λ − iǫ +� +dλ. +(7.17) +It follows from (7.6)-(7.8) that +���� +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(λ)φ− +k,ǫ(y, λ) +� +1 +b(y) − λ + iǫ − +1 +b(y) − λ − iǫ +� +dλ +���� ≲ |y − y∗|7/4Mk. +(7.18) +For γ ∈ (1, ∞) to be fixed below, by considering the three ranges (I) |λ − b(y∗)| ≲ |y − y∗|2, +(II) |λ − b(y∗)| ≥ γ|y − y∗|2, and (III) |y − y∗|2 ≪ |λ − b(y∗)| < γ|y − y∗|2, and using Lemma +6.2 and Lemma 6.39, we get that +���� +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(y) +� +ψk(y, λ) +b(y) − λ + iǫ +� +dλ +���� +≲ +� +|y − y∗|7/4� +1 + |k|1/2|y − y∗|1/2� ++ +1 +|k|t(|k|1/2 + γ−1/8|y − y∗|−1/4) + γ7/8|y − y∗|7/4� +Mk. +(7.19) +In the above, we used integration by part to get decay in t in range (II). Optimizing in γ, we +get that for t ≥ 1, +(i) if t|y − y∗|2 ≲ 1, +���� +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(y) +� +ψk(y, λ) +b(y) − λ + iǫ +� +dλ +���� +≲ +� +t−1 + |k|1/2|y − y∗|7/4 + t−7/8� +(7.20) + +30 +ALEXANDRU D. IONESCU, SAMEER IYER, AND HAO JIA +(ii) if t|y − y∗|2 ≫ 1, +���� +1 +2πi lim +ǫ→0+ +� +R +e−ikλtΦ∗(y) +� +ψk(y, λ) +b(y) − λ + iǫ +� +dλ +���� +≲ +� +|y − y∗|7/4� +1 + |k|1/2|y − y∗|1/2� ++ +1 +|k|1/2t7/8 + |y − y∗|7/4� +Mk. +(7.21) +The desired bounds (7.16), (7.18), (7.20)-(7.21). Theorem 1.2 is now proved. +References +[1] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, New York, 1998. +[2] J. Bedrossian and N. Masmoudi Inviscid damping and the asymptotic stability of planar shear flows in the +2D Euler equations, Publ. Math. Inst. Hautes Etudes Sci. 122 (2015), 195-300. +[3] J. Bedrossian, M. Coti Zelati, and V. 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Wei, Diffusion and mixing in fluid flow via the resolvent estimate, Science China Mathematics, volume +64, pages 507-518 (2021) +[34] C. Zillinger, Linear inviscid damping for monotone shear flows, Trans. Amer. Math. Soc. 369 (2017), +8799-8855. +[35] C. Zillinger, Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, +blow-up and critical Sobolev regularity, Arch. Ration. Mech. Anal. 221 (2016), 1449-1509. +[36] M. Coti Zelati and C. Zillinger, On degenerate circular and shear flows: the point vortex and power law +circular flows, Communications in Partial Differential Equations, 2019, 44:2, 110-155 +Princeton University +Email address: aionescu@math.princeton.edu +University of California, Davis +Email address: sameer@math.ucdavis.edu +University of Minnesota +Email address: jia@umn.edu + diff --git a/6tAyT4oBgHgl3EQfcvc9/content/tmp_files/load_file.txt b/6tAyT4oBgHgl3EQfcvc9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0c62c85f1a8c0856edd85935b54ceb244c19ca32 --- /dev/null +++ b/6tAyT4oBgHgl3EQfcvc9/content/tmp_files/load_file.txt @@ -0,0 +1,1248 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf,len=1247 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='00288v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='AP] 31 Dec 2022 LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We give a proof of linear inviscid damping and vorticity depletion for non-monotonic shear flows with one critical point in a bounded periodic channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In particular, we obtain quantitative depletion rates for the vorticity function without any symmetry assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Dedicated to Carlos Kenig, on the occasion of his 70th birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Key Words: Inviscid damping, vorticity depletion, non-monotonic shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Mathematics Subject Classification: 35B40, 35Q31, 35P25 Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Spectral property and representation formula 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on the Green’s function and modified Green’s function 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The limiting absorption principle 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on ψι k,ǫ: the non-degenerate case 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on ψι k,ǫ: the degenerate case 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 27 References 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Introduction The study of stability problems in mathematical analysis of fluid dynamics has a long and distinguished history, dating back to the work of Kelvin [18], Orr [25] and Rayleigh [26] among many others, and continuing to the present day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Hydrodynamical stability problems can be considered in both two and three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this paper we work with two dimensional inviscid flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For the Euler equations, there are significant recent progresses on the asymptotic stability of monotonic shear flows and vortices, assuming spectral stability, see for example [9, 30, 34, 35, 14, 16, 3, 17, 22, 28] for linear results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main mechanism of stabilization is the so called “inviscid damping”, which refers to the transfer of energy of vorticity to higher and higher frequencies leading to decay of the stream and velocity functions, as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Extending the linearized stability analysis for inviscid fluid equations to the full nonlinear setting is a challenging problem, and the only available results are on spectrally stable monotonic shear The first author was supported in part by NSF grant DMS-2007008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The second author is partially supported by a UC Davis startup grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The third author was supported in part by NSF grant DMS-1945179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1 2 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA flows [2, 23, 10, 12], and on point vortices [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We refer also to the recent review article [13] for a more in-depth discussion of recent developments of both linear and nonlinear inviscid damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Many physically important shear flows are not monotonic, such as Poiseuille flow and Kol- mogorov flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For such flows on the linear inviscid level, there is an additional significant physical phenomenon called “vorticity depletion” which refers to the asymptotic vanishing of vorticity as t → ∞ near the critical point where the derivative of the shear flow is zero, first predicted in Bouchet and Morita [5], and proved rigorously in Wei-Zhang-Zhao [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' A similar phenomenon was proved in Bedrossian-Coti Zelati-Vicol [3] for the case of vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' See also [17] by the first and third author for a refined description of the dynamics in Gevrey spaces as a step towards proving nonlinear vortex symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In [31] by Wei-Zhang-Zhao, sharp linear inviscid damping estimates and quantitative deple- tion estimates were obtained for an important class of “symmetric shear flows” in a periodic channel (see also [32] by Wei-Zhang-Zhao for a similar result for Kolmogorov flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' When no symmetry is assumed, only qualitative bounds are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Heuristically the general case should be similar to the symmetric one, since the main vorticity depletion mechanism is completely local and asymptotically all shear flows approach symmetric ones at the (non- degenerate) critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' However there are significant difficulties in using the approach of [31] to extend the quantitative depletion bounds of [31] to the general case, as the argument in [31] relies heavily on decomposition of functions into odd and even parts, which are specific to symmetric shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this paper we prove linear inviscid damping estimates and quantitative vorticity depletion estimates for a class of stable non-monotonic shear flows with one non-degenerate critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main new features of our results are that we do not need symmetry condition on the background shear flow, and that our formulation on quantitative depletion for vorticity function seem to be new even for general symmetric shear flows (see however Wei-Zhang-Zhao [32] which contains a sharp depletion rate at the critical points for Kolmogorov flow), see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 below for the precise statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We begin with the description of our main equations and theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Main equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Consider the two dimensional Euler equation linearized around a shear flow (b(y), 0), in the periodic channel (x, y, t) ∈ T × [0, 1] × [0, ∞): ∂tω + b(y)∂xω − b′′(y)uy = 0, div u = 0 and ω = −∂yux + ∂xuy, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) with the natural non-penetration boundary condition uy|y=0,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For the linearized flow, � T×[0, 1] ux(x, y, t) dxdy and � T×[0, 1] ω(x, y, t) dxdy are conserved quan- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this paper, we will assume that � T×[0,1] ux 0(x, y) dxdy = � T×[0,1] ω0 dxdy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' These assumptions can be dropped by adjusting b(y) with a linear shear flow C0y + C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then one can see from the divergence free condition on u that there exists a stream function ψ(t, x, y) with ψ(t, x, 0) = ψ(t, x, 1) ≡ 0, such that ux = −∂yψ, uy = ∂xψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 3 The stream function ψ can be solved through ∆ψ = ω, ψ|y=0,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) We summarize our equations as follows \uf8f1 \uf8f2 \uf8f3 ∂tω + b(y)∂xω − b′′(y)∂xψ = 0, ∆ψ(t, x, y) = ω(t, x, y), ψ(t, x, 0) = ψ(t, x, 1) = 0, (ux, uy) = (−∂yψ, ∂xψ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) for t ≥ 0, (x, y) ∈ T × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Our goal is to understand the long time behavior of ω(t) as t → ∞, with Sobolev regular initial vorticity ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We describe more precisely the main assumptions and our main conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main conditions we shall assume on the shear flow b(y) ∈ C4([0, 1]) are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We assume that the background flow b(y) ∈ C4([0, 1]) satisfies the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1) S := {y ∈ [0, 1] : b′(y) = 0} = {y∗} ⊂ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) In addition, b′′(y∗) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2) For k ∈ Z\\{0}, the linearized operator Lk : L2(0, 1) → L2(0, 1) defined as Lkg(y) := b(y)g(y) + b′′(y) � 1 0 Gk(y, z)g(z) dz (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) has no discrete eigenvalues nor generalized embedded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In the above Gk is the Green’s function for k2 − d2 dy2 on the interval (0, 1) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We refer to section 2 below for the definition and more discussion about generalized embed- ded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Our main result is the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that ω(t, ·) ∈ C([0, ∞), H4(T×[0, 1])) with the associated stream func- tion ψ(t, ·) is the unique solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4), with initial data ω0 ∈ H4(T × [0, 1]) satisfying for all y ∈ [0, 1], � T ω0(x, y) dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) Then we have the following bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (i) Inviscid damping estimates: ∥ψ(t, ·)∥L2(T×[0,1]) ≲ 1 ⟨t⟩2 ∥ω0∥H4(T×[0,1]), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) ∥ux(t, ·)∥L2(T×[0,1]) ≲ 1 ⟨t⟩∥ω0∥H4(T×[0,1]), ∥uy(t, ·)∥L2(T×[0,1]) ≲ 1 ⟨t⟩2 ∥ω0∥H4(T×[0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) (ii) Vorticity depletion estimates: there exists a decomposition ω(t, x, y) := ωloc(t, x, y) + ωnloc(t, x, y), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) 4 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA where for (x, y, t) ∈ T × [0, 1] × [0, ∞), |ωloc(t, x, y)| ≲ |y − y∗|7/4∥ω0∥H4(T×[0,1]), |ωnloc(t, x, y)| ≲ 1 ⟨t⟩7/8 ∥ω0∥H4(T×[0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Remarks and main ideas of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We have the following remarks on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Firstly, in the above theorem we have not tracked the minimal regularity required for the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) to hold, and a more careful argument can probably significantly reduce the number of derivatives needed on the initial data ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Secondly, we note also that the argument here can be applied to non-monotonic shear flows with multiple non-degenerate points, although the presentation will be more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Thirdly, a more sophisticated analysis may yield a sharper rate of vorticity depletion with rate |ωloc(t, x, y)| ≲ |y − y∗|2−, |ωnloc(t, x, y)| ≲ ⟨t⟩−1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It is not clear to us though if one can reach the optimal rates of |y − y∗|2 and ⟨t⟩−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We briefly explain the main ideas of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By a standard spectral representation formula, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7), it suffices to study the spectral density functions and the associated Rayleigh equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' There are two main cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' When the spectral parameter λ is not close to the critical value b(y∗), the situation is similar to monotonic shear flows and can be treated as in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main new case is when the spectral parameter λ is close to the critical value b(y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this case, the Rayleigh equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) is very singular, and the potential term b′′(y) b(y)−λ+iǫ has a quadratic singularity roughly of the form 2 (y−y∗)2+(λ−b(y∗))+iǫ for y close to y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The key observation here, as in [17], is that the potential term b′′(y) b(y)−λ+iǫ is critically singular and has real part with a favorable sign for 1 ≫ |y − y∗| ≫ |λ − b(y∗)|1/2, which needs to be incorporated as part of the main term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We therefore define a modified Green’s function for the main term, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13), which has strong vanishing conditions near y = y∗, leading ultimately to vorticity depletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' After extracting the main terms in the Rayleigh equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), the rest of the terms can be treated as compact perturbations, and can be bounded using a limiting absorption principle, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4, thanks to the spectral assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The limiting absorption principle provides preliminary bounds on the spectral density func- tions ψι k,ǫ(y, λ) with ι ∈ {±}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' To obtain the desired quantitative decay rates, we take up to two derivatives in λ of the spectral density functions, and again use the limiting absorption principle to estimate the resulting derivatives, after extracting the main singular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The procedure is more or less straightforward but the calculations are quite lengthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We refer to [14] also for similar calculations in a simpler setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lastly, we note that there are important cancellations between ψ+ k,ǫ(y, λ) and ψ− k,ǫ(y, λ) in the limit ǫ → 0+, which is the reason why we need two versions of the limiting absorption principle, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4, with different weighted spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We summarize here some notations that are specific for this paper for the reader’s conveniences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For positive numbers α, β, we set α ∧ β := min{α, β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We denote for d > 0, Σd := {b(y) : y ∈ [y∗ − d, y∗ + d]}, Sd := [y∗ − d, y∗ + d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We also denote Σ := {b(y) : y ∈ [0, 1]} and I := [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For k ∈ Z\\{0}, we define for f ∈ H1(I) the norm ∥f∥H1 k(I) := ∥f∥L2(I) + |k|−1∥f ′∥L2(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Spectral property and representation formula Taking Fourier transform in x in the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) for ω, we obtain that ∂tωk + ikb(y)ωk − ikb′′(y)ψk = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) for k ∈ Z, t ≥ 0, y ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In the above, ωk and ψk are the k-th Fourier coefficients of ω, ψ in x respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For each k ∈ Z\\{0}, recall from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) that for any g ∈ L2(0, 1), Lkg(y) = b(y)g(y) + b′′(y) � 1 0 Gk(y, z)g(z)dz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) where Gk is the Green’s function for the operator k2− d2 dy2 on (0, 1) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) can be reformulated abstractly as ∂tωk + ikLkωk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) In contrast to the spectral property of the linearized operator around monotonic shear flows, the spectral property of Lk is less understood, especially on the generation of discrete eigen- values and embedded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' From general spectral theory, we know that the spectrum of Lk consists of the continuous spectrum Σ := � b(y) : y ∈ [0, 1] � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) together with some discrete eigenvalues with nonzero imaginary part which can only accumulate at the set of continuous spectrum Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Unlike the case of monotonic shear flows where the discrete eigenvalues can accumulate only at inflection points of the background shear flow, there appears no simple characterization of the possible accumulation points for non-monotonic shear flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall that λ ∈ Σ is called an embedded eigenvalue if there exists a nontrivial g ∈ L2(0, 1), such that Lkg = λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) For non-monotonic shear flows, this definition is too restrictive, as accumulation points of discrete eigenvalues may no longer be embedded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' To capture the discrete eigen- values, we recall the following definition of “generalized embedded eigenvalues”, which can be found already in [31], adapted to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We call λ ∈ Σ a generalized embedded eigenvalue, if one of the following conditions is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ is an embedded eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ ̸= b(y∗) and there exists a nontrivial ψ ∈ H1 0(0, 1) : (0, 1) → C such that in the sense of distributions on (0, 1), (k2 − ∂2 y)ψ(y) + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='b′′(y)ψ(y) b(y) − λ + iπ � z∈[0,1], b(z)=λ b′′(z)ψ(z) |b′(z)| δ(y − z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) We remark that our assumption that the critical point y∗ of b(y) being non-degenerate implies that the sum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) is finite, and that the spectral assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 is satisfied if b′′ > 0 on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 6 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Suppose that k ∈ Z\\{0} and ωk 0 ∈ L2([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then the stream function ψk(t, y) for k ∈ Z\\{0}, y ∈ [0, 1], t ≥ 0 has the representation ψk(t, y) = − 1 2πi lim ǫ→0+ � Σ e−ikλt � ψ− k,ǫ(y, λ) − ψ+ k,ǫ(y, λ) � dλ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) where ψι k,ǫ(y, λ) for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ, k ∈ Z\\{0}, and sufficiently small ǫ ∈ [−1/4, 1/4]\\{0}, are the solutions to −k2ψι k,ǫ(y, λ) + d2 dy2 ψι k,ǫ(y, λ) − b′′(y) b(y) − λ + iιǫψι k,ǫ(y, λ) = −ωk 0(y) b(y) − λ + iιǫ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By standard theory of spectral projection, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3), we obtain that for y ∈ [0, 1], ωk(t, y) = 1 2πi lim ǫ→0+ � Σ eiλt �� (λ + kLk − iǫ)−1 − (λ + kLk + iǫ)−1� ωk 0 � (y) dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) We then obtain for y ∈ [0, 1], ψk(t, y) = − 1 2πi lim ǫ→0+ � Σ e−ikλt � 1 0 Gk(y, z) × �� (−λ + Lk − iǫ)−1 − (−λ + Lk + iǫ)−1� ωk 0 � (z) dzdλ = − 1 2πi lim ǫ→0+ � Σ e−ikλt � ψ− k,ǫ(y, λ) − ψ+ k,ǫ(y, λ) � dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) In the above, for y ∈ [0, 1] and λ ∈ Σ, ψ+ k,ǫ(y, λ) := � 1 0 Gk(y, z) � (−λ + Lk + iǫ)−1ωk 0 � (z) dz, ψ− k,ǫ(y, λ) := � 1 0 Gk(y, z) � (−λ + Lk − iǫ)−1ωk 0 � (z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) Therefore for ι ∈ {+, −}, y ∈ [0, 1], λ ∈ Σ, � k2 − d2 dy2 � ψι k,ǫ(y, y0) = (−λ + Lk + iιǫ)−1ωk 0(y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) which implies ωk 0(y) =(−λ + Lk + iιǫ) � k2 − d2 dy2 � ψι k,ǫ(y, λ) =(b(y) − λ + iιǫ) � k2 − d2 dy2 � ψι k,ǫ(y, λ) + b′′(y)ψι k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) that ψ+ k,ǫ(y, λ), ψ− k,ǫ(y, λ) satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The proposition is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The existence of ψι k,ǫ for sufficiently small ǫ ̸= 0 follows from our spectral assumptions, which imply the solvability of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) for sufficiently small ǫ ̸= 0, see also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on the Green’s function and modified Green’s function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Elementary properties of the standard Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For integers k ∈ Z\\{0}, recall that the Green’s function Gk(y, z) solves − d2 dy2 Gk(y, z) + k2Gk(y, z) = δ(y − z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) with Dirichlet boundary conditions Gk(0, z) = Gk(1, z) = 0, z ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Gk has the explicit formula Gk(y, z) = 1 k sinh k � sinh(k(1 − z)) sinh(ky) if y ≤ z, sinh(kz) sinh(k(1 − y)) if y ≥ z, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) and the symmetry Gk(y, z) = Gk(z, y), for k ∈ Z\\{0}, y, z ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) We note the following bounds for Gk sup y∈[0,1],|A|≤10 � |k|2��Gk(y, z)(log |z − A|)m�� L1(z∈[0,1]) + |k| ��∂y,zGk(y, z)(log |z − A|)m�� L1(z∈[0,1]) � + sup y∈[0,1],α∈{0,1} � |k|3/2−α ��∂α y,zGk(y, z) �� L2(z∈[0,1]) � ≲ | log ⟨k⟩|m, for m ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) Define Fk(y, z) = 1 sinh k � −k cosh (k(1 − z)) cosh (ky), 0 ≤ y ≤ z ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' −k cosh (kz) cosh (k(1 − y)), 1 ≥ y > z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) We note that ∂y∂zGk(y, z) = ∂z∂yGk(y, z) = δ(y − z) + Fk(y, z), for y, z ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) By direct computation, we see Fk satisfies the bounds sup y∈[0,1],|A|≤10 ���Fk(y, z)(log |z − A|)m�� L1(z∈[0,1]) + |k|−1��∂y,zFk(y, z)(log |z − A|)m�� L1(z∈[0,1]) � + sup y∈[0,1],α∈{0,1} � |k|−1/2−α ��∂α y,zFk(y, z) �� L2(z∈[0,1]) � ≲ | log ⟨k⟩|m, for m ∈ {0, 1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) The bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) can be proved by explicit calculations and are useful in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on the modified Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It follows from Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 that there exists a δ0 ∈ (0, 1/8) such that inf{|y∗|, |y∗ − 1|} > 10δ0 and sup y∈(y∗−4δ0,y∗+4δ0) |b′′′(y)|δ0 < |b′′(y∗)|/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) Define the set Σδ0 := {b(y) : y ∈ [y∗ − δ0, y∗ + δ0]}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) 8 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA and fix a standard smooth cutoff function ϕ ∈ C∞ c (−2, 2) satisfying ϕ ≡ 1 on [−3/2, 3/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For simplicity of notations, we denote I := (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) To simplify notations we define also for d ∈ (0, 1/10), Sd := [y∗ − d, y∗ + d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) For applications below, we also need to study the “modified Green’s function” Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ+iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\\{0}, which satisfies for y, z ∈ (0, 1), (k2−∂2 y)Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ+iǫ)+ b′′(y) b(y) − λ + iǫ � ϕ �y − y∗ δ0 � −ϕ �y − y∗ δ(λ) �� Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ+iǫ) = δ(y−z), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) with the boundary condition Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|y∈{0,1} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) In the above, we have used the notation that δ(λ) := 8 � |λ − b(y∗)|/b′′(y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) Define the weight ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\\{0} as ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) :=|λ − b(y∗)|1/2 + |ǫ|1/2 + |y − y∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='15) The crucial bounds we need for the modified Green’s function Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Let Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\\{0} be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then we have the identity for y, z ∈ [0, 1], Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) = Gk(z, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16) and the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (i) We have the bounds sup y∈[0,1], |y−z|≤min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),1/|k|} |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|}, sup y∈[0,1], |y−z|≤min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),1/|k|} |∂yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) (ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≳ 1/|k|, we have the bounds with α ∈ {0, 1} |∂α y Gk(y1, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ � |k| + ̺−1(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �α e−|k||y1−y2| � |k| � [y2−1/|k|,y2+1/|k|]∩I |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18) (iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≪ 1/|k|, we have the bounds with α ∈ {0, 1} |∂α y Gk(y1, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ � |k| + ̺−1(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �α min �̺2(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺2(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19) where M := � 1 ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � [y2−̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),y2+̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ)]∩I |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The proof is based on energy estimates and “entanglement inequalities”, as in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' See also the earlier work [33] where this type of inequality was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Step 1: the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We first establish the bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For simplicity of notation, we suppress the dependence on z, λ + iǫ and set for y ∈ [0, 1], h(y) := Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), V (y) := b′′(y) b(y) − λ + iǫ � ϕ �y − y∗ δ0 � − ϕ �y − y∗ δ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21) Multiplying h to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) and integrating over [0, 1], we obtain that � 1 0 |∂yh(y)|2 + |k|2|h(y)|2 dy + � 1 0 b′′(y) b(y) − λ + iǫ � ϕ �y − y∗ δ0 � − ϕ �y − y∗ δ �� |h(y)|2 dy = h(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='22) Note that for y ∈ [0, 1], ℜV (y) ≥ 0, and in addition, for y ∈ Sδ0 and |y − y∗| > C0 � |λ − b(y∗)|1/2 + |ǫ|1/2� with sufficiently large C0 ≫ 1, 1 + ℜV (y) ≳ 1 ̺2(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='22) that � 1 0 |∂yh(y)|2 + |k|2|h(y)|2 dy + � y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2) 1 � ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �2 |h(y)|2 dy ≲ |h(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='24) Using the Sobolev type inequality ∥h∥L∞(J) ≲ ∥h∥L2(J∗)|J|−1/2 + ∥∂yh∥L2(J)|J|1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='25) for any interval J, J∗ with J∗ ⊆ J and |J∗| ≳ |J|, and choosing the interval J ⊂ I as an interval containing z with length of the size C1 min{1/|k|, ̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)}, we obtain from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='24) that � 1 0 |∂yh(y)|2 + |k|2|h(y)|2 dy + � y∈Sδ0, |y−y∗|>C0(δ+|ǫ|1/2) 1 � ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �2 |h(y)|2 dy ≲ min{1/|k|, ̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26) The desired bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='25), and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Step 2: the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denote M1 := � |k| � [y2−1/|k|,y2+1/|k|]∩I |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='27) For the sake of concreteness, we assume that y1 > z (so y2 ∈ [z, y1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We shall also assume that y1 − y2 ≫ 1/|k| as the other case is analogous but easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For ϕ ∈ C1 p([y2, 1]), the space of piecewise C1 functions, with ϕ(y2) = 0, we multiply ϕ2h to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) and integrate over 10 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA [y2, 1] to obtain that � 1 y2 |∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='28) Taking the real part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='28) and using Cauchy-Schwarz inequality, we get that � 1 y2 � |∂yϕ(y)|2 − |k|2|ϕ(y)|2� |h(y)|2 dy ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29) We now choose ϕ more specifically as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We require that ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + 1/|k|], ϕ(y2 + 1/|k|) = 1, ϕ′(y) = |k|ϕ(y) for y ∈ [y2 + 1/|k|, y1 − 1/|k|], ϕ′(y) = 0 for y ∈ [y1 − 1/|k|, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30) that � 1 y1−1/|k| |k|2ϕ2(y)|h(y)|2 dy ≲ |k|M2 1 , ϕ(y) ≈ e|k||y1−y2| for y ∈ [y1 − 1/|k|, y1 + 1/|k|] ∩ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='31) The desired bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18) follow from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='31) and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Step 3: the the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For the sake of concreteness, we assume that y1 > z (and so y2 ∈ [z, y1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We shall also assume that y1 − y2 ≫ ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) and that y2 > y∗ + δ + |ǫ|1/2 as the other cases are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For ϕ ∈ C1 p([y2, 1]) with ϕ(y2) = 0, we multiply ϕ2h to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) and integrate over [y2, 1] to obtain that � 1 y2 |∂yh(y)|2ϕ2(y) + 2∂yh(y)h(y)ϕ(y)∂yϕ(y) + |k|2ϕ2(y)|h(y)|2 + V (y)|h(y)|2ϕ2(y) dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='32) Write for y ∈ [y2, 1] h(y) = (y − y∗)1/2h∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33) Simple calculations show that � 1 y2 (y − y∗)|∂yh∗(y)|2ϕ2(y) + 2(y − y∗)∂yϕ(y)ϕ(y)∂yh∗(y)h∗(y) + 1 4(y − y∗)|h∗(y)|2ϕ2(y) + |k|2|h(y)|2ϕ2(y) + (y − y∗)V (y)ϕ2(y)|h∗(y)|2 dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) Therefore � 1 y2 � 1 4(y − y∗) + (y − y∗)ℜVy∗(y) � ϕ2(y)|h∗(y)|2 dy ≤ � 1 y2 (y − y∗)(∂yϕ)2(y)|h∗(y)|2 dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='35) which implies that � 1 y2 1 y − y∗ �� (y − y∗)∂yϕ �2(y) − � 1/4 + (y − y∗)2ℜV (y) � ϕ2(y) � |h∗(y)|2 dy ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 11 We notice the pointwise bounds for y ∈ [y2, 1], 1/4 + (y − y∗)2ℜV (y) ≥ max � 0, 9/4 − C2 ̺2(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) (y − y∗)2 − C2|y − y∗| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) Now we choose ϕ ∈ C1 p([y2, 1]) more precisely as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We require that ϕ(y2) = 0, ϕ′′(y) = 0 for y ∈ [y2, y2 + ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)], ϕ(y2 + ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)) = 1, (y − y∗)ϕ′(y) = � 1/4 + (y − y∗)2ℜV (y) �1/2ϕ(y) for y ∈ [y2 + ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), y1 − ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)], and ϕ′(y) = 0 for y ∈ [y1 − ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='38) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='38) that � y1 y1−̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ) 1 ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ2(y)|h∗(y)|2 dy ≲ M2/̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), ϕ(y) ≈ (y1 − y∗)3/2 ̺3/2(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) for y ∈ [y1 − ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), y1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='39) The desired bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19) follow from the change of variable (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33), the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='39) and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ As a corollary of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1, we have the following additional bounds on the modified Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Let Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0, k ∈ Z\\{0} and ǫ ∈ [−1/8, 1/8]\\{0} be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) for δ = δ(λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Define h := 10(δ + |ǫ|1/2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='40) and also for y, z ∈ [0, 1], Hk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) := � ∂z + ϕ �y − y∗ h � ∂y � Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='41) Then the following statements hold for z ∈ S4δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (i) We have the bounds sup y∈[0,1], |y−z|≤min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),1/|k|} |Hk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ 1, sup y∈[0,1], |y−z|≤min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),1/|k|} |∂yHk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ 1/ min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='42) (ii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≳ 1/|k|, we have the bounds with α ∈ {0, 1} � min{̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|} �α|∂α y Hk(y1, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ e−|k||y1−y2| min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|} � |k| � [y2−1/|k|,y2+1/|k|]∩I |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='43) 12 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA (iii) For y1, y2 ∈ [0, 1] with y2 ∈ [min{y1, z}, max{y1, z}] and ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≪ 1/|k|, we have the bounds with α ∈ {0, 1} � min{̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|} �α|∂α y Hk(y1, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ 1 min{̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), 1/|k|} min �̺2(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺2(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺(y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='44) where M := � 1 ̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � [y2−̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ),y2+̺(y2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='λ+iǫ)]∩I |Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)|2 dy �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='45) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denote with a slight abuse of notation for y ∈ [0, 1], ϕ†(y) := ϕ �y − y∗ h � , V (y) := b′′(y) b(y) − λ + iǫ � ϕ �y − y∗ δ0 � − ϕ �y − y∗ δ(λ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='46) Then Hk,j(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) satisfies for y ∈ [0, 1], z ∈ S4δ, (k2 − ∂2 y)Hk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) + V (y)Hk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) = −∂2 yϕ†(y)∂yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) − ∂yV (y)ϕ†(y)Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) − 2∂yϕ†(y)∂2 yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='47) The desired bounds then follow from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='47), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and standard elliptic regu- larity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ The bounds in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 are quite sharp, since we can exploit the decay coming from both k2 and b′′(y) b(y)−λ+iǫ � ϕ � y−y∗ δ0 � − ϕ � y−y∗ δ(λ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It is however somewhat complicated to formulate a concrete bound that is easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Instead, the following simple bounds are more often used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Let Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) for y, z ∈ [0, 1], λ ∈ Σδ0 and ǫ ∈ [−1/8, 1/8]\\{0} be defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then we have the following bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (i) For y, z ∈ [0, 1], we have the bounds with α ∈ {0, 1} � |k| + ̺−1(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �−α |∂α y Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ 1 |k| + ̺−1(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) min � e−|k||y−z|, ̺2(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), ̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='48) (iii) For y ∈ [0, 1], z ∈ S4δ, we have the bounds with α ∈ {0, 1, 2} � |k| + ̺−1(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �−α |∂α y Hk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)| ≲ min � e−|k||y−z|, ̺2(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺2(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), ̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='49) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The desired bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='48)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='49) follow directly from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2, by choosing, if necessary, another point y′ between y and z such that ̺(y′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≈ 1/|k|, and applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='48)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='49) on intervals [min{z, y′}, max{z, y′}] and [min{y′, y}, max{y′, y}] succes- sively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The limiting absorption principle In this section we study the solvability of the main Rayleigh equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It turns out that the situation is very different for the spectral range λ ∈ Σ\\Σδ0/2 (the non-degenerate case) and λ ∈ Σδ0 (the degenerate case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We first consider the non-degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The non-degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Fix ǫ ∈ [−1/4, 1/4]\\{0}, λ ∈ Σ\\Σδ0/2, k ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Define for each g ∈ L2(0, 1) the operator Tk,λ,ǫg(y) := � 1 0 Gk(y, z) b′′(z)g(z) b(z) − λ + iǫdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) For applications below, we fix a smooth cutoff function Φ ∈ C∞ 0 (y∗ − δ0/3, y∗ + δ0/3) with Φ ≡ 1 on [y∗ − δ0/4, y∗ + δ0/4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' To obtain the optimal dependence on the frequency variable k, we define ∥g∥H1 k(I) := ∥g∥L2(I) + |k|−1∥g′∥L2(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For ǫ ∈ [−1/4, 1/4]\\{0}, λ ∈ Σ\\Σδ0/2, k ∈ Z\\{0}, the operator Tk,λ,ǫ satisfies the bound ∥Tk,λ,ǫg∥H1 k(I) ≲ |k|−1/3∥g∥H1 k(I), for all g ∈ H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) In addition, we have the more precise regularity structure ����∂yTk,λ,ǫg(y) + b′′(y)(1 − Φ(y))g(y) b′(y) log (b(y) − λ + iǫ) ���� W 1,1(R) ≲ ⟨k⟩4/3∥g∥H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We can decompose for y ∈ [0, 1], Tk,λ,ǫg(y) := T 1 k,λ,ǫg(y) + T 2 k,λ,ǫg(y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) where T 1 k,λ,ǫg(y) := � 1 0 Gk(y, z)Φ(z)b′′(z)g(z) b(z) − λ − iǫ dz, T 2 k,λ,ǫg(y) := � 1 0 Gk(y, z)(1 − Φ(z))b′′(z)g(z) b(z) − λ + iǫ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) It follows from the definition of Φ that T 1 k,λ,ǫg(y) satisfies the bound ∥T 1 k,λ,ǫg(y)∥H1 k(I) ≲ |k|−1/3∥g∥H1 k(I), ∥∂yT 1 k,λ,ǫg(y)∥W 1,1(I) ≲ ⟨k⟩4/3∥g∥H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) To bound T 2 k,λ,ǫg(y), we follow the approach in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using integration by parts, we obtain that T 2 k,λ,ǫg(y) = � 1 0 Gk(y, z)(1 − Φ(z))b′′(z)g(z) b′(z) ∂z log(b(z) − λ + iǫ) dz = − � 1 0 ∂zGk(y, z)(1 − Φ(z))b′′(z)g(z) b′(z) log(b(z) − λ + iǫ) dz − � 1 0 Gk(y, z)∂z �(1 − Φ(z))b′′(z)g(z) b′(z) � log(b(z) − λ + iǫ) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) The desired bounds follow from the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4), the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ We now prove the limiting absorption principle, using the assumption that there is no discrete or generalized embedded eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 14 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' There exist ǫ0, κ > 0, such that the following statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For all λ ∈ Σ\\Σδ0/2, k ∈ Z\\{0}, 0 < |ǫ| < ǫ0 and any g ∈ H1 k(I), we have the bound ∥g + Tk,λ,ǫg∥H1 k(I) ≥ κ∥g∥H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that there exist for j ≥ 1, a sequence of num- bers kj ∈ Z\\{0}, λj ∈ Σ\\Σδ0/2, ǫj ∈ R\\{0} → 0 and functions gj ∈ H1 kj(I) with ∥gj∥H1 kj (I) = 1, satisfying kj → k∗ ∈ (Z\\{0}) ∪ {±∞}, λj → λ∗ ∈ Σ\\Σδ0 as j → ∞, such that ��gj + Tkj,λj,ǫjgj �� H1 kj (I) → 0, as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) The bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) imply that |kj| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Thus k∗ ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using ∥gj∥H1 kj (I) = 1, the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) and the compact embedding W 1,1(I) → L2(I), we conclude that by passing to a subsequence, Tkj,λj,ǫjgj converges in H1(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) we can assume that gj → g in H1(I), where ∥g∥H1 k∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1), we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) that for y ∈ I, g(y) + lim j→∞ � 1 0 Gk∗(y, z) b′′(z)g(z) b(z) − λ + iǫj dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) Applying k2 ∗ − d2 dy2 to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11), we get that for y ∈ I, k2 ∗g(y) − g′′(y) + lim j→∞ (b(y) − λ∗)b′′(y)g(y) (b(y) − λ∗)2 + ǫ2 j + iπ � z∈[0,1],b(z)=λ b′′(z)g(z) |b′(z)| δ(y − z) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) in the sense of distributions for y ∈ (0, 1), which contradicts our spectral assumption that λ∗ is not a generalized embedded eigenvalue for Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The lemma is then proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The degenerate case λ ∈ Σδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) for δ = δ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For λ ∈ Σδ0, k ∈ Z\\{0}, y ∈ I and ǫ ∈ [−1/8, 1/8]\\{0}, we denote dk(λ, ǫ) := � |λ − b(y∗)|1/2 + |ǫ|1/2� ∧ 1 |k|, ̺k(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) := ̺(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ∧ 1 |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) Define the weight and the associated weighted Sobolev spaces XN,̺k and XL,̺k as ∥g∥XN,̺k (I) := � α∈{0,1} (δ + |ǫ|1/2)−1/2��� � dk(λ, ǫ) �(−7/4+α)∂α y g ��� L2(S3(δ+|ǫ|1/2)) + � α∈{0,1} ∥̺−7/4+α k (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)∂α y g∥L∞(I\\S3(δ+|ǫ|1/2)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) and ∥g∥XL,̺k (I) := � α∈{0,1} (δ + |ǫ|1/2)−1/2��dα k(λ, ǫ)∂α y g �� L2(S3(δ+|ǫ|1/2)) + � α∈{0,1} ��dk(λ, ǫ)−1̺α+1 k (·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)∂α y g �� L∞(I\\S3(δ+|ǫ|1/2)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='15) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 15 Fix ǫ ∈ [−1/4, 1/4]\\{0}, λ ∈ Σδ0, k ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) for δ = δ(λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Define for each g ∈ L2(0, 1) the operator T ∗ k (λ + iǫ)g(y) := � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � 1 − ϕ �y − y∗ δ0 � + ϕ �y − y∗ δ �� b′′(z)g(z) b(z) − λ + iǫdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16) Then we have the following bounds for T ∗ k (λ + iǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For ǫ ∈ [−1/4, 1/4]\\{0}, λ ∈ Σδ0, k ∈ Z\\{0}, the operator T ∗ k (λ + iǫ) satisfies the bound for X ∈ {XN,̺k(I), XL,̺k(I)} ∥T ∗ k (λ + iǫ)g∥X ≲ (1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2))−1/4∥g∥X, for all g ∈ H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We provide the detailed proof only for the case X = XN,̺k(I) as the other case is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Since k, λ, ǫ are fixed, for simplicity of notations, we suppress the dependence on k, λ, ǫ to write T ∗ as T ∗ k (λ + iǫ), and decompose for y ∈ I, T ∗g(y) := T ∗ 1 g(y) + T ∗ 2 g(y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18) where T ∗ 1 g(y) := � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � 1 − ϕ �z − y∗ δ0 �� b′′(z)g(z) b(z) − λ + iǫdz, T ∗ 2 g(y) := � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ �z − y∗ δ � b′′(z)g(z) b(z) − λ + iǫdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19) It follows from the bounds on modified Green’s function Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ), see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1, that ��T ∗ 1 g �� XN,̺k(I) ≲ |k|−1/2��g �� XN,̺k (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20) To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17), it suffices to prove ∥T ∗ 2 g∥XN,̺k (I) ≲ � 1 + |k|(δ + |ǫ|1/2) �−1/4∥g∥XN,̺k (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21) We assume momentarily that |ǫ| ≲ |λ − b(y∗)| and explain how to remove this assumption at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We decompose further for y ∈ I, T ∗ 2 g(y) = � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ �z − y∗ δ′ � ϕ �z − y∗ δ � b′′(z)g(z) b(z) − λ + iǫdz + � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) � 1 − ϕ �z − y∗ δ′ �� ϕ �z − y∗ δ � b′′(z)g(z) b(z) − λ + iǫdz := T ∗ 2,Rg(y) + T ∗ 2,Sg(y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='22) where we have chosen δ′ = δ/C3 with a large constant C3 so that |b(y) − λ| ≈ |λ − b(y∗)| for |y − y∗| < δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It suffices to prove for ⋄ ∈ {R, S} ∥T ∗ 2,⋄g∥XN,̺k (I) ≲ � 1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) �−1/4∥g∥XN,̺k (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We first prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) with ⋄ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this case for |z − y∗| ≲ δ and |y − y∗| ≲ 1 we have the bound ��Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �� ≲ δ2 + |ǫ| |y − y∗| + δ + |ǫ|1/2 , ��∂yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �� ≲ δ2 + |ǫ| (|y − y∗| + δ + |ǫ|1/2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='24) 16 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA It follows from the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='24) that ∥T ∗ 2,Rg∥XN,̺k (I) ≲ � 1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) �−1/4∥g∥XN,̺k (I) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='25) Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this case, we have for |z − y∗| ≲ δ and |y − y∗| ≲ 1 that ��Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �� + |k|−1��∂yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) �� ≲ |k|−1e−|k||y−z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26) The desired bound ∥T ∗ 2,Rg∥XN,̺k (I) ≲ � 1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) �−1/4∥g∥XN,̺k (I) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='27) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We now turn to the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) with ⋄ = S and still consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Case I: 1/|k| > |λ − b(y∗)|1/2 + |ǫ|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denoting for y ∈ I, ϕ∗�y − y∗ δ � := � 1 − ϕ �z − y∗ δ′ �� ϕ �z − y∗ δ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='28) we can rewrite T ∗ 2,Sg(y) = � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) ∂z log b(z) − λ + iǫ δ2 = − � 1 0 ∂z � Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) � log b(z) − λ + iǫ δ2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29) As a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29), we also have ∂y � T ∗ 2,Sg(y) � = ∂y � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) ∂z log b(z) − λ + iǫ δ2 dz = − � 1 0 � ∂y(∂z + ∂y)Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ, ǫ)ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) � log b(z) − λ + iǫ δ2 dz + � 1 0 � ∂2 yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) � log b(z) − λ + iǫ δ2 dz − � 1 0 ∂yGk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)∂z � ϕ∗�z − y∗ δ �b′′(z)g(z) b′(z) � log b(z) − λ + iǫ δ2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30) Note that on the support of ϕ∗(z−y∗ δ ), we have |b′(z)| ≈ δ, ̺(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) ≈ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='31) The desired bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) for ⋄ = S follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30), Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2, and we have, in addition, (δ + |ǫ|1/2)−1/2 ����∂y � ∂yT ∗ 2,Sg(y) + ϕ∗�y − y∗ δ �b′′(y)g(y) b′(y) log b(y) − λ + iǫ δ2 ����� L2(S3(δ+|ǫ|1/2),j) ≲ δ−1/4� 1 + |k|(|λ − b(y∗)|1/2 + |ǫ|1/2) �−1/4 ∥g∥XN,̺k (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='32) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 17 Case II: 1/|k| ≪ |λ − b(y∗)|1/2 + |ǫ|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' This case is analogous to Case I, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Finally we turn to the assumption that |ǫ|1/2 ≲ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Suppose |ǫ|1/2 ≫ δ, then the factor 1 b(z)−λ+iǫ is not truly singular, and the desired bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21) follow directly from the bounds on the modified Green’s function Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Indeed, we have the stronger bound ∥T ∗ 2 g∥XN,̺k (I) ≲ δ � |ǫ| ∥g∥XN,̺k (I), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33) which will be useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ The following limiting absorption principle plays an essential role in establishing the vorticity depletion phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' There exist positive numbers ǫ0, κ such that the following statement holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For ǫ ∈ [−ǫ0, ǫ0]\\{0}, λ ∈ Σδ0, k ∈ Z\\{0}, and X ∈ {XN,̺k(I), XL,̺k(I)}, ∥(I + T ∗ k (λ + iǫ))g∥XN,̺k (I) ≥ κ∥g∥XN,̺k (I), for all g ∈ H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We only consider the case X = XN,̺k(I) as the other case is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) by a contradiction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) does not hold for any ǫ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then there exist for ℓ ∈ Z ∩ [1, ∞), λℓ → λ∗ ∈ Σδ0, ǫℓ ̸= 0 with ǫℓ → 0, kℓ → k∗ ∈ (Z\\{0}) ∪ {±∞}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='35) and functions gℓ satisfying ∥gℓ∥XN,̺kℓ (I) = 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36) such that ��(I + T ∗ kℓ(λℓ + iǫℓ))gℓ �� XN,̺kℓ (I) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) We can assume that λ∗ = b(y∗), otherwise the proof follows from the argument in the non- degenerate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We consider several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Case I: lim supℓ→∞ ∥gℓ∥H1(I\\Sδ0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20), we can assume that k∗ ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37), we can assume (passing to a subsequence if necessary) that gℓ → g, in H1 loc(I\\{y∗}) as ℓ → ∞, g(0) = g(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='38) Then it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) that g satisfies |g(y)| ≲ |y − y∗|7/4, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='39) and for y ∈ (0, 1), (k2 ∗ − ∂2 y)g(y) + b′′(y) b(y) − b(y∗)g(y) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='40) which imply that b(y∗) is an embedded eigenvalue for Lk, a contradiction to the spectral assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Case II: lim supℓ→∞ ∥gℓ∥H1(I\\Sδ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) we can assume that |kℓ|(δℓ + |ǫℓ|1/2) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' In this case, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37), we obtain that (passing to a subsequence if necessary) ��(|λℓ − b(y∗)| + |ǫ|)−9/8gℓ �� L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2]) + ��(|λℓ − b(y∗)| + |ǫ|)−5/8∂ygℓ �� L2([y∗−δℓ−|ǫℓ|1/2, y∗+δℓ+|ǫℓ|1/2]) ≥ σ > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='41) 18 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA where we recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) that δℓ ≈ |λℓ − b(y∗)|1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='42) We divide into several subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Subcase II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1: |ǫℓ|1/2 ≈ δℓ for a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Define the change of variables for ℓ ≥ 1, y ∈ I, y − y∗ = δℓY, gℓ(y) := (|λℓ − b(y∗)| + |ǫℓ|)−7/8Hℓ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='43) It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='32) that we can extract a nontrivial limit H ∈ H1(R) of Hℓ satisfying for Y ∈ R, (β2 − ∂2 Y )H(Y ) + b′′(y∗) b′′(y∗)Y 2/2 + γ + iαH(Y ) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='44) where β ∈ R, α, γ ∈ R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' This is impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Subcase II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2: |ǫℓ|1/2 = o(δℓ) for a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Passing to a subsequence and using rescaling as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='43) we can extract a nontrivial limit H ∈ H1(R), such that (β2 − ∂2 Y )H(Y ) + lim ǫ→0 b′′(y∗) b′′(y∗)Y 2/2 + γ + iǫH(Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='45) This is again impossible since the shear flow (b′′(y∗)Y 2/2, 0), Y ∈ R is spectrally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Subcase II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3: δℓ = o(|ǫℓ|1/2) for a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' This case is not possible thanks to the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The lemma is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on ψι k,ǫ: the non-degenerate case In this section we obtain bounds on ψι k,ǫ(y, λ) in the non-degenerate case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' when λ ∈ Σ\\Σδ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Since the arguments are analogous to those in [14], we will be brief in the proofs, and provide only comments on the main ideas involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We begin with the following preliminary bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For λ ∈ Σ\\Σδ0/2, k ∈ Z\\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the bounds ∥ψι k,ǫ(·, λ)∥H1 k(I) ≲ |k|−1/2∥ω0k∥H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The desired bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) follow directly from the Rayleigh equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2, once we use the Green’s function Gk to invert k2 − ∂2 y and formulate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) as an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ To obtain control on ∂λψι k,ǫ(·, λ) for λ ∈ Σ\\Σδ0/2, we take derivative in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), and obtain that (k2 − ∂2 y)∂λψι k,ǫ(y, λ) + b′′(y)∂λψι k,ǫ(y, λ) b(y) − λ + iιǫ = ωk 0(y) (b(y) − λ + iιǫ)2 − b′′(y)ψι k,ǫ(z, λ) (b(y) − λ + iιǫ)2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 19 for y ∈ I with zero boundary value at y ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Reformulating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) as an integral equation, we obtain that ∂λψι k,ǫ(y, λ) + � 1 0 Gk(y, z) b′′(z)∂λψι k,ǫ(z, λ) b(z) − λ + iιǫ dz = � 1 0 Gk(y, z) ωk 0(z) (b(z) − λ + iιǫ)2 dz − � 1 0 Gk(y, z) b′′(z)ψι k,ǫ(z, λ) (b(z) − λ + iιǫ)2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) Recall the definition of the smooth cutoff function Φ below (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We have the following bounds for ∂λψι k,ǫ(y, λ) when λ ∈ Σ\\Σδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For λ ∈ Σ\\Σδ0/2, k ∈ Z\\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, ∂λψι k,ǫ(y, λ) satisfies the following decomposition ∂λψι k,ǫ(y, λ) = �b′(y0)ωk 0(y) |b′(y)|2 − b′′(y)ψι k,ǫ(y, λ) |b′(y)|2 � (1 − Φ(y)) log (b(y) − λ + iιǫ) + � σ=0,1 ωk 0(σ)Ψι k,σ,ǫ(y, λ) log (b(σ) − λ + iιǫ) + Rι σ,k,y0,ǫ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) In the above for σ ∈ {0, 1}, ι ∈ {±}, 0 < ǫ < ǫ0, and λ ∈ Σ\\Σδ0/2, ��Rι σ,k,y0,ǫ �� H1 k(I) ≲ |k|1/2∥ω0k∥H2 k(I), ��Ψι k,σ,ǫ(·, λ) �� H1 k(I) ≲ |k|−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The basic idea is to expand the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) using integration by parts, and apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 after removing the most singular parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Indeed, denoting schematically, U := � 1 0 Gk(y, z) ωk 0(z) (b(z) − λ + iιǫ)2 dz − � 1 0 Gk(y, z) b′′(z)ψι k,ιǫ(z, λ) (b(z) − λ + iιǫ)2 dz, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) we note that ∂λψι k,ǫ(y, λ) − U satisfies the equation (recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) for the definition of Tk,λ,ιǫ), (I + Tk,λ,ιǫ) � ∂λψι k,ǫ(y, λ) � = −Tk,λ,ιǫU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) The term Tk,λ,ιǫU ∈ H1 k(I) (noting however that for the boundary terms we need to track the singular coefficient log (b(σ) − λ + iιǫ), σ ∈ {0, 1}), and we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) in order to obtain the desired conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We refer to [14] for the detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ To obtain bounds on ∂2 λψι k,ǫ(y, λ) for λ ∈ Σ\\Σδ0/2, we take two derivatives in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) and obtain that (k2 − ∂2 y)∂2 λψι k,ǫ(y, λ) + b′′(y)∂2 λψι k,ǫ(y, λ) b(y) − λ + iιǫ = 2 ωk 0(y) (b(y) − λ + iιǫ)3 − 2 b′′(y)ψι k,ǫ(z, λ) (b(y) − λ + iιǫ)3 + b′′(y)∂λψι k,ǫ(z, λ) (b(y) − λ + iιǫ)2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) 20 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA for y ∈ I with zero boundary value at y ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We can reformulate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) in the integral form for y ∈ I, as ∂2 λψι k,ǫ(y, λ) + � 1 0 Gk(y, z) b′′(z)∂2 λψι k,ǫ(z, λ) b(z) − λ + iιǫ dz = � 1 0 Gk(y, z) � 2 ωk 0(z) (b(z) − λ + iιǫ)3 − 2 b′′(z)ψι k,ǫ(z, λ) (b(z) − λ + iιǫ)3 + b′′(z)∂λψι k,ǫ(z, λ) (b(z) − λ + iιǫ)2 � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) We have the following bounds on ∂2 λψι k,ǫ(y, λ) for λ ∈ Σ\\Σδ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' For k ∈ Z\\{0}, ι ∈ {±} and 0 < ǫ < ǫ0, we have the following bound ����∂2 λψι k,ǫ(y, λ) − ωk 0(1)Φ1ι k,ǫ(y, λ) b(1) − λ + iιǫ − ωk 0(0)Φ0ι k,ǫ(y, λ) b(0) − λ + iιǫ − b′′(y)ψι k,ǫ(y, λ) − ωk 0(y) |b′(y)|2(b(y) − λ + iιǫ) ���� L2(y∈I,λ∈Σ\\Σδ0/2) ≲ |k|3/2∥ω0k∥H3 k(I) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) In the above the functions Φσι k,ǫ, σ ∈ {0, 1} satisfy the equation for y ∈ I (I + Tk,λ,ιǫ)Φ1ι k,ǫ = sinh (ky) |b′(1)|2 sinh k, (I + Tk,λ,ιǫ)Φ0ι k,ǫ = sinh (k(1 − y)) |b′(0)|2 sinh k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The main idea of the proof is to expand the right side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) and apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 after removing the most singular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Indeed, denoting schematically, U∗ := � 1 0 Gk(y, z) � 2 ωk 0(z) (b(z) − λ + iιǫ)3 − 2 b′′(z)ψι k,ιǫ(z, λ) (b(z) − λ + iιǫ)3 + b′′(z)∂λψι k,ιǫ(z, λ) (b(z) − λ + iιǫ)2 � dz, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) we have (I + Tk,λ,ιǫ) � ∂2 λψι k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗� = � Tk,λ,ιǫ �2U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) We note that ∂2 λψι k,ǫ(y, λ) − U∗ + Tk,λ,ιǫU∗ ∈ H1 k(I) (however we again need to track the singularities in λ in the boundary terms, involving log(b(σ) − λ + iιǫ) and 1/(b(σ) − λ + iιǫ) for σ ∈ {0, 1}), and we can apply Lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) in order to obtain the desired conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We refer to [14] for the detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Bounds on ψι k,ǫ: the degenerate case In this section we use the limiting absorption principle to study the Rayleigh equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) for λ ∈ Σδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' More precisely, write for k ∈ Z\\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0, (recall the definition of ǫ0 from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) ψι k,ǫ(y, λ) = φι k,ǫ(y, λ) + Ψ(y) 1 b′′(y)ω0k(y), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) where Ψ ∈ C∞ c (S3δ0) and Ψ ≡ 1 on S2δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall that Sd = [y∗ −d, y∗ +d] for d > 0 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then φι k,ǫ(y, λ) satisfies for y ∈ I, (k2 − ∂2 y)φι k,ǫ(y, λ) + b′′(y) b(y) − λ + iιǫφι k,ǫ(y, λ) = gι k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 21 where for k ∈ Z\\{0}, ι ∈ {±}, λ ∈ Σδ0, 0 < ǫ < ǫ0 gι k,ǫ(y, λ) := 1 − Ψ(y) b(y) − λ + iιǫω0k(y) − (k2 − ∂2 y) � Ψ(y) b′′(y)ω0k(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) Our main results are bounds for the functions φι k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We begin with the following pre- liminary bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that k ∈ Z\\{0}, λ ∈ Σδ0 and let φι k,ǫ(y, λ) with ι ∈ {±}, ǫ ∈ (0, ǫ0) be as defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) that δ := δ(λ) = 8 � |λ − b(y∗)|/|b′′(y∗)|, dk = dk(λ, ǫ) := � |λ − b(y∗)|1/2 + |ǫ|1/2� ∧ 1 |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) We have the bounds for k ∈ Z\\{0}, ǫ ∈ (0, ǫ0), ι ∈ {±}, λ ∈ Σδ0, � α∈{0,1} ��d−7/4+α k ∂α y φι k,ǫ(y, λ) �� L2� [y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)] �(δ + |ǫ|1/2)−1/2 + � α∈{0,1} ��(|y − y∗| ∧ dk)−7/4+α∂α y φι k,ǫ(y, λ) �� L∞� [0,1]\\[y∗−3(δ+|ǫ|1/2),y∗+3(δ+|ǫ|1/2)] � ≲ |k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) Define for y ∈ [0, 1], k ∈ Z\\{0}, λ ∈ Σδ0\\{b(y∗)}, ψk(y, λ) := lim ǫ→0+ � ψ+ k,ǫ(y, λ) − ψ− k,ǫ(y, λ) � = lim ǫ→0+ � φ+ k,ǫ(y, λ) − φ− k,ǫ(y, λ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) Then we have the bounds for λ ∈ Σδ0\\{b(y∗)}, � α∈{0,1} ��(δ ∧ |k|−1)−7/4+α∂α y ψk(y, λ) �� L2([y∗−3δ,y∗+3δ])δ−1/2 + � α∈{0,1} ��(δ ∧ |k|−1)−11/4(|y − y∗| ∧ 1 |k|)1+α∂α y ψk(y, λ) �� L∞([0,1]\\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) and our assumptions on the initial data ω0k that we have the bound for k ∈ Z\\{0}, ι ∈ {±}, 0 < ǫ < ǫ0 and λ ∈ Σδ0, ��gι k,ǫ(y, λ) �� C2 k(I) ≲ |k|1/2∥ω0k∥H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) We can reformulate equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) in the integral form as (recall the definition of T ∗(λ + iǫ) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16)) φι k,ǫ(y, λ) + T ∗ k (λ + iιǫ)φι k,ǫ(y, λ) = � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)gι k,ǫ(z, λ)dz, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) for y ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4, we obtain the bound ��φι k,ǫ(·, λ) �� XN,̺k(I) ≲ ��� � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)gι k,ǫ(z, λ)dz ��� XN,̺k ≲ |k|5/2∥ω0k∥H3 k(I), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) which, by the definition of the space XN,̺k, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14), implies the desired bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 22 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA For applications below on isolating the singularity at λ = b(y), we fix ϕδ(y) ∈ C∞ c (S2δ) as ϕδ(y) := ϕ(y δ ) � 1 − ϕ( y δ′ ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) for y ∈ I, with δ′ := δ/M and an M ≫ 1 sufficiently large such that |b(y) − λ| ≈ |λ − b(y∗)| for |y − y∗| < δ/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' To prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7), we note from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) that φ+ k,ǫ(y, λ) − φ− k,ǫ(y, λ) satisfies the equation for y ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (k2 − ∂2 y) � φ+ k,ǫ(y, λ) − φ− k,ǫ(y, λ) � + b′′(y) b(y) − λ + iǫ � φ+ k,ǫ(y, λ) − φ− k,ǫ(y, λ) � = g+ k,ǫ(y, λ) − g− k,ǫ(y, λ) − � b′′(y) b(y) − λ + iǫ − b′′(y) b(y) − λ − iǫ � φ− k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) Denote for λ ∈ Σδ0\\{b(y∗)}, ǫ ∈ (0, ǫ0) and y ∈ I the function hk,ǫ(y, λ) as the solution to (k2 − ∂2 y)hk,ǫ(y, λ) + b′′(y) b(y) − λ + iǫhk,ǫ(y, λ) = ϕδ(y) � b′′(y) b(y) − λ − iǫ − b′′(y) b(y) − λ + iǫ � φ− k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Then it is clear that for λ ∈ Σδ0\\{b(y∗)}, y ∈ I, ψk(y, λ) = lim ǫ→0+ hk,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) We can reformulate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) as the following integral equation for λ ∈ Σδ0\\{b(y∗)}, y ∈ I, hk,ǫ(y, λ) + T ∗ k (λ + iǫ)hk,ǫ(y, λ) = − � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕδ(z) � b′′(z) b(z) − λ + iǫ − b′′(z) b(z) − λ − iǫ � φ− k,ǫ(z, λ) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='15) It follows from the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) that for |ǫ| ≲ (δ ∧ 1 |k|)4, ���� � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕδ(z) � b′′(z) b(z) − λ + iǫ − b′′(z) b(z) − λ − iǫ � φ− k,ǫ(z, λ) dz ���� XL,̺k ≲ (δ ∧ 1 |k|)7/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16) The desired bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) then follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4 with X = XL,̺k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ To obtain higher order regularity bounds (in λ) of φι k,ǫ(·, λ), we take the derivative ∂λ in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' It follows that ∂λφι k,ǫ(y, λ) satisfies for y ∈ I, � k2 − ∂2 y + b′′(y) b(y) − λ + iιǫ � ∂λφι k,ǫ(y, λ) = − b′′(y) (b(y) − λ + iιǫ)2 φι k,ǫ(y, λ) + ∂λgι k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall the definition of ϕδ from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We have the following bounds on ∂λφι k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that k ∈ Z\\{0}, λ ∈ Σδ0\\{b(y∗)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Let ψι k,ǫ(y, λ) and φι k,ǫ(y, λ) with ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} be as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) that δ := δ(λ) = 8 � |λ − b(y∗)|/b′′(y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 23 Denote for y ∈ [0, 1], ι ∈ {±}, λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, Λι 1,ǫ(y, λ) := φι k,ǫ(y, λ)ϕδ(y) b′′(y) (b′(y))2 log b(y) − λ + iιǫ δ2 , Λ1(y, λ) := ψk(y, λ)ϕδ(y) b′′(y) (b′(y))2 log b(y) − λ δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19) We have the bounds for 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, ι ∈ {±}, and λ ∈ Σδ0 that � α∈{0,1} ���(δ ∧ |k|−1)1/4+α∂α y � ∂λφι k,ǫ(y, λ) − Λι 1,ǫ(y, λ) ���� L2([y∗−3δ,y∗+3δ])δ−1/2 + � α∈{0,1} ���(δ ∧ |k|−1)2(|y − y∗| ∧ 1 |k|)−7/4+α∂α y ∂λφι k,ǫ(y, λ) ��� L∞([0,1]\\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20) In addition, we have the bounds for λ ∈ Σδ0\\{b(y∗)} and k ∈ Z\\{0}, � α∈{0,1} ��(δ ∧ |k|−1)1/4+α∂α y � ∂λψk(y, λ) − Λ1(y, λ) ���� L2([y∗−3δ,y∗+3δ])δ−1/2 + � α∈{0,1} ��(δ ∧ |k|−1)−3/4(|y − y∗| ∧ 1 |k|)1+α∂α y ∂λψk(y, λ) �� L∞([0,1]\\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Define for k ∈ Z\\{0}, ι ∈ {±}, λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I, ∂λφι k,ǫ(y, λ) := φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) + � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ) � −b′′(z) (b(z) − λ + iιǫ)2 φι k,ǫ(z, λ) + ∂λgι k,ǫ(z, λ) � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='22) It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) that φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) satisfies for y ∈ I, φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) + T ∗ k (λ + iιǫ)φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) = −T ∗ k (λ + iιǫ) � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ) � − b′′(z) (b(z) − λ + iιǫ)2 φι k,ǫ(z, λ) + ∂λgι k,ǫ(z, λ) � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='23) Denote for k ∈ Z\\{0}, ι ∈ {±}, λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, z ∈ I, hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) := b′′(z) (b(z) − λ + iιǫ)2 ϕδ(z)φι k,ǫ(z, λ), hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) := b′′(z) (b(z) − λ + iιǫ)2 (1 − ϕδ(z))φι k,ǫ(z, λ), hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 3) := ∂λgι k,ǫ(z, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='24) It follows from the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 that for j ∈ {2, 3} ��T ∗ k (λ + iιǫ) � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz �� XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2∥ω0k∥H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='25) 24 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA Using integration by parts argument similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30), we have also ����T ∗ k (λ + iιǫ) � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) dz ���� XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26) It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='25)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='26) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4 that for λ\\{b(y∗)}, ��φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) �� XN,̺k ≲ (δ ∧ |k|−1)−2|k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='27) The desired bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20) follows, as a consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='27) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17), we get that for y ∈ I, � k2 − ∂2 y + b′′(y) b(y) − λ + iǫ �� ∂λφ+ k,ǫ(y, λ) − ∂λφ− k,ǫ(y, λ) � = − � b′′(y) (b(y) − λ + iǫ)2 φ+ k,ǫ(y, λ) − b′′(y) (b(y) − λ − iǫ)2 φ− k,ǫ(y, λ) � + � ∂λg+ k,ǫ(y, λ) − ∂λg− k,ǫ(y, λ) � − � b′′(y) b(y) − λ + iǫ − b′′(y) b(y) − λ − iǫ � ∂λφ− k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='28) with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denoting for λ ∈ Σδ0\\{b(y∗)} and y ∈ I, Dφk,ǫ(y, λ) as the solution to � k2 − ∂2 y + b′′(y) b(y) − λ + iιǫ � Dφk,ǫ(y, λ) = −ϕδ(y) � b′′(y) (b(y) − λ + iǫ)2 φ+ k,ǫ(y, λ) − b′′(y) (b(y) − λ − iǫ)2 φ− k,ǫ(y, λ) � − ϕδ(y) � b′′(y) b(y) − λ + iιǫ − b′′(y) b(y) − λ − iιǫ � ∂λφ− k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29) for y ∈ I with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We notice the identity that for y ∈ I, λ ∈ Σδ0\\{b(y∗)}, ∂λψk(y, λ) = lim ǫ→0+ Dφk,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30) We can reformulate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29) as the integral equation for y ∈ I, Dφk,ǫ(y, λ) + T ∗ k (λ + iǫ)Dφk,ǫ(y, λ) = − � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕδ(z) � b′′(z) (b(z) − λ + iǫ)2 φ+ k,ǫ(z, λ) − b′′(z) (b(z) − λ − iǫ)2 φ− k,ǫ(z, λ) � dz − � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕδ(z) � b′′(z) b(z) − λ + iǫ − b′′(z) b(z) − λ − iιǫ � ∂λφ− k,ǫ(z, λ) dz := Rk,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='31) We can write for y ∈ I, λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, Dφk,ǫ(y, λ) := Rk,ǫ(y, λ) + Dφk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='32) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 25 Then Dφk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) satisfies for y ∈ I, λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, the equation Dφk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) + T ∗ k (λ + iǫ)Dφk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 1) = −T ∗ k (λ + iǫ)Rk,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33) The desired bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) follow from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='31)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='33), and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 with X = XL,̺k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' □ Lastly we turn to the highest order derivative ∂2 λψι k,ǫ(y, λ) that we need to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' To study ∂2 λψι k,ǫ(y, λ), we take the derivative ∂λ in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) and obtain that � k2 − ∂2 y + b′′(y) b(y) − λ + iιǫ � ∂2 λφι k,ǫ(·, λ) = − 2b′′(y) (b(y) − λ + iιǫ)2 ∂λφι k,ǫ(·, λ) − 2b′′(y) (b(y) − λ + iιǫ)3 φι k,ǫ(y, λ) + ∂2 λgι k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that k ∈ Z\\{0}, λ ∈ Λδ0\\{b(y∗)} and let φι k,ǫ(y, λ) with ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} be as defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Recall that δ := δ(λ) = 8 � |λ − b(y∗)|/b′′(y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='35) Denoting for y ∈ [0, 1], λ ∈ Λδ0\\{b(y∗)}, Λ2(y, λ) := − ψk(y, λ)ϕδ(y) b′′(y) (b′(y))2 lim ǫ→0+ 1 b(y) − λ + iǫ − ϕδ(y) b′′(y) (b′(y))2 lim ǫ→0+ � 1 b(y) − λ + iǫ − 1 b(y) − λ − iǫ � φ− k,ǫ(y, λ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='36) then we have the bounds for λ ∈ Λδ0\\{b(y∗)}, � α∈{0,1} ���(δ ∧ |k|−1)9/4� ∂2 λψk(y, λ) − Λ2(y, λ) ���� L2([y∗−3δ,y∗+3δ])δ−1/2 + � α∈{0,1} ���(δ ∧ |k|−1)5/4(|y − y∗| ∧ 1 |k|)∂2 λψk(y, λ) ��� L∞([0,1]\\[y∗−3δ,y∗+3δ])) ≲ |k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denote for k ∈ Z\\{0}, λ ∈ Λδ0\\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and y ∈ I, hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 4) := − 2b′′(z) (b(z) − λ − iιǫ)2 ϕδ(z)∂λφι k,ǫ(z, λ), hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 5) = − 2b′′(z) (b(z) − λ − iιǫ)3 ϕδ(z)φι k,ǫ(z, λ) hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 6) := − b′′(z) (b(z) − λ − iιǫ)2 (1 − ϕδ(z))∂λφι k,ǫ(z, λ), hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 7) = − 2b′′(z) (b(z) − λ − iιǫ)3 (1 − ϕδ(z))φι k,ǫ(z, λ), hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 8) := ∂2 λgι k,ǫ(z, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='38) 26 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA Define for k ∈ Z\\{0}, λ ∈ Λδ0\\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0} and z ∈ I, ∂2 λφι k,ǫ(y, λ) := φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) + 8 � j=4 � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz − 8 � j=4 T ∗ k (λ + iιǫ) � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='39) It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34) that φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) satisfies for y ∈ I, φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) + T ∗ k (λ + iιǫ)φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) = 8 � j=4 � T ∗ k (λ + iιǫ) �2 � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='40) It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 that for j ∈ {6, 7, 8} ��� � T ∗ k (λ + iǫ) �2 � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz ��� XN,̺k ≲ (δ ∧ |k|−1)−4|k|5/2∥ω0k∥H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='41) Using integration by parts argument similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='29)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='30), we have also for j ∈ {4, 5}, ���� � T ∗ k (λ + iǫ) �2 � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iιǫ)hι k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) dz ���� XN,̺k ≲ (δ ∧ |k|−1)−4|k|5/2��ω0k �� H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='42) It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='38)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='42) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4 that for λ ∈ Λδ0\\{b(y∗)}, ι ∈ {±}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, ��φι k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) �� XN,̺k ≲ (δ ∧ |k|−1)−4|k|5/2��ω0k �� H1 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='43) Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='34), we get that for y ∈ I, � k2 − ∂2 y + b′′(y) b(y) − λ + iǫ �� ∂2 λφ+ k,ǫ(y, λ) − ∂2 λφ− k,ǫ(y, λ) � = 8 � j=4 � h+ k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) − h− k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='44) Denoting D2φk,ǫ(y, λ), h ∈ I, λ ∈ Λδ0\\{b(y∗)}, as the solution to � k2 − ∂2 y + b′′(y) b(y) − λ + iιǫ � D2φk,ǫ(y, λ) = 5 � j=4 � h+ k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) − h− k,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='45) for y ∈ I with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We note the identity that for y ∈ I, λ ∈ Σδ0\\{b(y∗)}, ∂2 λψk(y, λ) = lim ǫ→0+ D2φk,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='46) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 27 We can reformulate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='45) as the integral equation for y ∈ I D2φk,ǫ(y, λ) + T ∗ k (λ + iǫ)D2φk,ǫ(y, λ) = � 1 0 Gk(y, z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' λ + iǫ)ϕδ(z) 5 � j=4 � h+ k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) − h− k,ǫ(z, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' j) � dz := R∗ k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='47) We can write for λ ∈ Σδ0\\{b(y∗)}, 0 < ǫ < min{|λ − b(y∗)|, ǫ0}, y ∈ I, D2φk,ǫ(y, λ) := D2φk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) + R∗ k,ǫ(y, λ) − T ∗ k (λ + iǫ)R∗ k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='48) Then D2φk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) satisfies for y ∈ I, λ ∈ Σδ0\\{b(y∗)}, D2φk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) + T ∗ k (λ + iǫ)D2φk,ǫ(y, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 2) = � T ∗ k (λ + iǫ) �2R∗ k,ǫ(y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='49) The desired bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='37) follow from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='47)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='49), and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 with X = XL,̺k, using also the bound ��� T ∗ k (λ + iǫ) �2R∗ k,ǫ(·, λ) �� XL,̺k ≲ � δ ∧ 1 |k| �−4 |k|5/2∥ω0k∥H3 k(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='50) □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We can assume that t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We first give the proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using the representation formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7), we have ψk(t, y) = 1 2πi lim ǫ→0+ � Σ e−ikλt� ψ+ k,ǫ(y, λ) − ψ− k,ǫ(y, λ) � dλ = − 1 2πik2t2 lim ǫ→0+ � Σ e−ikλt� ∂2 λψ+ k,ǫ(y, λ) − ∂2 λψ− k,ǫ(y, λ) � dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1) Fix Φ∗ ∈ C∞ 0 (Σδ0) with Φ∗ ≡ 1 on Σ2δ0/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We can decompose for t ≥ 1, y ∈ [0, 1], ψk(t, y) := ψ1 k(t, y) + ψ2 k(t, y), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2) where ψ1 k(t, y) := − 1 2πik2t2 lim ǫ→0+ � Σ e−ikλt(1 − Φ∗(λ)) � ∂2 λψι k,ǫ(y, λ) − ∂2 λψ− k,ǫ(y, λ) � dλ, ψ2 k(t, y) := − 1 2πik2t2 lim ǫ→0+ � Σ e−ikλtΦ∗(λ) � ∂2 λψι k,ǫ(y, λ) − ∂2 λψ− k,ǫ(y, λ) � dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3) For (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), it suffices to prove that for σ ∈ {1, 2}, k ∈ Z\\{0} and t ≥ 1, ��ψσ k(t, ·) �� L2([0,1]) ≲ |k|3 t2 ∥ω0k∥H3 k([0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) The case σ = 1 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) corresponding to the non-degenerate case is analogous to the case of monotonic shear flows, see [14], and follow from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1-Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We focus on the main new case σ = 2 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Denote for k ∈ Z\\{0}, Mk := |k|5/2∥ω0k∥H3 k([0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='5) Our main tools are Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='3, which imply the following bounds for y ∈ [0, 1], λ ∈ Σδ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' 28 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA If |λ − b(y∗)|1/2 < |y − y∗|/20, then |ψk(y, λ)| ≲ � min � |λ − b(y∗)|1/2, |k|−1��11/4(|y − y∗|−1 + |k|)Mk, |∂2 λψk(y, λ) ≲ � min � |λ − b(y∗)|1/2, |k|−1��−5/4(|y − y∗|−1 + |k|)Mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6) If |y − y∗|/20 < |λ − b(y∗)|1/2 < 20|y − y∗|, then |ψk(y, λ)| ≲ � min � |λ − b(y∗)|1/2, |k|−1��5/4|λ − b(y∗)|1/4Mk, |ψk(y, λ) − ψk(y, b(y))| ≲ |λ − b(y)|1/2|λ − b(y∗)|3/8Mk, ��∂2 λψk(·, λ) − Λ2(·, λ) �� L2(|y−y∗|≈|λ−b(y∗)|1/2) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7) If |λ − b(y∗)|1/2 > 20|y − y∗|, then |ψk(y, λ)| ≲ |λ − b(y∗)|1/4� min � |λ − b(y∗)|1/2, |k|−1��5/4Mk, ��∂2 λψk(·, λ) − Λ2(·, λ) �� L2(|y−y∗|<|λ−b(y∗)|1/2/20) ≲ (|λ − b(y∗)|−1/2 + |k|)9/4|λ − b(y∗)|1/4Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6)-(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) that for y ∈ [0, 1], t ≥ 1, ��� � R e−ikλtΦ∗(λ)Λ2(y, λ)dλ ��� ≲ |y − y∗|−1/4 max � 1, |k|1/2|y − y∗|1/2� Mk, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) and, by considering the cases |λ − b(y∗)| ≪ |y − y∗|2, |λ − b(y∗)| ≈ |y − y∗|2 and |λ − b(y∗)| ≫ |y − y∗|2, also that for y ∈ [0, 1], t ≥ 1, ��� � R e−ikλtΦ∗(λ) � ∂2 λψk(y, λ) − Λ2(y, λ) � dλ ��� L2([0,1]) ≲ |k|9/4Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10) The desired bound (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='4) for σ = 2 follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9)-(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' The proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='9) is similar to the proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We now turn to the proof of the depletion bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Assume that k ∈ Z\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Applying −k2 + ∂2 y to ψk(t, y) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='7), and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8), we get that for y ∈ [0, 1], t ≥ 1, ωk(t, y) = ω∗ k(t, y) + ω∗∗ k (t, y), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='11) where ω∗ k(t, y) := 1 2πi lim ǫ→0+ � Σ e−ikλt(1 − Φ∗(y)) �b′′(y)ψ+ k,ǫ(y, λ) − ω0k(y) b(y) − λ + iǫ − b′′(y)ψ− k,ǫ(y, λ) − ω0k(y) b(y) − λ − iǫ � dλ, ω∗∗ k (t, y) := 1 2πi lim ǫ→0+ � Σ e−ikλtΦ∗(y) �b′′(y)ψ+ k,ǫ(y, λ) − ω0k(y) b(y) − λ + iǫ − b′′(y)ψ− k,ǫ(y, λ) − ω0k(y) b(y) − λ − iǫ � dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='12) We have the bound for t ≥ 1, ��ω∗ k(t, y) �� L∞([0,1]) ≲ |k|2Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='13) LINEAR INVISCID DAMPING AND VORTICITY DEPLETION FOR NON-MONOTONIC SHEAR FLOWS 29 For |y − y∗| < δ0/10, t ≥ 1, since (b(y) − λ + iιǫ) with ι ∈ {±} is not singular in this case, we have in addition by integration by parts that |ω∗ k(t, y)| ≲ |k|2 1 t Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='14) We now turn to ω∗∗ k (t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='1), we can write for y ∈ [0, 1], t ≥ 1, 2πi ω∗∗ k (t, y) = lim ǫ→0+ � R e−ikλtΦ∗(λ) �φ+ k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y) b(y) − λ + iǫ − φ− k,ǫ(y, λ) − (1 − Ψ(y))ω0k(y) b(y) − λ − iǫ � dλ = lim ǫ→0+ � R e−ikλtΦ∗(λ) � φ+ k,ǫ(y, λ) b(y) − λ + iǫ − φ− k,ǫ(y, λ) b(y) − λ − iǫ � dλ + Wk(t, y), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='15) where Wk(t, y) satisfies the bound for t ≥ 1, ∥Wk(t, ·)∥L∞([0,1]) ≲ t−1Mk, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16) which follows from simple integration by parts argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' We decompose for y ∈ [0, 1]\\{y∗}, ω∗∗ k (t, y) − Wk(t, y) 2πi = 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(λ) � ψk(y, λ) b(y) − λ + iǫ � dλ + 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(λ)φ− k,ǫ(y, λ) � 1 b(y) − λ + iǫ − 1 b(y) − λ − iǫ � dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='17) It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='6)-(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='8) that ���� 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(λ)φ− k,ǫ(y, λ) � 1 b(y) − λ + iǫ − 1 b(y) − λ − iǫ � dλ ���� ≲ |y − y∗|7/4Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18) For γ ∈ (1, ∞) to be fixed below, by considering the three ranges (I) |λ − b(y∗)| ≲ |y − y∗|2, (II) |λ − b(y∗)| ≥ γ|y − y∗|2, and (III) |y − y∗|2 ≪ |λ − b(y∗)| < γ|y − y∗|2, and using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='39, we get that ���� 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(y) � ψk(y, λ) b(y) − λ + iǫ � dλ ���� ≲ � |y − y∗|7/4� 1 + |k|1/2|y − y∗|1/2� + 1 |k|t(|k|1/2 + γ−1/8|y − y∗|−1/4) + γ7/8|y − y∗|7/4� Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='19) In the above, we used integration by part to get decay in t in range (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Optimizing in γ, we get that for t ≥ 1, (i) if t|y − y∗|2 ≲ 1, ���� 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(y) � ψk(y, λ) b(y) − λ + iǫ � dλ ���� ≲ � t−1 + |k|1/2|y − y∗|7/4 + t−7/8� (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20) 30 ALEXANDRU D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' IONESCU, SAMEER IYER, AND HAO JIA (ii) if t|y − y∗|2 ≫ 1, ���� 1 2πi lim ǫ→0+ � R e−ikλtΦ∗(y) � ψk(y, λ) b(y) − λ + iǫ � dλ ���� ≲ � |y − y∗|7/4� 1 + |k|1/2|y − y∗|1/2� + 1 |k|1/2t7/8 + |y − y∗|7/4� Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21) The desired bounds (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='16), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='18), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='20)-(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tAyT4oBgHgl3EQfcvc9/content/2301.00288v1.pdf'} +page_content='2 is now proved.' 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a/E9FJT4oBgHgl3EQfCizA/content/tmp_files/2301.11430v1.pdf.txt b/E9FJT4oBgHgl3EQfCizA/content/tmp_files/2301.11430v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cba0346cf2d30e98b4948312091950e05705589 --- /dev/null +++ b/E9FJT4oBgHgl3EQfCizA/content/tmp_files/2301.11430v1.pdf.txt @@ -0,0 +1,1472 @@ +arXiv:2301.11430v1 [math.AP] 26 Jan 2023 +Vortex sheet solutions for the Ginzburg-Landau system +in cylinders: symmetry and global minimality +Radu Ignat∗ +Mircea Rus† +January 30, 2023 +Abstract +We consider the Ginzburg-Landau energy Eε for RM-valued maps defined in a +cylinder shape domain BN ×(0, 1)n satisfying a degree-one vortex boundary condition +on ∂BN × (0, 1)n in dimensions M ≥ N ≥ 2 and n ≥ 1. +The aim is to study +the radial symmetry of global minimizers of this variational problem. We prove the +following: if N ≥ 7, then for every ε > 0, there exists a unique global minimizer +which is given by the non-escaping radially symmetric vortex sheet solution uε(x, z) = +(fε(|x|) x +|x|, 0RM−N), ∀x ∈ BN that is invariant in z ∈ (0, 1)n. If 2 ≤ N ≤ 6 and M ≥ +N + 1, the following dichotomy occurs between escaping and non-escaping solutions: +there exists εN > 0 such that +• if ε ∈ (0, εN), then every global minimizer is an escaping radially symmetric +vortex sheet solution of the form R˜uε where ˜uε(x, z) = ( ˜fε(|x|) x +|x|, 0RM−N−1, gε(|x|)) +is invariant in z-direction with gε > 0 in (0, 1) and R ∈ O(M) is an orthogonal +transformation keeping invariant the space RN × {0RM−N}; +• if ε ≥ εN, then the non-escaping radially symmetric vortex sheet solution +uε(x, z) = (fε(|x|) x +|x|, 0RM−N), ∀x ∈ BN, z ∈ (0, 1)n is the unique global minimizer; +moreover, there are no bounded escaping solutions in this case. +We also discuss the problem of vortex sheet SM−1-valued harmonic maps. +Keywords: +vortex, uniqueness, symmetry, minimizers, Ginzburg-Landau equation, +harmonic maps. +MSC: 35A02, 35B06, 35J50. +Contents +1 +Introduction and main results +2 +1.1 +Minimality of the RN-valued vortex sheet solution +. . . . . . . . . . . . . . +2 +1.2 +Escaping RM-valued vortex sheet solutions when M ≥ N + 1 . . . . . . . . +5 +∗Institut de Math´ematiques de Toulouse & Institut Universitaire de France, UMR 5219, Universit´e de +Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email: Radu.Ignat@math.univ-toulouse.fr +†Department of Mathematics, Technical University of Cluj-Napoca, 400027 Cluj-Napoca, Romania. +Email: rus.mircea@math.utcluj.ro +1 + +2 +The non-escaping vortex sheet solution. Proof of Theorems 1 and 3 +7 +3 +Properties of escaping vortex sheet solutions when M ≥ N + 1 +11 +3.1 +Minimality of escaping vortex sheet solutions . . . . . . . . . . . . . . . . . +11 +3.2 +Escaping radial profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.3 +Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +A Appendix. Vortex sheet SM−1-valued harmonic maps in cylinders +17 +1 +Introduction and main results +In this paper, we consider the following Ginzburg-Landau type energy functional +Eε(u) = +� +Ω +�1 +2|∇u|2 + 1 +2ε2 W(1 − |u|2) +� +dX, +(1) +where ε > 0, X = (x, z) ∈ Ω = BN × (0, 1)n is a cylinder shape domain with BN the unit +ball in RN, n ≥ 1, N ≥ 2 and the potential W ∈ C2((−∞, 1]; R) satisfies +W(0) = 0, W(t) > 0 for all t ∈ (−∞, 1] \ {0} and W is convex. +(2) +(The prototype potential is W(t) = t2 +2 for t ≤ 1.) We investigate the global minimizers of +the energy Eε in the set of RN-valued maps: +AN := {u ∈ H1(Ω; RN) : u(x, z) = x for every x ∈ ∂BN = SN−1, z ∈ (0, 1)n}. +The boundary assumption u(x, z) = x for every x ∈ SN−1 and every z ∈ (0, 1)n is referred +in the literature as the degree-one vortex boundary condition. +The direct method in the calculus of variations yields the existence of a global minimizer +uε of Eε over AN for all range of ε > 0. Moreover, any minimizer uε satisfies |uε| ≤ 1 in +Ω, uε belongs to C1(Ω; RN) and solves the system of PDEs (in the sense of distributions) +with mixed Dirichlet-Neumann boundary conditions: + + + +−∆uε = 1 +ε2uε W ′(1 − |uε|2) +in Ω, +∂uε +∂z = 0 +on BN × ∂(0, 1)n, +u(x, z) = x +on ∂BN × (0, 1)n. +(3) +1.1 +Minimality of the RN-valued vortex sheet solution +The first goal of this paper is to prove the uniqueness and radial symmetry of the global +minimizer of Eε in AN for all ε > 0 in dimensions N ≥ 7 and n ≥ 1. In fact, in these +dimensions, we show that the global minimizer of Eε in AN is unique and given by the +following radially symmetric critical point of Eε that is invariant in z: 1 +uε(x, z) = fε(|x|) x +|x| +for all x ∈ BN and z ∈ (0, 1)n, +(4) +1If n = 0 and N ≥ 2, then SO(N) induces a group action on AN given by u(x) �→ R−1u(Rx) for every +x ∈ BN, R ∈ SO(N) and u ∈ AN under which the energy Eε and the vortex boundary condition are +invariant. Then every bounded critical point of Eε in AN that is invariant under this SO(N) group action +has the form (4), see e.g. [8, Lemma A.4]. +2 + +where the radial profile fε : [0, 1] → R in r = |x| is the unique solution to the ODE: +� −f ′′ +ε − N−1 +r f ′ +ε + N−1 +r2 fε = 1 +ε2fε W ′(1 − f 2 +ε ) +for r ∈ (0, 1), +fε(0) = 0, fε(1) = 1. +(5) +We recall that the unique radial profile fε satisfies fε > 0 and f ′ +ε > 0 in (0, 1) (see +e.g. [7, 9, 8]). Note that the zero set of uε is given by the n-dimensional vortex sheet +{0RN } × (0, 1)n in Ω (in particular, if n = 0, it is a vortex point, while for n = 1, it is a +vortex filament); therefore, uε in (4) is called (radially symmetric) vortex sheet solution to +the Ginzburg-Landau system (3). +Theorem 1. Assume that W satisfies (2) and n ≥ 1. If N ≥ 7, then uε given in (4) is +the unique global minimizer of Eε in AN for every ε > 0. +The proof is reminiscent of the works of Ignat-Nguyen-Slastikov-Zarnescu [12, 11] +studying uniqueness and symmetry of minimizers of the Ginzburg-Landau functionals for +RM-valued maps defined on smooth N-dimensional domains, where M is not necessarily +equal to N. The idea is to analyze Eε(u) for an arbitrary map u and to exploit the convex- +ity of W to lower estimate the excess energy w.r.t. Eε(uε) by a suitable quadratic energy +functional depending on u − uε. This quadratic functional comes from the linearized PDE +at uε and can be handled by a factorization argument. The positivity of the excess energy +then follows by a Hardy-type inequality holding true only in high dimensions N ≥ 7. This +is similar to the result of J¨ager and Kaul [14] on the minimality of the equator map for +the harmonic map problem in dimension N ≥ 7 that is proved using a certain inequality +involving the sharp constant in the Hardy inequality. +We expect that our result remains valid in dimensions 2 ≤ N ≤ 6: +Open problem 2. Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6. Is it true that +for every ε > 0, uε given in (4) is the unique global minimizer of Eε in AN? +It is well known that the uniqueness of uε holds true for large enough ε > 0 in any +dimension N ≥ 2. Indeed, denoting by λ1 the first eigenvalue of −∆x in BN with zero +Dirichlet boundary condition, then for any ε > +� +W ′(1)/λ1, Eε is strictly convex in AN +(see e.g., [1, Theorem VIII.7], [12, Remark 3.3]) and thus has a unique critical point in +AN that is the global minimizer of our problem. We improve this result as follows: for +the radial profile fε in (5), we denote by ℓ(ε) the first eigenvalue of the operator +Lε = −∆x − 1 +ε2 W ′(1 − f 2 +ε ) +(6) +acting on maps defined in BN with zero Dirichlet boundary condition. It is proved in [8, +Lemma 2.3] that if 2 ≤ N ≤ 6 and W ∈ C2((−∞, 1]) satisfies (2), then the first eigenvalue +ℓ(ε) is a continuous function in ε and there exists εN ∈ (0, ∞) such that +ℓ(ε) < 0 in (0, εN), +ℓ(εN) = 0 +and +ℓ(ε) > 0 in (εN, ∞). +(7) +3 + +Note that2 0 = ℓ(εN) > λ1 − +1 +ε2 +N W ′(1) yielding +εN < +� +W ′(1)/λ1. +Theorem 3. Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6. If ε ≥ εN, then uε +given in (4) is a global minimizer of Eε in AN. Moreover, if either ε > εN, or (ε = εN +and W is in addition strictly convex), then uε is the unique global minimizer of Eε in AN. +The case ε < εN is still not solved as stated in Open Problem 2. Let us summarize +some known results: +I. The case of n = 0 and Ω = BN (we also discuss here the problem for Ω = RN). In +this case, the above question was raised in dimension N = 2 for the disk Ω = B2 in +the seminal book of Bethuel, Brezis and H´elein [1, Problem 10, page 139], and in general +dimensions N ≥ 2 and also for the blow-up limiting problem around the vortex point +(when the domain Ω is the whole space RN and by rescaling, ε can be assumed equal to 1) +in an article of Brezis [3, Section 2]. For sufficiently small ε > 0 and for the disk domain +Ω = B2, Pacard and Rivi`ere [20, Theorem 10.2] showed that Eε has a unique critical point +in A2 and so, it is given by the radially symmetric solution uε in (4) (for n = 0). For +N ≥ 7, Ω = BN and any ε > 0, it is proved in [11] that Eε has a unique minimizer in AN +which is given by the radially symmetric solution uε in (4) (for n = 0). For 2 ≤ N ≤ 6 +and Ω = BN, Ignat-Nguyen [8] proved that for any ε > 0, uε is a local minimizer of Eε +in A (which is an extension of the result of Mironescu [18] in dimension N = 2). Also, +Mironescu [19] showed in dimension N = 2 that, when B2 is replaced by R2 and ε = 1, a +local minimizer of Eε satisfying a degree-one boundary condition at infinity is unique (up +to translation and suitable rotation). This was extended in dimension N = 3 by Millot and +Pisante [17] and in dimensions N ≥ 4 by Pisante [21] in the case of the blow-up limiting +problem on RN and ε = 1. All these results (holding for n = 0) are related to the study of +the limit problem obtained by sending ε → 0 when the Ginzburg-Landau problem on the +unit ball ‘converges’ to the harmonic map problem from BN into the unit sphere SN−1. +For that harmonic map problem, the vortex boundary condition yields uniqueness of the +minimizing harmonic SN−1-valued map x �→ +x +|x| if N ≥ 3; this is proved by Brezis, Coron +and Lieb [4] in dimension N = 3 and by Lin [15] in any dimension N ≥ 3; we also mention +J¨ager and Kaul [14] in dimension N ≥ 7 for the equator map x ∈ BN �→ ( x +|x|, 0) ∈ SN. +II. The case of n ≥ 1 and Ω = BN × (0, 1)n. As we explain in Remark 6 below, for some +ε > 0, if the minimality of the radially symmetric solution uε in (4) holds in the case n = 0 +(so, for Ω = BN), then this implies the minimality of uε in Ω = BN ×(0, 1)n also for every +dimension n ≥ 1. In particular, the result of Pacard-Rivi`ere [20, Theorem 10.2] for n = 0 +and N = 2 yields the minimality of uε in (4) defined in B2×(0, 1)n for every n ≥ 1 if ε > 0 +is sufficiently small. Also, the result of Ignat-Nguyen-Slastikov-Zarnescu [11, Theorem 1] +2Indeed, if v ∈ H1 +0(BN) is a first eigenfunction of LεN in BN such that ∥v∥L2(BN ) = 1 then +λ1 ≤ +� +BN |∇xv|2 dx = 1 +ε2 +N +� +BN W ′(1 − f 2 +εN )v2 dx < W ′(1) +ε2 +N +because ℓ(εN) = 0, 0 < fεN < 1 in (0, 1) and (2) implies W ′(0) = 0 and W ′(t) > 0 for t ∈ (0, 1]. +4 + +for n = 0, N ≥ 7 and any ε > 0 generalizes to dimension n ≥ 1 for Ω = BN × (0, 1)n (see +the proof of Theorem 1). We also mention the work of Sandier-Shafrir [24] where they +treat the case of topologically trivial R2-valued solutions in the domain Ω = R3 (see also +[5, 22] for vortex filament solutions). +1.2 +Escaping RM-valued vortex sheet solutions when M ≥ N + 1 +In dimension 2 ≤ N ≤ 6 and for ε < εN given in (7), a different type of radially symmetric +vortex sheet solution appears provided that the target space has dimension M ≥ N + 1. +More precisely, we consider the energy functional Eε in (1) over the set of RM-valued maps +A := {u ∈ H1(Ω; RM) : u(x, z) = (x, 0RM−N ) on ∂BN = SN−1 ⊂ RM, z ∈ (0, 1)n}. +(8) +If M ≥ N + 1, the prototype of radially symmetric critical points of Eε in A has the +following form (invariant in z-direction): 3 +˜uε(x, z) = ( ˜fε(r) x +|x|, 0RM−N−1, gε(r)) ∈ A , +x ∈ BN, z ∈ (0, 1)n, r = |x|, +(9) +where ( ˜fε, gε) satisfies the system of ODEs +− ˜f ′′ +ε − N − 1 +r +˜f ′ +ε + N − 1 +r2 +˜fε = 1 +ε2 W ′(1 − ˜f 2 +ε − g2 +ε) ˜fε +in (0, 1), +(10) +−g′′ +ε − N − 1 +r +g′ +ε = 1 +ε2 W ′(1 − ˜f 2 +ε − g2 +ε)gε +in (0, 1), +(11) +˜fε(1) = 1 and gε(1) = 0. +(12) +We distinguish two type of radial profiles: +• the non-escaping radial profile ( ˜fε = fε, gε = 0) with the unique radial profile fε given +in (5); in this case, we say that ˜uε = (uε, 0RM−N ) is a non-escaping (radially symmetric) +vortex sheet solution where uε is given in (4). +• the escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1); in this case, we call an +escaping (radially symmetric) vortex sheet solution ˜uε in (9). In this case, ˜fε ̸= fε and +obviously, ( ˜fε, −gε) is another radial profile to (9)-(12). +The properties of such radial profiles (e.g., existence, uniqueness, minimality, mono- +tonicity) are analyzed in Theorem 9 below and are based on ideas developed by Ignat- +Nguyen [8]. +Our main result proves the radial symmetry of global minimizers of Eε in A . More +precisely, the following dichotomy occurs at εN defined in (7): if ε < εN, then escaping +radially symmetric vortex sheet solutions exist and determine (up to certain orthogonal +transformations) the full set of global minimizers of Eε in A ; if instead ε ≥ εN, then the +non-escaping radially symmetric vortex sheet solution is the unique global minimizer of +Eε in A and no escaping radially symmetric vortex sheet solutions exist in this case. +3If M = N + 1, then ˜uε(x, z) = ( ˜fε(r) x +|x|, gε(r)) for every x ∈ BN and z ∈ (0, 1)n. In fact, if n = 0 (so, +for Ω = BN), every bounded critical point of Eε in A that is invariant under the action of a special group +(isomorphic to SO(N)) has the form of ˜uε, see [8, Definition A.1, Lemma A.5]. +5 + +Theorem 4. Let n ≥ 1, 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be +strictly convex. Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7). Then there exists an +escaping radially symmetric vortex sheet solution ˜uε in (9) with gε > 0 in (0, 1) if and +only if 0 < ε < εN. Moreover, +1. if 0 < ε < εN, the escaping radially symmetric vortex sheet solution ˜uε is a global +minimizer of Eε in A and all global minimizers of Eε in A are radially symmetric +given by R˜uε where R ∈ O(M) is an orthogonal transformation of RM satisfying +Rp = p for all p ∈ RN × {0RM−N }. +In this case, the non-escaping vortex sheet +solution (uε, 0RM−N ) in (4) is an unstable critical point of Eε in A . +2. if ε ≥ εN, the non-escaping vortex sheet solution (uε, 0RM−N ) in (4) is the unique +global minimizer of Eε in A . Furthermore, there are no bounded critical points wε +of Eε in A that escape in some direction e ∈ SM−1 (i.e., wε · e > 0 a.e. in Ω). +The result above holds also if n = 0, i.e., Ω = BN and the vortex sheets corresponding +to the above solutions become vortex points (see Theorem 10). It generalizes [12, Theorem +1.1] that was proved in the case N = 2 and M = 3 (without identifying the meaning of +the dichotomy parameter εN in (7)). The dichotomy in Theorem 4 happens in dimensions +2 ≤ N ≤ 6 because of the phenomenology occurring for the limit problem ε → 0. More +precisely, if M ≥ N + 1, then minimizing SM−1-valued harmonic maps in A are smooth +and escaping in a direction of SM−1 provided that N ≤ 6; if N ≥ 7, then there is a unique +minimizing SM−1-valued harmonic maps in A , non-escaping and singular, the singular +set being given by a vortex sheet of dimension n in Ω (see Theorem 11 in Appendix +below). This suggests why in dimension N ≥ 7 and for any ε > 0, there is no escaping +radially symmetric vortex sheet critical point ˜uε of Eε in A while the non-escaping vortex +sheet solution (uε, 0RM−N ) is the unique global minimizer of Eε in A (see Theorem 5 and +Remark 8 below). +The paper is meant to be self-contained and it is organized as follows. In Section 2, we +prove the minimality and the uniqueness results for the non-escaping radially symmetric +solution in Theorems 1 and 3; this is done in a more general setting by considering the +target dimension M ≥ N for the set of configurations A instead of AN. Section 3 is +devoted to characterize escaping vortex sheet solutions. First, we prove the minimality +of such bounded solutions stated in Theorem 7. Second, we prove existence, minimality +and uniqueness results for the escaping radial profile in Theorem 9. Finally, we prove our +main result on the dichotomy between escaping / non-escaping radially symmetric vortex +sheet solutions in Theorem 4. In Appendix, we prove the corresponding dichotomy result +for SM−1-valued harmonic maps in Theorem 11 which again is based on the minimality of +escaping SM−1-valued harmonic maps in Theorem 12. +Acknowledgment. R.I. is partially supported by the ANR projects ANR-21-CE40-0004 +and ANR-22-CE40-0006-01. He also thanks for the hospitality of the Hausdorff Research +Institute for Mathematics in Bonn during the trimester “Mathematics for Complex Ma- +terials”. +6 + +2 +The non-escaping vortex sheet solution. Proof of Theo- +rems 1 and 3 +Theorem 1 will be obtained as a consequence of a stronger result on the uniqueness of +global minimizers of the RM-valued Ginzburg-Landau functional with M ≥ N ≥ 7. For +that, we consider the energy functional Eε in (1) over the set A defined in (8). The aim +is to prove the minimality and uniqueness of the vortex sheet solution (uε, 0RM−N ) where +uε given in (4) with the obvious identification uε ≡ (uε, 0RM−N ) if M = N, following the +ideas of Ignat-Nguyen-Slastikov-Zarnescu [12, 11]. +Theorem 5. Assume that W satisfies (2) and n ≥ 1. If M ≥ N ≥ 7, then for every +ε > 0, (uε, 0RM−N ) given in (4) is the unique global minimizer of Eε in A . +Proof. To simplify notation, we identify +uε ≡ (uε, 0RM−N ) +when +M ≥ N. +(13) +The proof will be done in several steps following the strategy in [12, Theorem 1.7], [11, +Theorem 1]. +First, for an arbitrary competitor uε + v, we consider the excess energy +Eε(uε + v) − Eε(uε) for the critical point uε defined in (4) and show a lower estimate +by a quadratic energy functional Fε(v) coming from the operator Lε in (6). Second, we +show that Fε(v) ≥ 0 using the properties of the radial profile fε in (5) and a Hardy +decomposition method; this proves in particular that uε is a global minimizer of Eε over +A . Finally, by analyzing the zero excess energy states, we conclude to the uniqueness of +the global minimizer uε. +Step 1: Excess energy. For any v ∈ H1 +0(BN × Rn; RM), we have +Eε(uε + v) − Eε(uε) = +� +Ω +� +∇uε · ∇v + 1 +2|∇v|2� +dxdz ++ 1 +2ε2 +� +Ω +� +W(1 − |uε + v|2) − W(1 − |uε|2) +� +dxdz. +Note that for every u ∈ A , uε−u can be extended to v ∈ H1 +0(BN ×Rn; RM). In particular, +v(·, z) ∈ H1 +0(BN, RM) for a.e. z ∈ (0, 1)n. The convexity of W yields +W(1 − |uε + v|2) − W(1 − |uε|2) ≥ −W ′(1 − |uε|2)(|uε + v|2 − |uε|2). +(14) +Combining the above relations, we obtain the following lower bound for the excess energy: +Eε(uε + v) − Eε(uε) ≥ +� +Ω +� +∇uε · ∇v − 1 +ε2 W ′(1 − f 2 +ε )uε · v +� +dxdz ++ +� +Ω +�1 +2|∇v|2 − 1 +2ε2 W ′(1 − f 2 +ε )|v|2� +dxdz += +� +Ω +1 +2|∇zv|2 dxdz + +� +(0,1)n +1 +2Fε(v(·, z)) dz, +(15) +7 + +where we used the PDE (3) and introduced the quadratic functional +Fε(Ψ) = +� +BN +� +|∇xΨ|2 − 1 +ε2 W ′(1 − f 2 +ε )|Ψ|2� +dx, +for all Ψ ∈ H1 +0(BN; RM). Note that the L2-gradient of Fε represents a part of the lin- +earization of the PDE (3) at uε and it is given by the operator Lε in (6). The rest of the +proof is devoted to show that for N ≥ 3: +Fε(ψ) ≥ +�(N − 2)2 +4 +− (N − 1) +� � +BN +ψ2 +r2 dx, +∀ψ ∈ H1 +0(BN) +yielding the conclusion for N ≥ 7 and also the inequality for the first eigenvalue ℓ(ε) of +the operator Lε in (6) in BN: 4 +ℓ(ε) ≥ (N − 2)2 +4 +− (N − 1) > 0, +∀ε > 0 +and +N ≥ 7. +To keep the paper self-contained, we explain in the following the simple idea used in +[12, 11]. +Step 2: A factorization argument. As fε > 0 is a smooth positive radial profile in (0, 1), +we decompose every scalar test function ψ ∈ C∞ +c (BN \ {0}; R) as follows +ψ(x) = fε(r)w(x), +∀x ∈ BN \ {0}, r = |x|, +where w ∈ C∞ +c (BN \ {0}; R). Integrating by parts (see e.g. [10, Lemma A.1]), we deduce: +Fε(ψ) = +� +BN Lεψ · ψ dx = +� +BN w2(Lεfε · fε) dx + +� +BN f 2 +ε |∇xw|2 dx += +� +BN f 2 +ε +� +|∇xw|2 − N − 1 +r2 +w2 +� +dx, +because Lεfε · fε = − N−1 +r2 f 2 +ε in BN by (5). Furthermore, we decompose +w = ϕg +in +BN \ {0} +with ϕ = |x|− N−2 +2 +satisfying +−∆xϕ = (N − 2)2 +4|x|2 +ϕ +in RN \ {0} +and g ∈ C∞ +c (BN \ {0}; R). Then +|∇xw|2 = |∇xg|2ϕ2 + |∇xϕ|2g2 + 1 +2∇x(ϕ2) · ∇x(g2). +4Observe the difference between dimension N ≥ 7 and the case of dimension 2 ≤ N ≤ 6 where we have +ℓ(ε) < 0 for ε < εN in (7); moreover, if N ≤ 6, then ℓ(ε) blows up as − 1 +ε2 as ε → 0 (see [8, Lemma 2.3]). +8 + +As |∇xϕ|2 = (N−2)2 +4|x|2 ϕ2 and ϕ2 is harmonic in BN \ {0} (recall that N ≥ 7), integration by +parts yields +Fε(ψ) = +� +BN f 2 +ε +� +|∇xg|2ϕ2 + (N − 2)2 +4r2 +ϕ2g2 − N − 1 +r2 +ϕ2g2 +� +dx − 1 +2 +� +BN ∇x(ϕ2) · ∇x(f 2 +ε )g2 dx +≥ +� +BN f 2 +ε |∇xg|2ϕ2 dx + +�(N − 2)2 +4 +− (N − 1) +� � +BN +f 2 +ε +r2 ϕ2g2 dx +≥ +�(N − 2)2 +4 +− (N − 1) +� � +BN +ψ2 +r2 dx ≥ 0, +(16) +where we used N ≥ 7 and 1 +2∇x(ϕ2)·∇x(f 2 +ε ) = 2ϕϕ′fεf ′ +ε ≤ 0 in BN \{0} because ϕ, fε, f ′ +ε > +0 and ϕ′ < 0 in (0, 1) (see e.g. [7, 9, 8]). +Step 3: We prove that Fε(Ψ) ≥ 0 for every Ψ ∈ H1 +0(BN; RM); moreover, Fε(Ψ) = 0 if and +only if Ψ = 0. Let Ψ ∈ H1 +0(BN; RM). As a point in RN has zero H1 capacity, a standard +density argument implies the existence of a sequence Ψk ∈ C∞ +c (BN \ {0}; RM) such that +Ψk → Ψ in H1(BN, RM) and a.e. in BN. On the one hand, by definition of Fε, since +W ′(1 − f 2 +ε ) ∈ L∞, we deduce that Fε(Ψk) → Fε(Ψ) as k → ∞. On the other hand, by +(16) and Fatou’s lemma, we deduce +lim inf +k→∞ Fε(Ψk) ≥ +�(N − 2)2 +4 +− (N − 1) +� +lim inf +k→∞ +� +BN +|Ψk|2 +r2 +dx +≥ +�(N − 2)2 +4 +− (N − 1) +� � +BN +|Ψ|2 +r2 dx. +Therefore, we conclude that +Fε(Ψ) ≥ +�(N − 2)2 +4 +− (N − 1) +� � +BN +|Ψ|2 +r2 dx ≥ 0, +∀Ψ ∈ H1 +0(BN; RM). +Moreover, Fε(Ψ) = 0 if and only if Ψ = 0. +Step 4: Conclusion. By (15) and Step 3, we deduce that uε is a global minimizer of Eε +over A . For uniqueness, assume that ˆuε is another global minimizer of Eε over A . If +v := ˆuε − uε, then v can be extended in H1 +0(BN × Rn; RM) and by Steps 1 and 3, we have +that +0 = Eε(ˆuε) − Eε(uε) ≥ +� +Ω +1 +2|∇zv|2 dxdz + +� +(0,1)n +1 +2Fε(v(·, z)) dz ≥ 0, +which yields ∇zv = 0 a.e. in Ω and Fε(v(·, z)) = 0 for a.e. z ∈ (0, 1)n. In other words, +v = v(x) and Step 3 implies that v = 0, i.e., ˆuε = uε in Ω. +Remark 6. Theorem 5 reveals the following fact: if for n = 0 (i.e., Ω = BN) and some +ε > 0, a (radially symmetric) critical point ˆuε : BN → RM of Eε in A is proved to be a +global minimizer (and additionally, if one proves that it is the unique global minimizer), +then for any dimensions n ≥ 1 (i.e., Ω = BN ×(0, 1)n), this z-invariant solution ˆuε of (3) +9 + +in BN × (0, 1)n is also a global minimizer (and additionally, it is the unique minimizer) +of Eε in A . This is because for every u : BN × (0, 1)n → RM with u ∈ A , then u(·, z) +satisfies the degree-one vortex boundary condition on ∂BN for every z ∈ (0, 1)n yielding +Eε(u) = +� +Ω +1 +2|∇zu|2 dxdz + +� +(0,1)n Eε(u(·, z)) dz +≥ +� +(0,1)n Eε(ˆuε) dz = Eε(ˆuε); +the equality occurs only when u is z-invariant. Thus, if the uniqueness of the global mini- +mizer ˆuε holds in BN (i.e., n = 0), then this yields uniqueness of the global minimizer ˆuε +in Ω = BN × (0, 1)n (as a map independent of z-variable) for every n ≥ 1. +Proof of Theorem 3. We prove the result in the more general setting of RM-valued maps +u belonging to A for M ≥ N using the same identification (13). By Step 1 in the proof +of Theorem 5 (see (15)), the excess energy is estimated for every v ∈ H1 +0(BN × Rn; RM): +Eε(uε + v) − Eε(uε) ≥ +� +Ω +1 +2|∇zv|2 dxdz + 1 +2 +� +(0,1)n < Lεv(·, z), v(·, z) > dz, +where Lε is the operator in (6) and < ·, · > denotes the duality pairing (H−1, H1 +0) in BN. +If ε ≥ εN, then ℓ(ε) ≥ 0 (by [8, Lemma 2.3]) and therefore, 5 +< Lεv(·, z), v(·, z) > ≥ ℓ(ε)∥v(·, z)∥2 +L2(BN ) ≥ 0 +for a.e. z ∈ (0, 1)n, +(17) +where we used that v(·, z) ∈ H1 +0(BN; RM) for a.e. z ∈ (0, 1)n. Thus, uε is a minimizer +of Eε over A . It remains to prove uniqueness of the global minimizer. For that, if ˆuε is +another global minimizer of Eε over A , setting v := ˆuε − uε, then v can be extended in +H1 +0(BN × Rn; RM) and +0 = Eε(ˆuε) − Eε(uε) ≥ +� +Ω +1 +2|∇zv|2 dxdz + ℓ(ε) +2 +� +(0,1)n +� +BN |v(x, z)|2 dxdz ≥ 0 +(18) +because ℓ(ε) ≥ 0 for ε ≥ εN. Thus, equality holds in the above inequalities. +Case 1: ε > εN. In this case, ℓ(ε) > 0 and we conclude that v = 0 in Ω, i.e., ˆuε = uε in Ω. +5 Indeed, for a scalar function v ∈ C∞ +c (BN \ {0}, R), if ψ = ψ(r) > 0 is a radial first eigenfunction of +Lε in BN with zero Dirichlet data, i.e., Lεψ = ℓ(ε)ψ in BN, then the duality pairing (H−1, H1 +0) term in +BN writes (see e.g. [10, Lemma A.1]): +< Lεv, v > = +� +BN ψ2|∇( v +ψ )|2 dx + +� +BN ( v +ψ )2Lεψ · ψ dx = +� +BN ψ2|∇( v +ψ )|2 dx + ℓ(ε)∥v∥2 +L2(BN ). +By a density argument, Fatou’s lemma yields for every scalar function v ∈ H1 +0(BN, R), +< Lεv, v > ≥ +� +BN ψ2|∇( v +ψ )|2 dx + ℓ(ε)∥v∥2 +L2(BN ). +10 + +Case 2: ε = εN and W is in addition strictly convex. In this case, ℓ(ε) = 0 and by (18), v +is invariant in z, i.e., v = v(x) and equality holds in (17) and in (15), thus, equality holds +in (14). Note that by footnote 5 the equality in (17) holds if and only if v = λψ for some +λ ∈ RM, where ψ = ψ(r) is a radial first eigenfunction of Lε in BN with zero Dirichlet +data, in particular ψ > 0 in [0, 1) and ψ(1) = 0. Also, by the strict convexity of W, the +equality (14) is achieved if and only if |uε + v| = |uε| a.e. in Ω, that is, |v|2 + 2v · uε = 0 +a.e. in BN. It yields +|λ|2ψ2 + 2fε(|x|)( x +|x|, 0RM−N ) · λψ = 0 +for every x ∈ BN. +(19) +Dividing by ψ in BN, the continuity up to the boundary ∂BN leads to 2fε(|x|)(x, 0RM−N )· +λ = 0 for every x ∈ ∂BN since ψ = 0 on ∂BN. As fε(1) = 1, it follows that the first N +components of λ vanish. Coming back to (19), we conclude that |λ|2ψ2 = 0 in BN, i.e., +λ = 0 and so, v = 0 and ˆuε = uε in Ω. +3 +Properties of escaping vortex sheet solutions when M ≥ +N + 1 +3.1 +Minimality of escaping vortex sheet solutions +In this section, we require the additional assumption of strict convexity of W in order to +determine the set of global minimizers of Eε over A in (8). However, W is assumed to be +only C1 not C2. We prove that every bounded solution to (3) escaping in some direction +is a global minimizer of Eε over A ; moreover, such global minimizer is unique up to an +orthogonal transformation of RM keeping invariant the space RN × {0RM−N }. +Theorem 7. We consider the dimensions n ≥ 1 and M > N ≥ 2, the potential W ∈ +C1((−∞, 1], R) satisfying (2) and an escaping direction e ∈ SM−1. Fix any ε > 0 and let +wε ∈ H1 ∩ L∞(Ω, RM) be a critical point of the energy Eε in the set A which is positive +in the direction e inside Ω: +wε · e > 0 a.e. in Ω. +(20) +Then wε is a global minimizer of Eε in A . If in addition W is strictly convex, then all +minimizers of Eε in A are given by Rwε where R ∈ O(M) is an orthogonal transformation +of RM satisfying Rp = p for all p ∈ RN × {0RM−N }. +This result is reminiscent from [12, Theorem 1.3]. However, it doesn’t apply directly +as the domain Ω is not smooth here and the boundary condition is a mixed Dirichlet- +Neumann condition (w.r.t. Dirichlet boundary condition in [12]). +Proof. In the following, we denote the variable X = (x, z) ∈ Ω = BN × (0, 1)n. As a +critical point of Eε in the set A , wε : Ω → RM satisfies + + + +−∆wε = 1 +ε2wε W ′(1 − |wε|2) +in Ω, +∂wε +∂z = 0 +on BN × ∂(0, 1)n, +wε(x, z) = (x, 0RM−N ) +on ∂BN × (0, 1)n. +(21) +11 + +In particular, ∆wε ∈ L∞(Ω) (as W ′ is continuous and wε ∈ L∞(Ω)); then standard elliptic +regularity for the mixed boundary conditions in (21) yields wε ∈ C1(¯Ω, RM). Thus, (20) +implies wε·e ≥ 0 in ¯Ω and the vortex boundary condition in A implies that e is orthogonal +to RN × {0RM−N }. By the invariance of the energy and the vortex boundary condition +under the transformation wε(X) �→ Rwε(X) for any R ∈ O(M) satisfying Rp = p for all +p ∈ RN × {0RM−N }, we know that Rwε is also a critical point of Eε over A ; thus, we can +assume that +e := eM = (0, . . . , 0, 1) ∈ RM. +(22) +We prove the result in several steps. +Step 1: Excess energy. +By Step 1 in the proof of Theorem 5, we have for any v ∈ +H1 +0(BN × Rn, RM): +Eε(wε + v) − Eε(wε) ≥ +� +Ω +�1 +2|∇v|2 − 1 +2ε2 W ′(1 − |wε|2)|v|2� +dX =: 1 +2Gε(v) +(23) +(note that Gε(v) is larger than the integration of Fε(v) in (15) over (0, 1)n as it contains +also the integration of |∇zv|2). If in addition W is strictly convex, then equality holds +above if and only if |wε(X) + v(X)| = |wε(X)| a.e. X ∈ Ω (by (14)). +Step 2: Global minimality of wε. It is enough to show that the quadratic energy Gε(v) +defined in (23) is nonnegative for any v ∈ H1 +0(BN ×Rn, RM). Denoting the M-component +of wε by φ := wε · eM, we know that φ ∈ C1(¯Ω), φ ≥ 0 in Ω (by (20)) and satisfies the +Euler-Lagrange equation in the sense of distributions: + + + +−∆φ − 1 +ε2W ′(1 − |wε|2)φ = 0 in Ω, +φ = 0 on ∂BN × (0, 1)n, +∂φ +∂z = 0 on BN × ∂(0, 1)n. +(24) +Note that by strong maximum principle, φ > 0 in Ω (as φ cannot be identically 0 in Ω +by (20)). Moreover, Hopf’s lemma yields φ > 0 on BN × ∂(0, 1)n as ∂φ +∂z vanishes there. +Now, for any smooth map v ∈ C∞ +c (BN × Rn; RM), we can define Ψ = v +φ ∈ C1(¯Ω; RM) +with Ψ = 0 in a neighborhood of ∂BN × (0, 1)n and integration by parts yields for every +component vj = φΨj with 1 ≤ j ≤ M (as in [10, Lemma A.1.]): +Gε(vj) = +� +Ω +� +|∇vj|2 − 1 +ε2 W ′(1 − |wε|2)φ · φΨ2 +j +� +dX +(24) += +� +Ω +� +|∇(φΨj)|2 − ∇φ · ∇(φ Ψ2 +j) +� +dX = +� +Ω +φ2|∇Ψj|2 dX. +As Gε is continuous in strong H1(Ω) topology (since W ′(1 − |wε|2) ∈ L∞(Ω)), by density +of C∞ +c (BN × Rn; RM) in H1 +0(BN × Rn; RM), Fatou’s lemma yields +Gε(v) ≥ +� +Ω +φ2|∇ +�v +φ +� +|2 dX ≥ 0, +∀v ∈ H1 +0(BN × Rn; RM). +12 + +As a consequence of (23), we deduce that wε is a minimizer of Eε over A . Moreover, +Gε(v) = 0 if and only if there exists a (constant) vector λ ∈ RM such that v = λφ for a.e. +x ∈ Ω. +Step 3: Set of global minimizers. From now on, we assume that W is strictly convex and +denote wε = (wε,1, . . . , wε,M). Note that the map +˜wε := (wε,1, . . . , wε,N, 0RM−N−1, +� +w2 +ε,N+1 + · · · + w2 +ε,M) +(25) +belongs to A , | ˜wε| = |wε| and |∇ ˜wε| ≤ |∇wε| in Ω, so Eε(wε) ≥ Eε( ˜wε) and +� +w2 +ε,N+1 + · · · + w2 +ε,M ≥ wε,M = φ > 0 +in +Ω. +Hence, ˜wε is a minimizer of Eε on A (as wε minimizes Eε over A by Step 2). Therefore, +up to interchanging wε and ˜wε, we may assume +� wε,N+1 = · · · = wε,M−1 ≡ 0 in Ω +wε,M = φ +(20) +> 0 in Ω. +We now consider another minimizer Uε of Eε over A and denote v := Uε −wε ∈ H1 +0(BN × +Rn; RM) after a suitable extension. From Steps 1 and 2 we know that Eε(Uε) = Eε(v + +wε) = Eε(wε), Gε(v) = 0, |v+wε| = |wε| a.e. in Ω and v = λφ for some λ = (λ1, . . . , λM) ∈ +RM where we recall that φ = wε·eM. By continuity of wε and φ, the relation |v+wε| = |wε| +a.e. in Ω implies 2wε · v + |v|2 = 0 everywhere in Ω. Since v = λφ, dividing by φ > 0 in +Ω, we obtain +2λ · wε + φ|λ|2 = 0 in Ω +(26) +and by continuity, the equality holds also on ∂Ω. As for every (x, z) ∈ ∂BN × (0, 1)n, +φ(x, z) = 0 and wε(x, z) = (x, 0RM−N ), we deduce that λ · (x, 0RM−N ) = 0 for every +x ∈ ∂BN. It follows that λ1 = λ2 = · · · = λN = 0 and therefore, recalling that wε,N+1 = +· · · = wε,M−1 = 0 in Ω, we have by (26): +2λMφ + (λ2 +N+1 + · · · + λ2 +M)φ = 0 in Ω. +As φ > 0 in Ω, we obtain +λ2 +N+1 + · · · + λ2 +M−1 + (λM + 1)2 = 1; +hence we can find R ∈ O(M) such that Rp = p for all p ∈ RN × {0RM−N } and +ReM = (0, . . . , 0, λN+1, . . . , λM−1, λM + 1). +This implies Uε = wε+v = wε+λφ = Rwε as required. The converse statement is obvious: +if wε is a minimizer of Eε over A and R ∈ O(M) is a transformation fixing all points of +RN ×{0RM−N }, then Rwε is also a minimizer of Eε over A (because Eε and the boundary +condition in A are invariant under such orthogonal transformation R). +13 + +Remark 8. Note that if n ≥ 1, M > N ≥ 7 and W satisfies (2) (not necessarily strictly +convex), then there are no bounded critical points of the energy Eε in the set A escaping in +a direction e ∈ SM−1. Indeed, if such an escaping critical point of Eε in A exists, then by +Theorem 7, this solution would be a global minimizer of Eε in A which is a contradiction +with the uniqueness of the global minimizer (uε, 0RM−N ) in (4) (that is non-escaping) +proved in Theorem 5. +3.2 +Escaping radial profile +Let M ≥ N + 1. We give a necessary and sufficient condition for the existence of an +escaping radial profile ( ˜fε, gε > 0) in (0, 1) to the system (9)–(12); we also prove uniqueness, +minimality and monotonicity of the escaping radial profile. For that, in the context of Eε +defined over A , we introduce the functional +Iε(f, g) = +1 +|SN−1|Eε +� +(f(r) x +|x|, 0RM−N−1, g(r)) +� += 1 +2 +� 1 +0 +� +(f ′)2 + (g′)2 + N − 1 +r2 +f 2 + 1 +ε2 W(1 − f 2 − g2) +� +rN−1 dr +where (f, g) belongs to +B = +� +(f, g) : r +N−1 +2 f ′, r +N−3 +2 f, r +N−1 +2 g′, r +N−1 +2 g ∈ L2(0, 1), f(1) = 1, g(1) = 0 +� +. +(27) +The following result is reminiscent from Ignat-Nguyen [8, Theorem 2.4] (for ˜W ≡ 0). +The proof of [8, Theorem 2.4] is rather complicated (as it is proved for some general +potentials ˜W). We present here a simple proof that works in our context: +Theorem 9. Let 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be strictly +convex. Consider εN ∈ (0, ∞) in (7) such that ℓ(εN) = 0. Then the system (9)–(12) has +an escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1) if and only if 0 < ε < εN. Moreover, +in the case 0 < ε < εN, +1. ( ˜fε, gε > 0) is the unique escaping radial profile of (9)–(12) and +˜fε +r , gε ∈ C2([0, 1]), +˜f 2 +ε + g2 +ε < 1, ˜fε > 0, ˜f ′ +ε > 0, g′ +ε < 0 in (0, 1); +2. there are exactly two minimizers of Iε in B given by ( ˜fε, ±gε); +3. the non-escaping radial profile (fε, 0) is an unstable critical point of Iε in B where +fε is the unique radial profile in (5). +Recall that for ε ≥ εN, the non-escaping radial profile (fε, 0) is the unique global +minimizer of Iε in B (by Theorem 3 whose proof yields the minimality of (uε, 0RM−N ) of +Eε in A ). +Proof of Theorem 9. First, we focus on the existence of escaping radial profiles of (9)–(12). +Note that the direct method in calculus of variations implies that Iε admits a minimizer +14 + +( ˜fε, gε) ∈ B. +Since ( ˜fε, gε) ∈ B, ( ˜fε, gε) ∈ C((0, 1]). +It follows that ( ˜fε, gε) satisfies +(10)–(12) in the weak sense, and so ˜fε, gε ∈ C2((0, 1]). Since (| ˜fε|, |gε|) is also a minimizer +of Iε in B, the above argument also shows that | ˜fε|, |gε| ∈ C2((0, 1]) satisfies (10)–(12). +Since | ˜fε|, |gε| ≥ 0 and ˜fε(1) = 1, the strong maximum principle yields | ˜fε| > 0 in (0, 1), +and either |gε| > 0 in (0, 1) or gε ≡ 0 in (0, 1). It follows that ˜fε > 0 in (0, 1), and there +are three alternatives: gε > 0 in (0, 1), gε < 0 in (0, 1) or gε ≡ 0 in (0, 1). Clearly, when +gε ≡ 0, ˜fε is equal to the unique radial profile fε in (5). By considering ( ˜fε, −gε) instead +of ( ˜fε, gε) if necessary, we assume in the sequel that gε ≥ 0. +Claim: if 0 < ε < εN, then gε > 0 in (0, 1) and (fε, 0) is an unstable critical point of Iε in +B. +Proof of Claim: We define the second variation of Iε at (fε, 0) as +Qε(α, β) = d2 +dt2 +���� +t=0 +Iε +� +(fε, 0) + t(α, β) +� += +� +BN +� +Lεα · α + Lεβ · β + N − 1 +r2 +α2 + 2 +ε2 W ′′(1 − f 2 +ε )f 2 +ε α2� +dx, +for α, β ∈ C∞ +c ((0, 1)) which extends by density to the Hilbert space +H = {(α, β) : (fε + α, β) ∈ B} with the norm +∥(α, β)∥H := ∥(α x +|x|, β)∥H1(BN,RN+1). +As ε ∈ (0, εN), we have ℓ(ε) < 0 by (7). Taking β ∈ H1 +0(BN) to be any first eigenfunction +of Lε in BN, which is radially symmetric, we have r +N−1 +2 β′, r +N−1 +2 β ∈ L2(0, 1), β(1) = 0 and +Qε(0, β) = +� +BN Lεβ · β dx = ℓ(ε) +� +BN β2 dx < 0. +So, (fε, 0) is an unstable critical point of Iε in B if ε < εN. In particular, (fε, 0) is not +minimizing Iε in B and therefore, by the above construction of the minimizer ( ˜fε, gε) of +Iε in B, we deduce that gε > 0. This proves the above Claim. +Moreover, by [8, Lemmas 2.7 and A.5, Proposition 2.9] (for ˜W ≡ 0), we deduce that +˜fε +r , gε ∈ C2([0, 1]), ˜f 2 +ε + g2 +ε < 1, ˜f ′ +ε > 0 and g′ +ε < 0 in (0, 1). +To conclude, we distinguish two cases: +Case 1: if ε ∈ (0, εN), Claim yields the existence of an escaping radial profile ( ˜fε, gε > 0). +By [8, Lemmas 2.7], every escaping radial profile ( ˜fε, gε > 0) is bounded (i.e., ˜f 2 +ε + g2 +ε < 1 +in (0, 1)) and therefore, by Theorem 7, the corresponding (bounded) escaping critical point +˜uε in (9) is a global minimizer of Eε over A and the set of minimizers of Eε over A is then +given by {R˜uε : R ∈ O(M), Rp = p, ∀p ∈ RN × {0RM−N }}. Therefore, ( ˜fε, ±gε) are the +only two minimizers of Iε in B. In particular, this proves the uniqueness of the escaping +radial profile ( ˜fε, gε > 0). +Case 2: if ε ≥ εN, by the proof of Theorem 3, the non-escaping vortex sheet solution +uε(x) ≡ (fε(|x|) x +|x|, 0RM−N ) (by (13)) is the unique minimizer of Eε over A . In particular, +(fε, 0) is the unique minimizer of Iε in B, i.e., in the above construction of the minimizer +15 + +( ˜fε, gε) of Iε in B, we have ˜fε = fε and gε = 0 in (0, 1). We claim that no escaping +radial profile ( ˆfε, ˆgε > 0) exists if ε ≥ εN. Assume by contradiction that such an escaping +radial profile ( ˆfε, ˆgε > 0) exists. The same argument presented in Case 1 would imply +that ( ˆfε, ˆgε > 0) is a minimizer of Iε in B which contradicts the uniqueness of the global +minimizer (fε, 0). +3.3 +Proof of Theorem 4 +We now prove the main result: +Proof of Theorem 4. By Theorem 9, the existence of an escaping radially symmetric so- +lution ˜uε in (9) is equivalent to ε ∈ (0, εN). Moreover, in that case, the escaping radial +profile ( ˜fε, gε > 0) is unique and bounded, i.e., ˜f 2 +ε + g2 +ε < 1 in (0, 1). +Case 1: if ε ∈ (0, εN), Theorem 7 implies that the (bounded) escaping radially symmetric +critical point ˜uε in (9) is a global minimizer of Eε over A and every minimizer of Eε over +A has the form R˜uε for some orthogonal transformation R ∈ O(M) keeping invariant +the space RN × {0RM−N }. Moreover, by Theorem 9, the non-escaping radial profile (fε, 0) +is proved to be an unstable critical point of Iε in B, so the non-escaping vortex sheet +solution (uε, 0RM−N ) is an unstable critical point of Eε in A . +Case 2: if ε ≥ εN, the proof of Theorem 3 implies that the non-escaping radially symmetric +vortex sheet solution uε(x) ≡ (fε(|x|) x +|x|, 0RM−N ) (by (13)) is the unique minimizer of Eε +over A . In this case, there is no bounded critical point wε of Eε over A that escapes in +some direction e ∈ SM−1; indeed, if such (bounded) escaping solution wε satisfying (20) +exists, then Theorem 7 would imply that wε is a global minimizer of Eε over A which +contradicts that the non-escaping vortex sheet solution uε is the unique global minimizer +of Eε over A . +Theorem 4 holds also for the “degenerate” dimension n = 0. In this case, Ω = BN and +vortex sheets are vortex points, +Eε(u) = +� +BN +�1 +2|∇u|2 + 1 +2ε2 W(1 − |u|2) +� +dx, +A := {u ∈ H1(BN; RM) : u(x) = (x, 0RM−N ) on ∂BN = SN−1} +and radially symmetric vortex critical points of Eε in A have the corresponding form in +(9): +˜uε(x) = ( ˜fε(r) x +|x|, 0RM−N−1, gε(r)) ∈ A , +x ∈ BN, r = |x|, +(28) +where the radial profiles ( ˜fε, gε) satisfy the system (10)-(12) and are described in Theo- +rem 9; the non-escaping radially symmetric vortex solution is given here by +uε(x) = (fε(|x|) x +|x|, 0RM−N ) +for all x ∈ BN, +(29) +where the radial profile fε is the unique solution to (5). We obtain the following result +which generalizes [12, Theorem 1.1] that was proved in the case N = 2 and M = 3 (without +identifying the meaning of the dichotomy parameter εN in (7)). +16 + +Theorem 10. Let 2 ≤ N ≤ 6, M ≥ N + 1, Ω = BN, W ∈ C2((−∞, 1]) satisfy (2) and +be strictly convex. Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7). Then there exists an +escaping radially symmetric vortex solution ˜uε in (28) with the radial profile ( ˜fε, gε > 0) +given in Theorem 9 if and only if 0 < ε < εN. Moreover, +1. if 0 < ε < εN, ˜uε is a global minimizer of Eε in A and all global minimizers of +Eε in A are radially symmetric given by R˜uε where R ∈ O(M) is an orthogonal +transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }. In this case, the +non-escaping vortex solution uε in (29) is an unstable critical point of Eε in A . +2. if ε ≥ εN, the non-escaping vortex solution uε in (29) is the unique global minimizer +of Eε in A . Furthermore, there are no bounded critical points wε of Eε in A that +escape in a direction e ∈ SM−1, i.e., wε · e > 0 a.e. in Ω. +The proof follows by the same argument used for Theorem 4, the main difference is +that in the ball Ω = BN, a critical point wε of Eε in A satisfies the PDE system with +Dirichlet boundary condition (instead of the mixed Dirichlet-Neumann condition in (21)): +−∆wε = 1 +ε2 wε W ′(1 − |wε|2) +in BN, +wε(x) = (x, 0RM−N ) +on ∂BN. +A +Appendix. Vortex sheet SM−1-valued harmonic maps in +cylinders +In dimensions M > N ≥ 2 and n ≥ 1, for the cylinder shape domain Ω = BN × (0, 1)n, +we consider the harmonic map problem for SM−1-valued maps u ∈ H1(Ω; SM−1) ∩ A +associated to the Dirichlet energy +E(u) = 1 +2 +� +Ω +|∇u|2 dxdz. +Any critical point u : Ω → SM−1 of this problem satisfies + + + +−∆u = u |∇u|2 +in Ω, +∂u +∂z = 0 +on BN × ∂(0, 1)n, +u(x, z) = (x, 0RM−N ) +on ∂BN × (0, 1)n. +(30) +We will focus on radially symmetric vortex sheet SM−1-valued harmonic maps having the +following form (invariant in z-direction): +u(x, z) = (f(r) x +|x|, 0RM−N−1, g(r)) ∈ A , +x ∈ BN, z ∈ (0, 1)n, r = |x|, +(31) +where the radial profile (f, g) satisfies +f 2 + g2 = 1 +in +(0, 1), +(32) +17 + +and the system of ODEs: +−f ′′ − N − 1 +r +f ′ + N − 1 +r2 +f = Γ(r)f +in +(0, 1), +(33) +−g′′ − N − 1 +r +g′ = Γ(r)g +in +(0, 1), +(34) +f(1) = 1 and g(1) = 0, +(35) +where +Γ(r) = (f ′)2 + N − 1 +r2 +f 2 + (g′)2 +is the Lagrange multiplier due to the unit length constraint in (32). As for the Ginzburg- +Landau system, we distinguish two type of radial profiles: +• the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) yielding the non-escaping (radially +symmetric) vortex sheet SM−1-valued harmonic map (also called “equator” map): +¯u(x, z) = ( x +|x|, 0RM−N ) +x ∈ BN, z ∈ (0, 1)n. +(36) +Note that ¯u is singular and the singular set of this map is the vortex sheet {0RM−N }×(0, 1)n +of dimension n in Ω. Also, observe that ¯u ∈ H1(Ω, SM−1) if and only if N ≥ 3. +• the escaping radial profile (f, g) with g > 0 in (0, 1); in this case, it holds f(0) = 0, +g(0) = 1 and we say that u in (31) is an escaping (radially symmetric) vortex sheet SM−1- +valued harmonic map. Note that u is smooth for every dimension M > N ≥ 2 and n ≥ 1 +and the zero set of (u1, . . . , uN) is the vortex sheet {0RM−N } × (0, 1)n of dimension n in +Ω. Obviously, (f, −g < 0) is another radial profile satisfying (32)-(35). +The properties of such radial profiles are proved in [14] (see also [8, Theorem 2.6] for +˜W ≡ 0 in those notations). More precisely, +(a) If N ≥ 7, the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) is the unique minimizer of +I(f, g) = +1 +|SN−1|E +� +(f(r) x +|x|, 0RM−N−1, g(r)) +� += 1 +2 +� 1 +0 +� +(f ′)2 + (g′)2 + N − 1 +r2 +f 2� +rN−1 dr, +where (f, g) belongs to B ∩ +� +(f, g) : f 2 + g2 = 1 +� +with B defined in (27). Moreover, +the system (32)–(35) has no escaping radial profile (f, g) with g > 0 in (0, 1). +(b) If 2 ≤ N ≤ 6, then there exists a unique escaping radial profile (f, g) with g > 0 +satisfying (32)–(35). Moreover, (f, ±g) are the only two global minimizers of I in +B ∩ +� +(f, g) : f 2 + g2 = 1 +� +, f +r , g ∈ C∞([0, 1]), f(0) = 0, g(0) = 1, f > 0, f ′ > 0 and +g′ < 0 in (0, 1). In addition, for 3 ≤ N ≤ 6, the non-escaping solution ( ¯f ≡ 1, ¯g ≡ 0) +is an unstable critical point of I in B ∩ +� +(f, g) : f 2 + g2 = 1 +� +.6 +6For N = 2, (1, 0) /∈ B; however, we can define the second variation of I at (1, 0) along directions (0, q) +compactly supported in (0, 1): +Q(0, q) = +� 1 +0 +� +(q′)2 − N − 1 +r2 +q2� +rN−1 dr, +and one can prove the existence of q ∈ Lipc(0, 1) such that Q(0, q) < 0 (see e.g. [8, Remark 2.16]). +18 + +There is a large number of articles studying existence, uniqueness, regularity and +stability of radially symmetric SM−1-valued harmonic maps (e.g., [13, 14, 25, 26, 23, 16, +12]). We summarize here the main result for our problem in the cylinder shape domain +Ω = BN × (0, 1)n: if N ≤ 6, then minimizing SM−1-valued harmonic maps in A are +smooth, radially symmetric and escaping in one-direction; if N ≥ 7, then there is a unique +minimizing SM−1-valued harmonic map in A which is singular and given by the equator +map ¯u in (36). 7 +Theorem 11. Let n ≥ 1, N ≥ 2, M ≥ N + 1 and Ω = BN × (0, 1)n. Then +1. if 2 ≤ N ≤ 6, then the escaping radially symmetric vortex sheet solution u in (31) +with g > 0 is a minimizing SM−1-valued harmonic map in A and all minimizing +SM−1-valued harmonic maps in A are smooth radially symmetric given by Ru where +R ∈ O(M) satisfies Rp = p for all p ∈ RN ×{0RM−N }. In this case, the equator map +¯u in (36) is an unstable SM−1-valued harmonic map in A . +2. if N ≥ 7, the non-escaping vortex sheet solution ¯u in (36) is the unique minimizing +SM−1-valued harmonic map in A . Moreover, there is no SM−1-valued harmonic +map w in A escaping in a direction e ∈ SM−1, i.e., w · e > 0 a.e. in Ω. +The main ingredient is the following result yielding minimality of escaping SM−1-valued +harmonic maps. This is reminiscent from Sandier-Shafrir [23] (see also [12, Theorem 1.5]). +Theorem 12. Let n ≥ 1, M > N ≥ 2 and Ω = BN × (0, 1)n. +Assume that w ∈ +A ∩ H1(Ω, SM−1) is a SM−1-valued harmonic map satisfying (30) and +w · e > 0 a.e. in Ω +(37) +in an escaping direction e ∈ SM−1. Then w is a minimizing SM−1-valued harmonic map +in A and all minimizing SM−1-valued harmonic maps in A are of the form Rw where R ∈ +O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }. +Proof of Theorem 12. We give here a simple proof based on the argument in [12] that +avoids the regularity results used in [23]. By the H1/2-trace theorem applied for w ∈ +H1(Ω, SM−1), (37) implies that w · e ≥ 0 on ∂BN × (0, 1)n. Combined with the vortex +boundary condition in (30), we deduce that the escaping direction e has to be orthogonal +to RN × {0RM−N } and up to a rotation, we can assume that e = eM (as in (22)). Then +φ = w · eM > 0 a.e. in Ω satisfies +− ∆φ = |∇w|2φ in Ω, ∂φ +∂z = 0 on BN × ∂(0, 1)n, φ = 0 on ∂BN × (0, 1)n. +(38) +We consider configurations8 ˜w = w + v : Ω → SM−1 with v ∈ H1 +0(BN × Rn, RM) (in +particular, |v| ≤ 2 in Ω). Then +2w · v + |v|2 = 0 +a.e. in Ω. +(39) +7We mention the paper of Bethuel-Brezis-Coleman-H´elein [2] about a similar phenomenology in a do- +main Ω = (B2 \ Bρ) × (0, 1) ⊂ R3 where Bρ ⊂ R2 is the disk centered at 0 of radius ρ. +8Note that for any ˜w ∈ A ∩ H1(Ω, SM−1), the map ˜w − w has an extension in H1 +0(BN × Rn, RM). +19 + +Using (30) and (39), we obtain +2 +� +Ω +∇w · ∇v = 2 +� +Ω +|∇w|2w · v dx = − +� +Ω +|∇w|2|v|2 dx, +yielding9 +� +Ω +|∇(w + v)|2 dx − +� +Ω +|∇w|2 dx = +� +Ω +|∇v|2 − |∇w|2|v|2 dx =: Q(v). +(40) +To show that w is minimizing, we prove that Q(v) ≥ 0 for all v ∈ H1 +0(BN × Rn, RM) ∩ +L∞(Ω; RM) (note that this is a class larger than what we need, as we do not require +that v satisfy the pointwise constraint (39)). For that, we take an arbitrary map ˜v ∈ +C∞ +c (BN × Rn, RM) of support ω and decompose it as ˜v = φΨ in Ω. This decomposition +makes sense as φ ≥ δ > 0 in ω ∩ Ω for some δ > 0 (which may depend on ω). Indeed, by +(37) and (38), φ is a superharmonic function (i.e., −∆φ ≥ 0 in Ω) that belongs to H1(Ω). +As ∂φ +∂z = 0 on BN × ∂(0, 1)n, φ can be extended by even mirror symmetry to the domain +˜Ω = BN × (−1, 2)n so that φ is superharmonic in ˜Ω. Thus, the weak Harnack inequality +(see e.g. [6, Theorem 8.18]) implies that on the compact set ω ∩Ω in ˜Ω, we have φ ≥ δ > 0 +for some δ. So, ˜v = φΨ in Ω with Ψ = (Ψ1, . . . , ΨM) ∈ H1 ∩ L∞(Ω; RM) vanishing in a +neighborhood of ∂BN × (0, 1)n. Then integration by parts yields for 1 ≤ j ≤ M: +Q(˜vj) = +� +Ω +|∇˜vj|2 − |∇w|2φ · φΨ2 +j dx +(38) += +� +Ω +|∇(φΨj)|2 − ∇φ · ∇(φ Ψ2 +j) dx = +� +Ω +φ2|∇Ψj|2 dx ≥ 0 +for all ˜v ∈ C∞ +c (BN × Rn, RM). Then for every v ∈ H1 +0(BN × Rn, RM) ∩ L∞(Ω; RM), there +exists a sequence ˜vk ∈ C∞ +c (BN × Rn, RM) such that ˜vk → v and ∇˜vk → ∇v in L2 and +a.e. in BN × Rn and |˜vk| ≤ ∥v∥L∞(Ω) + 1 in Ω for every k. In particular, by dominated +convergence theorem, we have Q(˜vk) → Q(v) thanks to (40). Thus, we deduce that for +every compact ω ⊂ ˜Ω = BN × (−1, 2)n, +Q(v) = lim +k→∞ Q(˜vk) ≥ lim inf +k→∞ +� +ω∩Ω +φ2|∇ +�˜vk +φ +� +|2 dx ≥ +� +ω∩Ω +φ2|∇ +� v +φ +� +|2 dx ≥ 0, +where we used Fatou’s lemma. In particular, w is a minimizing SM−1-valued harmonic +map by (40) and Q(v) = 0 yields the existence of a vector λ ∈ RM such that v = λφ a.e. +in Ω. Then the classification of the minimizing SM−1-valued harmonic maps follows by +(39) as in the Step 3 of the proof of Theorem 7. +Proof of Theorem 11. 1. This part concerning the dimension 2 ≤ N ≤ 6 follows from +Theorem 12 and the instability of the radial profile (1, 0) for I in B∩ +� +(f, g) : f 2+g2 = 1 +� +as explained above. +9Note that the functional Q represents the second variation of E at w, but here the map v is not +necessarily orthogonal to w. +20 + +2. This part for dimension N ≥ 7 follows the ideas in [14]. More precisely, calling +X = (x, z) the variable in Ω, we have as in the proof of Theorem 12 for every v ∈ +H1 +0(BN × Rn, RM) with |v + ¯u| = 1 in Ω: +� +Ω +|∇(¯u + v)|2 dX− +� +Ω +|∇¯u|2 dX = +� +Ω +� +|∇v|2 − |∇¯u|2|v|2� +dX += +� +Ω +|∇zv|2 dX + +� +(0,1)n dz +� +BN +� +|∇xv|2 − N − 1 +|x|2 |v|2� +dx +≥ +� +Ω +|∇zv|2 dX + +�(N − 2)2 +4 +− (N − 1) +� � +Ω +|v|2 +|x|2 dX ≥ 0 +where we used the Hardy inequality for v(·, z) ∈ H1 +0(BN, RM) for a.e. z ∈ (0, 1)n. This +proves that ¯u is the unique minimizing SM−1-valued harmonic map in A . +Combined +with Theorem 12, we conclude that there is no escaping SM−1-valued harmonic map w in +A . +References +[1] F. Bethuel, H. Brezis and F. H´elein, Ginzburg-Landau vortices, Progress in Nonlinear +Differential Equations and their Applications, 13. Birkh¨auser Boston Inc., Boston, +MA, 1994. +[2] F. Bethuel, H. Brezis, B.D. Coleman and F. 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Differential +Geometry 17 (1982), 307-335. +23 + diff --git a/E9FJT4oBgHgl3EQfCizA/content/tmp_files/load_file.txt b/E9FJT4oBgHgl3EQfCizA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..abde0440c21e42a008c5b58d001f0ba1a32cb234 --- /dev/null +++ b/E9FJT4oBgHgl3EQfCizA/content/tmp_files/load_file.txt @@ -0,0 +1,874 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf,len=873 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='11430v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='AP] 26 Jan 2023 Vortex sheet solutions for the Ginzburg-Landau system in cylinders: symmetry and global minimality Radu Ignat∗ Mircea Rus† January 30, 2023 Abstract We consider the Ginzburg-Landau energy Eε for RM-valued maps defined in a cylinder shape domain BN ×(0, 1)n satisfying a degree-one vortex boundary condition on ∂BN × (0, 1)n in dimensions M ≥ N ≥ 2 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The aim is to study the radial symmetry of global minimizers of this variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We prove the following: if N ≥ 7, then for every ε > 0, there exists a unique global minimizer which is given by the non-escaping radially symmetric vortex sheet solution uε(x, z) = (fε(|x|) x |x|, 0RM−N), ∀x ∈ BN that is invariant in z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If 2 ≤ N ≤ 6 and M ≥ N + 1, the following dichotomy occurs between escaping and non-escaping solutions: there exists εN > 0 such that if ε ∈ (0, εN), then every global minimizer is an escaping radially symmetric vortex sheet solution of the form R˜uε where ˜uε(x, z) = ( ˜fε(|x|) x |x|, 0RM−N−1, gε(|x|)) is invariant in z-direction with gε > 0 in (0, 1) and R ∈ O(M) is an orthogonal transformation keeping invariant the space RN × {0RM−N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if ε ≥ εN, then the non-escaping radially symmetric vortex sheet solution uε(x, z) = (fε(|x|) x |x|, 0RM−N), ∀x ∈ BN, z ∈ (0, 1)n is the unique global minimizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' moreover, there are no bounded escaping solutions in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We also discuss the problem of vortex sheet SM−1-valued harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Keywords: vortex, uniqueness, symmetry, minimizers, Ginzburg-Landau equation, harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' MSC: 35A02, 35B06, 35J50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Contents 1 Introduction and main results 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1 Minimality of the RN-valued vortex sheet solution .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2 Escaping RM-valued vortex sheet solutions when M ≥ N + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 5 ∗Institut de Math´ematiques de Toulouse & Institut Universitaire de France, UMR 5219, Universit´e de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Email: Radu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='Ignat@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='fr †Department of Mathematics, Technical University of Cluj-Napoca, 400027 Cluj-Napoca, Romania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Email: rus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='mircea@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='utcluj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='ro 1 2 The non-escaping vortex sheet solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theorems 1 and 3 7 3 Properties of escaping vortex sheet solutions when M ≥ N + 1 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1 Minimality of escaping vortex sheet solutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2 Escaping radial profile .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3 Proof of Theorem 4 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 16 A Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Vortex sheet SM−1-valued harmonic maps in cylinders 17 1 Introduction and main results In this paper, we consider the following Ginzburg-Landau type energy functional Eε(u) = � Ω �1 2|∇u|2 + 1 2ε2 W(1 − |u|2) � dX, (1) where ε > 0, X = (x, z) ∈ Ω = BN × (0, 1)n is a cylinder shape domain with BN the unit ball in RN, n ≥ 1, N ≥ 2 and the potential W ∈ C2((−∞, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' R) satisfies W(0) = 0, W(t) > 0 for all t ∈ (−∞, 1] \\ {0} and W is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (2) (The prototype potential is W(t) = t2 2 for t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=') We investigate the global minimizers of the energy Eε in the set of RN-valued maps: AN := {u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RN) : u(x, z) = x for every x ∈ ∂BN = SN−1, z ∈ (0, 1)n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The boundary assumption u(x, z) = x for every x ∈ SN−1 and every z ∈ (0, 1)n is referred in the literature as the degree-one vortex boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The direct method in the calculus of variations yields the existence of a global minimizer uε of Eε over AN for all range of ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, any minimizer uε satisfies |uε| ≤ 1 in Ω, uε belongs to C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RN) and solves the system of PDEs (in the sense of distributions) with mixed Dirichlet-Neumann boundary conditions: \uf8f1 \uf8f2 \uf8f3 −∆uε = 1 ε2uε W ′(1 − |uε|2) in Ω, ∂uε ∂z = 0 on BN × ∂(0, 1)n, u(x, z) = x on ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1 Minimality of the RN-valued vortex sheet solution The first goal of this paper is to prove the uniqueness and radial symmetry of the global minimizer of Eε in AN for all ε > 0 in dimensions N ≥ 7 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In fact, in these dimensions, we show that the global minimizer of Eε in AN is unique and given by the following radially symmetric critical point of Eε that is invariant in z: 1 uε(x, z) = fε(|x|) x |x| for all x ∈ BN and z ∈ (0, 1)n, (4) 1If n = 0 and N ≥ 2, then SO(N) induces a group action on AN given by u(x) �→ R−1u(Rx) for every x ∈ BN, R ∈ SO(N) and u ∈ AN under which the energy Eε and the vortex boundary condition are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then every bounded critical point of Eε in AN that is invariant under this SO(N) group action has the form (4), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [8, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2 where the radial profile fε : [0, 1] → R in r = |x| is the unique solution to the ODE: � −f ′′ ε − N−1 r f ′ ε + N−1 r2 fε = 1 ε2fε W ′(1 − f 2 ε ) for r ∈ (0, 1), fε(0) = 0, fε(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (5) We recall that the unique radial profile fε satisfies fε > 0 and f ′ ε > 0 in (0, 1) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [7, 9, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that the zero set of uε is given by the n-dimensional vortex sheet {0RN } × (0, 1)n in Ω (in particular, if n = 0, it is a vortex point, while for n = 1, it is a vortex filament);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' therefore, uε in (4) is called (radially symmetric) vortex sheet solution to the Ginzburg-Landau system (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume that W satisfies (2) and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If N ≥ 7, then uε given in (4) is the unique global minimizer of Eε in AN for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The proof is reminiscent of the works of Ignat-Nguyen-Slastikov-Zarnescu [12, 11] studying uniqueness and symmetry of minimizers of the Ginzburg-Landau functionals for RM-valued maps defined on smooth N-dimensional domains, where M is not necessarily equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The idea is to analyze Eε(u) for an arbitrary map u and to exploit the convex- ity of W to lower estimate the excess energy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Eε(uε) by a suitable quadratic energy functional depending on u − uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This quadratic functional comes from the linearized PDE at uε and can be handled by a factorization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The positivity of the excess energy then follows by a Hardy-type inequality holding true only in high dimensions N ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This is similar to the result of J¨ager and Kaul [14] on the minimality of the equator map for the harmonic map problem in dimension N ≥ 7 that is proved using a certain inequality involving the sharp constant in the Hardy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We expect that our result remains valid in dimensions 2 ≤ N ≤ 6: Open problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Is it true that for every ε > 0, uε given in (4) is the unique global minimizer of Eε in AN?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It is well known that the uniqueness of uε holds true for large enough ε > 0 in any dimension N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Indeed, denoting by λ1 the first eigenvalue of −∆x in BN with zero Dirichlet boundary condition, then for any ε > � W ′(1)/λ1, Eε is strictly convex in AN (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', [1, Theorem VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='7], [12, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3]) and thus has a unique critical point in AN that is the global minimizer of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We improve this result as follows: for the radial profile fε in (5), we denote by ℓ(ε) the first eigenvalue of the operator Lε = −∆x − 1 ε2 W ′(1 − f 2 ε ) (6) acting on maps defined in BN with zero Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It is proved in [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3] that if 2 ≤ N ≤ 6 and W ∈ C2((−∞, 1]) satisfies (2), then the first eigenvalue ℓ(ε) is a continuous function in ε and there exists εN ∈ (0, ∞) such that ℓ(ε) < 0 in (0, εN), ℓ(εN) = 0 and ℓ(ε) > 0 in (εN, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (7) 3 Note that2 0 = ℓ(εN) > λ1 − 1 ε2 N W ′(1) yielding εN < � W ′(1)/λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume that W satisfies (2), n ≥ 1 and 2 ≤ N ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If ε ≥ εN, then uε given in (4) is a global minimizer of Eε in AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, if either ε > εN, or (ε = εN and W is in addition strictly convex), then uε is the unique global minimizer of Eε in AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The case ε < εN is still not solved as stated in Open Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let us summarize some known results: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The case of n = 0 and Ω = BN (we also discuss here the problem for Ω = RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, the above question was raised in dimension N = 2 for the disk Ω = B2 in the seminal book of Bethuel, Brezis and H´elein [1, Problem 10, page 139], and in general dimensions N ≥ 2 and also for the blow-up limiting problem around the vortex point (when the domain Ω is the whole space RN and by rescaling, ε can be assumed equal to 1) in an article of Brezis [3, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For sufficiently small ε > 0 and for the disk domain Ω = B2, Pacard and Rivi`ere [20, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2] showed that Eε has a unique critical point in A2 and so, it is given by the radially symmetric solution uε in (4) (for n = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For N ≥ 7, Ω = BN and any ε > 0, it is proved in [11] that Eε has a unique minimizer in AN which is given by the radially symmetric solution uε in (4) (for n = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For 2 ≤ N ≤ 6 and Ω = BN, Ignat-Nguyen [8] proved that for any ε > 0, uε is a local minimizer of Eε in A (which is an extension of the result of Mironescu [18] in dimension N = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Also, Mironescu [19] showed in dimension N = 2 that, when B2 is replaced by R2 and ε = 1, a local minimizer of Eε satisfying a degree-one boundary condition at infinity is unique (up to translation and suitable rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This was extended in dimension N = 3 by Millot and Pisante [17] and in dimensions N ≥ 4 by Pisante [21] in the case of the blow-up limiting problem on RN and ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' All these results (holding for n = 0) are related to the study of the limit problem obtained by sending ε → 0 when the Ginzburg-Landau problem on the unit ball ‘converges’ to the harmonic map problem from BN into the unit sphere SN−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For that harmonic map problem, the vortex boundary condition yields uniqueness of the minimizing harmonic SN−1-valued map x �→ x |x| if N ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' this is proved by Brezis, Coron and Lieb [4] in dimension N = 3 and by Lin [15] in any dimension N ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' we also mention J¨ager and Kaul [14] in dimension N ≥ 7 for the equator map x ∈ BN �→ ( x |x|, 0) ∈ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The case of n ≥ 1 and Ω = BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As we explain in Remark 6 below, for some ε > 0, if the minimality of the radially symmetric solution uε in (4) holds in the case n = 0 (so, for Ω = BN), then this implies the minimality of uε in Ω = BN ×(0, 1)n also for every dimension n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, the result of Pacard-Rivi`ere [20, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2] for n = 0 and N = 2 yields the minimality of uε in (4) defined in B2×(0, 1)n for every n ≥ 1 if ε > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Also, the result of Ignat-Nguyen-Slastikov-Zarnescu [11, Theorem 1] 2Indeed, if v ∈ H1 0(BN) is a first eigenfunction of LεN in BN such that ∥v∥L2(BN ) = 1 then λ1 ≤ � BN |∇xv|2 dx = 1 ε2 N � BN W ′(1 − f 2 εN )v2 dx < W ′(1) ε2 N because ℓ(εN) = 0, 0 < fεN < 1 in (0, 1) and (2) implies W ′(0) = 0 and W ′(t) > 0 for t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 4 for n = 0, N ≥ 7 and any ε > 0 generalizes to dimension n ≥ 1 for Ω = BN × (0, 1)n (see the proof of Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We also mention the work of Sandier-Shafrir [24] where they treat the case of topologically trivial R2-valued solutions in the domain Ω = R3 (see also [5, 22] for vortex filament solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2 Escaping RM-valued vortex sheet solutions when M ≥ N + 1 In dimension 2 ≤ N ≤ 6 and for ε < εN given in (7), a different type of radially symmetric vortex sheet solution appears provided that the target space has dimension M ≥ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' More precisely, we consider the energy functional Eε in (1) over the set of RM-valued maps A := {u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) : u(x, z) = (x, 0RM−N ) on ∂BN = SN−1 ⊂ RM, z ∈ (0, 1)n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (8) If M ≥ N + 1, the prototype of radially symmetric critical points of Eε in A has the following form (invariant in z-direction): 3 ˜uε(x, z) = ( ˜fε(r) x |x|, 0RM−N−1, gε(r)) ∈ A , x ∈ BN, z ∈ (0, 1)n, r = |x|, (9) where ( ˜fε, gε) satisfies the system of ODEs − ˜f ′′ ε − N − 1 r ˜f ′ ε + N − 1 r2 ˜fε = 1 ε2 W ′(1 − ˜f 2 ε − g2 ε) ˜fε in (0, 1), (10) −g′′ ε − N − 1 r g′ ε = 1 ε2 W ′(1 − ˜f 2 ε − g2 ε)gε in (0, 1), (11) ˜fε(1) = 1 and gε(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (12) We distinguish two type of radial profiles: the non-escaping radial profile ( ˜fε = fε, gε = 0) with the unique radial profile fε given in (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in this case, we say that ˜uε = (uε, 0RM−N ) is a non-escaping (radially symmetric) vortex sheet solution where uε is given in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' the escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in this case, we call an escaping (radially symmetric) vortex sheet solution ˜uε in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, ˜fε ̸= fε and obviously, ( ˜fε, −gε) is another radial profile to (9)-(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The properties of such radial profiles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', existence, uniqueness, minimality, mono- tonicity) are analyzed in Theorem 9 below and are based on ideas developed by Ignat- Nguyen [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Our main result proves the radial symmetry of global minimizers of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' More precisely, the following dichotomy occurs at εN defined in (7): if ε < εN, then escaping radially symmetric vortex sheet solutions exist and determine (up to certain orthogonal transformations) the full set of global minimizers of Eε in A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if instead ε ≥ εN, then the non-escaping radially symmetric vortex sheet solution is the unique global minimizer of Eε in A and no escaping radially symmetric vortex sheet solutions exist in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 3If M = N + 1, then ˜uε(x, z) = ( ˜fε(r) x |x|, gε(r)) for every x ∈ BN and z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In fact, if n = 0 (so, for Ω = BN), every bounded critical point of Eε in A that is invariant under the action of a special group (isomorphic to SO(N)) has the form of ˜uε, see [8, Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 5 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let n ≥ 1, 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then there exists an escaping radially symmetric vortex sheet solution ˜uε in (9) with gε > 0 in (0, 1) if and only if 0 < ε < εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if 0 < ε < εN, the escaping radially symmetric vortex sheet solution ˜uε is a global minimizer of Eε in A and all global minimizers of Eε in A are radially symmetric given by R˜uε where R ∈ O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN × {0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, the non-escaping vortex sheet solution (uε, 0RM−N ) in (4) is an unstable critical point of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if ε ≥ εN, the non-escaping vortex sheet solution (uε, 0RM−N ) in (4) is the unique global minimizer of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Furthermore, there are no bounded critical points wε of Eε in A that escape in some direction e ∈ SM−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', wε · e > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The result above holds also if n = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', Ω = BN and the vortex sheets corresponding to the above solutions become vortex points (see Theorem 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It generalizes [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1] that was proved in the case N = 2 and M = 3 (without identifying the meaning of the dichotomy parameter εN in (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The dichotomy in Theorem 4 happens in dimensions 2 ≤ N ≤ 6 because of the phenomenology occurring for the limit problem ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' More precisely, if M ≥ N + 1, then minimizing SM−1-valued harmonic maps in A are smooth and escaping in a direction of SM−1 provided that N ≤ 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if N ≥ 7, then there is a unique minimizing SM−1-valued harmonic maps in A , non-escaping and singular, the singular set being given by a vortex sheet of dimension n in Ω (see Theorem 11 in Appendix below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This suggests why in dimension N ≥ 7 and for any ε > 0, there is no escaping radially symmetric vortex sheet critical point ˜uε of Eε in A while the non-escaping vortex sheet solution (uε, 0RM−N ) is the unique global minimizer of Eε in A (see Theorem 5 and Remark 8 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The paper is meant to be self-contained and it is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In Section 2, we prove the minimality and the uniqueness results for the non-escaping radially symmetric solution in Theorems 1 and 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' this is done in a more general setting by considering the target dimension M ≥ N for the set of configurations A instead of AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Section 3 is devoted to characterize escaping vortex sheet solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' First, we prove the minimality of such bounded solutions stated in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Second, we prove existence, minimality and uniqueness results for the escaping radial profile in Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Finally, we prove our main result on the dichotomy between escaping / non-escaping radially symmetric vortex sheet solutions in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In Appendix, we prove the corresponding dichotomy result for SM−1-valued harmonic maps in Theorem 11 which again is based on the minimality of escaping SM−1-valued harmonic maps in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' is partially supported by the ANR projects ANR-21-CE40-0004 and ANR-22-CE40-0006-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' He also thanks for the hospitality of the Hausdorff Research Institute for Mathematics in Bonn during the trimester “Mathematics for Complex Ma- terials”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 6 2 The non-escaping vortex sheet solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theo- rems 1 and 3 Theorem 1 will be obtained as a consequence of a stronger result on the uniqueness of global minimizers of the RM-valued Ginzburg-Landau functional with M ≥ N ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For that, we consider the energy functional Eε in (1) over the set A defined in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The aim is to prove the minimality and uniqueness of the vortex sheet solution (uε, 0RM−N ) where uε given in (4) with the obvious identification uε ≡ (uε, 0RM−N ) if M = N, following the ideas of Ignat-Nguyen-Slastikov-Zarnescu [12, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume that W satisfies (2) and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If M ≥ N ≥ 7, then for every ε > 0, (uε, 0RM−N ) given in (4) is the unique global minimizer of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' To simplify notation, we identify uε ≡ (uε, 0RM−N ) when M ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (13) The proof will be done in several steps following the strategy in [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='7], [11, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' First, for an arbitrary competitor uε + v, we consider the excess energy Eε(uε + v) − Eε(uε) for the critical point uε defined in (4) and show a lower estimate by a quadratic energy functional Fε(v) coming from the operator Lε in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Second, we show that Fε(v) ≥ 0 using the properties of the radial profile fε in (5) and a Hardy decomposition method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' this proves in particular that uε is a global minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Finally, by analyzing the zero excess energy states, we conclude to the uniqueness of the global minimizer uε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 1: Excess energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For any v ∈ H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM), we have Eε(uε + v) − Eε(uε) = � Ω � ∇uε · ∇v + 1 2|∇v|2� dxdz + 1 2ε2 � Ω � W(1 − |uε + v|2) − W(1 − |uε|2) � dxdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that for every u ∈ A , uε−u can be extended to v ∈ H1 0(BN ×Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, v(·, z) ∈ H1 0(BN, RM) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The convexity of W yields W(1 − |uε + v|2) − W(1 − |uε|2) ≥ −W ′(1 − |uε|2)(|uε + v|2 − |uε|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (14) Combining the above relations, we obtain the following lower bound for the excess energy: Eε(uε + v) − Eε(uε) ≥ � Ω � ∇uε · ∇v − 1 ε2 W ′(1 − f 2 ε )uε · v � dxdz + � Ω �1 2|∇v|2 − 1 2ε2 W ′(1 − f 2 ε )|v|2� dxdz = � Ω 1 2|∇zv|2 dxdz + � (0,1)n 1 2Fε(v(·, z)) dz, (15) 7 where we used the PDE (3) and introduced the quadratic functional Fε(Ψ) = � BN � |∇xΨ|2 − 1 ε2 W ′(1 − f 2 ε )|Ψ|2� dx, for all Ψ ∈ H1 0(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that the L2-gradient of Fε represents a part of the lin- earization of the PDE (3) at uε and it is given by the operator Lε in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The rest of the proof is devoted to show that for N ≥ 3: Fε(ψ) ≥ �(N − 2)2 4 − (N − 1) � � BN ψ2 r2 dx, ∀ψ ∈ H1 0(BN) yielding the conclusion for N ≥ 7 and also the inequality for the first eigenvalue ℓ(ε) of the operator Lε in (6) in BN: 4 ℓ(ε) ≥ (N − 2)2 4 − (N − 1) > 0, ∀ε > 0 and N ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' To keep the paper self-contained, we explain in the following the simple idea used in [12, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 2: A factorization argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As fε > 0 is a smooth positive radial profile in (0, 1), we decompose every scalar test function ψ ∈ C∞ c (BN \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' R) as follows ψ(x) = fε(r)w(x), ∀x ∈ BN \\ {0}, r = |x|, where w ∈ C∞ c (BN \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Integrating by parts (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [10, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1]), we deduce: Fε(ψ) = � BN Lεψ · ψ dx = � BN w2(Lεfε · fε) dx + � BN f 2 ε |∇xw|2 dx = � BN f 2 ε � |∇xw|2 − N − 1 r2 w2 � dx, because Lεfε · fε = − N−1 r2 f 2 ε in BN by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Furthermore, we decompose w = ϕg in BN \\ {0} with ϕ = |x|− N−2 2 satisfying −∆xϕ = (N − 2)2 4|x|2 ϕ in RN \\ {0} and g ∈ C∞ c (BN \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then |∇xw|2 = |∇xg|2ϕ2 + |∇xϕ|2g2 + 1 2∇x(ϕ2) · ∇x(g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 4Observe the difference between dimension N ≥ 7 and the case of dimension 2 ≤ N ≤ 6 where we have ℓ(ε) < 0 for ε < εN in (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' moreover, if N ≤ 6, then ℓ(ε) blows up as − 1 ε2 as ε → 0 (see [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 8 As |∇xϕ|2 = (N−2)2 4|x|2 ϕ2 and ϕ2 is harmonic in BN \\ {0} (recall that N ≥ 7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' integration by parts yields Fε(ψ) = � BN f 2 ε � |∇xg|2ϕ2 + (N − 2)2 4r2 ϕ2g2 − N − 1 r2 ϕ2g2 � dx − 1 2 � BN ∇x(ϕ2) · ∇x(f 2 ε )g2 dx ≥ � BN f 2 ε |∇xg|2ϕ2 dx + �(N − 2)2 4 − (N − 1) � � BN f 2 ε r2 ϕ2g2 dx ≥ �(N − 2)2 4 − (N − 1) � � BN ψ2 r2 dx ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (16) where we used N ≥ 7 and 1 2∇x(ϕ2)·∇x(f 2 ε ) = 2ϕϕ′fεf ′ ε ≤ 0 in BN \\{0} because ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' fε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' f ′ ε > 0 and ϕ′ < 0 in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1) (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [7, 9, 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 3: We prove that Fε(Ψ) ≥ 0 for every Ψ ∈ H1 0(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' moreover, Fε(Ψ) = 0 if and only if Ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let Ψ ∈ H1 0(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As a point in RN has zero H1 capacity, a standard density argument implies the existence of a sequence Ψk ∈ C∞ c (BN \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) such that Ψk → Ψ in H1(BN, RM) and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' On the one hand, by definition of Fε, since W ′(1 − f 2 ε ) ∈ L∞, we deduce that Fε(Ψk) → Fε(Ψ) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' On the other hand, by (16) and Fatou’s lemma, we deduce lim inf k→∞ Fε(Ψk) ≥ �(N − 2)2 4 − (N − 1) � lim inf k→∞ � BN |Ψk|2 r2 dx ≥ �(N − 2)2 4 − (N − 1) � � BN |Ψ|2 r2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Therefore, we conclude that Fε(Ψ) ≥ �(N − 2)2 4 − (N − 1) � � BN |Ψ|2 r2 dx ≥ 0, ∀Ψ ∈ H1 0(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, Fε(Ψ) = 0 if and only if Ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 4: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By (15) and Step 3, we deduce that uε is a global minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For uniqueness, assume that ˆuε is another global minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If v := ˆuε − uε, then v can be extended in H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) and by Steps 1 and 3, we have that 0 = Eε(ˆuε) − Eε(uε) ≥ � Ω 1 2|∇zv|2 dxdz + � (0,1)n 1 2Fε(v(·, z)) dz ≥ 0, which yields ∇zv = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω and Fε(v(·, z)) = 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In other words, v = v(x) and Step 3 implies that v = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', ˆuε = uε in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 5 reveals the following fact: if for n = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', Ω = BN) and some ε > 0, a (radially symmetric) critical point ˆuε : BN → RM of Eε in A is proved to be a global minimizer (and additionally, if one proves that it is the unique global minimizer), then for any dimensions n ≥ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', Ω = BN ×(0, 1)n), this z-invariant solution ˆuε of (3) 9 in BN × (0, 1)n is also a global minimizer (and additionally, it is the unique minimizer) of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This is because for every u : BN × (0, 1)n → RM with u ∈ A , then u(·, z) satisfies the degree-one vortex boundary condition on ∂BN for every z ∈ (0, 1)n yielding Eε(u) = � Ω 1 2|∇zu|2 dxdz + � (0,1)n Eε(u(·, z)) dz ≥ � (0,1)n Eε(ˆuε) dz = Eε(ˆuε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' the equality occurs only when u is z-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, if the uniqueness of the global mini- mizer ˆuε holds in BN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', n = 0), then this yields uniqueness of the global minimizer ˆuε in Ω = BN × (0, 1)n (as a map independent of z-variable) for every n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We prove the result in the more general setting of RM-valued maps u belonging to A for M ≥ N using the same identification (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By Step 1 in the proof of Theorem 5 (see (15)), the excess energy is estimated for every v ∈ H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM): Eε(uε + v) − Eε(uε) ≥ � Ω 1 2|∇zv|2 dxdz + 1 2 � (0,1)n < Lεv(·, z), v(·, z) > dz, where Lε is the operator in (6) and < ·, · > denotes the duality pairing (H−1, H1 0) in BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If ε ≥ εN, then ℓ(ε) ≥ 0 (by [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3]) and therefore, 5 < Lεv(·, z), v(·, z) > ≥ ℓ(ε)∥v(·, z)∥2 L2(BN ) ≥ 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0, 1)n, (17) where we used that v(·, z) ∈ H1 0(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, uε is a minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It remains to prove uniqueness of the global minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For that, if ˆuε is another global minimizer of Eε over A , setting v := ˆuε − uε, then v can be extended in H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) and 0 = Eε(ˆuε) − Eε(uε) ≥ � Ω 1 2|∇zv|2 dxdz + ℓ(ε) 2 � (0,1)n � BN |v(x, z)|2 dxdz ≥ 0 (18) because ℓ(ε) ≥ 0 for ε ≥ εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, equality holds in the above inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Case 1: ε > εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, ℓ(ε) > 0 and we conclude that v = 0 in Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', ˆuε = uε in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 5 Indeed, for a scalar function v ∈ C∞ c (BN \\ {0}, R), if ψ = ψ(r) > 0 is a radial first eigenfunction of Lε in BN with zero Dirichlet data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', Lεψ = ℓ(ε)ψ in BN, then the duality pairing (H−1, H1 0) term in BN writes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [10, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1]): < Lεv, v > = � BN ψ2|∇( v ψ )|2 dx + � BN ( v ψ )2Lεψ · ψ dx = � BN ψ2|∇( v ψ )|2 dx + ℓ(ε)∥v∥2 L2(BN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By a density argument, Fatou’s lemma yields for every scalar function v ∈ H1 0(BN, R), < Lεv, v > ≥ � BN ψ2|∇( v ψ )|2 dx + ℓ(ε)∥v∥2 L2(BN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 10 Case 2: ε = εN and W is in addition strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, ℓ(ε) = 0 and by (18), v is invariant in z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', v = v(x) and equality holds in (17) and in (15), thus, equality holds in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that by footnote 5 the equality in (17) holds if and only if v = λψ for some λ ∈ RM, where ψ = ψ(r) is a radial first eigenfunction of Lε in BN with zero Dirichlet data, in particular ψ > 0 in [0, 1) and ψ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Also, by the strict convexity of W, the equality (14) is achieved if and only if |uε + v| = |uε| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω, that is, |v|2 + 2v · uε = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It yields |λ|2ψ2 + 2fε(|x|)( x |x|, 0RM−N ) · λψ = 0 for every x ∈ BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (19) Dividing by ψ in BN, the continuity up to the boundary ∂BN leads to 2fε(|x|)(x, 0RM−N )· λ = 0 for every x ∈ ∂BN since ψ = 0 on ∂BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As fε(1) = 1, it follows that the first N components of λ vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Coming back to (19), we conclude that |λ|2ψ2 = 0 in BN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', λ = 0 and so, v = 0 and ˆuε = uε in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 3 Properties of escaping vortex sheet solutions when M ≥ N + 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1 Minimality of escaping vortex sheet solutions In this section, we require the additional assumption of strict convexity of W in order to determine the set of global minimizers of Eε over A in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' However, W is assumed to be only C1 not C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We prove that every bounded solution to (3) escaping in some direction is a global minimizer of Eε over A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' moreover, such global minimizer is unique up to an orthogonal transformation of RM keeping invariant the space RN × {0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We consider the dimensions n ≥ 1 and M > N ≥ 2, the potential W ∈ C1((−∞, 1], R) satisfying (2) and an escaping direction e ∈ SM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Fix any ε > 0 and let wε ∈ H1 ∩ L∞(Ω, RM) be a critical point of the energy Eε in the set A which is positive in the direction e inside Ω: wε · e > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (20) Then wε is a global minimizer of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If in addition W is strictly convex, then all minimizers of Eε in A are given by Rwε where R ∈ O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN × {0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This result is reminiscent from [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' However, it doesn’t apply directly as the domain Ω is not smooth here and the boundary condition is a mixed Dirichlet- Neumann condition (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Dirichlet boundary condition in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In the following, we denote the variable X = (x, z) ∈ Ω = BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As a critical point of Eε in the set A , wε : Ω → RM satisfies \uf8f1 \uf8f2 \uf8f3 −∆wε = 1 ε2wε W ′(1 − |wε|2) in Ω, ∂wε ∂z = 0 on BN × ∂(0, 1)n, wε(x, z) = (x, 0RM−N ) on ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (21) 11 In particular, ∆wε ∈ L∞(Ω) (as W ′ is continuous and wε ∈ L∞(Ω));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' then standard elliptic regularity for the mixed boundary conditions in (21) yields wε ∈ C1(¯Ω, RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, (20) implies wε·e ≥ 0 in ¯Ω and the vortex boundary condition in A implies that e is orthogonal to RN × {0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By the invariance of the energy and the vortex boundary condition under the transformation wε(X) �→ Rwε(X) for any R ∈ O(M) satisfying Rp = p for all p ∈ RN × {0RM−N }, we know that Rwε is also a critical point of Eε over A ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' thus, we can assume that e := eM = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , 0, 1) ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (22) We prove the result in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 1: Excess energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By Step 1 in the proof of Theorem 5, we have for any v ∈ H1 0(BN × Rn, RM): Eε(wε + v) − Eε(wε) ≥ � Ω �1 2|∇v|2 − 1 2ε2 W ′(1 − |wε|2)|v|2� dX =: 1 2Gε(v) (23) (note that Gε(v) is larger than the integration of Fε(v) in (15) over (0, 1)n as it contains also the integration of |∇zv|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' If in addition W is strictly convex, then equality holds above if and only if |wε(X) + v(X)| = |wε(X)| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' X ∈ Ω (by (14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 2: Global minimality of wε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It is enough to show that the quadratic energy Gε(v) defined in (23) is nonnegative for any v ∈ H1 0(BN ×Rn, RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Denoting the M-component of wε by φ := wε · eM, we know that φ ∈ C1(¯Ω), φ ≥ 0 in Ω (by (20)) and satisfies the Euler-Lagrange equation in the sense of distributions: \uf8f1 \uf8f2 \uf8f3 −∆φ − 1 ε2W ′(1 − |wε|2)φ = 0 in Ω, φ = 0 on ∂BN × (0, 1)n, ∂φ ∂z = 0 on BN × ∂(0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (24) Note that by strong maximum principle, φ > 0 in Ω (as φ cannot be identically 0 in Ω by (20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, Hopf’s lemma yields φ > 0 on BN × ∂(0, 1)n as ∂φ ∂z vanishes there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Now, for any smooth map v ∈ C∞ c (BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM), we can define Ψ = v φ ∈ C1(¯Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) with Ψ = 0 in a neighborhood of ∂BN × (0, 1)n and integration by parts yields for every component vj = φΨj with 1 ≤ j ≤ M (as in [10, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' ]): Gε(vj) = � Ω � |∇vj|2 − 1 ε2 W ′(1 − |wε|2)φ · φΨ2 j � dX (24) = � Ω � |∇(φΨj)|2 − ∇φ · ∇(φ Ψ2 j) � dX = � Ω φ2|∇Ψj|2 dX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As Gε is continuous in strong H1(Ω) topology (since W ′(1 − |wε|2) ∈ L∞(Ω)), by density of C∞ c (BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) in H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM), Fatou’s lemma yields Gε(v) ≥ � Ω φ2|∇ �v φ � |2 dX ≥ 0, ∀v ∈ H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 12 As a consequence of (23), we deduce that wε is a minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, Gε(v) = 0 if and only if there exists a (constant) vector λ ∈ RM such that v = λφ for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Step 3: Set of global minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' From now on, we assume that W is strictly convex and denote wε = (wε,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , wε,M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that the map ˜wε := (wε,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , wε,N, 0RM−N−1, � w2 ε,N+1 + · · · + w2 ε,M) (25) belongs to A , | ˜wε| = |wε| and |∇ ˜wε| ≤ |∇wε| in Ω, so Eε(wε) ≥ Eε( ˜wε) and � w2 ε,N+1 + · · · + w2 ε,M ≥ wε,M = φ > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Hence, ˜wε is a minimizer of Eε on A (as wε minimizes Eε over A by Step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Therefore, up to interchanging wε and ˜wε, we may assume � wε,N+1 = · · · = wε,M−1 ≡ 0 in Ω wε,M = φ (20) > 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We now consider another minimizer Uε of Eε over A and denote v := Uε −wε ∈ H1 0(BN × Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) after a suitable extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' From Steps 1 and 2 we know that Eε(Uε) = Eε(v + wε) = Eε(wε), Gε(v) = 0, |v+wε| = |wε| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω and v = λφ for some λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , λM) ∈ RM where we recall that φ = wε·eM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By continuity of wε and φ, the relation |v+wε| = |wε| a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω implies 2wε · v + |v|2 = 0 everywhere in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Since v = λφ, dividing by φ > 0 in Ω, we obtain 2λ · wε + φ|λ|2 = 0 in Ω (26) and by continuity, the equality holds also on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As for every (x, z) ∈ ∂BN × (0, 1)n, φ(x, z) = 0 and wε(x, z) = (x, 0RM−N ), we deduce that λ · (x, 0RM−N ) = 0 for every x ∈ ∂BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It follows that λ1 = λ2 = · · · = λN = 0 and therefore, recalling that wε,N+1 = · · = wε,M−1 = 0 in Ω, we have by (26): 2λMφ + (λ2 N+1 + · · · + λ2 M)φ = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As φ > 0 in Ω, we obtain λ2 N+1 + · · · + λ2 M−1 + (λM + 1)2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' hence we can find R ∈ O(M) such that Rp = p for all p ∈ RN × {0RM−N } and ReM = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , 0, λN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , λM−1, λM + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This implies Uε = wε+v = wε+λφ = Rwε as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The converse statement is obvious: if wε is a minimizer of Eε over A and R ∈ O(M) is a transformation fixing all points of RN ×{0RM−N }, then Rwε is also a minimizer of Eε over A (because Eε and the boundary condition in A are invariant under such orthogonal transformation R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 13 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that if n ≥ 1, M > N ≥ 7 and W satisfies (2) (not necessarily strictly convex), then there are no bounded critical points of the energy Eε in the set A escaping in a direction e ∈ SM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Indeed, if such an escaping critical point of Eε in A exists, then by Theorem 7, this solution would be a global minimizer of Eε in A which is a contradiction with the uniqueness of the global minimizer (uε, 0RM−N ) in (4) (that is non-escaping) proved in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='2 Escaping radial profile Let M ≥ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We give a necessary and sufficient condition for the existence of an escaping radial profile ( ˜fε, gε > 0) in (0, 1) to the system (9)–(12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' we also prove uniqueness, minimality and monotonicity of the escaping radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For that, in the context of Eε defined over A , we introduce the functional Iε(f, g) = 1 |SN−1|Eε � (f(r) x |x|, 0RM−N−1, g(r)) � = 1 2 � 1 0 � (f ′)2 + (g′)2 + N − 1 r2 f 2 + 1 ε2 W(1 − f 2 − g2) � rN−1 dr where (f, g) belongs to B = � (f, g) : r N−1 2 f ′, r N−3 2 f, r N−1 2 g′, r N−1 2 g ∈ L2(0, 1), f(1) = 1, g(1) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (27) The following result is reminiscent from Ignat-Nguyen [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='4] (for ˜W ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The proof of [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='4] is rather complicated (as it is proved for some general potentials ˜W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We present here a simple proof that works in our context: Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let 2 ≤ N ≤ 6, M ≥ N + 1, W ∈ C2((−∞, 1]) satisfy (2) and be strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Consider εN ∈ (0, ∞) in (7) such that ℓ(εN) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then the system (9)–(12) has an escaping radial profile ( ˜fε, gε) with gε > 0 in (0, 1) if and only if 0 < ε < εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, in the case 0 < ε < εN, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' ( ˜fε, gε > 0) is the unique escaping radial profile of (9)–(12) and ˜fε r , gε ∈ C2([0, 1]), ˜f 2 ε + g2 ε < 1, ˜fε > 0, ˜f ′ ε > 0, g′ ε < 0 in (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' there are exactly two minimizers of Iε in B given by ( ˜fε, ±gε);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' the non-escaping radial profile (fε, 0) is an unstable critical point of Iε in B where fε is the unique radial profile in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Recall that for ε ≥ εN, the non-escaping radial profile (fε, 0) is the unique global minimizer of Iε in B (by Theorem 3 whose proof yields the minimality of (uε, 0RM−N ) of Eε in A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' First, we focus on the existence of escaping radial profiles of (9)–(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that the direct method in calculus of variations implies that Iε admits a minimizer 14 ( ˜fε, gε) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Since ( ˜fε, gε) ∈ B, ( ˜fε, gε) ∈ C((0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It follows that ( ˜fε, gε) satisfies (10)–(12) in the weak sense, and so ˜fε, gε ∈ C2((0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Since (| ˜fε|, |gε|) is also a minimizer of Iε in B, the above argument also shows that | ˜fε|, |gε| ∈ C2((0, 1]) satisfies (10)–(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Since | ˜fε|, |gε| ≥ 0 and ˜fε(1) = 1, the strong maximum principle yields | ˜fε| > 0 in (0, 1), and either |gε| > 0 in (0, 1) or gε ≡ 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' It follows that ˜fε > 0 in (0, 1), and there are three alternatives: gε > 0 in (0, 1), gε < 0 in (0, 1) or gε ≡ 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Clearly, when gε ≡ 0, ˜fε is equal to the unique radial profile fε in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By considering ( ˜fε, −gε) instead of ( ˜fε, gε) if necessary, we assume in the sequel that gε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Claim: if 0 < ε < εN, then gε > 0 in (0, 1) and (fε, 0) is an unstable critical point of Iε in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Claim: We define the second variation of Iε at (fε, 0) as Qε(α, β) = d2 dt2 ���� t=0 Iε � (fε, 0) + t(α, β) � = � BN � Lεα · α + Lεβ · β + N − 1 r2 α2 + 2 ε2 W ′′(1 − f 2 ε )f 2 ε α2� dx, for α, β ∈ C∞ c ((0, 1)) which extends by density to the Hilbert space H = {(α, β) : (fε + α, β) ∈ B} with the norm ∥(α, β)∥H := ∥(α x |x|, β)∥H1(BN,RN+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As ε ∈ (0, εN), we have ℓ(ε) < 0 by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Taking β ∈ H1 0(BN) to be any first eigenfunction of Lε in BN, which is radially symmetric, we have r N−1 2 β′, r N−1 2 β ∈ L2(0, 1), β(1) = 0 and Qε(0, β) = � BN Lεβ · β dx = ℓ(ε) � BN β2 dx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' So, (fε, 0) is an unstable critical point of Iε in B if ε < εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, (fε, 0) is not minimizing Iε in B and therefore, by the above construction of the minimizer ( ˜fε, gε) of Iε in B, we deduce that gε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This proves the above Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, by [8, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='7 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='9] (for ˜W ≡ 0), we deduce that ˜fε r , gε ∈ C2([0, 1]), ˜f 2 ε + g2 ε < 1, ˜f ′ ε > 0 and g′ ε < 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' To conclude, we distinguish two cases: Case 1: if ε ∈ (0, εN), Claim yields the existence of an escaping radial profile ( ˜fε, gε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By [8, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='7], every escaping radial profile ( ˜fε, gε > 0) is bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', ˜f 2 ε + g2 ε < 1 in (0, 1)) and therefore, by Theorem 7, the corresponding (bounded) escaping critical point ˜uε in (9) is a global minimizer of Eε over A and the set of minimizers of Eε over A is then given by {R˜uε : R ∈ O(M), Rp = p, ∀p ∈ RN × {0RM−N }}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Therefore, ( ˜fε, ±gε) are the only two minimizers of Iε in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, this proves the uniqueness of the escaping radial profile ( ˜fε, gε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Case 2: if ε ≥ εN, by the proof of Theorem 3, the non-escaping vortex sheet solution uε(x) ≡ (fε(|x|) x |x|, 0RM−N ) (by (13)) is the unique minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, (fε, 0) is the unique minimizer of Iε in B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', in the above construction of the minimizer 15 ( ˜fε, gε) of Iε in B, we have ˜fε = fε and gε = 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We claim that no escaping radial profile ( ˆfε, ˆgε > 0) exists if ε ≥ εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume by contradiction that such an escaping radial profile ( ˆfε, ˆgε > 0) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The same argument presented in Case 1 would imply that ( ˆfε, ˆgε > 0) is a minimizer of Iε in B which contradicts the uniqueness of the global minimizer (fε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='3 Proof of Theorem 4 We now prove the main result: Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By Theorem 9, the existence of an escaping radially symmetric so- lution ˜uε in (9) is equivalent to ε ∈ (0, εN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, in that case, the escaping radial profile ( ˜fε, gε > 0) is unique and bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', ˜f 2 ε + g2 ε < 1 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Case 1: if ε ∈ (0, εN), Theorem 7 implies that the (bounded) escaping radially symmetric critical point ˜uε in (9) is a global minimizer of Eε over A and every minimizer of Eε over A has the form R˜uε for some orthogonal transformation R ∈ O(M) keeping invariant the space RN × {0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, by Theorem 9, the non-escaping radial profile (fε, 0) is proved to be an unstable critical point of Iε in B, so the non-escaping vortex sheet solution (uε, 0RM−N ) is an unstable critical point of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Case 2: if ε ≥ εN, the proof of Theorem 3 implies that the non-escaping radially symmetric vortex sheet solution uε(x) ≡ (fε(|x|) x |x|, 0RM−N ) (by (13)) is the unique minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, there is no bounded critical point wε of Eε over A that escapes in some direction e ∈ SM−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' indeed, if such (bounded) escaping solution wε satisfying (20) exists, then Theorem 7 would imply that wε is a global minimizer of Eε over A which contradicts that the non-escaping vortex sheet solution uε is the unique global minimizer of Eε over A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 4 holds also for the “degenerate” dimension n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, Ω = BN and vortex sheets are vortex points, Eε(u) = � BN �1 2|∇u|2 + 1 2ε2 W(1 − |u|2) � dx, A := {u ∈ H1(BN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) : u(x) = (x, 0RM−N ) on ∂BN = SN−1} and radially symmetric vortex critical points of Eε in A have the corresponding form in (9): ˜uε(x) = ( ˜fε(r) x |x|, 0RM−N−1, gε(r)) ∈ A , x ∈ BN, r = |x|, (28) where the radial profiles ( ˜fε, gε) satisfy the system (10)-(12) and are described in Theo- rem 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' the non-escaping radially symmetric vortex solution is given here by uε(x) = (fε(|x|) x |x|, 0RM−N ) for all x ∈ BN, (29) where the radial profile fε is the unique solution to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We obtain the following result which generalizes [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='1] that was proved in the case N = 2 and M = 3 (without identifying the meaning of the dichotomy parameter εN in (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 16 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let 2 ≤ N ≤ 6, M ≥ N + 1, Ω = BN, W ∈ C2((−∞, 1]) satisfy (2) and be strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Consider εN ∈ (0, ∞) such that ℓ(εN) = 0 in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then there exists an escaping radially symmetric vortex solution ˜uε in (28) with the radial profile ( ˜fε, gε > 0) given in Theorem 9 if and only if 0 < ε < εN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if 0 < ε < εN, ˜uε is a global minimizer of Eε in A and all global minimizers of Eε in A are radially symmetric given by R˜uε where R ∈ O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, the non-escaping vortex solution uε in (29) is an unstable critical point of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if ε ≥ εN, the non-escaping vortex solution uε in (29) is the unique global minimizer of Eε in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Furthermore, there are no bounded critical points wε of Eε in A that escape in a direction e ∈ SM−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', wε · e > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The proof follows by the same argument used for Theorem 4, the main difference is that in the ball Ω = BN, a critical point wε of Eε in A satisfies the PDE system with Dirichlet boundary condition (instead of the mixed Dirichlet-Neumann condition in (21)): −∆wε = 1 ε2 wε W ′(1 − |wε|2) in BN, wε(x) = (x, 0RM−N ) on ∂BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' A Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Vortex sheet SM−1-valued harmonic maps in cylinders In dimensions M > N ≥ 2 and n ≥ 1, for the cylinder shape domain Ω = BN × (0, 1)n, we consider the harmonic map problem for SM−1-valued maps u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' SM−1) ∩ A associated to the Dirichlet energy E(u) = 1 2 � Ω |∇u|2 dxdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Any critical point u : Ω → SM−1 of this problem satisfies \uf8f1 \uf8f2 \uf8f3 −∆u = u |∇u|2 in Ω, ∂u ∂z = 0 on BN × ∂(0, 1)n, u(x, z) = (x, 0RM−N ) on ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (30) We will focus on radially symmetric vortex sheet SM−1-valued harmonic maps having the following form (invariant in z-direction): u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z) = (f(r) x |x|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 0RM−N−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' g(r)) ∈ A ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' x ∈ BN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1)n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' r = |x|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (31) where the radial profile (f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' g) satisfies f 2 + g2 = 1 in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (32) 17 and the system of ODEs: −f ′′ − N − 1 r f ′ + N − 1 r2 f = Γ(r)f in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (33) −g′′ − N − 1 r g′ = Γ(r)g in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (34) f(1) = 1 and g(1) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (35) where Γ(r) = (f ′)2 + N − 1 r2 f 2 + (g′)2 is the Lagrange multiplier due to the unit length constraint in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As for the Ginzburg- Landau system, we distinguish two type of radial profiles: the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) yielding the non-escaping (radially symmetric) vortex sheet SM−1-valued harmonic map (also called “equator” map): ¯u(x, z) = ( x |x|, 0RM−N ) x ∈ BN, z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (36) Note that ¯u is singular and the singular set of this map is the vortex sheet {0RM−N }×(0, 1)n of dimension n in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Also, observe that ¯u ∈ H1(Ω, SM−1) if and only if N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' the escaping radial profile (f, g) with g > 0 in (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in this case, it holds f(0) = 0, g(0) = 1 and we say that u in (31) is an escaping (radially symmetric) vortex sheet SM−1- valued harmonic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Note that u is smooth for every dimension M > N ≥ 2 and n ≥ 1 and the zero set of (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , uN) is the vortex sheet {0RM−N } × (0, 1)n of dimension n in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Obviously, (f, −g < 0) is another radial profile satisfying (32)-(35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The properties of such radial profiles are proved in [14] (see also [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='6] for ˜W ≡ 0 in those notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' More precisely, (a) If N ≥ 7, the non-escaping radial profile ( ¯f ≡ 1, ¯g ≡ 0) is the unique minimizer of I(f, g) = 1 |SN−1|E � (f(r) x |x|, 0RM−N−1, g(r)) � = 1 2 � 1 0 � (f ′)2 + (g′)2 + N − 1 r2 f 2� rN−1 dr, where (f, g) belongs to B ∩ � (f, g) : f 2 + g2 = 1 � with B defined in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, the system (32)–(35) has no escaping radial profile (f, g) with g > 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (b) If 2 ≤ N ≤ 6, then there exists a unique escaping radial profile (f, g) with g > 0 satisfying (32)–(35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, (f, ±g) are the only two global minimizers of I in B ∩ � (f, g) : f 2 + g2 = 1 � , f r , g ∈ C∞([0, 1]), f(0) = 0, g(0) = 1, f > 0, f ′ > 0 and g′ < 0 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In addition, for 3 ≤ N ≤ 6, the non-escaping solution ( ¯f ≡ 1, ¯g ≡ 0) is an unstable critical point of I in B ∩ � (f, g) : f 2 + g2 = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='6 6For N = 2, (1, 0) /∈ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' however, we can define the second variation of I at (1, 0) along directions (0, q) compactly supported in (0, 1): Q(0, q) = � 1 0 � (q′)2 − N − 1 r2 q2� rN−1 dr, and one can prove the existence of q ∈ Lipc(0, 1) such that Q(0, q) < 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [8, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 18 There is a large number of articles studying existence, uniqueness, regularity and stability of radially symmetric SM−1-valued harmonic maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', [13, 14, 25, 26, 23, 16, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We summarize here the main result for our problem in the cylinder shape domain Ω = BN × (0, 1)n: if N ≤ 6, then minimizing SM−1-valued harmonic maps in A are smooth, radially symmetric and escaping in one-direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if N ≥ 7, then there is a unique minimizing SM−1-valued harmonic map in A which is singular and given by the equator map ¯u in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 7 Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let n ≥ 1, N ≥ 2, M ≥ N + 1 and Ω = BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if 2 ≤ N ≤ 6, then the escaping radially symmetric vortex sheet solution u in (31) with g > 0 is a minimizing SM−1-valued harmonic map in A and all minimizing SM−1-valued harmonic maps in A are smooth radially symmetric given by Ru where R ∈ O(M) satisfies Rp = p for all p ∈ RN ×{0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In this case, the equator map ¯u in (36) is an unstable SM−1-valued harmonic map in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' if N ≥ 7, the non-escaping vortex sheet solution ¯u in (36) is the unique minimizing SM−1-valued harmonic map in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Moreover, there is no SM−1-valued harmonic map w in A escaping in a direction e ∈ SM−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', w · e > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' The main ingredient is the following result yielding minimality of escaping SM−1-valued harmonic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This is reminiscent from Sandier-Shafrir [23] (see also [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Let n ≥ 1, M > N ≥ 2 and Ω = BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Assume that w ∈ A ∩ H1(Ω, SM−1) is a SM−1-valued harmonic map satisfying (30) and w · e > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω (37) in an escaping direction e ∈ SM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then w is a minimizing SM−1-valued harmonic map in A and all minimizing SM−1-valued harmonic maps in A are of the form Rw where R ∈ O(M) is an orthogonal transformation of RM satisfying Rp = p for all p ∈ RN ×{0RM−N }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' We give here a simple proof based on the argument in [12] that avoids the regularity results used in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' By the H1/2-trace theorem applied for w ∈ H1(Ω, SM−1), (37) implies that w · e ≥ 0 on ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Combined with the vortex boundary condition in (30), we deduce that the escaping direction e has to be orthogonal to RN × {0RM−N } and up to a rotation, we can assume that e = eM (as in (22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then φ = w · eM > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω satisfies − ∆φ = |∇w|2φ in Ω, ∂φ ∂z = 0 on BN × ∂(0, 1)n, φ = 0 on ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (38) We consider configurations8 ˜w = w + v : Ω → SM−1 with v ∈ H1 0(BN × Rn, RM) (in particular, |v| ≤ 2 in Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then 2w · v + |v|2 = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (39) 7We mention the paper of Bethuel-Brezis-Coleman-H´elein [2] about a similar phenomenology in a do- main Ω = (B2 \\ Bρ) × (0, 1) ⊂ R3 where Bρ ⊂ R2 is the disk centered at 0 of radius ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 8Note that for any ˜w ∈ A ∩ H1(Ω, SM−1), the map ˜w − w has an extension in H1 0(BN × Rn, RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 19 Using (30) and (39), we obtain 2 � Ω ∇w · ∇v = 2 � Ω |∇w|2w · v dx = − � Ω |∇w|2|v|2 dx, yielding9 � Ω |∇(w + v)|2 dx − � Ω |∇w|2 dx = � Ω |∇v|2 − |∇w|2|v|2 dx =: Q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' (40) To show that w is minimizing, we prove that Q(v) ≥ 0 for all v ∈ H1 0(BN × Rn, RM) ∩ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) (note that this is a class larger than what we need, as we do not require that v satisfy the pointwise constraint (39)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' For that, we take an arbitrary map ˜v ∈ C∞ c (BN × Rn, RM) of support ω and decompose it as ˜v = φΨ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This decomposition makes sense as φ ≥ δ > 0 in ω ∩ Ω for some δ > 0 (which may depend on ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Indeed, by (37) and (38), φ is a superharmonic function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=', −∆φ ≥ 0 in Ω) that belongs to H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' As ∂φ ∂z = 0 on BN × ∂(0, 1)n, φ can be extended by even mirror symmetry to the domain ˜Ω = BN × (−1, 2)n so that φ is superharmonic in ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, the weak Harnack inequality (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' [6, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='18]) implies that on the compact set ω ∩Ω in ˜Ω, we have φ ≥ δ > 0 for some δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' So, ˜v = φΨ in Ω with Ψ = (Ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' , ΨM) ∈ H1 ∩ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM) vanishing in a neighborhood of ∂BN × (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then integration by parts yields for 1 ≤ j ≤ M: Q(˜vj) = � Ω |∇˜vj|2 − |∇w|2φ · φΨ2 j dx (38) = � Ω |∇(φΨj)|2 − ∇φ · ∇(φ Ψ2 j) dx = � Ω φ2|∇Ψj|2 dx ≥ 0 for all ˜v ∈ C∞ c (BN × Rn, RM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then for every v ∈ H1 0(BN × Rn, RM) ∩ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' RM), there exists a sequence ˜vk ∈ C∞ c (BN × Rn, RM) such that ˜vk → v and ∇˜vk → ∇v in L2 and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in BN × Rn and |˜vk| ≤ ∥v∥L∞(Ω) + 1 in Ω for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, by dominated convergence theorem, we have Q(˜vk) → Q(v) thanks to (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Thus, we deduce that for every compact ω ⊂ ˜Ω = BN × (−1, 2)n, Q(v) = lim k→∞ Q(˜vk) ≥ lim inf k→∞ � ω∩Ω φ2|∇ �˜vk φ � |2 dx ≥ � ω∩Ω φ2|∇ � v φ � |2 dx ≥ 0, where we used Fatou’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' In particular, w is a minimizing SM−1-valued harmonic map by (40) and Q(v) = 0 yields the existence of a vector λ ∈ RM such that v = λφ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Then the classification of the minimizing SM−1-valued harmonic maps follows by (39) as in the Step 3 of the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This part concerning the dimension 2 ≤ N ≤ 6 follows from Theorem 12 and the instability of the radial profile (1, 0) for I in B∩ � (f, g) : f 2+g2 = 1 � as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 9Note that the functional Q represents the second variation of E at w, but here the map v is not necessarily orthogonal to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This part for dimension N ≥ 7 follows the ideas in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' More precisely, calling X = (x, z) the variable in Ω, we have as in the proof of Theorem 12 for every v ∈ H1 0(BN × Rn, RM) with |v + ¯u| = 1 in Ω: � Ω |∇(¯u + v)|2 dX− � Ω |∇¯u|2 dX = � Ω � |∇v|2 − |∇¯u|2|v|2� dX = � Ω |∇zv|2 dX + � (0,1)n dz � BN � |∇xv|2 − N − 1 |x|2 |v|2� dx ≥ � Ω |∇zv|2 dX + �(N − 2)2 4 − (N − 1) � � Ω |v|2 |x|2 dX ≥ 0 where we used the Hardy inequality for v(·, z) ∈ H1 0(BN, RM) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' z ∈ (0, 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' This proves that ¯u is the unique minimizing SM−1-valued harmonic map in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Combined with Theorem 12, we conclude that there is no escaping SM−1-valued harmonic map w in A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Bethuel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' Brezis and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9FJT4oBgHgl3EQfCizA/content/2301.11430v1.pdf'} +page_content=' H´elein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, 13.' metadata={'source': 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b/ENE0T4oBgHgl3EQfgwE0/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b1dfe34debd29fde60b0b80cc10e4ed8daab63ca564b150c29a2ad7b5b57c228 +size 82516 diff --git a/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/2301.03793v1.pdf.txt b/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/2301.03793v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ebb904eca3300f8eb8f8f02223929408fadfe42 --- /dev/null +++ b/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/2301.03793v1.pdf.txt @@ -0,0 +1,715 @@ +ESTIMATION OF USER’S WORLD MODEL USING GRAPH2VEC ∗ +Tatsuya Sakai, +Takayuki Nagai +Graduate School of Engineering Science, Osaka University +1-3, Machikaneyama, Toyonaka, Osaka, Japan +nagai@sys.es.osaka-u.ac.jp +ABSTRACT +To obtain advanced interaction between autonomous robots and users, robots should be able to +distinguish their state space representations (i.e., world models). Herein, a novel method was +proposed for estimating the user’s world model based on queries. In this method, the agent learns +the distributed representation of world models using graph2vec and generates concept activation +vectors that represent the meaning of queries in the latent space. Experimental results revealed that +the proposed method can estimate the user’s world model more efficiently than the simple method of +using the “AND” search of queries. +Keywords Autonomous robot · Explainability · Representation learning · User’s world model +1 +Introduction +Autonomous robots are increasingly being used in numerous applications. Currently, they assist humans in performing +tasks by executing commands. For autonomous robots performing sophisticated decisions, the blind execution of +commands is not always the best strategy. Moreover, in many situations, fully executing commands is difficult. These +autonomous robots should be able to explain the reasons for their decisions to gain user trust. Explainable autonomous +robots (XAR) are defined as robots that have these explanatory capabilities. The following four requirements have been +identified for the XARs [1]: +(1) Owning an interpretable decision-making space +(2) Estimation of the model of others +(3) Estimation of the information required for a user to estimate the policy of the robot +(4) Presentation to the user of explanation +This explanation mechanism is a mutual process between the robot and the user and is displayed in Fig. 1. The world +model refers to the correspondence between actions and state changes, that is, the internal model [2] that represents the +dynamics of the environment, and we do not distinguish between the world model and the environment in this paper. +“Policy sharing” in Fig.1 is a spontaneous presentation of information of a policy (e.g., presentation of a sequence of +actions to be taken), and an explanation is generated when a query to this information presentation is requested by +others. +Among the four requirements, the estimation of the other’s world model is particularly crucial for providing a user- +specific explanation. In the context of human–robot interaction, the importance of estimating the user’s internal state +has already been recognized. Gao et al. [3] and Clair et al. [4] proposed a framework for estimating plausible action +strategies based on the actions and interaction history of users. Huang et al. [5] and Lage et al. [6] focused on restoring +explanations to policies and advocated the importance of appropriately estimating the restoration algorithm of users +requesting explanation. These studies have focused on internal states, particularly policies, and planning algorithms. +∗This paper is an extended (English translated) version of “T.Sakai, T.Horii, T.Nagai, Representation Learning of World Models +and Estimation of World Model of Others Using Graph2vec, Journal of RSJ, 40, 2, pp.166-169, 2022,” (in Japanese) ISSN 1884-7145, +Print ISSN 0289-1824, https://doi.org/10.7210/jrsj.40.166 +arXiv:2301.03793v1 [cs.RO] 10 Jan 2023 + +Preprint +Figure 1: Explanation process as communication. To clarify elements of the explanation, observations/actions, and +interactions between world models of others are segregated in the figure. +Figure 2: Simplified explanation process considered in this paper. We only consider unidirectional explanation process +from robot to user. +However, real-world robots are designed to exhibit desired behavior in terms of algorithms and policies, and they +share the same final objective with their users. In these situations, the results of action decisions typically stem from +discrepancies in environmental awareness. +In this study, a novel method was proposed for estimating the user’s world model based on the robot’s world model +and the questions (queries) posed by the user. The proposed method can identify differences in the world models and +generate an explanation that resolves the discrepancy between the perception of the environment by the robot and the +user. In this study, we simplified the mechanism of Fig.1 as in Fig.2 2 and focused on estimating the world model of the +explained person as the user’s world model. +2 +Proposed method +In the proposed method, the world model of the user is estimated by the following procedure (Fig. 3). +(1) Acquisition of a distributed representation of the world model: A distributed representation of each world +model is obtained using a graph-structured world model. +(2) Acquisition of the query vector: Based on the query given by the user, the system acquires a direction vector +that represents the meaning of the query in the representation space of the world model. +(3) Estimation of user’s world model: The distributed representations of the world model and query vector are +used to estimate the user’s world model based on cosine similarity. +2When the model of others outputs an explanation, the content of the explanation is determined by considering information +received from the world model. +2 + +Preprint +Why donʼt you do +Agentʼs +1 +2 +3 +4 +5 +6 +7 +8 +9 +Userʼs +Figure 3: Schematic of our proposed method. +… +… +… += +1 +2 +⋮ += +1 +2 +⋮ +Figure 4: Learning of a distributed representation of the world model. The experience on the environment is used to +obtain graph-based world models. Then, after converting to WL labels, a distributed representation of each environment +is obtained. +The robot and user are assumed to share the same state space and measures; only the connection relation between states +is assumed to be unshared. As a distributed representation of the world model, the parameters of a model representing a +continuous state space, for example, [2], could be used. However, presenting the differences of the world model to the +user requires the discretization of the state transition structure; the parameter space of the model does not necessarily +represent the similarity of the state transition structure of the world model. Therefore, the world model is considered to +be a discretized graph for obtaining a distributed representation. The generation of explanations is outside the scope of +this study. +2.1 +Acquisition of a distributed representation of the world model +The learning process of a distributed representation of the world model is shown in Fig. 4. The robot learns a +representation space representing the similarity of environments based on its experiences. First, the robot acquires an +undirected graph representing the state transitions of each environment; simultaneously, it acquires policies through +reinforcement learning. Specifically, the robot assumes that the states whose transitions were observed during the policy +learning search are adjacent to each other and adds edges3. +After acquiring the undirected graphs of all environments, graph2vec is applied to acquire the distributed representation +of the graph of each environment [8]. Graph2vec is a method in which doc2vec [9] is used to acquire distributed +representations of graphs, and instead of predicting the occurrence of words, the occurrence of labels that represent +each subgraph is presented. Labels representing subgraphs are obtained using the Weisfeiler–Lehman (WL) relabeling +process [10]. In this process, the label of the next layer is determined by considering the labels of neighboring nodes. +Higher layer labels represent more global information regarding the graph. +3This method is assumed to be used in a discrete state space; discretization of the state space is required for application to a real +robot. However, this measure is beyond the scope of this study. For discretization, the method proposed in [7], wherein the policy +implications of each state is considered, can be used. +3 + +------Preprint +Graph2vec allows graphs in which the same subgraphs occur, that is, graphs in environments with similar state transition +structures, to be embedded closer together in the representation space. Efficient search based on user queries becomes +possible by acquiring the representation space of the world model in advance. +When acquiring a world model, we explicitly provide an environment label to indicate the environment wherein the +experience occurs. If no environment label is given, a world model is obtained using temporally continuous experiences. +Furthermore, we consider that models with similar distributed representations represent the same environment and +merging multiple models may be effective in obtaining a world model with higher accuracy. +2.2 +Acquisition of the query vector +Based on the query given by the user, the robot acquires a direction vector that represents the meaning of the query in +the representation space of the world model. Kim et al. [11] focused on the middle layer of the neural network and +generated concept vectors (CAV: concept activation vectors) by calculating the difference in latent representations when +features that satisfy a specific concept and features that do not satisfy the concept are input. In this study, this method +was applied to define a query vector vquery as Eq. (1) by considering the difference of distributed representations +between environments that satisfy the query and those that do not. The query assumes the form “action aquery should +be selected in state squery.” +vquery = vpos − vneg, +(1) +where +vpos = +� +i vi · P(aquery|vi, squery) +� +i P(aquery|vi, squery) +, +vneg = +� +i vi · (1 − P(aquery|vi, squery)) +� +i(1 − P(aquery|vi, squery)) +. +(2) +Here, vi represents a distributed representation of the i-th environment. To correspond to a policy in which actions +are selected probabilistically, the probability value of selecting an action is used as the coefficient of each distributed +representation vi. If necessary, the coefficients can be expressed as the binary values of {0, 1}. Note that if the state +Squery is not included in the undirected graph of the environment i, the CAV is calculated by excluding vi4. +2.3 +Estimation of user’s world model +Using the distributed representation of the world model and the query vector, the user’s world model was estimated +using cosine similarity. The likelihood of an environment i as an estimated environment is expressed by Eq. (3). +S(vi, vobs, Vq) = +� +j +similarity(vj +query, vi − vobs) − λ · distance(vi, vobs), +(3) +where vobs is a distributed representation of the environment currently observed by the agent, Vq is any number of query +vectors, and vj +query ∈ Vq is the j-th query vector. Furthermore, similarity(a, b) and distance(a, b) are functions that +output the cosine similarity [−1, 1] and the distance between vectors a and b in the representation space, respectively. +When reasoning in a real environment, the robot and the user observe almost the same environment. Therefore, because +their world models are similar, the similarity between the direction of each environment and the direction of the query +vector, as seen from vobs, is used as the estimation criterion. The coefficient λ is a hyperparameter that determines +how much distance between vectors is considered and represents the strength of the assumption that the observed +environment of the robot and the user are similar. +Using the definitions, the world model of others to be estimated is expressed by Eq. (4). +Env_est = arg max +i +S(vi, vobs, Vq) +(4) +In this study, we assume that the importance of all queries are equivalent and designed the evaluation function S as the +sum of the similarities for each query vector vj +query. However, in the real world, the importance of each query may +differ, in which case the similarity should be multiplied by a coefficient ρj. +4When estimating the user’s world model, the likelihood S(vi, vobs, Vq) is computed for all environments i, including those that +do not contain the state squery. +4 + +Preprint +2.3.1 +User vectors +In addition to the query vector, the user vector that represents “what kind of environment with the distributed repre- +sentation the user is likely to retain as a world model” can also be defined. The user vector is a vector that represents +how the user tends to misunderstand the environment and plays a role in adding this tendency to the result of the user’s +world model estimation. +vuser = vu_pos − vu_neg, +(5) +where +vu_pos = +� +i vi · P(vi) +� +i P(vi) +, +vu_neg = +� +i vi · (1 − P(vi)) +� +i(1 − P(vi)) +. +(6) +P(vi) is the probability that the user estimates the environment corresponding to the distributed representation vi as +the current world model when no information about the current environment is given to the user. In this paper, P(vi) is +assumed to be known, and the estimation of P(vi) is outside the scope of this research. +2.4 +Explanation by language +Using a pre-trained language vector Eq. (7) in the representation space, the relationship between the world model +maintained by the agent and the user’s world model can also be explained by language. +vword(x) = average(vm − vn), +(7) +where vm and vn are distributed representations of environment pairs whose relation is represented by the language +x. By averaging the differences of distributed representations of environment pairs vm and vn that satisfy a specific +language description x, we can obtain a semantic vector represented by the language description x in the representation +space. +Using the language vector vword, the language describing the relationship between the world model maintained by the +agent and the user’s world model is represented by Eq. (8). +Explanation = arg max +x +similarity(vword(x), vEnv_est − vobs)). +(8) +By means of Eq. (8), the language x that represents the closest relationship between the current world models is selected +as the explanation. +3 +Experiment +The proposed method was applied to an agent that acquires action strategies by proximal policy optimization (PPO) in a +simulation environment, and its usefulness was evaluated. A partially modified version of the grid environment [12] +with multiple objects was used for the experiments (Fig. 5). In this environment, the agent (triangle) obtains a reward by +taking a key, opening a door, and reaching a goal in the lower right corner. The position of the goal remains unchanged, +but the positions of the key and the door change every trial. The agent has five actions, namely go straight, turn left, +turn right, take the key from the grid in front of it, and open the door. The agent observes the absolute position of the +key (x, y coordinates), the absolute position of the door (x, y coordinates), its own absolute position (x, y coordinates) +and orientation, holding/not holding the key, and opening/closing the door, for a total of nine dimensions. +3.1 +Experiment 1: Acquisition of a distributed representation of the world model +A graph representing the state transitions of each environment was obtained simultaneously with the learning of policies +by PPO, and graph2vec was used to obtain a 16-dimensional distributed representation. An example of the acquired +graph is shown in Fig.6. There are three edges that transition from the group of states before key acquisition to the +group of states after key acquisition because keys can be acquired from three directions. On the other hand, there is +only one edge that transitions from the group of states where the door is not open to the group of states where the door +is open, because the door can be opened from only one state. +5 + +Preprint +Why donʼt you pick up +Figure 5: Estimation results of the user’s world model. +Table 1: Cumulative frequency and average value of the order in which the optimal environment appears. The number +of trials is 100 for each method. +Method +Order +1 +2 +3 +Order Average +Our method +35 +49 +60 +7.1 +Random +6 +8 +11 +23.1 +The representation space was compressed to eight dimensions using independent component analysis, and the visualized +results are displayed in Fig. 7. In this experiment, the absolute positions of the key and door were used as environment +labels, and the five-dimensional observations excluding them were used as node features. The experimental results +revealed that clusters are formed in the representation space based on the absolute positions of the key and door. In +particular, clusters related to the position of the door are apparent, which can be attributed to the fact that the surrounding +state transition relationship changes considerably compared with that of the key. In this experiment, the absolute +positions of keys and doors are used only for identifying the environmental graph to be updated and are not embedded +in the graph itself. Therefore, graph2vec created a representation space that appropriately reflects the differences in the +location information of keys and doors expressed in the state transition structure. +3.2 +Experiment 2: Estimation of the user’s world model +The query vector was applied to the obtained representation space to estimate the world model of others. In this +verification, we assume that questions were asked in the situations of acquiring a key and opening a door, and the query +“In the state Squery, we should {take the key/open the door}” was considered. Figure 5 displays the results of estimating +the others’ world model given the reference world model and query. The environment that satisfies the query has the +highest evaluation value, and the environment in which the key is located on the opposite side of the grid environment +has the lowest evaluation value. This result suggests that the obtained query vector is appropriate for estimating the +world model. +The optimal environment is defined as the agent’s world model with minimal modifications for satisfying the query. For +example, given the query “In state Squery, the agent should take the key,” the optimal environment is the environment +in which only the position of the key is changed to satisfy the query, whereas the position of the door is left unchanged. +For each randomly selected agent world model/query pair, the evaluation values for each environment obtained using +Eq. (3) were sorted in descending order to obtain the order of appearance of the optimal environment (Table 1). The +order of appearance of the environments that satisfy the query when sorted randomly is displayed for comparison. The +experimental results confirmed that the proposed method ranks the optimal environments higher. +6 + +FoFo +-- +ooPreprint +Figure 6: An example graph of the acquired world model. The blue nodes represent the state before the key acquisition. +The orange nodes represent the state after the key acquisition when the door is not open, and the green nodes represent +the state when the door is open. +Although the direct manipulation of the positions of keys and doors can change the order of the appearance of the +optimal environment to one, direct manipulation is not always possible in cases in which directly manipulatable +information is not given as a query. The proposed method estimates the optimal environment at an early stage, although +the state transition relations to be changed are not explicitly given. This property of the proposed method is crucial. +3.3 +Experiment 3: Validation with multiple queries +An explanatory agent A and an explained agent B are prepared, and the number of queries required for A to accurately +estimate B’s world model is evaluated. This experiment assumes a situation in which the user is asked to confirm the +correctness of the estimation results of the other’s world model through the presentation of an action sequence, which +improves the estimation accuracy (Fig. 8). The outline of the experiment is as follows: +(1) A and B share an initial state (absolute position and orientation of the agent, and the state of the key and door) +and an environment-policy pair that specifies the strategy to be used in specific environments. They have +arbitrary world models with different key and door positions. +(2) Here, A sets its own world model as the initial value of the other’s world model and presents the optimal +sequence of actions in that model in turn (policy sharing) 5. +(3) B adds the query “In state squery, action aquery should be chosen” when the given action differs from the +optimal action in its measure. The existing query is not deleted. +(4) A updates the other’s world model based on the query and presents the action sequences in the updated other’s +world model again in sequence with Squery as the initial state. +(5) Repeat (3) and (4) to evaluate the number of environment updates required for A to obtain B’s world model as +the other’s world model. +The environment selected once is not selected, and the second or subsequent candidate is adopted. If the same +environment has not been obtained after all the action sequences are presented, the environment is continuously updated +5Policy sharing in this experiment (Fig. 8) is performed to confirm the estimation results of the other’s world model and differs +from the presentation of information about one’s own policy in Fig. 1 and Fig. 2. +7 + +Preprint +Figure 7: Results of latent space visualization. Each data point is illustrated according to (a) X-coordinate of the key, +(b) Y-coordinate of the key, (c) X-coordinate of the door, and (d) Y-coordinate of the door. +Figure 8: Schematic of experiment 3. The agent transmits information on its policy to the user and updates the user’s +world model based on queries. +without increasing the number of queries. In this verification, the proposed method was compared with the “AND” +search of queries as a method of updating the world model of others. In practice, the following three methods are +compared. +Proposed method: +Select a plausible environment using Eq. (4). +AND search 1: +Randomly selects an environment from among the environments that satisfy the query. +AND search 2: +The environment is randomly selected with the constraint that “the optimal behavior for the policy B +is selected in all states from the initial state until reaching state squery”. Thus, it adds constraints and increases +the information provided compared with the two update methods described. +Experimental results revealed that the proposed method can estimate others’ world models with the fewest number +of updates (Table 2). The results of the corresponding two-tailed t-test revealed that t(100) = 8.07 and p < .01 for +the proposed method and AND search 1, and t(100) = 4.59 and p < .01 for the proposed method and AND search 2. +Thus, both significant differences were confirmed. +8 + +V +V +M +V +V +V +V +V +会 +VI +V +V +V +V +V +V +V +V +V +V +△ +W +V +V +V +W +△ +V +△ +V +V +V +V +V +V +V +V +V +V +V +△ +1 +口 +I +I +V +△△ +口 +V +口 +口 +口 +I +口 +口 +I +1 +口 +1 +口口 +口 +口口 +口 +口 +V +V +V +V +V +V +V +V +VV +V口 +0VX +W +V +V +口 +口 +口 +口 +口 +■ +口 +口 +口Preprint +Table 2: Number of updates required to estimate the user’s world model. +Method +Number of updates +Standard deviation +Our method +5.53 +4.83 +AND search 1 +20.72 +17.43 +AND search 2 +8.69 +4.71 +Table 3: Comparison with the use of probabilistic evaluation. +Method +Nuber of updates +Standard deviation +Our method (λ = 0.05) +3.93 +5.38 +Our method (λ = 0) +5.75 +7.11 +Probabilistic value (λ = 0.05) +7.13 +7.79 +The most common information given directly is “AND search2”. By contrast, the proposed method can reduce the +number of updates by vectorizing queries and utilizing prior knowledge embedded in the representation space as +additional information. +3.4 +Experiment 4: Comparison with the use of probabilistic evaluation +We compare the proposed method with the case where the probability value of each environment satisfying the query is +used as the evaluation function instead of CAV. The proposed method uses Eq. (3) as the evaluation function, while the +comparison method uses Eq. (9). +S(vi, vobs, Vq) = +� +j +P(aquery|vi, squery) − λ · distance(vi, vobs). +(9) +The procedure for this experiment is the same as in experiment 3; however, the initial world model is not completely +random, and the coordinates of either the key or the door are assumed to be identical. This condition replicates the +assumption that the agent’s world model and the user’s world model are similar6. +Experimental results showed that the proposed method, which takes into account the distance in the representation +space (λ = 0.05), was able to estimate the user’s world model with the fewest number of updates (Table3). The results +of the corresponding two-tailed t test showed that the proposed method with λ = 0.05 compared to λ = 0 showed +t(99) = 4.59 and p < .005, while the proposed method with λ = 0.05 compared to the method using probability values +showed t(99) = 4.44 and p < .005, both of which are significantly different from each other7. +The proposed method, which takes into account the distance in the representation space, was able to estimate the +environment with a significantly smaller number of updates than the method using the same coordinates for either the +key or the door. Compared to the method using probability value as the evaluation function, the proposed method +was able to absorb small errors in probability values, resulting in a significantly smaller number of updates. When the +distance in the representation space corresponds to the similarity of the state transition structure between environments, +as in the present verification, it is effective to use CAV to obtain environments that are perpendicular to the virtual +separation boundary as the user’s world models. +3.5 +Experiment 5: Number of samples and accuracy of CAV +We evaluate the number of queries required to correctly estimate the world model when the number of samples (number +of environments) used to compute the CAV is reduced. The evaluation procedure is the same as in Experiment 4. In this +experiment, the maximum number of samples is 300 because 300 environments are embedded in the representation +space. We also set λ = 0.05. +6Without this assumption, the world model with minimal modification to satisfy the query (the optimal environment) is not +necessarily the user’s world model. However, theoretically, when the evaluation value is calculated using Eq. (3), other environments +that satisfy the query will have a lower evaluation value compared to the optimal environment. Therefore, if the assumption that the +world models of the agent and user are similar cannot be made, it is desirable to use Eq. (9). However, in a real environment, the +world models of the agent and user are not completely independent, and similarity can be assumed. +7Because the t test was applied twice in this verification, a significant difference was found at the significance level α = 0.01 +based on the Bonferroni method. +9 + +Preprint +Table 4: Relationship between the number of CAV samples and the order of appearance of the optimal environment. +Each value represents the cumulative frequency, and the number of trials is 100 for each. +Nuber of samples +Order of appearance +1 +2 +3 +300 +40 +62 +69 +250 +44 +61 +64 +200 +42 +51 +62 +150 +40 +51 +55 +100 +35 +46 +54 +50 +9 +18 +26 +Prior distribution 1 +Prior distribution 2 +P(door_y = 1) = 0.4 +P(door_y = 2) = 0.3 +P(door_y = 3) = 0.2 +P(door_y = 4) = 0.1 +P(door_y = 5) = 0.0 +P(door_y = 6) = 0.0 +P(door_y = 1) = 0.0 +P(door_y = 2) = 0.1 +P(door_y = 3) = 0.4 +P(door_y = 4) = 0.4 +P(door_y = 5) = 0.1 +P(door_y = 6) = 0.0 +Figure 9: The prior distribution used in experiment 6. The prior distribution for the y-coordinate of the door (in the +vertical direction) is defined. +The experimental results show that the accuracy deteriorates much more slowly up to 100 samples than when the CAV +is generated with 300 samples (Table 4). This suggests that a specific level of estimation accuracy can be maintained +even when the number of samples is reduced. The fact that the user’s world model is estimated taking into account +the distance in the representation space may also contribute to maintaining accuracy. On the other hand, the accuracy +dropped drastically when the number of samples was 50. This may be because of an increase in the number of trials in +which the number of positive data (the number of data satisfying the query) in the sample is very small. +3.6 +Experiment 6: Use of the user vector +We test whether the estimation accuracy of the other-world model can be improved by using the user vector defined +by Eq. (5), as opposed to using only CAV. In this verification, two types of prior distributions are defined for the +y-coordinate of the door (Fig. 9), and the user vector is calculated for each of them. The user vector is treated as one +of the query vectors in the evaluation value calculation for each environment. Note that λ = 0 is assumed in this +verification because there is no assumption that the world models held by agents A and B are similar8. +The estimation results of the other-world model for the same reference environment and query are shown in Fig. 10. +User vectors 1 and 2 correspond to prior distributions 1 and 2, respectively. It can be seen that while the results of the +inference of the door location are unstable when only the query vector is applied, the results of the inference using the +other vectors show that the door coordinates are concentrated in locations that have high probability values in the prior +distribution. +The same validation as in experiment 3 was conducted by applying the prior distribution shown in Fig. 9, and the +number of queries required for both distributions 1 and 2 was lower when using the user vector (Tab. 5). The results +8If this experiment is conducted under the assumption that the world models of agents A and B are similar, it is necessary to set +the number of objects that determine the state transition structure of the environment to three or more and that they are placed at the +same coordinates as the current environment. Under these conditions, it is desirable to set λ = 0.05. +10 + +Fo +-Preprint +Why don’t you +pick up the key +at the state? +Base environment +Query vector +Query vector ++User vector 1 +Query vector ++User vector 2 +Highest +Score +Lower +Figure 10: User vector and environments. +Table 5: Change in estimation accuracy when using user vectors. The number of trials is 100 for each. +Distribution 1 +Distribution 2 +Method +Number of updates +Standard deviation +Number of updates +Standard deviation +Query + User +5.31 +4.59 +4.69 +3.34 +Query +6.39 +7.13 +5.11 +3.46 +of the corresponding two-tailed t test showed that t(99) = 2.34 and p < .05 for distribution 1 and t(99) = 1.45 and +p = 0.15 for distribution 2. These results suggest that the number of queries required for estimation can be reduced by +using the user vector, although the size of the effect depends on the shape of the prior distribution. +3.7 +Experiment 7: Explanation by language +We test whether the language vectors learned in advance can be used to correctly output language that describes the +relationship between the agent’s world model and user’s world model. The language vectors used in this study are eight +different explanations, such as “In the world model assumed by the user, {key, door} is located at {upper, lower, right, +left} than in the world model maintained by the agent.” In the experiment, language explanations were first given to n +pairs of world models with different coordinates of keys or doors, and the language vectors were obtained using Eq. (7). +We then generated linguistic explanations for randomly selected pairs of world models using the same conditions as +those used to generate the language vectors and evaluated the percentage of the explanations that correctly explained +the relationships between the world models (i.e., the percentage of correct responses). +The percentage of correct responses after 100 trials is shown in Table 6. Note that if more than one linguistic explanation +correctly represented the relationship between the world models, both were considered as correct answers. For example, +if the key is located on the upper right, “the key is on the right” and “the key is on the top” are treated as correct +answers. The “1st” in the table indicates the percentage of correct explanations generated by Eq. (8). The “1st and 2nd” +represents the percentage of correctness of the two languages that were the first and second most similar. Note that “1st +and 2nd” was evaluated only in cases where more than one language explanation was considered to be a correct answer. +Although the number of world model pairs considered in this experiment is approximately 9000, the experimental +results show that language explanation generation is possible with high accuracy even when the number of data used +for language vector acquisition is n = 1000. The accuracy was also maintained even when the number of data was +extremely reduced to n = 100 and n = 50, suggesting that the learning in the representation space is effective. In this +experiment, only eight language vectors were set, but it is expected that more accurate explanation generation will +become possible by setting more detailed language vectors. However, it should be noted that in these cases, sufficient +teacher data is required. +11 + +-------Fo +-Preprint +Table 6: Accuracy of language description generation. +n +1st +1st and 2nd +5000 +0.89 +0.67 +3000 +0.89 +0.73 +1000 +0.88 +0.68 +500 +0.84 +0.63 +300 +0.80 +0.57 +100 +0.69 +0.54 +50 +0.60 +0.37 +4 +Conclusion +In this study, a novel method was proposed for estimating the user’s world model from the robot’s world model and the +query given by the user to obtain XAR. The proposed method can estimate others’ world models more efficiently than +using the “AND” search of queries. In the future, user vectors should be introduced, and the methods for generating +explanations using differences in world models should be devised. +Acknowledgments +This study was supported by the New Energy and Industrial Technology Development Organization (NEDO). +References +[1] Sakai, Tatsuya and Nagai, Takayuki, "Explainable Autonomous Robots: A Survey and Perspective," Advanced +Robotics, 36(5-6), pp.219-238, 2022. +[2] Ha, David and Schmidhuber, Jürgen, "Recurrent World Models Facilitate Policy Evolution," In Advances in +Neural Information Processing Systems 31, pp.2450-2462, 2018. +[3] Xiaofeng Gao, Ran Gong, Yizhou Zhao, et al. "Joint Mind Modeling for Explanation Generation in Complex +Human-Robot Collaborative Tasks," In Proceedings of 29th IEEE International Conference on Robot and Human +Interactive Communication (RO-MAN), pp.1119-1126, 2020. +[4] A. S. Clair and M. 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"graph2vec: Learning distributed representations +of graphs," ArXiv, abs/1707.05005, 2017. +[9] Quoc Le and Tomas Mikolov, "Distributed Representations of Sentences and Documents," In Proceedings of the +31st International Conference on Machine Learning, volume 32 of Proceedings of Machine Learning Research, +pp.1188-1196, 2014. +[10] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, et al. "Weisfeiler-Lehman Graph Kernels," Journal +of Machine Learning Research, 12(77), pp.2539–2561, 2011. +[11] Been Kim, M. Wattenberg, J. Gilmer, et al. "Interpretability Beyond Feature Attribution: Quantitative Testing +with Concept Activation Vectors (TCAV)," In ICML, 2018. +[12] Maxime Chevalier-Boisvert, Lucas Willems, and Suman Pal, "Minimalistic Gridworld Environment for OpenAI +Gym," 2018. +12 + diff --git a/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/load_file.txt b/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9fd150ac0184361408cad578739d1fa1cf4f2da6 --- /dev/null +++ b/ENE2T4oBgHgl3EQfSgdx/content/tmp_files/load_file.txt @@ -0,0 +1,396 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf,len=395 +page_content='ESTIMATION OF USER’S WORLD MODEL USING GRAPH2VEC ∗ Tatsuya Sakai, Takayuki Nagai Graduate School of Engineering Science, Osaka University 1-3, Machikaneyama, Toyonaka, Osaka, Japan nagai@sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='jp ABSTRACT To obtain advanced interaction between autonomous robots and users, robots should be able to distinguish their state space representations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=', world models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Herein, a novel method was proposed for estimating the user’s world model based on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this method, the agent learns the distributed representation of world models using graph2vec and generates concept activation vectors that represent the meaning of queries in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Experimental results revealed that the proposed method can estimate the user’s world model more efficiently than the simple method of using the “AND” search of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Keywords Autonomous robot · Explainability · Representation learning · User’s world model 1 Introduction Autonomous robots are increasingly being used in numerous applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Currently, they assist humans in performing tasks by executing commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' For autonomous robots performing sophisticated decisions, the blind execution of commands is not always the best strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Moreover, in many situations, fully executing commands is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' These autonomous robots should be able to explain the reasons for their decisions to gain user trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Explainable autonomous robots (XAR) are defined as robots that have these explanatory capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The following four requirements have been identified for the XARs [1]: (1) Owning an interpretable decision-making space (2) Estimation of the model of others (3) Estimation of the information required for a user to estimate the policy of the robot (4) Presentation to the user of explanation This explanation mechanism is a mutual process between the robot and the user and is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The world model refers to the correspondence between actions and state changes, that is, the internal model [2] that represents the dynamics of the environment, and we do not distinguish between the world model and the environment in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' “Policy sharing” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 is a spontaneous presentation of information of a policy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=', presentation of a sequence of actions to be taken), and an explanation is generated when a query to this information presentation is requested by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Among the four requirements, the estimation of the other’s world model is particularly crucial for providing a user- specific explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In the context of human–robot interaction, the importance of estimating the user’s internal state has already been recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [3] and Clair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [4] proposed a framework for estimating plausible action strategies based on the actions and interaction history of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [5] and Lage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [6] focused on restoring explanations to policies and advocated the importance of appropriately estimating the restoration algorithm of users requesting explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' These studies have focused on internal states, particularly policies, and planning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' ∗This paper is an extended (English translated) version of “T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='Sakai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='Horii, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='Nagai, Representation Learning of World Models and Estimation of World Model of Others Using Graph2vec, Journal of RSJ, 40, 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='166-169, 2022,” (in Japanese) ISSN 1884-7145, Print ISSN 0289-1824, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='7210/jrsj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='166 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='03793v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='RO] 10 Jan 2023 Preprint Figure 1: Explanation process as communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' To clarify elements of the explanation, observations/actions, and interactions between world models of others are segregated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Figure 2: Simplified explanation process considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' We only consider unidirectional explanation process from robot to user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, real-world robots are designed to exhibit desired behavior in terms of algorithms and policies, and they share the same final objective with their users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In these situations, the results of action decisions typically stem from discrepancies in environmental awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this study, a novel method was proposed for estimating the user’s world model based on the robot’s world model and the questions (queries) posed by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The proposed method can identify differences in the world models and generate an explanation that resolves the discrepancy between the perception of the environment by the robot and the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this study, we simplified the mechanism of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='2 2 and focused on estimating the world model of the explained person as the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2 Proposed method In the proposed method, the world model of the user is estimated by the following procedure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (1) Acquisition of a distributed representation of the world model: A distributed representation of each world model is obtained using a graph-structured world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (2) Acquisition of the query vector: Based on the query given by the user, the system acquires a direction vector that represents the meaning of the query in the representation space of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3) Estimation of user’s world model: The distributed representations of the world model and query vector are used to estimate the user’s world model based on cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2When the model of others outputs an explanation, the content of the explanation is determined by considering information received from the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2 Preprint Why donʼt you do Agentʼs 1 2 3 4 5 6 7 8 9 Userʼs Figure 3: Schematic of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' … … … = 1 2 ⋮ = 1 2 ⋮ Figure 4: Learning of a distributed representation of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The experience on the environment is used to obtain graph-based world models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Then, after converting to WL labels, a distributed representation of each environment is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The robot and user are assumed to share the same state space and measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' only the connection relation between states is assumed to be unshared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' As a distributed representation of the world model, the parameters of a model representing a continuous state space, for example, [2], could be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, presenting the differences of the world model to the user requires the discretization of the state transition structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' the parameter space of the model does not necessarily represent the similarity of the state transition structure of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Therefore, the world model is considered to be a discretized graph for obtaining a distributed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The generation of explanations is outside the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 Acquisition of a distributed representation of the world model The learning process of a distributed representation of the world model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The robot learns a representation space representing the similarity of environments based on its experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' First, the robot acquires an undirected graph representing the state transitions of each environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' simultaneously, it acquires policies through reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Specifically, the robot assumes that the states whose transitions were observed during the policy learning search are adjacent to each other and adds edges3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' After acquiring the undirected graphs of all environments, graph2vec is applied to acquire the distributed representation of the graph of each environment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Graph2vec is a method in which doc2vec [9] is used to acquire distributed representations of graphs, and instead of predicting the occurrence of words, the occurrence of labels that represent each subgraph is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Labels representing subgraphs are obtained using the Weisfeiler–Lehman (WL) relabeling process [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this process, the label of the next layer is determined by considering the labels of neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Higher layer labels represent more global information regarding the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3This method is assumed to be used in a discrete state space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' discretization of the state space is required for application to a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, this measure is beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' For discretization, the method proposed in [7], wherein the policy implications of each state is considered, can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3 ------Preprint Graph2vec allows graphs in which the same subgraphs occur, that is, graphs in environments with similar state transition structures, to be embedded closer together in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Efficient search based on user queries becomes possible by acquiring the representation space of the world model in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' When acquiring a world model, we explicitly provide an environment label to indicate the environment wherein the experience occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' If no environment label is given, a world model is obtained using temporally continuous experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Furthermore, we consider that models with similar distributed representations represent the same environment and merging multiple models may be effective in obtaining a world model with higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='2 Acquisition of the query vector Based on the query given by the user, the robot acquires a direction vector that represents the meaning of the query in the representation space of the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [11] focused on the middle layer of the neural network and generated concept vectors (CAV: concept activation vectors) by calculating the difference in latent representations when features that satisfy a specific concept and features that do not satisfy the concept are input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this study, this method was applied to define a query vector vquery as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (1) by considering the difference of distributed representations between environments that satisfy the query and those that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The query assumes the form “action aquery should be selected in state squery.” vquery = vpos − vneg, (1) where vpos = � i vi · P(aquery|vi, squery) � i P(aquery|vi, squery) , vneg = � i vi · (1 − P(aquery|vi, squery)) � i(1 − P(aquery|vi, squery)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (2) Here, vi represents a distributed representation of the i-th environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' To correspond to a policy in which actions are selected probabilistically, the probability value of selecting an action is used as the coefficient of each distributed representation vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' If necessary, the coefficients can be expressed as the binary values of {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Note that if the state Squery is not included in the undirected graph of the environment i, the CAV is calculated by excluding vi4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='3 Estimation of user’s world model Using the distributed representation of the world model and the query vector, the user’s world model was estimated using cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The likelihood of an environment i as an estimated environment is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' S(vi, vobs, Vq) = � j similarity(vj query, vi − vobs) − λ · distance(vi, vobs), (3) where vobs is a distributed representation of the environment currently observed by the agent, Vq is any number of query vectors, and vj query ∈ Vq is the j-th query vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Furthermore, similarity(a, b) and distance(a, b) are functions that output the cosine similarity [−1, 1] and the distance between vectors a and b in the representation space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' When reasoning in a real environment, the robot and the user observe almost the same environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Therefore, because their world models are similar, the similarity between the direction of each environment and the direction of the query vector, as seen from vobs, is used as the estimation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The coefficient λ is a hyperparameter that determines how much distance between vectors is considered and represents the strength of the assumption that the observed environment of the robot and the user are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Using the definitions, the world model of others to be estimated is expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Env_est = arg max i S(vi, vobs, Vq) (4) In this study, we assume that the importance of all queries are equivalent and designed the evaluation function S as the sum of the similarities for each query vector vj query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, in the real world, the importance of each query may differ, in which case the similarity should be multiplied by a coefficient ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 4When estimating the user’s world model, the likelihood S(vi, vobs, Vq) is computed for all environments i, including those that do not contain the state squery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 4 Preprint 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 User vectors In addition to the query vector, the user vector that represents “what kind of environment with the distributed repre- sentation the user is likely to retain as a world model” can also be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The user vector is a vector that represents how the user tends to misunderstand the environment and plays a role in adding this tendency to the result of the user’s world model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' vuser = vu_pos − vu_neg, (5) where vu_pos = � i vi · P(vi) � i P(vi) , vu_neg = � i vi · (1 − P(vi)) � i(1 − P(vi)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (6) P(vi) is the probability that the user estimates the environment corresponding to the distributed representation vi as the current world model when no information about the current environment is given to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this paper, P(vi) is assumed to be known, and the estimation of P(vi) is outside the scope of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='4 Explanation by language Using a pre-trained language vector Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (7) in the representation space, the relationship between the world model maintained by the agent and the user’s world model can also be explained by language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' vword(x) = average(vm − vn), (7) where vm and vn are distributed representations of environment pairs whose relation is represented by the language x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' By averaging the differences of distributed representations of environment pairs vm and vn that satisfy a specific language description x, we can obtain a semantic vector represented by the language description x in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Using the language vector vword, the language describing the relationship between the world model maintained by the agent and the user’s world model is represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Explanation = arg max x similarity(vword(x), vEnv_est − vobs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (8) By means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (8), the language x that represents the closest relationship between the current world models is selected as the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3 Experiment The proposed method was applied to an agent that acquires action strategies by proximal policy optimization (PPO) in a simulation environment, and its usefulness was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' A partially modified version of the grid environment [12] with multiple objects was used for the experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this environment, the agent (triangle) obtains a reward by taking a key, opening a door, and reaching a goal in the lower right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The position of the goal remains unchanged, but the positions of the key and the door change every trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The agent has five actions, namely go straight, turn left, turn right, take the key from the grid in front of it, and open the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The agent observes the absolute position of the key (x, y coordinates), the absolute position of the door (x, y coordinates), its own absolute position (x, y coordinates) and orientation, holding/not holding the key, and opening/closing the door, for a total of nine dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 Experiment 1: Acquisition of a distributed representation of the world model A graph representing the state transitions of each environment was obtained simultaneously with the learning of policies by PPO, and graph2vec was used to obtain a 16-dimensional distributed representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' An example of the acquired graph is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' There are three edges that transition from the group of states before key acquisition to the group of states after key acquisition because keys can be acquired from three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' On the other hand, there is only one edge that transitions from the group of states where the door is not open to the group of states where the door is open, because the door can be opened from only one state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 5 Preprint Why donʼt you pick up Figure 5: Estimation results of the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Table 1: Cumulative frequency and average value of the order in which the optimal environment appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The number of trials is 100 for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Method Order 1 2 3 Order Average Our method 35 49 60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 Random 6 8 11 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 The representation space was compressed to eight dimensions using independent component analysis, and the visualized results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this experiment, the absolute positions of the key and door were used as environment labels, and the five-dimensional observations excluding them were used as node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The experimental results revealed that clusters are formed in the representation space based on the absolute positions of the key and door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In particular, clusters related to the position of the door are apparent, which can be attributed to the fact that the surrounding state transition relationship changes considerably compared with that of the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this experiment, the absolute positions of keys and doors are used only for identifying the environmental graph to be updated and are not embedded in the graph itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Therefore, graph2vec created a representation space that appropriately reflects the differences in the location information of keys and doors expressed in the state transition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='2 Experiment 2: Estimation of the user’s world model The query vector was applied to the obtained representation space to estimate the world model of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this verification, we assume that questions were asked in the situations of acquiring a key and opening a door, and the query “In the state Squery, we should {take the key/open the door}” was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Figure 5 displays the results of estimating the others’ world model given the reference world model and query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The environment that satisfies the query has the highest evaluation value, and the environment in which the key is located on the opposite side of the grid environment has the lowest evaluation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This result suggests that the obtained query vector is appropriate for estimating the world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The optimal environment is defined as the agent’s world model with minimal modifications for satisfying the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' For example, given the query “In state Squery, the agent should take the key,” the optimal environment is the environment in which only the position of the key is changed to satisfy the query, whereas the position of the door is left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' For each randomly selected agent world model/query pair, the evaluation values for each environment obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3) were sorted in descending order to obtain the order of appearance of the optimal environment (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The order of appearance of the environments that satisfy the query when sorted randomly is displayed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The experimental results confirmed that the proposed method ranks the optimal environments higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 6 FoFo -- ooPreprint Figure 6: An example graph of the acquired world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The blue nodes represent the state before the key acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The orange nodes represent the state after the key acquisition when the door is not open, and the green nodes represent the state when the door is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Although the direct manipulation of the positions of keys and doors can change the order of the appearance of the optimal environment to one, direct manipulation is not always possible in cases in which directly manipulatable information is not given as a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The proposed method estimates the optimal environment at an early stage, although the state transition relations to be changed are not explicitly given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This property of the proposed method is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='3 Experiment 3: Validation with multiple queries An explanatory agent A and an explained agent B are prepared, and the number of queries required for A to accurately estimate B’s world model is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This experiment assumes a situation in which the user is asked to confirm the correctness of the estimation results of the other’s world model through the presentation of an action sequence, which improves the estimation accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The outline of the experiment is as follows: (1) A and B share an initial state (absolute position and orientation of the agent, and the state of the key and door) and an environment-policy pair that specifies the strategy to be used in specific environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' They have arbitrary world models with different key and door positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (2) Here, A sets its own world model as the initial value of the other’s world model and presents the optimal sequence of actions in that model in turn (policy sharing) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3) B adds the query “In state squery, action aquery should be chosen” when the given action differs from the optimal action in its measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The existing query is not deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (4) A updates the other’s world model based on the query and presents the action sequences in the updated other’s world model again in sequence with Squery as the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (5) Repeat (3) and (4) to evaluate the number of environment updates required for A to obtain B’s world model as the other’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The environment selected once is not selected, and the second or subsequent candidate is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' If the same environment has not been obtained after all the action sequences are presented, the environment is continuously updated 5Policy sharing in this experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 8) is performed to confirm the estimation results of the other’s world model and differs from the presentation of information about one’s own policy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 7 Preprint Figure 7: Results of latent space visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Each data point is illustrated according to (a) X-coordinate of the key, (b) Y-coordinate of the key, (c) X-coordinate of the door, and (d) Y-coordinate of the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Figure 8: Schematic of experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The agent transmits information on its policy to the user and updates the user’s world model based on queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' without increasing the number of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this verification, the proposed method was compared with the “AND” search of queries as a method of updating the world model of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In practice, the following three methods are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Proposed method: Select a plausible environment using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' AND search 1: Randomly selects an environment from among the environments that satisfy the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' AND search 2: The environment is randomly selected with the constraint that “the optimal behavior for the policy B is selected in all states from the initial state until reaching state squery”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Thus, it adds constraints and increases the information provided compared with the two update methods described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Experimental results revealed that the proposed method can estimate others’ world models with the fewest number of updates (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The results of the corresponding two-tailed t-test revealed that t(100) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='07 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='01 for the proposed method and AND search 1, and t(100) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='59 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='01 for the proposed method and AND search 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Thus, both significant differences were confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 8 V V M V V V V V 会 VI V V V V V V V V V V △ W V V V W △ V △ V V V V V V V V V V V △ 1 口 I I V △△ 口 V 口 口 口 I 口 口 I 1 口 1 口口 口 口口 口 口 V V V V V V V V VV V口 0VX W V V 口 口 口 口 口 ■ 口 口 口Preprint Table 2: Number of updates required to estimate the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Method Number of updates Standard deviation Our method 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='83 AND search 1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='72 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='43 AND search 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='71 Table 3: Comparison with the use of probabilistic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Method Nuber of updates Standard deviation Our method (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='38 Our method (λ = 0) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='11 Probabilistic value (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='79 The most common information given directly is “AND search2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' By contrast, the proposed method can reduce the number of updates by vectorizing queries and utilizing prior knowledge embedded in the representation space as additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='4 Experiment 4: Comparison with the use of probabilistic evaluation We compare the proposed method with the case where the probability value of each environment satisfying the query is used as the evaluation function instead of CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The proposed method uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3) as the evaluation function, while the comparison method uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' S(vi, vobs, Vq) = � j P(aquery|vi, squery) − λ · distance(vi, vobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (9) The procedure for this experiment is the same as in experiment 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' however, the initial world model is not completely random, and the coordinates of either the key or the door are assumed to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This condition replicates the assumption that the agent’s world model and the user’s world model are similar6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Experimental results showed that the proposed method, which takes into account the distance in the representation space (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05), was able to estimate the user’s world model with the fewest number of updates (Table3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The results of the corresponding two-tailed t test showed that the proposed method with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05 compared to λ = 0 showed t(99) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='59 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='005, while the proposed method with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05 compared to the method using probability values showed t(99) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='44 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='005, both of which are significantly different from each other7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The proposed method, which takes into account the distance in the representation space, was able to estimate the environment with a significantly smaller number of updates than the method using the same coordinates for either the key or the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Compared to the method using probability value as the evaluation function, the proposed method was able to absorb small errors in probability values, resulting in a significantly smaller number of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' When the distance in the representation space corresponds to the similarity of the state transition structure between environments, as in the present verification, it is effective to use CAV to obtain environments that are perpendicular to the virtual separation boundary as the user’s world models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='5 Experiment 5: Number of samples and accuracy of CAV We evaluate the number of queries required to correctly estimate the world model when the number of samples (number of environments) used to compute the CAV is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The evaluation procedure is the same as in Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this experiment, the maximum number of samples is 300 because 300 environments are embedded in the representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' We also set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 6Without this assumption, the world model with minimal modification to satisfy the query (the optimal environment) is not necessarily the user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, theoretically, when the evaluation value is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (3), other environments that satisfy the query will have a lower evaluation value compared to the optimal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Therefore, if the assumption that the world models of the agent and user are similar cannot be made, it is desirable to use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, in a real environment, the world models of the agent and user are not completely independent, and similarity can be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 7Because the t test was applied twice in this verification, a significant difference was found at the significance level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='01 based on the Bonferroni method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 9 Preprint Table 4: Relationship between the number of CAV samples and the order of appearance of the optimal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Each value represents the cumulative frequency, and the number of trials is 100 for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Nuber of samples Order of appearance 1 2 3 300 40 62 69 250 44 61 64 200 42 51 62 150 40 51 55 100 35 46 54 50 9 18 26 Prior distribution 1 Prior distribution 2 P(door_y = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='4 P(door_y = 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='3 P(door_y = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='2 P(door_y = 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 P(door_y = 5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='0 P(door_y = 6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='0 P(door_y = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='0 P(door_y = 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 P(door_y = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='4 P(door_y = 4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='4 P(door_y = 5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='1 P(door_y = 6) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='0 Figure 9: The prior distribution used in experiment 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The prior distribution for the y-coordinate of the door (in the vertical direction) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The experimental results show that the accuracy deteriorates much more slowly up to 100 samples than when the CAV is generated with 300 samples (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This suggests that a specific level of estimation accuracy can be maintained even when the number of samples is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The fact that the user’s world model is estimated taking into account the distance in the representation space may also contribute to maintaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' On the other hand, the accuracy dropped drastically when the number of samples was 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' This may be because of an increase in the number of trials in which the number of positive data (the number of data satisfying the query) in the sample is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='6 Experiment 6: Use of the user vector We test whether the estimation accuracy of the other-world model can be improved by using the user vector defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (5), as opposed to using only CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this verification, two types of prior distributions are defined for the y-coordinate of the door (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 9), and the user vector is calculated for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The user vector is treated as one of the query vectors in the evaluation value calculation for each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Note that λ = 0 is assumed in this verification because there is no assumption that the world models held by agents A and B are similar8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The estimation results of the other-world model for the same reference environment and query are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' User vectors 1 and 2 correspond to prior distributions 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' It can be seen that while the results of the inference of the door location are unstable when only the query vector is applied, the results of the inference using the other vectors show that the door coordinates are concentrated in locations that have high probability values in the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The same validation as in experiment 3 was conducted by applying the prior distribution shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 9, and the number of queries required for both distributions 1 and 2 was lower when using the user vector (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The results 8If this experiment is conducted under the assumption that the world models of agents A and B are similar, it is necessary to set the number of objects that determine the state transition structure of the environment to three or more and that they are placed at the same coordinates as the current environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Under these conditions, it is desirable to set λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 10 Fo Preprint Why don’t you pick up the key at the state?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Base environment Query vector Query vector +User vector 1 Query vector +User vector 2 Highest Score Lower Figure 10: User vector and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Table 5: Change in estimation accuracy when using user vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The number of trials is 100 for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Distribution 1 Distribution 2 Method Number of updates Standard deviation Number of updates Standard deviation Query + User 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='34 Query 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='46 of the corresponding two-tailed t test showed that t(99) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='34 and p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='05 for distribution 1 and t(99) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='45 and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='15 for distribution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' These results suggest that the number of queries required for estimation can be reduced by using the user vector, although the size of the effect depends on the shape of the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='7 Experiment 7: Explanation by language We test whether the language vectors learned in advance can be used to correctly output language that describes the relationship between the agent’s world model and user’s world model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The language vectors used in this study are eight different explanations, such as “In the world model assumed by the user, {key, door} is located at {upper, lower, right, left} than in the world model maintained by the agent.” In the experiment, language explanations were first given to n pairs of world models with different coordinates of keys or doors, and the language vectors were obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' We then generated linguistic explanations for randomly selected pairs of world models using the same conditions as those used to generate the language vectors and evaluated the percentage of the explanations that correctly explained the relationships between the world models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=', the percentage of correct responses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The percentage of correct responses after 100 trials is shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Note that if more than one linguistic explanation correctly represented the relationship between the world models, both were considered as correct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' For example, if the key is located on the upper right, “the key is on the right” and “the key is on the top” are treated as correct answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The “1st” in the table indicates the percentage of correct explanations generated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The “1st and 2nd” represents the percentage of correctness of the two languages that were the first and second most similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Note that “1st and 2nd” was evaluated only in cases where more than one language explanation was considered to be a correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Although the number of world model pairs considered in this experiment is approximately 9000, the experimental results show that language explanation generation is possible with high accuracy even when the number of data used for language vector acquisition is n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The accuracy was also maintained even when the number of data was extremely reduced to n = 100 and n = 50, suggesting that the learning in the representation space is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In this experiment, only eight language vectors were set, but it is expected that more accurate explanation generation will become possible by setting more detailed language vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' However, it should be noted that in these cases, sufficient teacher data is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' 11 -------Fo Preprint Table 6: Accuracy of language description generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' n 1st 1st and 2nd 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='67 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='73 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='68 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='63 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='57 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='54 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='37 4 Conclusion In this study, a novel method was proposed for estimating the user’s world model from the robot’s world model and the query given by the user to obtain XAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' The proposed method can estimate others’ world models more efficiently than using the “AND” search of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' In the future, user vectors should be introduced, and the methods for generating explanations using differences in world models should be devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' Acknowledgments This study was supported by the New Energy and Industrial Technology Development Organization (NEDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' References [1] Sakai, Tatsuya and Nagai, Takayuki, "Explainable Autonomous Robots: A Survey and Perspective," Advanced Robotics, 36(5-6), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content='219-238, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE2T4oBgHgl3EQfSgdx/content/2301.03793v1.pdf'} +page_content=' [2] Ha, David and Schmidhuber, Jürgen, "Recurrent World Models Facilitate Policy Evolution," In Advances in Neural Information Processing Systems 31, pp.' metadata={'source': 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machine learning. We derive a theory of human-like few- +shot learning from von-Neuman-Landauer’s principle. Modelling human learn- +ing is difficult as how people learn varies from one to another. Under com- +monly accepted definitions, we prove that all human or animal few-shot learn- +ing, and major models including Free Energy Principle and Bayesian Program +Learning that model such learning, approximate our theory, under Church- +Turing thesis. We find that deep generative model like variational autoencoder +(VAE) can be used to approximate our theory and perform significantly better +than baseline models including deep neural networks, for image recognition, +low resource language processing, and character recognition. +Introduction +During the past decade, fast progress in deep learning (1) has empowered computer speech +recognition, image processing, natural language processing, protein folding, game playing and +many other applications. However, these great progresses fell short when we try to understand +our own learning mechanism: How to model human learning (2), (3), (4)? +Species in nature learn quickly to survive. When a dragonfly is hatched, within hours it +firms up its wings and then flies to catch mosquitoes; a newborn does not need tons of repeated +examples or transfer learning to identify an apple. Most human or animal learning exhibits a +mixture of inherited intelligence, few-shot learning without prior knowledge, as well as long +term many-shot learning. It is interesting to note that these learning programs are encoded in +our genomes but they are not all the same, even for individuals within the same species. The +diversity of these learning algorithms is vividly expressed by Spearman’s "g" factor (2). +Work in progress. +1 +arXiv:2301.01047v1 [cs.LG] 3 Jan 2023 + +Unlike data-laden, model-heavy, and energy-hungry deep learning approaches, most human +learning appear to be simple and easy. Merely scaling up current deep learning approaches may +not be sufficient for achieving human level intelligence. We miss certain major components +when modelling human or animal learning. +Diversity is one of the missing part when modelling human or animal few-shot learning. +There are eight billion people on earth, each with a unique few-shot learning model (5). Even if +we just want to model one person, a single person often uses different parameters, features, and +perhaps different algorithms to deal with different learning tasks. Ideally we want a framework +that can cover the diversity in human and animal few-shot learning. Facing such a seemingly +formidable task, traditional thinking in machine learning will only lead us to various traps. To +avoid such traps we need to go back to the very first principles of physics. +Specifically, we start from an agreed-upon law in thermodynamics, to formally derive our +model for few-shot learning, and prove this is the optimal model within our framework in the +sense that all other models including human ones may be viewed as approximations to our +framework. We show a deep connection between our framework and the free energy principle (3) +and the Bayesian Program Learning model (4). By the end of this process, a component of data +compression during the inference phase of learning emerges as a key component of all few-shot +learning models. +First, we formalize our intuitive and commonly accepted concept of human-like few-shot +learning. For example, our definition below is consistent with what is used in (4), and in the +same spirit of (3). +Definition 1. Consider a universe Ω, partitioned into H disjoint concept classes: Ch, h = +1, 2, . . . , H. Few-shot (k-shot) learning is described as follows: +1. n elements in or outside Ω are given as unlabelled samples y1, . . . , yn; +2. There are k labelled examples for each class Ch, for small k; +3. The learning program, using a computable metric M, few-shot learns Ch, h = 1, 2, ...H, +if it uses the n unlabelled samples and k labelled samples and minimizes the objective +function: +H +� +h=1 +|Ch| +� +i=1 +M(xi, coreh) | y1, . . . , yn, xi ∈ Ch, +where coreh = ψ(k samples of Ch) representing a transformed representation of the k +labelled samples from Ch. +This definition covers most of our common sense few-shot learning scenarios and other +studies. In particular, this is used in one-shot learning by (4). As each independent individual, +we do not all use a same metric, or even similar metric, to few-shot learning. For example, +MN Hebart et al (6) identified 49 highly reproducible dimensions to 1854 objects to measure +2 + +their similarity. Different people can be equipped to better observe some of these dimensional +features. +We explain the intuition behind Definition 1 via a simple example. A human toddler may +have already seen many unlabelled samples of fruits which, for example, contains two classes: +apples and pears. Then given a new labelled sample from each class, the toddler learns how to +differentiate between these two fruits. The number of labelled data required for one to classify +may vary as people have different learning algorithms. +Current deep learning based approaches for few-shot learning generally depend on 1) many +auxiliary labelled training samples or task-specific data augmentation for transfer learning or +meta learning (7); or 2) very large scale self-supervised pre-training (8). These approaches thus +fall short to model few-shot learning in nature by humans and animals as they can hardly account +for the diversity in learning algorithms and they either neglect the unsupervised scenario that +humans are mostly exposed to or use the scale of unlabelled data and training parameters that +are far beyond creatures need. +Many attempts have been made to understand human learning through cognitive, biological, +and behavior sciences. Some studies have established basic principles a human learning model +should obey. One theory is the two-factor theory of intelligence by Charles Spearman in 1904 (2), +where the “g” factor is an indicator of the overall cognitive ability, and the “s” factor stands for +the aptitude that a person possesses in specific areas. As “g” factor is genetically-related (9), it +indicates the necessity of a learning theory that can account for the diversity in creatures’ learning +ability. Another theory is the Free Energy Principle by Karl Friston (3) that human (and all +biological systems) learning tends to minimize the free energy between internal understanding in +the sense of Bayesian (under internal perceived distribution p) and that of the environmental event +(under distribution q), measured by KL-divergence (10). In a similar spirit, Lake, Salakhutdinov +and Tenenbaum (4) proposed a Bayesian program learning (BPL) model, learning a probabilistic +model for each concept and achieve human-level performance. Two articles by Schmidhuber (11) +and by Chater and Vitanyi (12) linked simplicity to human cognition and appreciation of arts. +Instead of exploring a biological basis for few-shot learning, we think it is possible to +mathematically derive an optimal framework that can unify the above theories. We further +demonstrate by experiments that our new model indeed works significantly better than other +classical deep learning neural networks for few-shot learning. As a byproduct of our new model, +a new concept class of "interestingness" is learned; this class implies where our appreciation +of art, music, science and games comes from. Extending this observation, some aspects of +consciousness may be modelled as a set of few-shot learned concepts. Consequently, we +hypothesize the ability of labelling input data becomes a key step to acquiring some aspects of +consciousness. +3 + +A theory of few-shot learning +We mathematically derive an optimal few-shot learning model for Definition 1 that is effective +and is able to cover enormous diversities existed in different species. The task may appear to be +formidable because of conflicting and seemingly very general goals: each individual is allowed +to have a different learning model, yet our model has just one program to model everybody; +we do not yet exactly know the complete underlying biological mechanisms, yet we need to +implement the right functionality; there are infinite number of models, yet we need to choose +one that is optimal; we are not really interested in "proposing models" out of blue, yet we wish +our model to be a mathematical consequence of some basic laws of physics; the model needs to +be theoretically sound, yet practically useful. +For simplicity and readability, we begin with one-shot learning, k = 1 in Definition 1. Thus, +coreh in Definition 1 is just the single labelled sample xh. For larger k, coreh can be some form +of average of the k samples. As Definition 1 defined, some unlabelled objects are assumed and +it’s also possible to extend the definition by adding distribution, learnt from either unlabelled or +labelled data, to Ω. Using metric M that is responsible for k-shot learning of an individual, the +learning system seeks to minimize the energy function +H +� +h=1 +|Ch| +� +i=1 +M(xi, xh|y1, . . . , yn), +or, assuming H(y1, . . . , yn) is a pre-trained model of y1, . . . , yn, or other labelled samples, +capturing the distribution. +H +� +h=1 +|Ch| +� +i=1 +M(xi, xh|H(y1, . . . , yn)), +Now the question is, what sort of M should we use? Indeed, this varies from person to person. +Can we unify all such measures, algorithms and inferences? Let’s go back to the fundamentals. +Principle 1 (von-Neuman-Landauer Principle). Irreversibly processing 1 bit of information costs +1kT; reversible computation is free. +Then for two objects x, y, the minimum energy needed to convert between x and y in our +brain is: +EU(x, y) = min{|p| : U(x, p) = y, U(y, p) = x}, +where U is a universal Turing machine or our brain, assuming Church-Turing thesis. Since we +can prove a theorem showing all Universal Turing machines are equivalent modulo a constant and +efficiency, we will drop the index U (see (13)). To interpret, E(x, y) is the length of the shortest +program that reversibly converts between x and y. These bits used in the shortest program p +when they are erased will cost |p|kT of energy, according to the John von Neuman and Rolf +Landuaer’s law. This leads us to a fundamental theorem (14): +4 + +... +... +degree +degree +Figure 1: Bipartite Graph +Theorem 1. E(x, y) = max{K(x|y), K(y|x)} + O(1). +K(x|y) is the Kolmogorov complexity of x given y, or informally, the length of the shortest +program that outputs x given input y (details are shown in (13)). As this theorem was proved +thirty years ago and it is vital in our theory, to help our readers, we will provide an intuitive but +less formal proof here. +Proof. By the definition of E(x, y), it follows E(x, y) ≥ K(x|y) and E(x, y) ≥ K(y|x), thus +we have E(x, y) ≥ max{K(x|y), K(y|x)}. +To prove the other direction E(x, y) ≤ max{K(x|y), K(y|x)}, we need to construct a +program p such that p outputs y on input x and p outputs x on input y, and length of p is +bounded by max{K(x|y), K(y|x)} + O(1). +Let k1 = K(x|y), and k2 = K(y|x). Without loss of generality, assume k1 ≤ k2. We first +define a bipartite graph {X, Y, E}, where X, Y = {0, 1}∗, as shown in Figure 1 and E is a finite +set of edges defined between X and Y as follows: +E = {{u, v}, u ∈ X, v ∈ Y, K(u|v) ≤ k1, K(v|u) ≤ k2} +Note that a particular edge (x, y) is in E. If we find edge (x, y), then given x, p can output y, +and vice versa. So the idea of the proof is to partition E properly so that we can identify (x, y) +easily. Two edges are disjoint if they do not share nodes on either end. A matching in graph +theory is a set of disjoint edges in E. +Claim. E can be partitioned into at most 2k2+2 matchings. +Proof of Claim. Consider edge (u, v) ∈ E. The degree of a node u ∈ X is bounded by 2k2+1 +because there are at most 2k2+1 different strings v such that K(v|u) ≤ k2, accumulating possible +strings from i = 1 to i = k2 gives us �i=k2 +i=1 = 2k2+1 − 2. Hence u belongs to at most 2k2+1 +matchings. Similarly, node v ∈ Y belongs to at most 2k1+1 matchings. We just need to put edge +(u, v) in an unused matching. (End of Proof of Claim) +Let Mi be the matching that contains edge (x, y) We now construct our program p. p operates +as follows: +• Generate Mi following the proof of Claim, i.e. enumerating the matchings. This uses +information k1, k2, and i. K(i) ≤ k2 + O(1) +5 + +• Given x, p uses Mi to output y, and given y, p uses Mi to output x. +A conditional version of Theorem 1, using information in Definition 1, can be obtained +E(x, y|y1, . . . , yn) = max{K(x|y, y1, . . . , yn), K(y|x, y1, . . . , yn)}, conditioning on unlabelled +samples y1, . . . , yn. According to (14), this distance is universal, in the sense that E(x, y) is the +minimum among any other computable distances: +Theorem 2. For any computable metric D, there is a constant c, such that for all x, y, E(x, y) ≤ +D(x, y) + c. +This theorem implies: if D metric finds some similarity between x and y, so will E. Thus, +the above theorem implies, up to some constant O(H) +H +� +h=1 +|Ch| +� +i=1 +E(xi ∈ Ch, coreh|y1, . . . , yn) ≤ +H +� +h=1 +|Ch| +� +i=1 +M(xi ∈ Ch, coreh|y1, . . . , yn). +When unlabelled samples y1, . . . , yn plus other irrelevant historical labelled samples are modeled +by some model H such as a generative model (e.g., VAE), then the above inequality can be +rewritten as: +H +� +h=1 +|Ch| +� +i=1 +E(xi ∈ Ch, coreh|H) ≤ +H +� +h=1 +|Ch| +� +i=1 +M(xi ∈ Ch, coreh|H). +(1) +Thus, E gives optimal metric for few-shot learning algorithm. Other algorithms satisfied +Definition 1 are the approximation to this optimal solution. 1 +In addition, we show that our theory’s deep connection to two well-established principles +of learning in neuroscience and psychology. Friston’s Free Energy Principle (FEP) (3), derived +from Bayesian brain hypothesis (15), states that brain seeks to minimize surprises. Specifically, +it assumes the brain has its internal state (a.k.a. generative model) that implicitly models the +environment according to the sensory data. Hidden (latent) variables need to be defined for +the internal state, which are drawn from prior beliefs. Ideally, these prior knowledge is also +modelled, which is made possible by hierarchical generative models. The free energy principle +(FEP) is often interpreted as Bayesian optimization, using the Evidence Lower Bound (ELBO) +as ELBO = log p(x; θ) − D(q(z)∥p(z|x; θ) optimization function. Here the evidence log p(x; θ) +is the encoding length of x under probability p, and the Kullback-Leibler divergence term is the +p-expected encoding length difference. This is half of Theorem 1 and FEP is asymmetric if we +view it as a distance. However, the symmetry is important to few-shot learning. For example, a +scarlet king snake may look like a coral snake, but the latter certainly has more deadly features +the former lacks, one way compression K(ScarletKingSnake|CoralSnake) is not sufficient to +1Note that E is a metric: it is symmetric, and satisfies triangle inequality +6 + +Compressor +Unlabeled Data +Distribution +Test Instance +Figure 2: Illustration of our framework, dashed line indicates optional component when learning. +distinguish the two. Despite of the fact H. influnza with genome size 1.8 million and E. coli with +genome size 5 million they are sister species but E. coli would be much closer to a species with +zero genome G0 or just a covid-19 genome with this asymmetric measure (K(G0|E.coli) than +with H. influnza (K(H. influnza|E. coli)). A symmetric interpretation of Friston’s FEP can be +derived by requiring minimum conversion energy as we show in Theorem 1. +Different individuals may use different compression algorithms to do data abstraction and +inference. It can be viewed that these algorithms all approximate E(x, y). Some are more efficient +than others in different situations. The individuals with better compression algorithms have +bigger “g” factor. Diversified compression algorithms also guarantee better survival chances of a +community when facing a pandemic. As compression neural networks are genetically encoded, +the “g” factor is thus inheritable. This can be seen via Figure 2, compression algorithms vary +from one to another. The distribution of the data to be learnt is either implicitly or explicitly +captured by creatures. Those who can better utilize unlabelled data to capture distribution may +have a more efficient compression algorithm. +Experimental Results +Image Experiments +To approximate our universal few-shot learning model, we use a hierarchical VAE as our +underlying model H in Inequality 1 to model the unlabelled samples y1, . . . , yn. This hierarchical +structure coincides with our visual cortex and brain structure (16). According to integrated +information theory (17), an input y may come from all sensing terminals: vision, hearing, +smell, taste, sensation. Often, creatures are exposed to an unsupervised environment where +objects are unknown and unlabelled. Revisiting the negative ELBO, we can see it can be +interpreted as changing perceptions to minimize discrepancy (minimize KL divergence) or +changing observations to maximize evidence, in the context of FEP. When the creatures are +7 + +exposed to a “tree” and they do not fully realize what it is, the sensory information of the +objects are internalized with hidden states (inner belief) that can describes how it believes the +generation process of a “tree”. This process of generation, helps the creatures to identify the +latent similarities among objects that belong to the same category, without the full awareness. +This process of "unconsciously" training to generate helps the creatures to better categorize in +future. When the identity of a “tree” is finally revealed, they can generalize quickly. This explains +our rationale of using a VAE to process unlabelled samples. Consequently, the Kolmogorov +complexity terms in Inequality 1 are naturally approximated by a VAE based compressor (18). +To test the hypothesis, we carry out the experiment on five datasets, MNIST, KMNIST, +FashionMNIST, STL-10 and CIFAR-10. We first train a hierarchical VAE on unlabelled data +to learn to generate ˆx that’s as close to x as possible. This corresponds to the time when +creatures exposed to a environment without knowing the object, implicitly learning the latent +representation among objects. When the identity of objects are revealed, a VAE based universal +compressor can be used to identify the new objects. Specifically, after training a hierarchical +VAE unsupervisedly, we compare the E energy function between a labelled image and a test +image, as in Definition 1. In our experiment, we use 5 labelled samples per class to test the +accuracy of classification. The energy function E relies on a compressor to approximate. We thus +use the bits-back argument to directly use our trained VAE for the compressor in (18). Our result +shows that using only 5 samples, our method outperforms traditional supervised models like +SVM, CNN, VGG and Vision Transformer (ViT) on all five datasets. These supervised methods +are chosen to represent different model complexity with wide range of number of parameters. +As we can see, when labelled data are scarce, supervised methods are not effective: complex +models like VGG cannot perform better than SVM and this tendency is more obvious on ViT +without pre-training. The improvement that our method brings is more obvious on more complex +datasets like STL-10 and CIFAR-10. Similar result is also obtained in the recent work, across +different shot settings (19). +We also compare with using latent representation directly with k-Nearest-Neighbor classifier, +labelled as “Latent” in the table. The architecture and training procedure for “Latent” method +is exactly the same to our method — we train on unlabelled data to generate the sample and +then take the latent representation for classification. We can see using latent representation +outperforms all supervised methods on four out of five datasets. But the accuracy is still way +lower than our method, indicating our method can better utilize the generative models. +Text Experiments +Our theory is generally applicable, even without pre-training on unlabelled data. Here, we +demonstrate significant advantages of our approach with a simple compressor gzip over lower +resource languages. +Languages with Abundant Resources +We first test our method on datasets with abundant +resources. Specifically, we compare with three datasets — AG News, SogouNews and DBpedia. +8 + +MNIST +KMNIST +FashionMNIST +STL-10 +CIFAR-10 +SVM +69.4±2.2 +40.3±3.6 +67.1±2.1 +21.3±2.8 +21.1±1.9 +CNN +72.4±3.5 +41.2±1.9 +67.4±1.9 +24.8±1.5 +23.4±2.9 +VGG +69.4±5.7 +36.4±4.7 +62.8±4.1 +20.6±2.0 +22.2±1.6 +ViT (disc) +58.8±4.6 +35.8±4.1 +61.5±2.2 +24.2±2.5 +22.3±1.8 +Latent +73.6±3.1 +48.1±3.3 +69.5±3.5 +31.5±3.7 +22.2±1.6 +Ours +77.6±0.4 +55.4±4.3 +74.1±3.2 +39.6±3.1 +35.3±2.9 +Table 1: 5-shot image classification accuracy on five datasets. +AG News +SogouNews +DBpedia +fasttext +27.3±2.1 +54.5±5.3 +47.5±4.1 +Bi-LSTM+Attn +26.9±2.2 +53.4±4.2 +50.6±4.1 +HAN +27.4±2.4 +42.5±7.2 +35.0± 1.2 +W2V +38.8±18.6 +14.4±0.5 +32.5±11.3 +BERT +80.3±2.6 +22.1±4.1 +96.4±4.1 +Ours +58.7±4.8 +64.9±6.1 +62.2±2.2 +Table 2: 5-shot text classification accuracy on three datasets. +Similar to image classification, we compare with both supervised methods, including fasttext (20), +BiLSTM (21) with attention mechanism (22) and Hierarchical Attention Network (HAN) (23), +and non-parametric methods that use Word2Vec (W2V) (24) as representation. We also compare +with pre-trained language models like BERT (25) We use five labelled data for each class (5-shot) +for all the methods. +Surprisingly, even without any pre-training and with a simple compressor like gzip, our +method outperforms all non-pretrained supervised methods and non-parametric methods in +low data regime. This indicates that compressor serves as an efficient method to capture the +regularity and our information distance is effective in comparing the similarity based on the +essential information. When comparing with pre-trained models like BERT, we can see our +method is significantly higher on SogouNews, a special dataset that includes Pinyin — a phonetic +romanization of Chinese, which can be viewed as an Out-Of-Distributed (OOD) dataset as it +uses the same alphabet as english corpus. +Low-Resource Languages +Sufficiently pre-trained language models are exceptional few-shot +learners (8). However, when faced with low resource data or distributions that are significantly +different from any pre-trained data, those pre-trained language models lose their advantages +to our method. We compare our method with BERT on four different low-resource language +datasets - Kinyarwanda, Kirundi, Swahili and Filipino. These datasets are curated +to have the Latin alphabets, same as english corpus. BERT has performed extremely well as +9 + +Kinnews +Kirnews +Swahili +Filipino +BERT +24.0±6.0 +38.6±10.0 +39.6±9.6 +40.9±5.8 +mBERT +22.9±6.6 +32.4±7.1 +55.8±16.9 +46.5±4.8 +Ours +45.8±6.5 +54.1±5.6 +62.7±7.2 +65.2±4.8 +Table 3: 5-shot text classification accuracy on low-resource datasets +shown in Table 2 due to pre-training on billions of tokens. However, when facing low-resource +datasets, BERT perform significantly worse than our method only using gzip as we can see +in Table 3, no matter using multilingual pre-trained version or the original one. Note that mBERT +is pre-trained on 104 languages including Swahili and Tagalog (on which Filipino is based +on). As we can see on Swahili and Filipino, mBERT performs better than BERT, but still +significantly lower than our method. +Omniglot one-shot-classification dataset +Figure 3: Distance between two Bezier curves +In (4), +a one-shot learning framework +Bayesian program learning (BPL) was pro- +posed. It learns a simple probabilistic model +for each concept. Taking a negative logarithm +converts a Bayesian formula to a description +length paradigm, hence BPL can be viewed +as one particular approximation to our theory. +Here we provide another simple approxima- +tion of our theory for the Omniglot one-shot- +classification dataset of (4). +Our system first decompose a given char- +acter into strokes, then compute E(a, b) be- +tween characters a and b, using all their possi- +ble stroke decomposition. We provide how to +calculate E(a, b) here and details of decompo- +sition program is given in Appendix A. +1. Fit a stroke by a Bezier curve; +2. Ensure the number of points on two curves are same. This algorithm utilize equally split +method to select certain same number of points on each curve Figure 3; +3. Ensure the area of the convex hull and the barycenter of the compared characters are the +same; +10 + +0 +-20 +-40 +-60 +-80 +-100- +0 +20 +40 +60 +80 +1004. Use max Cartesian distance between parallel points on two Bezier curves to approximate +the minimum encoding distance between two Bezier curves, as shown in Figure 3; +5. Choose the character with minimum distance. +This simple implementation achieves 92.25% accuracy 20-way-1-shot on this dataset. The +point here is to demonstrate various approximations of our theory that work rather than com- +paring accuracy. At 96.75% (4) or at 92.25% might be two different individuals with different +compression algorithms. +Unification +Our framework can unify other popular deep neural networks for few-shot learning. +Siamese Network: Siamese network uses twin subnetwork to rank the similarity between +two inputs in order to learn useful features. M here is often a contrastive loss. This framework +shows strong performance in one-shot image recognition (26). +Prototypical Network: Prototypical networks (27) propose to optimize the distance metric +M directly by learning coreh in representation space. coreh are represented as the mean of +embedded support samples. +Bi-Encoder: In the context of natural language processing, one of the dominant structure +is the Bi-Encoder design with each encoder being a pre-trained language model. For example, +in information retrieval, Dense Passage Retrieval (DPR), with two encoders encoding query +and document respectively, has become the new state of the art. To capture semantic similarity, +sentenceBERT (28) also adopts the bi-encoder design and becoming one of the most prevalent +methods for semantic textual similarity. M in both cases can either be cosine similarity or +Euclidean distance between the representation learned through pre-trained models. +Information Distance based Methods: Hundreds of algorithms were published, before the +deep learning era, on parameter-free data mining, clustering, anomaly detection, classification +using information distance E (29–34), with a comprehensive list in (13). Recently (19) have +discovered using information distance with deep neural networks and leverage the generalizability +of few-shot image classification. This work shows that with the help of deep generative models, +unlabelled data can be better utilized for few-shot learning under our framework. +Conclusion and a discussion on consciousness +We have defined human-like few-shot learning and derived an optimal form of such few-shot +learning. Note there is an interesting difference between our theory and classical learning +theory. In classical learning theory, it is well-known that if we compress training data to a +smaller consistent description, whether it is a classical Bayesian network or a deep neural +networks (13,35), we would achieve learning. In this paper, we demonstrate that in the inference +11 + +stage, compression is also important, especially when there are not enough labelled data to +train a small model. On the biological side, compression circuits using predictive coding in +human cortex has been studied by (36). Experiments have also strongly supported our theory. +We expect to see more practical systems approximating our theory can be implemented to +solve commonplace few-shot learning problems when large amounts of labelled data for deep +learning is lacking. We now wish to explore two consequences of our few-shot learning model, +to consciousness. +A binary classifier of interestingness +Our few-shot learning model has a by-product. We have proved compression is a universal goal +that few-shot learning algorithms approximate. Thus this implies immediately a (subconscious) +binary classifier: if something is compressed, then something interesting happens, and attention +is given. It turns out that this "Interestingness" has been theoretically studied as logical depth first +proposed by Charles Bennett (13). According to Bennett, a structure is deep if it is superficially +random but subtly redundant. When few-shot learning happens, significant compression happens, +and these deep objects gain attention. Such a binary classifier might explain our appreciation +of arts, music, games, and science, since these all share a common feature of dealing with +non-trivially compressible objects: whether it is a shorter description of the data that gives rise +of Newton’s laws (13), or a piece of art or music that itself is compressible or that reminds us of +something we have experienced before, hence very compressible, we feel we understand it and +hence appreciate it. Science is nothing but compressing data into simpler descriptions of nature. +Consciousness and the ability of labelling data +Do other species have consciousness? It is difficult to answer this question as consciousness is +not testable. Thomas Nagel (37) made a comment: We will never know if a bat is conscious +because we are not bats. +Consider an alternative data-driven approach by asking what a species can do instead of how +they feel. That is, if we treat some aspects of consciousness as a collection of learned concepts, +then given a compression network, the ability of acquiring the relevant concepts becomes a matter +of labelling relevant data. We know learning and consciousness are both located at posterior +cortex region (38). This is in agreement with some injured patients when they lost consciousness. +This is also in agreement with “bistable perception” training results with monkeys (39). +Varieties of consciousness are being pragmatically studied (40). These include: 1) the ability +of consciously perceive the environment; 2) the ability of evaluating conscious emotions; 3) the +ability of having a unified conscious experience; 4) the ability of integrating across time as a +continuous stream, one moment flowing into the next; 5) the conscious awareness of oneself +as distinct from the world outside. Many of these abilities may be seen as a few-shot learnable +concepts, given properly labelled data. +12 + +Different animals have various levels of some of such consciousness by passing certain tests. +For example, chimpanzees, dolphins, Asian elephants, and magpies can recognize themselves +by passing some mirror-mark tests. The corvids display some emotions, and are able to plan +ahead. Octopus have powerful perceptual facilities obtaining and processing data independently +with each tentacle. Experimentally, awareness emerges when information travels back and forth +between brain areas (41) instead of a linear chain of command. +According to our theory, the brain really only needs to use a universal compressor to compress +information, regardless of one processor in the head or a few processors in the tentacle (in case +of Cephalopods). Thus we can conjecture that "consciousness” then is a matter of ability of +labelling the data from sensory terminals. Food or enemy in the environment are easy to label. +Emotional labelling requires some level of abstraction. Self-awareness of “me” and “others” +thus is just another binary classifier trainable depending on if the species is able to do “displaced +reference” mental labelling. Other than the human beings, only orangutans are known to have +limited displaced reference ability (42). +Thus we have just reduced the non-testable question of whether an animal has consciousness +in some aspects to if it is able to label the corresponding data properly. +Acknowledgement +We thank Dr. Hang Li for suggestions and bringing (43) to our attention and Dr. Amy Sun for +bringing (44) to our attention. The work is supported in part by Canada’s NSERC operating grant +OGP0046506, Canada Research Chair Program, and the Leading Innovative and Entrepreneur +teams program of Zhejiang, number 2019R02002, and NSFC grant 61832019. +References and Notes +1. Y. LeCun, Y. Bengio, G. Hinton, Nature 521, 436 (2015). +2. C. Spearman (1961). +3. K. Friston, Nature Review Neuroscience 11, 21 (2010). +4. B. M. Lake, R. Salakhutdinov, J. B. Tenenbaum, Science 350, 1332 (2015). +5. E. Stern, npj Science of Learning 2, 1 (2017). +6. M. Hebart, C. Zheng, F. Pereira, C. Baker, Nature, Human Behaviour pp. 1173–1185 (2020). +7. C. Finn, P. Abbeel, S. Levine, International conference on machine learning (PMLR, 2017), +pp. 1126–1135. +8. T. Brown, et al., Advances in neural information processing systems 33, 1877 (2020). +13 + +9. T. J. Bouchard Jr, Annals of Human Biology 36, 527 (2009). +10. M. N. Bernstein, mbernste.github.io/posts/elbo/ . +11. J. Schmidhuber, arXiv:0812.4360v2 [cs.AI] (2009). +12. N. Chater, P. Vitányi, Trends in Cognitive Sciences 7, 19 (2003). +13. M. Li, P. Vitányi, An Introduction to Kolmogorov Complexity and Its Applications (Springer- +Verlag, 1993, 1997, 2008, 2019). +14. C. Bennett, P. Gács, M. Li, P. Vitányi, W. Zurek, IEEE Trans. Inform. Theory 44, 1407 +(1998). +15. D. C. Knill, A. Pouget, TRENDS in Neurosciences 27, 712 (2004). +16. K. Friston, PLoS computational biology 4, e1000211 (2008). +17. C. Koch, G. Tononi, Scientific American 304 (2011). +18. J. Townsend, T. Bird, D. Barber, International Conference on Learning Representations +(2018). +19. Z. Jiang, Y. Dai, J. Xin, M. Li, J. Lin, Advances in Neural Information Processing Systems +(2022). +20. A. Joulin, E. Grave, P. B. T. Mikolov, EACL 2017 p. 427 (2017). +21. M. Schuster, K. K. Paliwal, IEEE transactions on Signal Processing 45, 2673 (1997). +22. Y. Wang, M. Huang, X. Zhu, L. Zhao, Proceedings of the 2016 conference on empirical +methods in natural language processing (2016), pp. 606–615. +23. Z. Yang, et al., Proceedings of the 2016 conference of the North American chapter of +the association for computational linguistics: human language technologies (2016), pp. +1480–1489. +24. T. Mikolov, K. Chen, G. Corrado, J. Dean, arXiv preprint arXiv:1301.3781 (2013). +25. J. Devlin, M.-W. Chang, K. Lee, K. Toutanova, Proceedings of the 2019 Conference of +the North American Chapter of the Association for Computational Linguistics: Human +Language Technologies, Volume 1 (Long and Short Papers) (2019), pp. 4171–4186. +26. G. Koch, R. Zemel, R. Salakhutdinov, et al., ICML deep learning workshop (Lille, 2015), +vol. 2, p. 0. +14 + +27. J. Snell, K. Swersky, R. Zemel, Advances in neural information processing systems 30 +(2017). +28. N. Reimers, I. Gurevych, Proceedings of the 2019 Conference on Empirical Methods +in Natural Language Processing and the 9th International Joint Conference on Natural +Language Processing (EMNLP-IJCNLP) (2019), pp. 3982–3992. +29. M. Li, et al., Bioinformatics 17, 149 (2001). +30. E. Keogh, S. Lonardi, C. A. Ratanamahatana, Proceedings of the tenth ACM SIGKDD +international conference on Knowledge discovery and data mining (2004), pp. 206–215. +31. C. H. Bennett, M. Li, B. Ma, Scientific American 288, 76 (2003). +32. M. Nykter, et al., Physical review letters 100, 058702 (2008). +33. D. Benedetto, E. Caglioti, V. Loreto, Physical Review Letters 88, 048702 (2002). +34. M. Nykter, et al., Proceedings of the National Academy of Sciences 105, 1897 (2008). +35. Y. Bengio, et al., Foundations and trends® in Machine Learning 2, 1 (2009). +36. R. P. Rao, D. H. Ballard, Nature neuroscience 2, 79 (1999). +37. T. Negel, Readings in philosophy of psychology (1974). +38. C. Koch, Scientific American. (2018). +39. G. Miller, Science 309, 79 (2005). +40. J. Birch, A. Schnell, N. Clayton, Trends in cognitive sciences (2020). +41. M. Boly, et al., Science 332 (May, 2011). +42. H. Lyn, et al., Animal Cognition 17 (2014). +43. Y. Ma, D. Tsao, H. Shum (2022). +44. F. Scherr, C. Stöckl, W. Maass, BioRxiv (2020). +15 + +A +Algorithm for extracting strokes from a character +Repeat until all pixels of a character are marked, by depth-first search: +(1) Extract its skeleton so that the stroke width is 1 pixel point. Then convert the image to +a graph and shrink adjacent cross points. (2) Randomly select an endpoint as starting point, +endpoint at top left has a greater chance of being selected. Walk until a cross point or endpoint. +If there is a circle then select a cross point of a top left point if there is no cross point. Record +this stroke and mark it on the character. Allow small number of marked pixel points to make +the decomposition more natural. (3) When meeting a cross point, then enumerate two situations +of pen-up and turning, randomly. Pen-up means end of a stroke, go to step (2) with the marked +graph. Turning means continuation hence repeat step (2). If walking to an endpoint, then attempt +to turn by going back to find a new unmarked pixels within some small number of pixels or +directly end the stroke and repeat step (2) with marked graph. +16 + diff --git a/HNAzT4oBgHgl3EQfHfvA/content/tmp_files/load_file.txt b/HNAzT4oBgHgl3EQfHfvA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d06fcdc0488301d901b8297c4286653cf5e6b089 --- /dev/null +++ b/HNAzT4oBgHgl3EQfHfvA/content/tmp_files/load_file.txt @@ -0,0 +1,700 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf,len=699 +page_content='A Theory of Human-Like Few-Shot Learning Zhiying Jiang1, Rui Wang2, Dongbo Bu2, Ming Li1∗ 1David Cheriton School of Computer Science, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada 2Institute of Computing Technology, Chinese Academy of Science, Beijing, China ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' E-mail: mli@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='ca We aim to bridge the gap between our common-sense few-sample human learn- ing and large-data machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We derive a theory of human-like few- shot learning from von-Neuman-Landauer’s principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Modelling human learn- ing is difficult as how people learn varies from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Under com- monly accepted definitions, we prove that all human or animal few-shot learn- ing, and major models including Free Energy Principle and Bayesian Program Learning that model such learning, approximate our theory, under Church- Turing thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We find that deep generative model like variational autoencoder (VAE) can be used to approximate our theory and perform significantly better than baseline models including deep neural networks, for image recognition, low resource language processing, and character recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Introduction During the past decade, fast progress in deep learning (1) has empowered computer speech recognition, image processing, natural language processing, protein folding, game playing and many other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' However, these great progresses fell short when we try to understand our own learning mechanism: How to model human learning (2), (3), (4)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Species in nature learn quickly to survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When a dragonfly is hatched, within hours it firms up its wings and then flies to catch mosquitoes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' a newborn does not need tons of repeated examples or transfer learning to identify an apple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Most human or animal learning exhibits a mixture of inherited intelligence, few-shot learning without prior knowledge, as well as long term many-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' It is interesting to note that these learning programs are encoded in our genomes but they are not all the same, even for individuals within the same species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The diversity of these learning algorithms is vividly expressed by Spearman’s "g" factor (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='01047v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='LG] 3 Jan 2023 Unlike data-laden, model-heavy, and energy-hungry deep learning approaches, most human learning appear to be simple and easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Merely scaling up current deep learning approaches may not be sufficient for achieving human level intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We miss certain major components when modelling human or animal learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Diversity is one of the missing part when modelling human or animal few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' There are eight billion people on earth, each with a unique few-shot learning model (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Even if we just want to model one person, a single person often uses different parameters, features, and perhaps different algorithms to deal with different learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Ideally we want a framework that can cover the diversity in human and animal few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Facing such a seemingly formidable task, traditional thinking in machine learning will only lead us to various traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' To avoid such traps we need to go back to the very first principles of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Specifically, we start from an agreed-upon law in thermodynamics, to formally derive our model for few-shot learning, and prove this is the optimal model within our framework in the sense that all other models including human ones may be viewed as approximations to our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We show a deep connection between our framework and the free energy principle (3) and the Bayesian Program Learning model (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' By the end of this process, a component of data compression during the inference phase of learning emerges as a key component of all few-shot learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' First, we formalize our intuitive and commonly accepted concept of human-like few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For example, our definition below is consistent with what is used in (4), and in the same spirit of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consider a universe Ω, partitioned into H disjoint concept classes: Ch, h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Few-shot (k-shot) learning is described as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' n elements in or outside Ω are given as unlabelled samples y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' There are k labelled examples for each class Ch, for small k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The learning program, using a computable metric M, few-shot learns Ch, h = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='H, if it uses the n unlabelled samples and k labelled samples and minimizes the objective function: H � h=1 |Ch| � i=1 M(xi, coreh) | y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn, xi ∈ Ch, where coreh = ψ(k samples of Ch) representing a transformed representation of the k labelled samples from Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This definition covers most of our common sense few-shot learning scenarios and other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' In particular, this is used in one-shot learning by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As each independent individual, we do not all use a same metric, or even similar metric, to few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For example, MN Hebart et al (6) identified 49 highly reproducible dimensions to 1854 objects to measure 2 their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Different people can be equipped to better observe some of these dimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We explain the intuition behind Definition 1 via a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' A human toddler may have already seen many unlabelled samples of fruits which, for example, contains two classes: apples and pears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Then given a new labelled sample from each class, the toddler learns how to differentiate between these two fruits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The number of labelled data required for one to classify may vary as people have different learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Current deep learning based approaches for few-shot learning generally depend on 1) many auxiliary labelled training samples or task-specific data augmentation for transfer learning or meta learning (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' or 2) very large scale self-supervised pre-training (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' These approaches thus fall short to model few-shot learning in nature by humans and animals as they can hardly account for the diversity in learning algorithms and they either neglect the unsupervised scenario that humans are mostly exposed to or use the scale of unlabelled data and training parameters that are far beyond creatures need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Many attempts have been made to understand human learning through cognitive, biological, and behavior sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Some studies have established basic principles a human learning model should obey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' One theory is the two-factor theory of intelligence by Charles Spearman in 1904 (2), where the “g” factor is an indicator of the overall cognitive ability, and the “s” factor stands for the aptitude that a person possesses in specific areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As “g” factor is genetically-related (9), it indicates the necessity of a learning theory that can account for the diversity in creatures’ learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Another theory is the Free Energy Principle by Karl Friston (3) that human (and all biological systems) learning tends to minimize the free energy between internal understanding in the sense of Bayesian (under internal perceived distribution p) and that of the environmental event (under distribution q), measured by KL-divergence (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' In a similar spirit, Lake, Salakhutdinov and Tenenbaum (4) proposed a Bayesian program learning (BPL) model, learning a probabilistic model for each concept and achieve human-level performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Two articles by Schmidhuber (11) and by Chater and Vitanyi (12) linked simplicity to human cognition and appreciation of arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Instead of exploring a biological basis for few-shot learning, we think it is possible to mathematically derive an optimal framework that can unify the above theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We further demonstrate by experiments that our new model indeed works significantly better than other classical deep learning neural networks for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As a byproduct of our new model, a new concept class of "interestingness" is learned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' this class implies where our appreciation of art, music, science and games comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Extending this observation, some aspects of consciousness may be modelled as a set of few-shot learned concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consequently, we hypothesize the ability of labelling input data becomes a key step to acquiring some aspects of consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 3 A theory of few-shot learning We mathematically derive an optimal few-shot learning model for Definition 1 that is effective and is able to cover enormous diversities existed in different species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The task may appear to be formidable because of conflicting and seemingly very general goals: each individual is allowed to have a different learning model, yet our model has just one program to model everybody;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' we do not yet exactly know the complete underlying biological mechanisms, yet we need to implement the right functionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' there are infinite number of models, yet we need to choose one that is optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' we are not really interested in "proposing models" out of blue, yet we wish our model to be a mathematical consequence of some basic laws of physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' the model needs to be theoretically sound, yet practically useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For simplicity and readability, we begin with one-shot learning, k = 1 in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thus, coreh in Definition 1 is just the single labelled sample xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For larger k, coreh can be some form of average of the k samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As Definition 1 defined, some unlabelled objects are assumed and it’s also possible to extend the definition by adding distribution, learnt from either unlabelled or labelled data, to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Using metric M that is responsible for k-shot learning of an individual, the learning system seeks to minimize the energy function H � h=1 |Ch| � i=1 M(xi, xh|y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn), or, assuming H(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn) is a pre-trained model of y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn, or other labelled samples, capturing the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' H � h=1 |Ch| � i=1 M(xi, xh|H(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn)), Now the question is, what sort of M should we use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Indeed, this varies from person to person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Can we unify all such measures, algorithms and inferences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Let’s go back to the fundamentals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Principle 1 (von-Neuman-Landauer Principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Irreversibly processing 1 bit of information costs 1kT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' reversible computation is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Then for two objects x, y, the minimum energy needed to convert between x and y in our brain is: EU(x, y) = min{|p| : U(x, p) = y, U(y, p) = x}, where U is a universal Turing machine or our brain, assuming Church-Turing thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Since we can prove a theorem showing all Universal Turing machines are equivalent modulo a constant and efficiency, we will drop the index U (see (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' To interpret, E(x, y) is the length of the shortest program that reversibly converts between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' These bits used in the shortest program p when they are erased will cost |p|kT of energy, according to the John von Neuman and Rolf Landuaer’s law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This leads us to a fundamental theorem (14): 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' degree degree Figure 1: Bipartite Graph Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' E(x, y) = max{K(x|y), K(y|x)} + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' K(x|y) is the Kolmogorov complexity of x given y, or informally, the length of the shortest program that outputs x given input y (details are shown in (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As this theorem was proved thirty years ago and it is vital in our theory, to help our readers, we will provide an intuitive but less formal proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' By the definition of E(x, y), it follows E(x, y) ≥ K(x|y) and E(x, y) ≥ K(y|x), thus we have E(x, y) ≥ max{K(x|y), K(y|x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' To prove the other direction E(x, y) ≤ max{K(x|y), K(y|x)}, we need to construct a program p such that p outputs y on input x and p outputs x on input y, and length of p is bounded by max{K(x|y), K(y|x)} + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Let k1 = K(x|y), and k2 = K(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Without loss of generality, assume k1 ≤ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We first define a bipartite graph {X, Y, E}, where X, Y = {0, 1}∗, as shown in Figure 1 and E is a finite set of edges defined between X and Y as follows: E = {{u, v}, u ∈ X, v ∈ Y, K(u|v) ≤ k1, K(v|u) ≤ k2} Note that a particular edge (x, y) is in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' If we find edge (x, y), then given x, p can output y, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' So the idea of the proof is to partition E properly so that we can identify (x, y) easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Two edges are disjoint if they do not share nodes on either end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' A matching in graph theory is a set of disjoint edges in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' E can be partitioned into at most 2k2+2 matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consider edge (u, v) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The degree of a node u ∈ X is bounded by 2k2+1 because there are at most 2k2+1 different strings v such that K(v|u) ≤ k2, accumulating possible strings from i = 1 to i = k2 gives us �i=k2 i=1 = 2k2+1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Hence u belongs to at most 2k2+1 matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Similarly, node v ∈ Y belongs to at most 2k1+1 matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We just need to put edge (u, v) in an unused matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' (End of Proof of Claim) Let Mi be the matching that contains edge (x, y) We now construct our program p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' p operates as follows: Generate Mi following the proof of Claim, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' enumerating the matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This uses information k1, k2, and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' K(i) ≤ k2 + O(1) 5 Given x, p uses Mi to output y, and given y, p uses Mi to output x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' A conditional version of Theorem 1, using information in Definition 1, can be obtained E(x, y|y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn) = max{K(x|y, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn), K(y|x, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn)}, conditioning on unlabelled samples y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' According to (14), this distance is universal, in the sense that E(x, y) is the minimum among any other computable distances: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For any computable metric D, there is a constant c, such that for all x, y, E(x, y) ≤ D(x, y) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This theorem implies: if D metric finds some similarity between x and y, so will E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thus, the above theorem implies, up to some constant O(H) H � h=1 |Ch| � i=1 E(xi ∈ Ch, coreh|y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn) ≤ H � h=1 |Ch| � i=1 M(xi ∈ Ch, coreh|y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When unlabelled samples y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn plus other irrelevant historical labelled samples are modeled by some model H such as a generative model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=', VAE), then the above inequality can be rewritten as: H � h=1 |Ch| � i=1 E(xi ∈ Ch, coreh|H) ≤ H � h=1 |Ch| � i=1 M(xi ∈ Ch, coreh|H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' (1) Thus, E gives optimal metric for few-shot learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Other algorithms satisfied Definition 1 are the approximation to this optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 1 In addition, we show that our theory’s deep connection to two well-established principles of learning in neuroscience and psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Friston’s Free Energy Principle (FEP) (3), derived from Bayesian brain hypothesis (15), states that brain seeks to minimize surprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Specifically, it assumes the brain has its internal state (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' generative model) that implicitly models the environment according to the sensory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Hidden (latent) variables need to be defined for the internal state, which are drawn from prior beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Ideally, these prior knowledge is also modelled, which is made possible by hierarchical generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The free energy principle (FEP) is often interpreted as Bayesian optimization, using the Evidence Lower Bound (ELBO) as ELBO = log p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' θ) − D(q(z)∥p(z|x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' θ) optimization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Here the evidence log p(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' θ) is the encoding length of x under probability p, and the Kullback-Leibler divergence term is the p-expected encoding length difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This is half of Theorem 1 and FEP is asymmetric if we view it as a distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' However, the symmetry is important to few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For example, a scarlet king snake may look like a coral snake, but the latter certainly has more deadly features the former lacks, one way compression K(ScarletKingSnake|CoralSnake) is not sufficient to 1Note that E is a metric: it is symmetric, and satisfies triangle inequality 6 Compressor Unlabeled Data Distribution Test Instance Figure 2: Illustration of our framework, dashed line indicates optional component when learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' distinguish the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Despite of the fact H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' influnza with genome size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 million and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' coli with genome size 5 million they are sister species but E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' coli would be much closer to a species with zero genome G0 or just a covid-19 genome with this asymmetric measure (K(G0|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='coli) than with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' influnza (K(H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' influnza|E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' coli)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' A symmetric interpretation of Friston’s FEP can be derived by requiring minimum conversion energy as we show in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Different individuals may use different compression algorithms to do data abstraction and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' It can be viewed that these algorithms all approximate E(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Some are more efficient than others in different situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The individuals with better compression algorithms have bigger “g” factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Diversified compression algorithms also guarantee better survival chances of a community when facing a pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As compression neural networks are genetically encoded, the “g” factor is thus inheritable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This can be seen via Figure 2, compression algorithms vary from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The distribution of the data to be learnt is either implicitly or explicitly captured by creatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Those who can better utilize unlabelled data to capture distribution may have a more efficient compression algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Experimental Results Image Experiments To approximate our universal few-shot learning model, we use a hierarchical VAE as our underlying model H in Inequality 1 to model the unlabelled samples y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' , yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This hierarchical structure coincides with our visual cortex and brain structure (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' According to integrated information theory (17), an input y may come from all sensing terminals: vision, hearing, smell, taste, sensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Often, creatures are exposed to an unsupervised environment where objects are unknown and unlabelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Revisiting the negative ELBO, we can see it can be interpreted as changing perceptions to minimize discrepancy (minimize KL divergence) or changing observations to maximize evidence, in the context of FEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When the creatures are 7 exposed to a “tree” and they do not fully realize what it is, the sensory information of the objects are internalized with hidden states (inner belief) that can describes how it believes the generation process of a “tree”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This process of generation, helps the creatures to identify the latent similarities among objects that belong to the same category, without the full awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This process of "unconsciously" training to generate helps the creatures to better categorize in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When the identity of a “tree” is finally revealed, they can generalize quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This explains our rationale of using a VAE to process unlabelled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consequently, the Kolmogorov complexity terms in Inequality 1 are naturally approximated by a VAE based compressor (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' To test the hypothesis, we carry out the experiment on five datasets, MNIST, KMNIST, FashionMNIST, STL-10 and CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We first train a hierarchical VAE on unlabelled data to learn to generate ˆx that’s as close to x as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This corresponds to the time when creatures exposed to a environment without knowing the object, implicitly learning the latent representation among objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When the identity of objects are revealed, a VAE based universal compressor can be used to identify the new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Specifically, after training a hierarchical VAE unsupervisedly, we compare the E energy function between a labelled image and a test image, as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' In our experiment, we use 5 labelled samples per class to test the accuracy of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The energy function E relies on a compressor to approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We thus use the bits-back argument to directly use our trained VAE for the compressor in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Our result shows that using only 5 samples, our method outperforms traditional supervised models like SVM, CNN, VGG and Vision Transformer (ViT) on all five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' These supervised methods are chosen to represent different model complexity with wide range of number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As we can see, when labelled data are scarce, supervised methods are not effective: complex models like VGG cannot perform better than SVM and this tendency is more obvious on ViT without pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The improvement that our method brings is more obvious on more complex datasets like STL-10 and CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Similar result is also obtained in the recent work, across different shot settings (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We also compare with using latent representation directly with k-Nearest-Neighbor classifier, labelled as “Latent” in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The architecture and training procedure for “Latent” method is exactly the same to our method — we train on unlabelled data to generate the sample and then take the latent representation for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We can see using latent representation outperforms all supervised methods on four out of five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' But the accuracy is still way lower than our method, indicating our method can better utilize the generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Text Experiments Our theory is generally applicable, even without pre-training on unlabelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Here, we demonstrate significant advantages of our approach with a simple compressor gzip over lower resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Languages with Abundant Resources We first test our method on datasets with abundant resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Specifically, we compare with three datasets — AG News, SogouNews and DBpedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 8 MNIST KMNIST FashionMNIST STL-10 CIFAR-10 SVM 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 CNN 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 VGG 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 ViT (disc) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 35.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 Latent 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 Ours 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 Table 1: 5-shot image classification accuracy on five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' AG News SogouNews DBpedia fasttext 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 Bi-LSTM+Attn 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 HAN 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='0± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 W2V 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3 BERT 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='3±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 Ours 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='7±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 Table 2: 5-shot text classification accuracy on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Similar to image classification, we compare with both supervised methods, including fasttext (20), BiLSTM (21) with attention mechanism (22) and Hierarchical Attention Network (HAN) (23), and non-parametric methods that use Word2Vec (W2V) (24) as representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We also compare with pre-trained language models like BERT (25) We use five labelled data for each class (5-shot) for all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Surprisingly, even without any pre-training and with a simple compressor like gzip, our method outperforms all non-pretrained supervised methods and non-parametric methods in low data regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This indicates that compressor serves as an efficient method to capture the regularity and our information distance is effective in comparing the similarity based on the essential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When comparing with pre-trained models like BERT, we can see our method is significantly higher on SogouNews, a special dataset that includes Pinyin — a phonetic romanization of Chinese, which can be viewed as an Out-Of-Distributed (OOD) dataset as it uses the same alphabet as english corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Low-Resource Languages Sufficiently pre-trained language models are exceptional few-shot learners (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' However, when faced with low resource data or distributions that are significantly different from any pre-trained data, those pre-trained language models lose their advantages to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We compare our method with BERT on four different low-resource language datasets - Kinyarwanda, Kirundi, Swahili and Filipino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' These datasets are curated to have the Latin alphabets, same as english corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' BERT has performed extremely well as 9 Kinnews Kirnews Swahili Filipino BERT 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='0±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='0 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 mBERT 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='4±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='9 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 Ours 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='1±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='7±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='2±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='8 Table 3: 5-shot text classification accuracy on low-resource datasets shown in Table 2 due to pre-training on billions of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' However, when facing low-resource datasets, BERT perform significantly worse than our method only using gzip as we can see in Table 3, no matter using multilingual pre-trained version or the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Note that mBERT is pre-trained on 104 languages including Swahili and Tagalog (on which Filipino is based on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' As we can see on Swahili and Filipino, mBERT performs better than BERT, but still significantly lower than our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Omniglot one-shot-classification dataset Figure 3: Distance between two Bezier curves In (4), a one-shot learning framework Bayesian program learning (BPL) was pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' It learns a simple probabilistic model for each concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Taking a negative logarithm converts a Bayesian formula to a description length paradigm, hence BPL can be viewed as one particular approximation to our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Here we provide another simple approxima- tion of our theory for the Omniglot one-shot- classification dataset of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Our system first decompose a given char- acter into strokes, then compute E(a, b) be- tween characters a and b, using all their possi- ble stroke decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We provide how to calculate E(a, b) here and details of decompo- sition program is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Fit a stroke by a Bezier curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Ensure the number of points on two curves are same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This algorithm utilize equally split method to select certain same number of points on each curve Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Ensure the area of the convex hull and the barycenter of the compared characters are the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 10 0 20 40 60 80 100- 0 20 40 60 80 1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Use max Cartesian distance between parallel points on two Bezier curves to approximate the minimum encoding distance between two Bezier curves, as shown in Figure 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Choose the character with minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This simple implementation achieves 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='25% accuracy 20-way-1-shot on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The point here is to demonstrate various approximations of our theory that work rather than com- paring accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' At 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='75% (4) or at 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content='25% might be two different individuals with different compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Unification Our framework can unify other popular deep neural networks for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Siamese Network: Siamese network uses twin subnetwork to rank the similarity between two inputs in order to learn useful features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' M here is often a contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This framework shows strong performance in one-shot image recognition (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Prototypical Network: Prototypical networks (27) propose to optimize the distance metric M directly by learning coreh in representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' coreh are represented as the mean of embedded support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Bi-Encoder: In the context of natural language processing, one of the dominant structure is the Bi-Encoder design with each encoder being a pre-trained language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For example, in information retrieval, Dense Passage Retrieval (DPR), with two encoders encoding query and document respectively, has become the new state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' To capture semantic similarity, sentenceBERT (28) also adopts the bi-encoder design and becoming one of the most prevalent methods for semantic textual similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' M in both cases can either be cosine similarity or Euclidean distance between the representation learned through pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Information Distance based Methods: Hundreds of algorithms were published, before the deep learning era, on parameter-free data mining, clustering, anomaly detection, classification using information distance E (29–34), with a comprehensive list in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Recently (19) have discovered using information distance with deep neural networks and leverage the generalizability of few-shot image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This work shows that with the help of deep generative models, unlabelled data can be better utilized for few-shot learning under our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Conclusion and a discussion on consciousness We have defined human-like few-shot learning and derived an optimal form of such few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Note there is an interesting difference between our theory and classical learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' In classical learning theory, it is well-known that if we compress training data to a smaller consistent description, whether it is a classical Bayesian network or a deep neural networks (13,35), we would achieve learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' In this paper, we demonstrate that in the inference 11 stage, compression is also important, especially when there are not enough labelled data to train a small model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' On the biological side, compression circuits using predictive coding in human cortex has been studied by (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Experiments have also strongly supported our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We expect to see more practical systems approximating our theory can be implemented to solve commonplace few-shot learning problems when large amounts of labelled data for deep learning is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We now wish to explore two consequences of our few-shot learning model, to consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' A binary classifier of interestingness Our few-shot learning model has a by-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We have proved compression is a universal goal that few-shot learning algorithms approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thus this implies immediately a (subconscious) binary classifier: if something is compressed, then something interesting happens, and attention is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' It turns out that this "Interestingness" has been theoretically studied as logical depth first proposed by Charles Bennett (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' According to Bennett, a structure is deep if it is superficially random but subtly redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' When few-shot learning happens, significant compression happens, and these deep objects gain attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Such a binary classifier might explain our appreciation of arts, music, games, and science, since these all share a common feature of dealing with non-trivially compressible objects: whether it is a shorter description of the data that gives rise of Newton’s laws (13), or a piece of art or music that itself is compressible or that reminds us of something we have experienced before, hence very compressible, we feel we understand it and hence appreciate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Science is nothing but compressing data into simpler descriptions of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consciousness and the ability of labelling data Do other species have consciousness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' It is difficult to answer this question as consciousness is not testable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thomas Nagel (37) made a comment: We will never know if a bat is conscious because we are not bats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Consider an alternative data-driven approach by asking what a species can do instead of how they feel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' That is, if we treat some aspects of consciousness as a collection of learned concepts, then given a compression network, the ability of acquiring the relevant concepts becomes a matter of labelling relevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' We know learning and consciousness are both located at posterior cortex region (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This is in agreement with some injured patients when they lost consciousness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' This is also in agreement with “bistable perception” training results with monkeys (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Varieties of consciousness are being pragmatically studied (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' These include: 1) the ability of consciously perceive the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 2) the ability of evaluating conscious emotions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 3) the ability of having a unified conscious experience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 4) the ability of integrating across time as a continuous stream, one moment flowing into the next;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 5) the conscious awareness of oneself as distinct from the world outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Many of these abilities may be seen as a few-shot learnable concepts, given properly labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 12 Different animals have various levels of some of such consciousness by passing certain tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' For example, chimpanzees, dolphins, Asian elephants, and magpies can recognize themselves by passing some mirror-mark tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The corvids display some emotions, and are able to plan ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Octopus have powerful perceptual facilities obtaining and processing data independently with each tentacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Experimentally, awareness emerges when information travels back and forth between brain areas (41) instead of a linear chain of command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' According to our theory, the brain really only needs to use a universal compressor to compress information, regardless of one processor in the head or a few processors in the tentacle (in case of Cephalopods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thus we can conjecture that "consciousness” then is a matter of ability of labelling the data from sensory terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Food or enemy in the environment are easy to label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Emotional labelling requires some level of abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Self-awareness of “me” and “others” thus is just another binary classifier trainable depending on if the species is able to do “displaced reference” mental labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Other than the human beings, only orangutans are known to have limited displaced reference ability (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Thus we have just reduced the non-testable question of whether an animal has consciousness in some aspects to if it is able to label the corresponding data properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Acknowledgement We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Hang Li for suggestions and bringing (43) to our attention and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Amy Sun for bringing (44) to our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' The work is supported in part by Canada’s NSERC operating grant OGP0046506, Canada Research Chair Program, and the Leading Innovative and Entrepreneur teams program of Zhejiang, number 2019R02002, and NSFC grant 61832019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' References and Notes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Bengio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Hinton, Nature 521, 436 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 15 A Algorithm for extracting strokes from a character Repeat until all pixels of a character are marked, by depth-first search: (1) Extract its skeleton so that the stroke width is 1 pixel point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Then convert the image to a graph and shrink adjacent cross points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' (2) Randomly select an endpoint as starting point, endpoint at top left has a greater chance of being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Walk until a cross point or endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' If there is a circle then select a cross point of a top left point if there is no cross point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Record this stroke and mark it on the character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Allow small number of marked pixel points to make the decomposition more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' (3) When meeting a cross point, then enumerate two situations of pen-up and turning, randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Pen-up means end of a stroke, go to step (2) with the marked graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' Turning means continuation hence repeat step (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' If walking to an endpoint, then attempt to turn by going back to find a new unmarked pixels within some small number of pixels or directly end the stroke and repeat step (2) with marked graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNAzT4oBgHgl3EQfHfvA/content/2301.01047v1.pdf'} diff --git a/HdE1T4oBgHgl3EQfFgNH/content/2301.02902v1.pdf b/HdE1T4oBgHgl3EQfFgNH/content/2301.02902v1.pdf new file mode 100644 index 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We systematically extend the elementary differential and Riemannian geometry +of classical U(1)-gauge theory to the setting of noncommutative differential geometry in the +sense of Connes by combining recent advances in noncommutative Riemannian geometry +with the theory of coherent 2-groups. We show that Hermitian line bimodules with Hermitian +bimodule connection over a unital pre-𝐶∗-algebra with ∗-exterior algebra form a coherent +2-group, and we prove that weak monoidal functors between coherent 2-groups canonically +define bar or involutive monoidal functors in the sense of Beggs–Majid and Egger, respectively. +Hence, we prove that a suitable Hermitian line bimodule with Hermitian bimodule connection +yields an essentially unique differentiable quantum principal U(1)-bundle with principal +connection and vice versa; here, U(1) is 𝑞-deformed for 𝑞 a numerical invariant of the bimodule +connection. From here, we compute moduli spaces of solutions to Maxwell’s equations and +we formulate and solve the interrelated lifting problems for noncommutative Riemannian +structure in terms of abstract Hodge star operators and formal spectral triples, respectively; +all the while, we account precisely for emergent modular phenomena of geometric nature. +On the one hand, it follows that no spectral triple on quantum CP1 lifts to a twisted spectral +triple for quantum SU(2) with the 3-dimensional calculus. On the other hand, we may use the +canonical lift of the spin Dirac spectral triple on quantum CP1 with respect to the 𝑞-monopole +connection to recover Kaad–Kyed’s compact quantum metric space on quantum SU(2) for a +canonical choice of parameters. +Contents +1. +Introduction +2 +2. +A coherent 2-group of noncommutative Hermitian line bundles with connection +5 +2.1. +Preliminaries on coherent 2-groups +5 +2.2. +The Picard 2-group of a nc topological space +9 +2.3. +The differential Picard 2-group of a noncommutative manifold +15 +2.4. +Canonical actions of the differential Picard group +20 +3. +Reconstruction of noncommutative principal U(1)-bundles with connection +25 +3.1. +Monoidal inversion and homomorphisms of coherent 2-groups +25 +3.2. +Generalised crossed products via homomorphisms of coherent 2-groups +30 +3.3. +Horizontal calculi as generalised crossed products +35 +3.4. +Reconstruction of total calculi +40 +4. +Lifting problems for noncommutative Riemannian structures +46 +4.1. +Basic noncommutative Hodge theory and moduli spaces of U(1)-instantons +46 +4.2. +The lifting problem for Riemannian structures via Hodge operators +53 +4.3. +Unbounded lifts of commutator representations +60 +4.4. +Twisted boundedness of lifted commutator representations +73 +References +82 +1 +arXiv:2301.01749v1 [math-ph] 4 Jan 2023 + +2 +BRANIMIR ĆAĆIĆ +1. Introduction +The primordial application of noncommutative (nc) geometry to theoretical physics is the +conceptually economical construction of physical models as classical physics on nc manifolds. +For example, in Bellissard–Van Elst–Schulz-Baldes’ model of the integer quantum hall effect +[15], the nc Brouillin zone accounts for both the magnetic field and disorder in the crystal, while +in particle physics [43] and cosmological models [69] using Chamseddine–Connes’s spectral +action principle [30], 0-dimensional nc fibres encode the particle content. The prototypical +such construction is Connes–Rieffel’s topologically non-trivial Yang–Mills gauge theory on +irrational nc 2-tori [35], the first of many nc field theories built from a range of seemingly +disparate variations on Connes’s nc differential geometry [31, 33]. Indeed, one can approach +various aspects or special cases of nc U(1)-gauge theory in terms of quantum principal bundles +[22, 39], principal U(1)-spectral triples [42, 19, 26], or even the spectral action principle [43]. +This fragmentary understanding of classical U(1)-gauge theory on nc manifolds is unsat- +isfactory for reasons beyond the obvious ones internal to nc geometry. For example, consider +mathematical analysis of the integer quantum Hall effect in terms of quantum adiabatic trans- +port, where one probes the qualitative behaviour of relevant observables by considering the +integer quantum Hall effect on general compact Riemann surfaces [5]. A satisfactory generali- +sation to nc compact Riemann surfaces would require a precise extension of the elementary +differential and Riemannian geometry of classical U(1)-gauge theory as a coherent whole to the +nc setting that is compatible with both nc Kähler geometry [79] and the framework of spectral +triples [32]. Our goal here is to effect just such an extension, which would also be applicable +to the study of electromagnetism on nc spacetimes [68] and to the differential-geometric +refinement of nc 𝑇-duality as applied to the bulk-edge correspondence [70]. +We construct this extension from the ground up in accordance with the philosophy of +quantum Riemannian geometry [14]. Thus, we view a nc manifold as consisting of a unital +pre-𝐶∗-algebra equipped with a ∗-exterior calculus, so that a nc Riemannian manifold is a nc +manifold in this sense equipped with a compatible nc Riemannian structure, whether it be an +abstract Hodge star operator or a spectral triple. This, in stark contrast with other areas of +nc geometry and operator algebras, requires working exclusively ‘on the nose’—at worst, up +to explicit isomorphism. Fortunately, in our setting, we may obviate any resulting algebraic +difficulties through the systematic use of coherent 2-groups [7], generalised groups whose +group law, unit object, and inversion satisfy the group axioms up to coherent isomorphisms. +Moreover, borrowing insights from relevant applications of unbounded 𝐾𝐾-theory [19, 48, 26], +we minimise the use of functional analysis and obviate further algebraic difficulties through +the systematic use of finite tight Parseval frames on (pre-)Hilbert modules [49]. +Our results are independent of Schwieger–Wagner’s cohomological classification of princi- +pal T𝑁-𝐶∗-algebras [93] and Saldaña’s Tannaka–Krein theorem [72] for differentiable quantum +principal bundles d’après Ðurđević [40]. However, the former presages the rôle of coherent +2-groups and their group-cohomological classification in the case of Abelian structure groups, +while the latter will be prototypical for any generalisation of our results to non-Abelian or +quantum structure groups. +Overview of results. We begin in §2 by developing the elementary theory of nc Hermitian +line bundle with unitary connection. Let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior algebra +(Ω𝐵, d𝐵). Building on a proposal of Beggs–Brzeziński [9], we define Hermitian line 𝐵-bimodules +with connection (up to formal refinement) to be strong Morita auto-equivalences of 𝐵 equipped +with extendable bimodule connections [13] with respect to (Ω𝐵, d𝐵). Thus, building on results +of Beggs–Majid [13], we prove that Hermitian line 𝐵-bimodules with connection form a + +NONCOMMUTATIVE U(1)-GAUGE THEORY +3 +coherent 2-group DPic(𝐵), the differential Picard 2-group of (𝐵; Ω𝐵, d𝐵). The isomorphism +classes of DPic(𝐵) still form a group DPic(𝐵), the differential Picard group, whose canonical +(and typically non-trivial) action on the graded centre Z(Ω𝐵) of Ω𝐵 will appear throughout. +By results of Beggs–Majid [13], this DPic(𝐵)-action admits a 1-cocycle of supreme importance: +the curvature 2-forms of Hermitian line 𝐵-bimodules with connection. Moreover, we can +characterise the fibres of the forgetful map from DPic(𝐵) to the 𝐾0-monoid V(𝐵) of 𝐵. +Next, in §3, we develop the corresponding elementary theory of nc principal U(1)-bundles +with principal connection. Given 𝜅 > 0, we synthesize a definition of 𝜅-differentiable quantum +principal U(1)-bundle with connection from relevant work of Brzeziński–Majid [22], Hajac +[52], Ðurđević [39], and Beggs–Majid [14]; here, the differential calculus on U(1) is deformed +to satisfy d𝑧 · 𝑧 = 𝜅𝑧 · d𝑧. Hence, we may define a functor that maps a 𝜅-differentiable +quantum principal U(1)-bundle with connection to its nc associated Hermitian line bundle +with unitary connection of winding number −1; in fact, we show that [𝐸, ∇𝐸] ∈ DPic(𝐵) +is contained in the essential range of this functor if and only if its curvature 2-form F[𝐸,∇𝐸] +satisfies F[𝐸,∇𝐸] ⊳ [𝐸, ∇𝐸] = 𝜅−1F[𝐸,∇𝐸] with respect to the DPic(𝐵)-action on Z(Ω𝐵). We +prove that this functor is indeed an equivalence of categories onto its essential range, thereby +generalising the familiar dictionary between Hermitian line bundles with unitary connection +and principal U(1)-bundles with principal connection. +Our proof ultimately depends on applying two apparently novel technical results on +coherent 2-groups to weak monoidal functors Z → DPic(𝐵), which typically output the nc +Hermitian line bundles with unitary connection associated to a nc principal U(1)-bundle +with principal connection. The first, that Z is the free coherent 2-group on one generator, +is a straightforward corollary of Joyal–Street’s group-cohomological classification of weak +monoidal functors between coherent 2-groups [57]. The second, that every weak monoidal +functor between coherent 2-groups is a bar functor or involutive monoidal functor in the sense +of Beggs–Majid [13] and Egger [44], respectively, is a non-trivial application of the coherence +theorem for coherent 2-groups of Ulbrich [95] and Laplaza [66]. We may view this pair of +results as an abstract distillation of Pimsner’s construction [81]—by applying them to weak +monoidal functors from Z to the coherent 2-group Pic(𝐵) of Hermitian line 𝐵-bimodules, one +may recover Arici–Kaad–Landi’s characterisation [4] of nc topological principal U(1)-bundles. +At last, in §4, we turn to the nc Riemannian geometry of nc principal U(1)-bundles +with principal connection. Note that the best-known nc 3-manifolds are total spaces of nc +principal U(1)-bundles with principal connection, whose nc Riemannian geometry therefore +has implications for the construction of nc Lorentzian 4-manifolds. However, 3-dimensional +quantum SU(2) poses fundamental challenges for all existing frameworks—for instance, it +cannot be faithfully represented by a spectral triple [90]. We therefore draw on a range of +advances in nc Riemannian geometry—unbounded 𝐾𝐾-theory [71, 59, 19], nc Kähler geometry +[79], and quantum Riemannian geometry [14]—to lift nc Riemannian geometry from well- +behaved nc base spaces to nc total spaces. Our guide is the commutative case: a principal +U(1)-bundle 𝜋 : 𝑋 → 𝑌 with principal connection Π admits a bijection between metrics on +𝑌 and U(1)-invariant metrics on 𝑋 that make Π orthogonal and the fibres have unit length, +which is defined by the constraint that 𝜋 become a Riemannian submersion [2, §4]. +First, in §§4.1 and 4.2, we consider the conceptually prior notion of nc Riemannian ge- +ometry via abstract Hodge operators. For us, a Riemannian geometry on an nc manifold +(𝐵; Ω𝐵, d𝐵) is a pair (★, 𝜏), where ★ generalises the Hodge star operator and 𝜏 is a faithful +state generalising integration against the Riemannian volume form. This suffices for the for- +mulation of (Euclidean) Maxwell’s equations, whose moduli spaces of solutions we construct +by combining the results of §2.4 with the relevant nc Hodge decomposition theorem. Hence, + +4 +BRANIMIR ĆAĆIĆ +we propose a similar definition of total Riemannian geometry for a 𝜅-differentiable quantum +principal U(1)-bundle with connection (𝑃; Ω𝑃, d𝑃; Π) on (𝐵; Ω𝐵, d𝐵), where failure of the +Hodge operator to be right 𝑃-linear and ∗-preserving is governed by a commuting pair of +modular automorphisms of Ω𝑃. We show that (★, 𝜏) lifts to at most one total Riemannian +geometry on (𝑃; Ω𝑃, d𝑃; Π), whose existence we characterize in terms of conformality of the +corresponding Hermitian line 𝐵-bimodule with connection. For example, the unique lift of +canonical Riemanian geometry on quantum CP1 as an nc Kähler manifold to the 𝑞-monopole +of Brzeziński–Majid [22] recovers a construction of Zampini [99] for a canonical choice of +parameters. +Next, in §4.3, we consider Connes’s familiar nc Riemannian geometry via spectral triples +[32], which, following Schmüdgen [90], we generalise to bounded commutator representations. +We propose a definition of projectable commutator representation, where represented 1-forms +are only locally bounded in a certain precise sense. We then use a formal version of the +unbounded Kasparov product [71, 59] to construct an equivalence of categories between faithful +bounded commutator representations of (𝐵; Ω𝐵, d𝐵) and faithful projectable commutator +representations of (𝑃; Ω𝑃, d𝑃; Π); isomorphism of the latter is given by U(1)-equivariant +unitary equivalence up to perturbation by a suitable relative remainder. Moreover, if (𝐵; Ω𝐵, d𝐵) +is equipped with a liftable Riemannian geometry and (𝑃; Ω𝑃, d𝑃; Π) is equipped with its unique +lift, then the resulting Hodge–de Rham commutator representation of (𝐵; Ω𝐵, d𝐵) lifts to the +resulting total Hodge–de Rham commutator representation of (𝑃; Ω𝑃, d𝑃; Π). +Finally, in §4.4, we draw on Connes–Moscovici’s formalism of twisted spectral triples [34] +to control unboundedness of represented 1-forms. We consider modular pairs (𝑁, 𝜈), where +𝜈 is a modular automorphism of Ω𝑃 and 𝑁 is a suitable unbounded operator satisfying +𝜈 = 𝑁−1(·)𝑁; let us say that (𝑁, 𝜈) dampens an unbounded operator 𝑆 whenever 𝑁𝑆𝑁 is +bounded. Hence, we define a vertical or horizontal twist for a faithful projectable commutator +representation to be a modular pair that dampens represented vertical or horizontal 1-forms, +respectively. There is a universal vertical twist, but the existence of horizontal twists is +non-trivial and characterizable using a conformal generalisation of metric equicontinuity à la +Bellissard–Marcolli–Reihani [16]; in particular, a total Hodge–de Rham representation always +admits a canonical horizontal twist of conformal origin. In the case of 3-dimensional quantum +SU(2), vertical and horizontal twists are unique but distinct, thereby excluding the existence +of non-pathological U(1)-equivariant twisted spectral triples. Still, they permit a geometric +derivation of Kaad–Kyed’s compact quantum metric space on quantum SU(2) for 𝑡 = 𝑞2 +[58] from the spin Dirac spectral triple on quantum CP1 of Dąbrowski–Sitarz [41] via the +𝑞-monopole. +Running examples. We consider three main running examples, which we index here for +the reader’s convenience: +(1) the commutative case—Exx. 2.12, 2.21, 2.33, 2.39, 3.12, 3.37, 4.3, 4.11, 4.15; +(2) the real multiplication instanton—Exx. 2.24, 2.28, 2.31, 2.41, 3.52, 4.9, 4.12, 4.27, 4.38, 4.62, 4.67; +(3) the 𝑞-monopole—Exx. 3.13, 3.23, 3.26, 3.33, 3.51, 4.4, 4.26, 4.32, 4.37, 4.55, 4.61, 4.68, 4.73, 4.77. +Acknowledgements. The author wishes to thank Edwin Beggs, Cole Dunphy, Viqar Hu- +sain, Andrey Krutov, Matilde Marcolli, Bram Mesland, Réamonn Ó Buachalla, Adam Rennie, +Karen Strung, Nicholas Touikan, and Alessandro Zampini for helpful conversations and cor- +respondence, and he especially thanks Timmavajjula Venkata Karthik for numerous technical +conversations over the last several years that have indelibly shaped this work. The author was +supported by nserc Discovery Grant rgpin-2017-04249 and a Harrison McCain Foundation +Young Scholar Award. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +5 +2. A coherent 2-group of noncommutative Hermitian line bundles with connection +In this section, we build on work of Beggs–Brzeziński [9] and Beggs–Majid [13] to construct +a coherent 2-group of nc Hermitian line bundles with unitary connection over a nc differen- +tiable manifold, the differential Picard 2-group, that makes curvature into a canonical group +1-cocycle. Moreover, we algebraically characterise the fibers of the forgetful functors passing +to nc Hermitian line bundles and nc Hermitian vector bundles, respectively. +Let us recall some category-theoretic terminology. A category is essentially small whenever +its hom-sets and its class of isomorphism classes are all sets. A concrete category is a category C +equipped with a faithful functor 𝑈 : C → Set to the category Set of sets and functions, which +we view as the forgetful functor mapping objects of C to their underlying sets and arrows +of C to their underlying functions. Likewise, we define a functor category to be a category C +equipped with a faithful functor 𝑈 : C → [A, B], where A and B are categories and [A, B] +is the usual functor category whose objects are functors 𝐹 : A → B and whose arrows are +natural transformations. Finally, a subcategory A of a category B is strictly full whenever it is +full—every arrow in B between objects of A is an arrow of A—and closed under isomorphism. +2.1. Preliminaries on coherent 2-groups. We begin by reviewing the elementary theory +of coherent 2-groups, which generalise ordinary groups by permitting the group law, unit, and +inversion to satisfy the group axioms up to coherent isomorphisms. In particular, we show +that Z is the free coherent 2-group on one generator. We follow the account of Baez–Lauda +[7] but with technical simplications drawn from Laplaza [66]. +Recall that a (weak) monoidal category is a category C with a bifunctor ⊗ : C × C → C, the +monoidal product, a distinguished object 1, the unit, and natural isomorphisms +𝜆 � (𝜆𝑎 : 1 ⊗ 𝑎 → 𝑎)𝑎∈Obj(C), +𝜌 � (𝜌𝑎 : 𝑎 ⊗ 1 → 𝑎)𝑎∈Obj(C), +𝛼 � �𝛼𝑎,𝑏,𝑐 : (𝑎 ⊗ 𝑏) ⊗ 𝑐 → 𝑎 ⊗ (𝑏 ⊗ 𝑐)� +(𝑎,𝑏,𝑐) ∈Obj(C)3 , +respectively, the left unitor, right unitor, and associator, that satisfy certain coherence diagrams +[7, pp. 428–9]; in particular, it is strict whenever its left unitor, right unitor, and associator +are given by identity arrows. Moreover, a monoidal subcategory of a monoidal category C is a +subcategory D of C that is closed under the monoidal product, contains the unit, and contains +all left unitor, right unitor, and associator arrows between its objects. +Example 2.1. Let 𝐵 be a unital associative algebra over C. The concrete category Bimod(𝐵) +of 𝐵-bimodules and 𝐵-bimodule homomorphisms defines a monoidal category with respect +to the usual balanced tensor product of 𝐵-bimodules and of 𝐵-bimodule homomorphisms. In +particular, the associator 𝛼𝐸,𝐹,𝐺 of 𝐵-bimodules 𝐸, 𝐹, and 𝐺 is given by +∀𝑒 ∈ 𝐸, ∀𝑓 ∈ 𝐹, ∀𝑔 ∈ 𝐺, +𝛼𝐸,𝐹,𝐺((𝑒 ⊗ 𝑓) ⊗ 𝑔) � 𝑒 ⊗ (𝑓 ⊗ 𝑔), +the unit object is the trivial 𝐵-bimodule 𝐵, and the left and right unitors of a 𝐵-bimodule are +induced by its left and right 𝐵-module structures, respectively. +Just as a group is a monoid with a notion of inversion, so too is a coherent 2-group a +monoidal category together with a notion of inversion up to coherent isomorphisms. +Definition 2.2 (Sính [94]; Laplaza [66, §4]; Baez–Lauda [7]). A coherent 2-group is an essentially +small monoidal category G in which every arrow is invertible together with: +(1) a function (𝑔 ↦→ 𝑔) : Obj(G) → Obj(G) called monoidal inversion; +(2) a family of arrows ev = (ev𝑔 : 𝑔 ⊗ 𝑔 → 1)𝑔∈Obj(G) in G called evaluation. +Hence, a sub-2-group of a coherent 2-group G is a monoidal subcategory H of G that is closed +under monoidal inversion and contains {ev𝑔 | 𝑔 ∈ Obj(H)}. + +6 +BRANIMIR ĆAĆIĆ +A group Γ defines a coherent 2-group: take the discrete category on its underlying set +with the strict monoidal structure given by the group law and monoidal inversion given by +inversion in the group. This example admits the following wide-ranging generalisation; for a +review of the relevant group cohomology, see [56, §2.1]. +Example 2.3 (see [56, §2.2]). Let Γ be a group, let 𝑀 be a Γ-module, and let 𝜔 ∈ 𝑍3(Γ, 𝑀) be a +normalised cocycle. The following defines a coherent 2-group 2Grp(Γ, 𝑀, 𝜔). +(1) The set of objects of 2Grp(Γ, 𝑀, 𝜔) is Γ. +(2) The set of arrows of 2Grp(Γ, 𝑀, 𝜔) is 𝑀 × Γ, where (𝑚, 𝛾) ∈ 𝑀 × Γ is an automorphism +of the object 𝛾; moreover, composition of arrows is induced by the group law of 𝑀, so +that the identity automorphism of an object 𝛾 is (1𝑀, 𝛾). +(3) The monoidal product on objects is given by the group law of Γ, the monoidal product on +arrows is given by the group law of 𝑀 ⋊ Γ, the monoidal unit is 1Γ, left unitors and right +unitors are identity arrows, and the associator is given by +∀𝛾1, 𝛾2, 𝛾3 ∈ Γ, +𝛼𝛾1,𝛾2,𝛾3 � (𝜔(𝛾1, 𝛾2, 𝛾3), 𝛾1𝛾2𝛾3) . +(4) Monoidal inversion is given by inversion in the group Γ, so that evaluation is induced by +the group law of Γ. +We now take a closer look at monoidal inversion. Let 𝑔 be an object of a monoidal category +G. Recall [45, Deff. 2.10.1 & 2.11.1] that an inverse for 𝑔 is a triple (ℎ, e, i) consisting of an object +ℎ of G and isomorphisms e : ℎ ⊗ 𝑔 → 1 and i : 1 → 𝑔 ⊗ ℎ in G that make the following +diagrams commute: +(𝑔 ⊗ ℎ) ⊗ 𝑔 +𝑔 ⊗ (ℎ ⊗ 𝑔) +1 ⊗ 𝑔 +𝑔 +𝑔 ⊗ 1 +𝛼𝑔,ℎ,𝑔 +𝜆𝑔 +𝜌𝑔 +i⊗id𝑔 +id𝑔 ⊗e +(2.1) +(ℎ ⊗ 𝑔) ⊗ ℎ +ℎ ⊗ (𝑔 ⊗ ℎ) +1 ⊗ ℎ +ℎ +ℎ ⊗ 1 +𝛼ℎ,𝑔,ℎ−1 +𝜆ℎ +𝜌ℎ +e⊗idℎ +idℎ ⊗i +(2.2) +Recall, moreover, that an isomorphism of inverses (ℎ1, e1, i1) and (ℎ2, e2, i2) for the object 𝑔 +is an isomorphism 𝑢 : ℎ1 → ℎ2 in G that makes the following diagrams commute: +ℎ1 ⊗ 𝑔 +ℎ2 ⊗ 𝑔 +𝑔 +𝑢⊗id𝑔 +e1 +e2 +(2.3) +𝑔 ⊗ ℎ1 +𝑔 ⊗ ℎ2 +𝑔 +id𝑔 ⊗𝑢 +i1 +i2 +(2.4) +It is well known that if an object 𝑔 of a monoidal category G has an inverse, then that inverse +is unique up to unique isomorphism in the above sense [45, Prop. 2.10.5]. +Theorem 2.4 (Laplaza [66, §4]). Let G be a coherent 2-group. +(1) Monoidal inversion in G uniquely extends to a functor G → G that makes evaluation in G +into a natural isomorphism. +(2) There exists a unique natural isomorphism coev = (coev𝑔 : 1G → 𝑔 ⊗ 𝑔)𝑔∈Obj(G), such that, +for every 𝑔 ∈ Obj(G), the triple (𝑔, ev𝑔, coev𝑔) defines an inverse for 𝑔. +(3) There exists a unique natural isomorphism bb = (bb𝑔 : 𝑔 → 𝑔)𝑔∈Obj(G), such that, for every +𝑔 ∈ Obj(G), the arrow bb𝑔 : 𝑔 → 𝑔 gives an isomorphism of the inverses (𝑔, coev−1 +𝑔 , ev−1 +𝑔 ) +and (𝑔, ev𝑔, coev𝑔) of 𝑔. +This robust functorial picture of monoidal inversion and evaluation permits a direct +statement for general coherent 2-groups of the following elementary structural result. +Corollary 2.5 (Sính [94], see [7, §8.3]). Let G be a coherent 2-group. Let 𝜋0(G) be the group +of isomorphisms classes in G with group law induced by the monoidal product, and let 𝜋1(G) be + +NONCOMMUTATIVE U(1)-GAUGE THEORY +7 +the group of automorphisms of the monoidal unit 1 of G. Then 𝜋1(G) is Abelian and defines a +𝜋0(G)-module with respect to the left action ⊲G given by +∀𝑔 ∈ Obj(G), ∀𝛼 ∈ 𝜋1(G), +[𝑔] ⊲G 𝛼 � coev−1 +𝑔 ◦(𝜌𝑔 ⊗ id𝑔) ◦ �(id𝑔 ⊗ 𝛼) ⊗ id𝑔 +� ◦ (𝜌−1 +𝑔 ⊗ id𝑔) ◦ coev𝑔 . +For example, a group Γ viewed as a coherent 2-group Γ satisfies 𝜋0(Γ) = Γ and 𝜋1(Γ) = 1. +More generally, given a group Γ, a Γ-module 𝑀, and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), +it follows that 𝜋0(2Grp(Γ, 𝑀, 𝜔)) = Γ and 𝜋1(2Grp(Γ, 𝑀, 𝜔)) = 𝑀 × {1Γ} � 𝑀, where the +𝜋0(2Grp(Γ, 𝑀, 𝜔))-module structure on 𝜋1(2Grp(Γ, 𝑀, 𝜔)) reduces to the given Γ-module +structure on 𝑀. +We now generalise group homomorphisms to the setting of coherent 2-groups. Suppose +that G and G′ are monoidal categories. A (weak) monoidal functor 𝐹 : G → G′ consists of a func- +tor 𝐹 : G → G′ together with an an isomorphism 𝐹 (0) : 𝐹(1) → 1 and a natural isomorphism +𝐹 (2) = +� +𝐹 (2) +𝑔,ℎ : 𝐹(𝑔 ⊗ ℎ) → 𝐹(𝑔) ⊗ 𝐹(ℎ) +� +(𝑔,ℎ) ∈Obj(G)2 satisfying certain coherence diagrams +[7, pp. 429–430]. Given monoidal functors 𝑃 : G → G′ and 𝑄 : G → G′, a natural transfor- +mation 𝜙 : 𝑃 ⇒ 𝑄 is monoidal whenever 𝑃 (0) = 𝑄(0) ◦ 𝜙1 and 𝜙𝑔⊗ℎ ◦ 𝑃 (2) +𝑔,ℎ = 𝑄(2) +𝑔,ℎ ◦ (𝜙𝑔 ⊗ 𝜙ℎ) +for all objects 𝑔 and ℎ of G. +Definition 2.6 (see [7, §3]). Let G and G′ be coherent 2-groups. Then Hom(G, G′) is the +essentially small functor category whose objects are monoidal functor 𝐹 : G → G′ and +whose arrows are monoidal natural transformations. A homomorphism from G to G′ is an +object of Hom(G, G′), while a 2-isomorphism between homomorphisms 𝑅, 𝑆 : G → G′ +is an arrow 𝜂 : 𝑅 ⇒ 𝑆 in Hom(G, G′). Hence, given a homomorphism 𝐹 : G → G′, +let 𝜋0(𝐹) : 𝜋0(G) → 𝜋0(G′) and 𝜋1(𝐹) : 𝜋1(G) → 𝜋1(G′) denote the respective group +homomorphisms induced by 𝐹. +For example, let Γ1 and Γ2 be groups. A homomorphism of coherent 2-groups 𝑓 : Γ1 → Γ2 is +simply a group homomorphism with 𝑓 (0) and 𝑓 (2) given by identity arrows, so that 𝜋0(𝑓) = 𝑓 +and 𝜋1(𝑓) = id1. Moreover, all 2-homomorphisms in Hom(Γ1,Γ2) are simply identity natural +isomorphisms. +It turns out that a composition of homomorphisms of coherent 2-groups is again a homo- +morphism of coherent 2-groups, making the assignments 𝜋0 and 𝜋1 functorial in the sense +of mapping compositions to compositions. Indeed, more generally, let G1, G2, and G3 be +monoidal categories, and let 𝑃 : G1 → G2 and 𝑄 : G2 → G3 be monoidal functors. Then +𝑄 ◦ 𝑃 : G1 → G3 defines a monoidal functor with respect to the natural transformations +(𝑄 ◦ 𝑃)(0) � 𝑄(0) ◦ 𝑄(𝑃 (0)) and (𝑄 ◦ 𝑃)(2) � +� +𝑄(2) +𝑃(𝑔),𝑃(ℎ) ◦ 𝑄(𝑃 (2) +𝑔,ℎ ) +� +(𝑔,ℎ) ∈Obj(G1)2. +We conclude by using the cohomological classification of coherent 2-groups and their +homomorphisms to show that Z is the free coherent 2-group on one generator. Recall that a +monoidal equivalence of monoidal categories G1 and G2 is a monoidal functor 𝑃 : G1 → G2 +for which there exist a monoidal functor 𝑄 : G2 → G1 and monoidal natural isomorphisms +𝑃 ◦ 𝑄 ⇒ idG2 and 𝑄 ◦ 𝑃 ⇒ idG1; in fact, it suffices that the underlying functor 𝑃 : G1 → G2 +be an equivalence of categories [45, Rem. 2.4.10]. Coherent 2-groups admit the following +classification up to monoidal equivalence. +Theorem 2.7 (Sính [94], see [7, §8.3]). Let G be a coherent 2-group. There exists a normalised co- +cycle 𝜔 ∈ 𝑍3(𝜋0(G), ⊲G, 𝜋1(G)), unique up to cohomology, such that G is monoidally equivalent to +2Grp(𝜋0(G), 𝜋1(G), 𝜔). Moreover, the coherent 2-group G is uniquely determined up to monoidal +equivalence by the resulting quadruple (𝜋0(G), 𝜋1(G), ⊲G, [𝜔]), where [𝜔] ∈ 𝐻3(𝜋0(G), 𝜋1(G)) +is the cohomology class of 𝜔. + +8 +BRANIMIR ĆAĆIĆ +Hence, the Sính invariant of a coherent 2-group G is the complete monoidal equivalence +invariant (𝜋0(G), 𝜋1(G), ⊲G, [𝜔]) constructed by Sính’s theorem; note that the Sính invariant +is referred to as the Postnikov invariant in some references. For example, the Sính invariant +of a group Γ viewed a strict 2-group is (Γ, 1,Γ × 1 → 1, 1). Given the additional data of a +Γ-module 𝑀 and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), the Sính invariant of 2Grp(Γ, 𝑀, 𝜔) +reduces to (Γ, 𝑀, ⊲, [𝜔]), where ⊲ is the given Γ-action on 𝑀. +Homomorphisms of coherent 2-groups now also admit a cohomological classification—for +simplicity, we give the relevant special case. +Theorem 2.8 (Joyal–Street [57, §6]; see [7, §8.3] and [55, §5.3]). Let 𝐺 and Γ be groups, let 𝑀 be +a Γ-module, and let 𝜔 ∈ Z3(Γ, 𝑀) be a normalised cocycle. Define a category H(𝐺;Γ, 𝑀, 𝜔) as +follows. +(1) An object is a pair (𝛼, 𝜅), where 𝛼 : 𝐺 → Γ is a group homomorphism and 𝜅 ∈ 𝐵2(𝐺, 𝑀) is a +normalised 2-cochain with respect to 𝛼 that satisfies d𝜅 = (𝛼∗𝜔)−1; +(2) Let (𝛼1, 𝜅1) and (𝛼2, 𝜅2) be objects. If 𝛼1 = 𝛼2, then an arrow 𝜛 : (𝛼1, 𝜅1) → (𝛼2, 𝜅2) consists +of a normalised 1-cochain 𝜛 ∈ 𝐵1(𝐺, 𝑀), such that d𝜇 = 𝜅1 · 𝜅−1 +2 ; else, there are no arrows +from (𝛼1, 𝜅1) to (𝛼2, 𝜅2). +(3) Let (𝜇1 : (𝛼1, 𝜅1) → (𝛼2, 𝜅2) and 𝜇2 : (𝛼2, 𝜅2) → (𝛼3, 𝜅3) be arrows with 𝛼1 = 𝛼2 = 𝛼3. +Then 𝜇2 ◦ 𝜇1 : (𝛼1, 𝜅2) → (𝛼3, 𝜅3) is given by 𝜇2 · 𝜇1. +(4) The identity arrow of an object (𝛼, 𝜅) is given by the trivial 1-cochain 1 : Γ → 𝜋1(G). +The following defines an equivalence Θ : H(𝐺;Γ, 𝑀, 𝜔) → Hom(𝐺, 2Grp(Γ, 𝑀, 𝜔)). +(1) Given an object (𝛼, 𝜅), define Θ(𝛼, 𝜅) : 𝐺 → 2Grp(Γ, 𝑀, 𝜔) by +∀𝑔 ∈ 𝐺, +Θ(𝛼, 𝜅)(𝑔) � 𝛼(𝑔); +Θ(𝛼, 𝜅)(0) � (1, 1); +∀𝑔, ℎ ∈ 𝐺, +Θ(𝛼, 𝜅)(2) +𝑔,ℎ � (𝜅(𝑔, ℎ), 𝑔ℎ). +(2) Given an arrow 𝜇 : (𝛼, 𝜅1) → (𝛼, 𝜅2), define Θ(𝜇) : Θ(𝛼, 𝜅1) ⇒ Θ(𝛼, 𝜅2) by +∀𝑔 ∈ 𝐺, +Θ(𝜇)𝑔 � (𝜇(𝑔), 𝛼(𝑔)). +Finally, given coherent 2-groups G and G′, each object 𝑔 of G yields a corresponding +evaluation functor 𝜖𝑔 : Hom(G, G′) → G′ defined by +∀𝑃 ∈ Obj(Hom(G, G′)), +𝜖𝑔(𝑃) � 𝑃(𝑔); +∀𝜂 ∈ Hom(Hom(G, G′)), +𝜖𝑔(𝜂) � 𝜂𝑔. +We now show that Z is indeed the free coherent 2-group on one generator. +Corollary 2.9. Let G be a coherent 2-group. The evaluation functor 𝜖1 : Hom(Z, G) → G +is an equivalence of categories. Hence, for every object 𝑔 of G, there exists an essentially unique +homomorphism 𝐹 : Z → G that satisfies 𝐹(1) � 𝑔. +Proof. By Theorem 2.7, we may assume without loss of generality that exist a group Γ, a +Γ-module 𝑀, and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), such that G = 2Grp(Γ, 𝑀, 𝜔). Hence, +let Θ : H(Z;Γ, 𝑀, 𝜔) → Hom(Z, G) be the equivalence of categories of Theorem 2.8. It +therefore suffices to show that 𝜖1 ◦ Θ : H(Z;Γ, 𝑀, 𝜔) → G is an equivalence of categories. +First, we show that the functor 𝜖1 ◦ Θ is essentially surjective. Let 𝛾 ∈ Γ = Obj(G) be +given, and set 𝛼𝛾 � (𝑘 ↦→ 𝛾𝑘). Since the group Z has cohomological dimension 1 [20, Ex. +2.4.(b)], the 3-cocycle 𝛼∗ +𝛾𝜔 on Z is necessarily trivial in cohomology, so that there exists a +normalised 2-cochain 𝜅𝛾 ∈ 𝐵2(Z, 𝑀) that satisfies d𝜅𝛾 · 𝛼∗ +𝛾𝜔 = 1. It now follows that (𝛼𝛾, 𝜅𝛾) +is a well-defined object of H(Z;Γ, 𝑀, 𝜔) satisfying 𝜖1 ◦ Θ(𝛼𝛾, 𝜅𝛾) = 𝛾. +Next, we show that 𝜖1 ◦ Θ is full. Let (𝑚, 𝛾) ∈ 𝑀 × Γ = Hom(G) be given, so that (𝑚, 𝛾) +is an automorphism of the object 𝛾. By the above argument, let (𝛼𝛾, 𝜅𝛾) be any preimage of + +NONCOMMUTATIVE U(1)-GAUGE THEORY +9 +the object 𝛾 under 𝜖1 ◦ Θ, and let 𝛽(𝑚,𝛾) ∈ 𝑍1(Z, 𝑀) be the unique normalised 1-cocycle with +respect to 𝛼𝛾 that satisfies 𝛽(𝑚,𝛾) (1) = 𝑚. Then 𝛽(𝑚,𝛾) : (𝛼𝛾, 𝜅𝛾) → (𝛼𝛾, 𝜅𝛾) is a well-defined +arrow of H(Z;Γ, 𝑀, 𝜔) satisfying 𝜖1 ◦ Θ(𝛽(𝑚,𝛾)) = (𝑚, 𝛾). +Finally, we show that the functor 𝜖1 ◦ Θ is faithful. Fix a homomorphism 𝛼 : Z → Γ +and normalised 2-cochains 𝜅, 𝜅′ ∈ 𝐵2(Z, 𝑀), such that d𝜅 = d𝜅′ = (𝛼∗𝜔)−1; suppose that +𝜇1, 𝜇2 : (𝛼, 𝜅) → (𝛼, 𝜅′) satisfy 𝜖1 ◦ Θ(𝜇1) = 𝜖1 ◦ Θ(𝜇2). This means that 𝜇1, 𝜇2 ∈ 𝐵1(Z, 𝑀) +are normalised chains, such that d𝜇1 = 𝜅 · (𝜅′)−1 = d𝜇2 and 𝜇1(1) = 𝜇2(1). It follows that +𝛽 � 𝜇1 · 𝜇−1 +2 is a normalised 1-cocycle on Z that satisfies 𝛽(1) = 1, but the only such 1-cocycle +is the trivial 1-cocycle 𝑚 ↦→ 1, which forces 𝜇1 = 𝜇2. +□ +2.2. The Picard 2-group of a nc topological space. Let 𝐵 be a given unital pre-𝐶∗-algebra, +which we view as a nc topological space. We now review the theory of nc Hermitian line +bundles over 𝐵, i.e., strong Morita auto-equivalences [88] passed through the algebraic lens +of Beggs–Brzeziński [9]. This is standard material with adaptations to the setting of pre-𝐶∗- +algebras; following Kajiwara–Watatani [60], we derive substantial technical simplifications +from the systematic use of finite pre-Hilbert module frames or bases. +Let 𝐸 be a right 𝐵-bimodule. Recall that a 𝐵-valued inner product on 𝐸 is a R-bilinear map +(·, ·) : 𝐸 × 𝐸 → 𝐵 that is right 𝐵-linear in the second argument and satisfies +∀𝑥, 𝑦 ∈ 𝐸, +(𝑦, 𝑥) = (𝑥, 𝑦)∗; +hence, we define a cobasis for (·, ·) to be finite family (𝜖𝑖)𝑛 +𝑖=1 in 𝐸, such that �𝑛 +𝑖=1(𝜖𝑖, 𝜖𝑖) = 1, and +we say that (·, ·) is strictly full whenever (·, ·) admits a cobasis. Note that a right 𝐵-bimodule +is faithful whenever it admits a strictly full 𝐵-valued inner product [60, Lemma 1.5]. +Definition 2.10 (Rieffel [88, §6], cf. Bass [8, §ii.5]). A Hermitian line 𝐵-bimodule is a 𝐵-bimodule +𝐸 together with strictly full inner products on both 𝐸 and 𝐸, respectively, such that +∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, +∥(𝑏𝑥, 𝑏𝑥)∥ ≤ ∥𝑏∥2∥(𝑥, 𝑥)∥, +(2.5) +∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, +∥(𝑥𝑏, 𝑥𝑏)∥ ≤ ∥𝑏∥2∥(𝑥, 𝑥)∥, +(2.6) +∀𝑏 ∈ 𝐵, ∀𝑥, 𝑦 ∈ 𝐸, +(𝑥, 𝑏𝑦) = (𝑏∗𝑥, 𝑦), +(2.7) +∀𝑏 ∈ 𝐵, ∀𝑥, 𝑦 ∈ 𝐸, +(𝑥, 𝑦𝑏) = (𝑥𝑏∗, 𝑦), +(2.8) +∀𝑥, 𝑦, 𝑧 ∈ 𝐸, +(𝑥, 𝑦)𝑧 = 𝑥(𝑦, 𝑧). +(2.9) +For example, the trivial Hermitian line 𝐵-bimodule is the trivial 𝐵-bimodule 𝐵 together with +the 𝐵-valued inner products on 𝐵 and 𝐵 defined, respectively, by +∀𝑏, 𝑐 ∈ 𝐵, +(𝑏, 𝑐) � 𝑏∗𝑐, +(𝑏, 𝑐) � 𝑏𝑐∗. +(2.10) +This example admits the following non-trivial generalisation. +Example 2.11. Let 𝜙 be an isometric ∗-automorphism of 𝐵. Let 𝐵𝜙 � {𝑏𝜙 | 𝑏 ∈ 𝐵} be 𝐵 as a +free left 𝐵-module together with the right 𝐵-module structure defined by +∀𝑏, 𝑐 ∈ 𝐵, +𝑏𝜙 · 𝑐 � (𝑏𝜙(𝑐))𝜙, +and the 𝐵-valued inner products on 𝐵𝜙 and 𝐵𝜙 respectively defined by +∀𝑏, 𝑐 ∈ 𝐵, +(𝑏𝜙, 𝑐𝜙) � 𝜙−1(𝑏∗𝑐), +(𝑏𝜙, 𝑐𝜙) � 𝑏𝑐∗. +Then 𝐵𝜙 is a Hermitian line 𝐵-bimodule with cobases 1𝜙 for 𝐵𝜙 and 1𝜙 for 𝐵𝜙. +Example 2.12. Let 𝑋 be a closed manifold. Recall that the commutative unital ∗-algebra +𝐶∞(𝑋) of smooth complex-valued functions on 𝑋 defines a unital pre-𝐶∗-algebra with respect + +10 +BRANIMIR ĆAĆIĆ +to the supremum norm. Given a Hermitian line bundle E → 𝑋, the balanced 𝐶∞(𝑋)- +bimodule Γ(E) of global smooth sections of Edefines a Hermitian line 𝐶∞(𝑋)-bimodule +with respect to the 𝐶∞(𝑋)-valued inner product on Γ(E) induced by the Hermitian metric +on Eand the 𝐶∞(𝑋)-valued inner product on Γ(E) � Γ(E) defined by +∀𝜎1, 𝜎2 ∈ Γ(E), +(𝜎1, 𝜎2) � (𝜎2, 𝜎1). +In particular, cobases for both of these 𝐶∞(𝑋)-valued inner products can be constructed +using an atlas of local trivialisations for E → 𝑋 together with a smooth partition of unity +subordinate to the corresponding open cover of 𝑋. +Our primary goal for this subsection is the following refinement of standard lore. +Theorem-Definition 2.13 (Rieffel [88, §6], Brown–Green–Rieffel [21]; cf. Bass [8, §ii.5]). The +Picard 2-group of 𝐵 is the coherent 2-group Pic(𝐵) defined as follows. +(1) As a category, Pic(𝐵) is the concrete category whose objects are Hermitian line 𝐵-bimod- +ules and whose arrows are 𝐵-bimodule isomorphisms 𝑢 : 𝐸 → 𝐹, such that +∀𝑥, 𝑦 ∈ 𝐸, +(𝑢(𝑥), 𝑢(𝑦)) = (𝑥, 𝑦). +(2.11) +(2) The monoidal product of objects 𝐸 and 𝐹 is the balanced tensor product 𝐸 ⊗𝐵 𝐹 together +with the 𝐵-valued inner products on 𝐸 ⊗𝐵 𝐹 and 𝐸 ⊗𝐵 𝐹 defined by +∀𝑥1, 𝑦1, 𝑥2, 𝑦2 ∈ 𝐸, +(𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) � (𝑦1, (𝑥1, 𝑥2) 𝑦2), +(2.12) +∀𝑥1, 𝑦1, 𝑥2, 𝑦2 ∈ 𝐸, +(𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) � (𝑥1, (𝑦1, 𝑦2)𝑥2), +(2.13) +respectively; moreover, the monoidal product of arrows is given by their monoidal product +in Bimod(𝐵). +(3) The unit object is the trivial Hermitian line 𝐵-bimodule 𝐵, and left unitors, right unitors, +and associators are given by the corresponding left unitors, right unitors, and associators +in Bimod(𝐵), respectively. +(4) The monoidal inverse of a Hermitian line 𝐵-bimodule 𝐸 is 𝐸 together with the given +𝐵-valued inner product on 𝐸 and the 𝐵-valued inner product on 𝐸 defined by +∀𝑥, 𝑦 ∈ 𝐸, +(𝑥, 𝑦) � (𝑥, 𝑦). +(2.14) +(5) The evaluation morphism for an object 𝐸 is the arrow ev𝐸 : 𝐸 ⊗𝐵 𝐸 → 𝐵 defined by +∀𝑒1, 𝑒2 ∈ 𝐸, +ev𝐸(𝑒1 ⊗ 𝑒2) � (𝑒1, 𝑒2). +(2.15) +Hence, the Picard group of 𝐵 is the group Pic(𝐵) � 𝜋0(Pic(𝐵)). +Example 2.14 (Bass [8, Prop. 5.2]). The following defines a homomorphism of coherent +2-groups 𝜏 : Aut(𝐵) → Pic(𝐵). +(1) Given 𝜙 ∈ Aut(𝐵), let 𝜏(𝜙) � 𝐵𝜙 be the Hermitian line 𝐵-bimodule of Example 2.11. +(2) Set 𝜏 (0) � id𝐵; given 𝜙, 𝜓 ∈ Aut(𝐵), define 𝜏 (2) +𝜙,𝜓 : 𝜏(𝜙) ⊗𝐵 𝜏(𝜓) → 𝜏(𝜙𝜓) by +∀𝑎, 𝑏 ∈ 𝐵, +𝜏 (2) +𝜙,𝜓 (𝑎𝜙 ⊗ 𝑏𝜓) � (𝑎𝜙(𝑏))𝜙𝜓 . +Recall [60, §1] that a basis for a right 𝐵-module 𝐸 with respect to a right 𝐵-valued inner +product (·, ·) is a finite family (𝑒𝑖)𝑛 +𝑖=1 in 𝐸, such that 𝑥 = �𝑛 +𝑖=1 𝑒𝑖(𝑒𝑖, 𝑥) for all 𝑥 ∈ 𝐸. Thus, we +define a right pre-Hilbert 𝐵-module of finite type to be a right 𝐵-module 𝐸 equipped with a +𝐵-valued inner product ⟨·, ·⟩ that admits a basis. In turn, we denote by Hilb(𝐵) the concrete +category whose objects are right pre-Hilbert 𝐵-modules of finite type and whose arrows are +isomorphisms of right 𝐵-modules satisfying (2.11). + +NONCOMMUTATIVE U(1)-GAUGE THEORY +11 +Example 2.15. Let 𝐵 be a unital pre-𝐶∗-algebra, let 𝑛 ∈ N, and let P ∈ 𝑀𝑛(𝐵) be an +orthogonal projection, which means that P2 = P = 𝑃∗. Then P· 𝐵𝑛 defines a right pre-Hilbert +𝐵-module of finite type with respect to the 𝐵-linear inner product defined by +∀(𝑥𝑖)𝑛 +𝑖=1, (𝑦𝑖)𝑛 +𝑖=1 ∈ P · 𝐵𝑛, +�(𝑥𝑖)𝑛 +𝑖=1, (𝑦𝑖)𝑛 +𝑖=1 +� � +𝑛 +∑︁ +𝑖=1 +𝑥∗ +𝑖 𝑦𝑖. +Note that if 𝐸 is a right pre-Hilbert 𝐵-module of finite type with 𝐵-valued inner product +(·, ·), then 𝐸 is necessarily finitely generated and projective as a right 𝐵-module and (·, ·) is +necessarily positive definite in the sense that +∀𝑥 ∈ 𝐸, +(𝑥, 𝑥) ≥ 0, +(2.16) +{𝑥 ∈ 𝐸 | (𝑥, 𝑥) = 0} = {0}. +(2.17) +Thus, every right pre-Hilbert 𝐵-module is isomorphic in Hilb(𝐵) to a right pre-Hilbert 𝐵- +module of finite type of the kind constructed in Example 2.15, so that the category Hilb(𝐵) is +essentially small. +Now, let 𝐸 be a a right pre-Hilbert 𝐵-module of finite type. By positive-definiteness of the +𝐵-valued inner product (·, ·) on 𝐸, the norm ∥ · ∥ on 𝐸 defined by +∀𝑥 ∈ 𝐸, +∥𝑥∥ � ∥(𝑥, 𝑥)∥1/2 +satisfies the following crucial inequalities: +∀𝑥 ∈ 𝐸, ∀𝑏 ∈ 𝐵, +∥𝑥𝑏∥ ≤ ∥𝑥∥ · ∥𝑏∥, +(2.18) +∀𝑥, 𝑦 ∈ 𝐸, +(𝑥, 𝑦)∗(𝑥, 𝑦) ≤ ∥ 𝑦∥2(𝑥, 𝑥). +(2.19) +Hence, one can show that the algebra L(𝐸) of all right 𝐵-linear maps 𝐸 → 𝐸 defines a unital +pre-𝐶∗-algebra with respect to the ∗-operation implicitly defined by +∀𝑇 ∈ L(𝐸), ∀𝑥, 𝑦 ∈ 𝐸, +(𝑥,𝑇∗ 𝑦) � (𝑇𝑥, 𝑦) +and the operator norm induced by the aforementioned norm ∥ · ∥ on 𝐸. At last, given a unital +pre-𝐶∗-algebra 𝐴, we define an (𝐴, 𝐵)-correspondence of finite type to be a right pre-Hilbert +𝐵-module of finite type 𝐸 equipped with a isometric unital ∗-homomorphism 𝐴 → L(𝐸); in +particular, when 𝐴 = 𝐵, we call 𝐸 a 𝐵-self-correspondence of finite type. +Proposition 2.16. Let 𝐸 be a Hermitian line 𝐵-bimodule equipped with 𝐵-valued inner products +(·, ·)𝐸 on 𝐸 and (·, ·)𝐸 on 𝐸, respectively. Then 𝐸 and 𝐸 define 𝐵-self-correspondences of finite +type with respect to (·, ·)𝐸 and (·, ·)𝐸, respectively, such that +∀𝑥 ∈ 𝐸, +∥(𝑥, 𝑥)𝐸∥ = ∥(𝑥, 𝑥)𝐸∥. +(2.20) +Lemma 2.17 (Rieffel [88, Lemma 6.22], Kajiwara–Watatani [60, Prop. 2.5]). Let 𝐵 be a unital +pre-𝐶∗-algebra, and let 𝐸 be a right pre-Hilbert 𝐵-module of finite type. There exists a unique +isomorphism of L(𝐸)-bimodules coev𝐸 : L(𝐸) → 𝐸 ⊗𝐵 𝐸, such that +∀𝑥, 𝑦, 𝑧 ∈ 𝐸, +coev−1 +𝐸 (𝑥 ⊗ 𝑦)𝑧 = 𝑥(𝑦, 𝑧). +Proof of Prop. 2.16. Fix cobases (𝜖𝑖)𝑚 +𝑖=1 and (𝑒𝑗)𝑛 +𝑗=1 for (·, ·)𝐸 and (·, ·)𝐸, respectively. Using (2.9), +one shows that (𝑒𝑗)𝑛 +𝑗=1 is a basis for 𝐸 with respect to (·, ·)𝐸 and that (𝜖𝑖)𝑚 +𝑖=1 is a basis for 𝐸 +with respect to (·, ·)𝐸, so that 𝐸 is a right pre-Hilbert 𝐵-module of finite type with respect to +(·, ·)𝐸, and 𝐸 is a right pre-Hilbert 𝐴-module of finite type with respect to (·, ·)𝐸. +Next, by (2.5) and (2.7), the map 𝜋𝐸 : 𝐵 → L(𝐸) defines a bounded ∗-homomorphism, +which is surjective by Lemma 2.17 together with strict fullness of (·, ·)𝐸 and injective by strict + +12 +BRANIMIR ĆAĆIĆ +fullness of (·, ·)𝐸. By symmetry, this also shows that 𝜋𝐸 : 𝐵 → L(𝐸) defines a bounded +bijective ∗-homomorphism 𝜋𝐸 : 𝐵 → L(𝐸). +Now, by the fact that (·, ·)𝐸 is positive definite together with the assumption that 𝐵 is a +pre-𝐶∗-algebra, the data �𝐸, (·, ·)𝐸, (·, ·)𝐸 +� yield a pre-imprimitivity (𝐵, 𝐵)-bimodule in the +usual sense [88, Def. 6.10] with left 𝐵-valued inner product induced by (·, ·)𝐸. Thus, Equation +2.20 follows from the corresponding result for pre-imprimitivity bimodules [86, Prop. 3.11]. +We now prove boundedness of 𝜋−1 +𝐸 and 𝜋−1 +𝐸 as follows. Let 𝑡 ∈ L(𝐸) be given. Using (2.9), one +shows that 𝜋−1 +𝐸 (𝑡) = �𝑛 +𝑗=1(𝑡𝑒𝑗, 𝑒𝑗), so that +∥𝜋−1 +𝐸 (𝑡)∥ ≤ +∑︁𝑛 +𝑗=1∥(𝑡𝑒𝑗, 𝑒𝑗)∥ ≤ +∑︁𝑛 +𝑗=1∥𝑡𝑒𝑗∥∥𝑒𝑗∥ = +∑︁𝑛 +𝑗=1∥𝑡𝑒𝑗∥∥𝑒𝑗∥ ≤ +�∑︁𝑛 +𝑗=1∥𝑒𝑗∥2� +∥𝑡∥, +by (2.19) together with (2.20). The same argument also shows that 𝜋−1 +𝐸 is bounded. Thus, the +maps 𝜋𝐸 and 𝜋𝐸 are bounded bijective ∗-homomorphisms between unital pre-𝐶∗-algebras +with bounded inverses, and hence are both isometric ∗-isomorphisms. +□ +It is easy to check that a 𝐵-self-correspondence of finite type admits at most one 𝐵-valued +inner product on 𝐸 making 𝐸 into a Hermitian line 𝐵-bimodule. Indeed, suppose that (·, ·)1 +and (·, ·)2 are two such 𝐵-valued inner products on 𝐸. Then, for all 𝑧 ∈ 𝐸, +((𝑥, 𝑦)1 − (𝑥, 𝑦)2)𝑧 = 𝑥(𝑦, 𝑧) − 𝑥(𝑦, 𝑧) = 0𝑧, +so that (𝑥, 𝑦)1 = (𝑥, 𝑦)2 by strict fullness of either of (·, ·)1 or (·, ·)2. Moreover, by Proposition +2.16, such a 𝐵-valued inner product on 𝐸 exists only if the left 𝐵-module structure 𝐵 → L(𝐸) +on 𝐸 is an isometric ∗-isomorphism. This turns out to be not only necessary but sufficient. +Corollary 2.18. Let 𝐸 be an 𝐵-self-correspondence of finite type with strictly full 𝐵-valued inner +product, and let 𝜋𝐸 : 𝐵 → L(𝐸) be the left 𝐵-module structure on 𝐸. There exists a 𝐵-valued +inner product on 𝐸 making 𝐸 into a Hermitian line 𝐵-bimodule if and only if 𝜋𝐸 is an isometric +∗-isomorphism. +Proof. Suppose that the left 𝐵-module structure 𝜋𝐸 is an isometric ∗-isomorphism. Note that +(2.5) and (2.7) are already satisfied. By Lemma 2.17 together with bijectivity of 𝜋𝐸, we may +define an 𝐵-valued inner product (·, ·) on 𝐸 satsifying (2.9) and (2.8) by (𝑥, 𝑦) � 𝜋−1 +𝐸 (𝑥 ⊗ 𝑦) +for 𝑥, 𝑦 ∈ 𝐸; indeed, this 𝐵-valued inner product is strictly full since any basis (𝑒𝑖)𝑛 +𝑖=1 for 𝐸 +yields a cobasis (𝑒𝑖)𝑛 +𝑖=1 for 𝐸. Finally, Equation 2.6 follows since, for all 𝑥, 𝑦 ∈ 𝐸 and 𝑏 ∈ 𝐵, by +positive definitness of ⟨·, ·⟩ on 𝐸, isometry of 𝜋𝐸, and equations 2.18 and 2.19, +∥(𝑥𝑏, 𝑥𝑏) 𝑦∥ = ∥𝑥𝑏(𝑥𝑏, 𝑦)∥ = ∥𝑥𝑏𝑏∗(𝑥, 𝑦)∥ ≤ ∥𝑥∥∥𝑏∥2∥(𝑥, 𝑦)∥ ≤ ∥𝑏∥2∥𝑥∥2∥ 𝑦∥. +□ +At last, we can prove Theorem-Definition 2.13 exactly as stated. +Proof of Theorem-Definition 2.13. First, by swapping out Hermitian line 𝐵-bimodules for 𝐵- +self-correspondences in the proposed definition of Pic(𝐵), we obtain a more familiar essen- +tially small monoidal concrete category Corr(𝐵) whose objects are 𝐵-self-correspondences of +finite type [23, §2.2]; note that essential smallness of Corr(𝐵) follows from essential smallness +of the category Hilb(𝐵). Corollary 2.18 now implies that the category Pic(𝐵) is well-defined as +a strictly full subcategory of Corr(𝐵), which clearly contains the monoidal unit 𝐵. Moreover, +Proposition 2.16 and Corollary 2.18 together show that monoidal inversion is well-defined as a +function Obj(Pic(𝐵)) → Obj(Pic(𝐵)). +Next, let 𝐸 and 𝐹 be Hermitian 𝐵-line modules, so that their tensor product 𝐸 ⊗𝐵 𝐹 in +Corr(𝐵) is a well-defined 𝐵-self-correspondence of finite type. On the one hand, the 𝐵- +valued inner product on 𝐸 ⊗𝐵 𝐹 is strictly full since cobases (𝜖𝑖)𝑛 +𝑖=1 and (𝜙𝑗)𝑞 +𝑗=1 for 𝐸 and 𝐹, + +NONCOMMUTATIVE U(1)-GAUGE THEORY +13 +respectively, yield a cobasis (𝜖𝑖 ⊗ 𝜙𝑗)1≤𝑖≤𝑛,1≤𝑗≤𝑞 for 𝐸 ⊗𝐵 𝐹. On the other hand, the 𝐵-valued +inner product on the tensor product 𝐹 ⊗𝐵 𝐸 in Corr(𝐵) pulls back under the canonical +isomorphism of 𝐵-bimodules (𝑥 ⊗ 𝑦 ↦→ 𝑦 ⊗ 𝑥) : 𝐸 ⊗𝐵 𝐹 → 𝐹 ⊗𝐵 𝐸 to the 𝐵-valued inner +product on 𝐸 ⊗𝐵 𝐹 of (2.13); this 𝐵-valued inner product is strictly full since strict cobases +(𝑒𝑖)𝑚 +𝑖=1 and (𝑓𝑗)𝑝 +𝑗=1 for 𝐸 and 𝐹, respectively, yield a cobasis (𝑒𝑖 ⊗ 𝑓𝑗)1≤𝑖≤𝑚,1≤𝑗≤𝑝 for 𝐸 ⊗𝐵 𝐹. +Equation (2.9) for 𝐸 ⊗𝐵 𝐹 now follows from repeated applications of (2.7), (2.8), and (2.9). +Finally, let 𝐸 be a Hermitian 𝐵-line module. By Lemma 2.17 together with Proposition 2.16, +the map ev𝐸 is an isomorphism of 𝐵-bimodules; that ev𝐸 satisfies (2.11) now follows from +observing that for all 𝑥1, 𝑥2 ∈ 𝐸 and 𝑦1, 𝑦2 ∈ 𝐸, +(𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) = (𝑦1, (𝑥1, 𝑥2) 𝑦2) = (𝑦1, 𝑥1(𝑥2, 𝑦2)) = (𝑥1, 𝑦1)∗(𝑥2, 𝑦2). +□ +This characterization of the monoidal category Pic(𝐵) as a monoidal subcategory of the +monoidal category Corr(𝐵) of 𝐵-self-correspondences of finite type yields, with superficial +changes, a right action of the Picard group Pic(𝐵) on the 𝐾0-monoid V(𝐵) of isomorphism +classes of right pre-Hilbert 𝐵-modules of finite type. Indeed, given a right pre-Hilbert 𝐵- +module of finite type 𝐸 and a Hermitian line 𝐵-bimodule 𝐹, set [𝐸] ⊳ [𝐹] � [𝐸 ⊗𝐵 𝐹], where +the balanced tensor product 𝐸 ⊗𝐵 𝐹 is equipped with the right 𝐵-valued inner product given +by (2.12). We may use this Pic(𝐵)-action to characterise the fibres of the obvious forgetful map +Pic(𝐵) → V(𝐵); in turn, this helps us understand the information lost when passing from +Pic(𝐵) to the 𝐾-theory of 𝐵 or its 𝐶∗-algebraic completion. +Proposition 2.19 (Bass [8, Propp. 5.2 & 5.3]). Let ΠV(𝐵) : Pic(𝐵) → V(𝐵) denote the set +function induced by the forgetful functor Pic(𝐵) → Hilb(𝐵). Let Pic(𝐵)[𝐵] denote the stabiliser +subgroup of Pic(𝐵) with respect to [𝐵] ∈ ran ΠV(𝐵). Then the homomorphism of coherent 2-groups +𝜏 : Aut(𝐵) → Pic(𝐵) of Example 2.14 yields into a exact sequence of groups +1 → U(Z(𝐵)) → U(𝐵) +𝑢↦→Ad𝑢 +−−−−−→ Aut(𝐵) +𝜋0(𝜏) +−−−−→ Pic(𝐵)[𝐵] → 1. +Note that this canonically identifies the outer automorphism group of 𝐵 with a subgroup +of Pic(𝐵). What is more surprising is that the entire Picard group Pic(𝐵) acts as isometric +∗-automorphisms on the centre of 𝐵. +Proposition-Definition 2.20 (Fröhlich [50, Thm. 2], Beggs–Brzeziński [9, §10]). The Fröhlich +homomorphism of 𝐵 is the unique group homomorphism Φ : Pic(𝐵) → Aut(Z(𝐵)), such that, +for every Hermitian line 𝐵-bimodule 𝐸, the Fröhlich automorphism Φ[𝐸] of [𝐸] satisfies +∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, +Φ[𝐸](𝑏)𝑥 = 𝑥𝑏. +(2.21) +Hence, the canonical left action of 𝜋0(Pic(𝐵)) � Pic(𝐵) on 𝜋1(Pic(𝐵)) = U(Z(𝐵)) is the +left action induced by Φ. +Proof. Relative to the references, it remains to show each Fröhlich automorphism is isometric. +Let 𝐸 be a Hermitian line 𝐵-bimodule, and let (𝑒𝑖)𝑛 +𝑖=1 be a cobasis for 𝐸. Then, +∀𝑏 ∈ 𝐵, +∥Φ−1 +[𝐸](𝑏)∥ = +��� +∑︁𝑛 +𝑖=1(𝑒𝑖, 𝑏𝑒𝑖) +��� ≤ +∑︁𝑛 +𝑖=1∥𝑒𝑖∥∥𝑏𝑒∥ ≤ +�∑︁𝑛 +𝑖=1∥𝑒𝑖∥2� +∥𝑏∥, +so that Φ−1 +[𝐸] is bounded. Using 𝐸 instead now shows that 𝜙[𝐸] = 𝜙−1 +[𝐸] is also bounded. +□ +Example 2.21. We continue from Example 2.12. Let Pic(𝑋) be the Picard group of isomor- +phism classes of complex line bundles over 𝑋, which admits a right action of Diff(𝑋) by +pullbacks. On the one hand, any two Hermitian metrics on a line bundle are unitarily equiva- +lent, so that ([E] ↦→ [Γ(E)]) : Pic(𝑋) → Pic(𝐶∞(𝑋)) is a well-defined homomorphism. On +the other hand, the map �𝑓 ↦→ (𝑓 −1)∗� : Diff(𝑋) → Aut(𝐶∞(𝑋)) is a group isomorphism [74], + +14 +BRANIMIR ĆAĆIĆ +so let Ψ : Pic(𝐶∞(𝑋)) → Diff(𝑋) be the resulting homomorphism induced by the Fröhlich +homomorphism of 𝐶∞(𝑋). Thus, Serre–Swan duality yields a split exact sequence +1 → Pic(𝑋) +[E]↦→[Γ(E)] +−−−−−−−−−−→ Pic(𝐶∞(𝑋)) +Ψ−→ Diff(𝑋) → 1 +with right splitting 𝜙 ↦→ 𝜋0(𝜏)((𝜙−1)∗). Moreover, given the resulting isomorphism +�(𝜙, [E]) ↦→ [Γ((𝜙−1)∗E)] · 𝜋0(𝜏)((𝜙−1)∗)� : Diff(𝑋) ⋉ Pic(𝑋) → Pic(𝐶∞(𝑋)), +we may identify the Fröhlich homomorphism of 𝐶∞(𝑋) with the quotient map +((𝜙, [E]) ↦→ 𝜙) : Diff(𝑋) ⋉ Pic(𝑋) → Diff(𝑋). +We conclude by noting certain simplications that arise when 𝐵 behaves sufficiently like a +𝐶∗-algebra. This will permit us to introduce our first main running example. +Let 𝑛 ∈ N, and let 𝑀𝑛(𝐵) denote the unital ∗-algebra of 𝑛 × 𝑛 matrices with entries in 𝐵, +which is defined by analogy with 𝑀𝑛(C); one calls 𝑀𝑛(𝐵) a matrix algebra over 𝐵. Recall that +𝐵𝑛 defines a right pre-Hilbert 𝐵-module of finite type by Example 2.15; hence observe that +matrix multiplication on left defines an injective ∗-homomorphism 𝑀𝑛(C) → L(𝐵𝑛). Thus, +the operator norm on L(𝐵𝑛) pulls back to a 𝐶∗-norm on 𝑀𝑛(𝐵). +Definition 2.22. We say that 𝐵 admits polar decompositions if, for every 𝑛 ∈ N and positive +𝑏 ∈ 𝑀𝑛(𝐵), there exists unique positive element +√ +𝑏 ∈ 𝑀𝑛(𝐵) that satisfies ( +√ +𝑏)2 = 𝑏 and +is invertible in 𝑀𝑛(𝐵) whenever 𝑏 is. In this case, given 𝑛 ∈ N, the polar decomposition of +invertible 𝑏 ∈ 𝑀𝑛(𝐵) is 𝑏 = sgn(𝑏)|𝑏|, where |𝑏| � +√ +𝑏∗𝑏 ∈ 𝑀𝑛(𝐵) is positive and invertible +and sgn(𝑏) � 𝑏|𝑏|−1 ∈ 𝑀𝑛(𝐵) is unitary. +For example, a unital 𝐶∗-algebra admits polar decompositions by the holomorphic func- +tional calculus. More generally, a unital pre-𝐶∗-algebra 𝐵 admits polar decompositions +whenever it and all its matrix algebras are closed under the holomorphic functional calculus +in their respective 𝐶∗-closures. +Finally, recall that a 𝐵-valued inner product on a right 𝐵-module 𝐸 is algebraically full +whenever it satisfies SpanC{(𝑥, 𝑦) | 𝑥, 𝑦 ∈ 𝐸} = 𝐵. +Proposition 2.23. Suppose that 𝐵 is unital pre-𝐶∗-algebra that admits polar decompositions. Let +𝐸 be a 𝐵-bimodule, let (·, ·)𝐸 be a 𝐵-valued inner product on 𝐸, and let (·, ·)𝐸 be a 𝐵-valued inner +product on 𝐸. Then 𝐸 defines a Hermitian line 𝐵-bimodule with respect to (·, ·)𝐸 and (·, ·)𝐸 if and +only if the following conditions are all satisfied: +(1) the 𝐵-valued inner product (·, ·)𝐸 is algebraically full and satisfies (2.16), (2.5) and (2.7); +(2) the 𝐵-valued inner product (·, ·)𝐸 is algebraically full and satisfies (2.16), (2.6) and (2.8); +(3) the 𝐵-valued inner products (·, ·)𝐸 and (·, ·)𝐸 respectively satisfy (2.9). +Proof. The forward implication is trivial, so we prove the backward implication. Suppose that +all three conditions are satisfied; it remains to show that both (·, ·)𝐸 and (·, ·)𝐸 are strictly full. +Since (·, ·)𝐸 is algebraically full, we may choose finite families (𝑥𝑖)𝑛 +𝑖=1 and (𝑦𝑖)𝑛 +𝑖=1 in 𝐸 that +satisfy �𝑛 +𝑖=1(𝑥𝑖, 𝑦𝑖)𝐸 = 1. Define 𝑋 ∈ 𝑀𝑛(𝐵) by 𝑋 � �(𝑥𝑖, 𝑥𝑗)𝐸 +�𝑛 +𝑖,𝑗=1, so that, by (2.9), +1 = +𝑛 +∑︁ +𝑖,𝑗=1 +(𝑦𝑖, 𝑥𝑖)𝐸(𝑥𝑗, 𝑦𝑗)𝐸 = +𝑛 +∑︁ +𝑖,𝑗=1 +(𝑦𝑖, 𝑥𝑖(𝑥𝑗, 𝑦𝑗)𝐸)𝐸 = +𝑛 +∑︁ +𝑖,𝑗=1 +(𝑦𝑖, (𝑥𝑖, 𝑥𝑗)𝐸 𝑦𝑗)𝐸 = +𝑛 +∑︁ +𝑖,𝑗=1 +(𝑦𝑖, 𝑋𝑖𝑗 𝑦𝑗)𝐸. +Applying to [88, Cor. 2.7] to 𝑋 as a bounded operator on 𝐵𝑛 with the 𝐵-valued inner product +of Example 2.15 shows that 𝑋 ≥ 0. By our hypothesis on 𝐵, there exists 𝑎 = (𝑎𝑖𝑗)𝑛 +𝑖,𝑗=1 ∈ 𝑀𝑛(𝐵), +such that 𝑎∗𝑎 = 𝑋; it now follows that ��𝑛 +𝑘=1 𝑎𝑖𝑘 𝑦𝑘 +�𝑛 +𝑖=1 is a cobasis for (·, ·)𝐸. An identical +argument shows that (·, ·)𝐸 is strictly full. +□ + +NONCOMMUTATIVE U(1)-GAUGE THEORY +15 +We now introduce our first main running example. Let 𝜃 ∈ R, so that the corresponding +(continuous) nc 2-torus is the universal 𝐶∗-algebra 𝐶𝜃(T2) generated by unitaries 𝑢 and 𝑣 +satisfying 𝑣𝑢 = e2𝜋i𝜃𝑢𝑣. The corresponding smooth nc 2-torus 𝐶∞ +𝜃 (T2) is the dense unital ∗- +subalgebra of 𝐶𝜃(T2) consisting of Laurent series in 𝑢 and 𝑣 with rapidly decaying coefficients, +which admits polar decompositions since it and all its matrix algebras are closed under the +holomorphic functional calculus [31]. +Example 2.24. Let 𝜃 ∈ R be a quadratic irrationality, so that the subgroup +Γ𝜃 � +� +𝑔 ∈ SL(2, Z) +��� 𝑔11𝜃+𝑔12 +𝑔21𝜃+𝑔22 = 𝜃, 𝑔21𝜃 + 𝑔22 > 0 +� +of SL(2, Z) is non-trivial and hence infinite cyclic [53, Thm 5.2.10]. Connes’s Heisenberg +modules [31] over the unital pre-𝐶∗-algebra 𝐶∞ +𝜃 (T2) yield, in particular, a homomorphism +𝐸 : Γ𝜃 → Pic(𝐶∞ +𝜃 (T2)) as follows. +(1) Given 𝑔 ∈ Γ𝜃, let 𝐸(𝑔) be the basic Heisenberg module of rank 𝑔21𝜃 + 𝑔22 and degree 𝑔21 +[84, §1.1], which defines a Hermitian line 𝐶∞ +𝜃 (T2)-bimodule by a result of Rieffel [89, Thm +2.15] together with Proposition 2.23. +(2) Since 𝐸(1) = 𝐶∞ +𝜃 (T2), set 𝐸(0) � id𝐶∞ +𝜃 (T2). +(3) Given 𝑔, ℎ ∈ Γ𝜃, let 𝐸(2) +𝑔,ℎ : 𝐸(𝑔) ⊗𝐴∞ +𝜃 𝐸(ℎ) → 𝐸(𝑔ℎ) be the isomorphism of 𝐶∞ +𝜃 (T2)- +bimodules constructed by Schwarz [91, §3] and Dieng–Schwarz [37], which is an isomor- +phism of Hermitian line 𝐶∞ +𝜃 (T2)-bimodules by a result of Vlasenko [97, Thm 6.1]. +In particular, that the functor 𝐸 is monodal with respect to 𝐸(0) and 𝐸(2) reduces to a result +of Polishchuk–Schwarz [84, Prop. 1.2]. +2.3. The differential Picard 2-group of a noncommutative manifold. At last, we build +on results of Beggs–Majid [13] to construct a coherent 2-group of Hermitian line bundles with +connection over a manifold, which we term the differential Picard 2-group. +We begin with preliminary definitions. Recall that a graded algebra is a unital complex +algebra Ω together with a vector space decomposition Ω = �∞ +𝑘=0 Ω𝑘, such that 1 ∈ Ω0 and +Ω𝑗 · Ω𝑗+𝑘 ⊆ Ω𝑗+𝑘 for all 𝑗, 𝑘 ∈ N0. Hence, a graded ∗-algebra is a graded algebra Ω = �∞ +𝑘=0 Ω𝑘 +with a unit- and grading-preserving complex-antilinear involution ∗ : Ω → Ω, such that +∀𝑗, 𝑘 ∈ N0, ∀𝛼 ∈ Ω𝑗, ∀𝛽 ∈ Ω𝑘, +(𝛼𝛽)∗ = (−1)𝑗𝑘𝛽𝛼. +At last, given a unital pre-𝐶∗-algebra 𝐵, a graded ∗-algebra over 𝐵 is a graded ∗-algebra Ω +together with a unital ∗-isomorphism 𝐵 → Ω0, which we suppress; in this case, we denote by +Aut(Ω) the group of all grading- and ∗-preserving automorphisms 𝜙 of Ω as a unital complex +algebra that restrict to an isometric ∗-automorphism of 𝐵. +Now, suppose that 𝐵 is a unital pre-𝐶∗-algebra. We define a ∗-quasi-differential graded +algebra or ∗-quasi-dga over 𝐵 to be a pair (Ω, d), where Ω is a graded ∗-algebra over 𝐵 and +d : Ω → Ω is a ∗-preserving complex linear map that satisfies d(Ω𝑘) ⊂ Ω𝑘+1 for all 𝑘 ∈ N0 +together with the graded Leibniz rule +∀𝑘 ∈ N0, ∀𝛼 ∈ Ω𝑘, ∀𝛽 ∈ Ω, +d𝐵(𝛼𝛽) = d𝐵(𝛼)𝛽 + (−1)𝑘𝛼d𝐵(𝛽); +hence, its graded centre, then, is the graded ∗-subalgebra Z(Ω) of Ω defined by +∀𝑚 ∈ N0, +Z(Ω)𝑚 � {𝜔 ∈ Ω𝑚 | ∀𝑛 ∈ N0, ∀𝜉 ∈ Ω𝑛 +𝐵, 𝜔𝜉 = (−1)𝑚𝑛𝜉𝜔}, +which is closed under d, and its dimension, if it exists, is the largest 𝑁 ∈ N such that Ω𝑁 ≠ 0 +and Ω𝑘 = 0 for all 𝑘 > 𝑁. At last, we call (Ω, d) a ∗-exterior algebra over 𝐵 whenever the +algebra Ω is generated by 𝐵 and d(𝐵) and the map d satisfies d2 = 0. + +16 +BRANIMIR ĆAĆIĆ +Finally, we may define a concrete category QDGA whose objects (𝐵; Ω, d) consist of a +unital pre-𝐶∗-algebra 𝐵 together with a choice of ∗-quasi-dga (Ω, d) over 𝐵 and whose +arrows 𝑓 : (𝐵1, Ω1, d1) → (𝐵2, Ω2, d2) consist of a grading- and ∗-preserving homomorphism +of unital complex algebras 𝑓 : Ω1 → Ω2 that restrict to a bounded (and hence necessarily +contractive) ∗-homomorphism 𝑓↾𝐵1: 𝐵1 → 𝐵2 and satisfy 𝑓 ◦d1 = d2 ◦𝑓. In particular, given a +∗-quasi-dga (Ω, d) over a unital pre-𝐶∗-algebra 𝐵, we denote by Aut(Ω, d) the automorphism +group of (𝐵; Ω, d) in this category. +From now on, let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior calculus (Ω𝐵, d𝐵), which +we view as a manifold. Given a 𝐵-bimodule 𝐸, we shall apply the following Sweedler-type +notation to elements of 𝐸 ⊗𝐵 Ω1 +𝐵 and Ω1 +𝐵 ⊗𝐵 𝐸, respectively: +∀𝜂 ∈ 𝐸 ⊗𝐵 Ω1 +𝐵, +𝜂⟨0⟩ ⊗ 𝜂⟨1⟩ � 𝜂; +∀𝜉 ∈ Ω1 +𝐵 ⊗𝐵 𝐸, +𝜉⟨−1⟩ ⊗ 𝜉⟨0⟩ � 𝜉. +We now recall the relevant generalization of unitary connection appropriate to our setting. +Let 𝐸 be a right pre-Hilbert 𝐵-module of finite type. Extend the 𝐵-valued inner product (·, ·) +on 𝐸 to a real-bilinear map (·, ·) : 𝐸 ⊗𝐵 Ω𝐵 × 𝐸 ⊗𝐵 Ω𝐵 → Ω𝐵 by setting +∀𝑥, 𝑦 ∈ 𝐸, ∀𝛼, 𝛽 ∈ Ω𝐵, +(𝑥 ⊗ 𝛼, 𝑦 ⊗ 𝛽) � 𝛼∗(𝑥, 𝑦)𝛽. +This extension satisfies +∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, ∀𝛽 ∈ Ω𝐵, +(𝜉, 𝜐𝛽) = (𝜉, 𝜐)𝛽, +(2.22) +∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, +(𝜐, 𝜉) = (−1) |𝜉 ||𝜐|(𝜉, 𝜐)∗. +(2.23) +Following Connes [33, Def. ii.18], one now defines right Hermitian connection on 𝐸 to be a +complex-linear map ∇ : 𝐸 → 𝐸 ⊗𝐵 Ω1 +𝐵, such that +∀𝑥 ∈ 𝐸, ∀𝑏 ∈ 𝐵, +∇(𝑥𝑏) = ∇𝑥𝑏 + 𝑥 ⊗ d𝐵𝑏, +(2.24) +∀𝑥, 𝑦 ∈ 𝐸, +d𝐵(𝑥, 𝑦) = (∇𝑥, 𝑦 ⊗ 1) + (𝑥 ⊗ 1, ∇𝑦). +(2.25) +One can now show that there exists unique complex-linear ∇ : 𝐸⊗𝐵 Ω𝐵 → 𝐸⊗𝐵 Ω𝐵 extending +the right Hermitian connection ∇, such that +∀𝜂 ∈ 𝐸⊗𝐵, ∀𝛽 ∈ Ω𝐵, +∇(𝜂𝛽) = ∇(𝜂)𝛽 + (−1) |𝜂|𝜂d𝛽, +∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, +d𝐵(𝜉, 𝜐) = (∇𝜉, 𝜂) + (−1) |𝜉 |(𝜉, ∇𝜂). +Definition 2.25 (Beggs–Majid [13, Def. 5.1 & §5.2]). Let 𝐸 be a 𝐵-self-correspondence of finite +type. A generalised braiding for 𝐸 is an isomorphism 𝜎 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 of graded +𝐵-bimodules that extends 𝜌−1 +𝐸 ◦ 𝜆𝐸 : 𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 𝐵 and satisfies +∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝑥 ∈ 𝐸, +𝜎 (𝛼 ⊗ 𝜎 (𝛽 ⊗ 𝑥) ⟨0⟩)𝜎 (𝛽 ⊗ 𝑥) ⟨1⟩ = 𝜎 (𝛼𝛽 ⊗ 𝑥). +(2.26) +Hence, a Hermitian bimodule connection on 𝐸 is a pair (𝜎, ∇), where 𝜎 is a Hermitian generalised +braiding on 𝐸 and ∇ is a Hermitian right connection on 𝐸, such that +∀𝛽 ∈ Ω𝐵, ∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, +∇(𝛽𝜉) = d𝐵(𝛽)𝜉 + (−1) |𝛽|𝛽∇𝜉, +(2.27) +where 𝐸 ⊗𝐵 Ω𝐵 carries the graded Ω𝐵-bimodule structure given by +∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, +𝛼𝜉𝛽 � 𝜎 (𝛼 ⊗ 𝜉⟨0⟩)𝜉⟨1⟩𝛽. +(2.28) +Example 2.26. The pair (𝜎𝐵, ∇𝐵) � (𝜆−1 +Ω𝐵 ◦ 𝜌Ω𝐵, 𝜆−1 +Ω𝐵 ◦ d) defines a Hermitian bimodule +connection on the trivial Hermitian line 𝐵-bimodule 𝐵; where convenient, we shall abuse +notation and identify (𝜎𝐵, ∇𝐵) with (idΩ𝐵, d). + +NONCOMMUTATIVE U(1)-GAUGE THEORY +17 +It is easy enough to check that if 𝐸 is a 𝐵-self-correspondence of finite type with right +Hermitian connection ∇𝐸, then there exists at most one Hermitian generalised braiding 𝜎𝐸 on +𝐸 that makes (𝜎𝐸, ∇𝐸) into a Hermitian bimodule connection. Moreover, in this case, +∀𝜉, 𝜐 ∈ Ω𝐵, +d𝐵(𝜉, 𝜐) = (∇𝐸𝜉, 𝜐) + (−1) |𝜉 |(𝜉, ∇𝐸𝜐), +(2.29) +∀𝛽 ∈ Ω𝐵, ∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, +(𝛼𝜉, 𝜐) = (𝜉, 𝛼∗𝜐); +(2.30) +by (2.26), it suffices to check (2.30) in the special case where 𝛽 ∈ d(𝐵) and 𝜉, 𝜐 ∈ 𝐸 ⊗ 1. +We shall use the following characterisation of Hermitian bimodule connections. +Proposition 2.27 (Beggs–Majid [13, Lemma 5.2]). Let 𝐸 be a 𝐵-self-correspondence of finite type, +let 𝜎 be a generalised braiding on 𝐸, and let ∇ be a Hermitian right connection on 𝐸. Then (𝜎, ∇) +defines a Hermitian bimodule connection on 𝐸 if and only if the following both hold: +∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, +∇(𝑏𝜉) = 𝜎 (d𝐵𝑏 ⊗ 𝑥) + 𝑏∇𝑥, +(2.31) +∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, +∇2(𝑏𝜉) = 𝑏∇2𝜉. +(2.32) +We now introduce our first nontrivial family of examples of Hermitian bimodule con- +nections on Hermitian line bimodules. Let 𝜃 ∈ R. Recall that the smooth 2-torus 𝐶∞ +𝜃 (T2) +admits a canonical ∗-exterior calculus (Ω𝜃(T2), d) due to Connes [31]. First, let 𝛿1 and 𝛿2 be +the unique ∗-derivations on 𝐶∞ +𝜃 (T2), such that, respectively +∀(𝑚, 𝑛) ∈ Z2, +𝛿1(𝑈𝑚𝑉 𝑛) = 2𝜋i𝑚𝑈𝑚𝑉 𝑛, +𝛿2(𝑈𝑚𝑉 𝑛) = 2𝜋i𝑛𝑈𝑚𝑉 𝑛; +Then let Ω𝜃(T2) be the graded ∗-algebra over 𝐶∞ +𝜃 (T2) generated by central self-adjoint el- +ements 𝑒1, 𝑒2 ∈ Ω1 +𝜃 (T2) satisfying (𝑒1)2 = (𝑒2)2 = 𝑒1𝑒2 + 𝑒2𝑒1 = 0, and let d be the unique +∗-derivation of degree 1 on Ω𝜃(T2), such that +∀𝑎 ∈ 𝐶∞ +𝜃 (T2), d𝑎 � 𝛿1(𝑎)𝑒1 + 𝛿2(𝑎)𝑒2; +d𝑒1 � 0; +d𝑒2 � 0. +In the case where 𝜃 is a quadratic irrationality, the basic Heisenberg modules of Example 2.24 +admit canonical Hermitian bimodule connections due to Connes. +Example 2.28 (Connes [31, Thm 7], Polishchuk–Schwarz [84, Prop. 2.1]). We continue from +Example 2.24. Let 𝑔 ∈ Γ𝜃 be given. Connes constructs maps ∇𝑔,1, ∇𝑔,2 : 𝐸(𝑔) → 𝐸(𝑔) that +yield a right Hermitian connection ∇𝑔 : 𝐸(𝑔) → 𝐸(𝑔) ⊗𝐵 Ω1 +𝐵 by setting +∀𝑝 ∈ 𝐸(𝑔), +∇𝑔(𝑝) � ∇𝑔,1(𝑝) ⊗ 𝑒1 + ∇𝑔,2(𝑝) ⊗ 𝑒2; +in particular, [31, Thm 7] implies that +∀𝑝 ∈ 𝐸(𝑔), +∇2 +𝑔(𝑝) = 𝑝 · 2𝜋i +𝑔21 +𝑔21𝜃 + 𝑔22 +𝑒1𝑒2. +Hence, by [84, Prop. 2.1] together with Proposition 2.27, the map ∇𝑔 defines a Hermitian +bimodule connection on 𝐸(𝑔) with respect to the generalised braiding 𝜎𝑔 given by +∀𝑖 ∈ {1, 2}, ∀𝑝 ∈ 𝐸(𝑔), +𝜎𝑔(𝑒𝑖 ⊗ 𝑝) � +1 +𝑔21𝜃 + 𝑔22 +𝑝 ⊗ 𝑒𝑖. +The primary technical advantage of bimodule connections is that they are compatible with +taking balanced tensor products of bimodules. In fact, they give rise to a monoidal category +of 𝐵-self-correspondence of finite type with Hermitian bimodule connection. +Theorem 2.29 (Beggs–Majid [13, Thm 5.3], cf. Beggs–Brzeziński [10, §2.4]). The following +defines an essentially small monoidal concrete category DCorr(𝐵). +(1) An object of DCorr(𝐵) is a triple (𝐸, 𝜎𝐸, ∇𝐸), where 𝐸 is a 𝐵-self-correspondence of finite +type and (𝜎𝐸, ∇𝐸) is a Hermitian bimodule connection on 𝐸; + +18 +BRANIMIR ĆAĆIĆ +(2) An arrow 𝑢 : (𝐸, 𝜎𝐸, ∇𝐸) → (𝐹, 𝜎𝐹, ∇𝐹) is a 𝐵-bimodule isomorphism 𝑓 : 𝐸 → 𝐹 satisfying +(2.11) and +∇𝐹 ◦ 𝑢 = (𝑢 ⊗ idΩ1 +𝐵) ◦ ∇𝐸. +(2.33) +(3) The tensor product of objects (𝐸, 𝜎𝐸, ∇𝐸) and (𝐹, 𝜎𝐹, ∇𝐹) is (𝐸 ⊗𝐵 𝐹, 𝜎𝐸⊗𝐵𝐹, ∇𝐸⊗𝐵𝐹), where +𝐸 ⊗𝐵 𝐹 is the balanced tensor product of 𝐵-bimodules equipped with the 𝐵-valued inner product +of (2.12), and where +𝜎𝐸⊗𝐵𝐹 � 𝛼−1 +𝐸,𝐹,Ω𝐵 ◦ (id𝐸 ⊗ 𝜎𝐹) ◦ 𝛼𝐸,Ω𝐵,𝐹 ◦ 𝜎𝐸 ⊗ id𝐹, +(2.34) +∇𝐸⊗𝐵𝐹 � 𝛼−1 +𝐸,𝐹,Ω𝐵 ◦ �(id𝐸 ⊗ 𝜎𝐹) ◦ 𝛼𝐸,Ω𝐵,𝐹 ◦ (∇𝐸 ⊗ id) + id𝐸 ⊗ ∇𝐹 +� ; +(2.35) +moreover, the monoidal product of arrows is given by the balanced tensor product of 𝐵-bimodule +homomorphisms, and the associators are given by the corresponding associators in Bimod(𝐵). +(4) The unit object of DCorr(𝐵) is (𝐵, 𝜎𝐵, ∇𝐵), where (𝜎𝐵, ∇𝐵) is the Hermitian bimodule con- +nection of Example 2.26; moreover, left unitors, and right unitors are given by the corresponding +left unitors and right unitors of Bimod(𝐵), respectively. +In addition, if 𝑢 : (𝐸, 𝜎𝐸, ∇𝐸) → (𝐹, 𝜎𝐹, ∇𝐹) is an arrow in DPic(𝐵), then +∇𝐹 ◦ (𝑢 ⊗ idΩ𝐵) = (𝑢 ⊗ idΩ𝐵) ◦ ∇𝐸, +(2.36) +𝜎𝐹 ◦ (idΩ𝐵 ⊗ 𝑢) = (𝑢 ⊗ idΩ𝐵) ◦ 𝜎𝐸. +(2.37) +Proof. Relative to [13, Thm 5.7] and the discussion before the proof of Theorem-Definition +2.13 (with minor changes), it suffices to check that the tensor product is indeed well-defined +on objects. Indeed, let 𝐸 be a 𝐵-self-correspondence of finite type with Hermitian bimodule +connection (𝜎𝐸, ∇𝐸), and let 𝐹 be a 𝐵-self-correspondence of finite type with Hermitian +bimodule connection (𝜎𝐹, ∇𝐹). A straightforward calculation shows that +∀𝑥, 𝑣 ∈ 𝐸, ∀𝜐, 𝜏 ∈ 𝐹 ⊗𝐵 Ω𝐵, +�(𝑥 ⊗ 𝜐⟨0⟩) ⊗ 𝜐⟨1⟩, (𝑣 ⊗ 𝜏⟨0⟩) ⊗ 𝜏⟨1⟩ +� = (𝜐, (𝑥, 𝑣)𝜏). +(2.38) +Relative to [13, Thm 5.7], it remains to check that 𝜎𝐸⊗𝐵𝐹 satisfies (2.30) and that ∇𝐸⊗𝐵𝐹 satisfies +(2.25), but this now follows by repeated application of (2.38), (2.22), and (2.23), as appropriate, +to the Hermitian bimodule connections (𝜎𝐸, ∇𝐸) and (𝜎𝐹, ∇𝐹). +□ +We now construct our coherent 2-group of Hermitian line bundles with unitary connection. +Theorem-Definition 2.30 (cf. Beggs–Majid [11, Thm 3.3]). The differential Picard 2-group of +the unital pre-𝐶∗-algebra 𝐵 is the coherent 2-group DPic(𝐵) defined as follows. +(1) As a monoidal category, DPic(𝐵) is the full monoidal subcategory of DCorr(𝐵), whose +objects are of the form (𝐸, 𝜎𝐸, ∇𝐸), where 𝐸 is a Hermitian line 𝐵-bimodule. +(2) The monoidal inverse of an object (𝐸, 𝜎𝐸, ∇𝐸) is given by (𝐸, 𝜎𝐸, ∇𝐸), where +∀𝛽 ∈ Ω𝐵, ∀𝑥 ∈ 𝐸, +𝜎𝐸(𝛽 ⊗ 𝑥) � ΥΩ𝐵,𝐸 +� +𝜎−1 +𝐸 (𝑥 ⊗ 𝛽∗) +� +, +(2.39) +∀𝑥 ∈ 𝐸, +∇𝐸𝑥 � ΥΩ𝐵,𝐸 +� +𝜎−1 +𝐸 (∇𝐸𝑥) +� +; +(2.40) +here, by abuse of notation, we let ΥΩ𝐵,𝐸 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 denote the isomorphism +of 𝐵-bimodules defined by +∀𝑥 ∈ 𝐸, ∀𝛽 ∈ Ω𝐵, +ΥΩ𝐵,𝐸(𝛽 ⊗ 𝑥) � 𝑥 ⊗ 𝛽∗. +(2.41) +(3) The evaluation arrows are given by the corresponding evaluation arrows in Pic(𝐵). +Hence, a Hermitian line 𝐵-bimodule with connection is an object of DPic(𝐵), and an isomorphism +of Hermitian line 𝐵-bimodules with connection is an arrow of DPic(𝐵). Finally, the differential +Picard group of 𝐵 is the group DPic(𝐵) � 𝜋0(DPic(𝐵)). + +NONCOMMUTATIVE U(1)-GAUGE THEORY +19 +Proof. Given Theorem 2.29 and Theorem-Definition 2.13, it remains to show that monoidal +inversion and evaluation in DPic(𝐵) are well-defined. Let 𝐸 be a Hermitian line 𝐵-bimodule +with Hermitian bimodule connection (𝜎𝐸, ∇𝐸). +Let us first show that 𝐸 admits the Hermitian bimodule connection (𝜎𝐸, ∇𝐸) defined by +(2.39) and (2.40). By a theorem of Beggs–Majid [11, Thm 3.3], suitably adapted, we know that 𝜎𝐸 +is a 𝐵-bimodule isomorphism, that ∇𝐸 satisfies (2.24), and that the pair (𝜎𝐸, ∇𝐸) satisfies (2.31). +By Proposition 2.27, it remains to show that 𝜎𝐸 satisfies (2.26) and that ∇𝐸 satisfies (2.25) and +(2.32). In turn, by construction of 𝜎𝐸 and ∇𝐸, it therefore suffices to show that, respectively, +for all 𝛼, 𝛽 ∈ Ω𝐵 and 𝑥, 𝑦, 𝑧 ∈ 𝐸, +𝜎−1 +𝐸 (𝑥 ⊗ 𝛽𝛼) = 𝜎−1 +𝐸 (𝑥 ⊗ 𝛽) ⟨−1⟩𝜎−1 +𝐸 +� +𝜎𝐸(𝑥 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼 +� +, +(2.42) +𝜎𝐸(d𝐵(𝑥, 𝑦) ⊗ 𝑧) = 𝜎𝐸 +�(∇𝐸𝑥, 𝑦 ⊗ 1) ⊗ 𝑧� + 𝜎𝐸 +�(𝑥 ⊗ 1, ∇𝐸 𝑦) ⊗ 𝑧�, +(2.43) +𝜎−1 +𝐸 +�∇2 +𝐸𝑥� = 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩𝜎−1 +𝐸 +� +∇𝐸(𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩) +� ++ d𝐵𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩. +(2.44) +First, we check (2.42). Let 𝛼, 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸 be given. Then, by (2.26) applied to 𝜎𝐸, +𝑒 ⊗ 𝛽𝛼 = 𝜎𝐸 +� +𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ +� +𝛼 += 𝜎𝐸 +� +𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩ ⊗ (𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼) ⟨0⟩ +� +(𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼) ⟨1⟩ += 𝜎𝐸 +� +𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩𝜎−1 +𝐸 +� +𝜎−1 +𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼 +�� +. +Next, we check (2.43). Let 𝑥, 𝑦, 𝑧 ∈ 𝐸 be given. Since (𝜎𝐸, ∇𝐸) is a Hermitian bimodule +connection, it follows that +𝜎𝐸(d𝐵(𝑥, 𝑦) ⊗ 𝑧) = ∇𝐸((𝑥, 𝑦)𝑧) − (𝑥, 𝑦)∇𝐸𝑧 = ∇𝐸(𝑥)(𝑦, 𝑧) + 𝑥 ⊗ (∇𝐸 𝑦, 𝑧), +On the one hand, by definition of ∇𝐸, we see that +𝜎𝐸 +�(∇𝐸𝑥, 𝑦) ⊗ 𝑧� = 𝜎𝐸 +� +𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩(𝑦, 𝑧) +� += ∇𝐸(𝑥)(𝑦, 𝑧). +On the other hand, by definition of ∇𝐸 together with (2.30) applied to 𝜎𝐸, +𝑥 ⊗ (∇𝐸 𝑦, 𝑧) = 𝑥 ⊗ +� +𝜎−1 +𝐸 (∇𝐸 𝑦) ⟨−1⟩(𝜎−1 +𝐸 (∇𝐸 𝑦) ⟨0⟩ ⊗ 1), 𝑧 ⊗ 1 +� += 𝑥 ⊗ +� +𝜎−1 +𝐸 (∇𝐸 𝑦) ⟨0⟩ ⊗ 1, 𝜎−1 +𝐸 (∇𝐸 𝑦) +∗ +⟨−1⟩ ⊗ 𝑧) +� += +� +𝑥, 𝜎−1 +𝐸 (∇𝐸 𝑦) ⟨0⟩ +� +𝜎𝐸(𝜎−1 +𝐸 (∇𝐸 𝑦) +∗ +⟨−1⟩ ⊗ 𝑧) += 𝜎𝐸 +�(𝑥 ⊗ 1, ∇𝐸(𝑦))) ⊗ 𝑧�. +Finally, we check (2.44). Let 𝑥 ∈ 𝐸 be given. By (2.26) and (2.27) applied to 𝜎𝐸 and(𝜎𝐸, ∇𝐸), +respectively, +∇2 +𝐸𝑥 = ∇𝐸 ◦ 𝜎𝐸 +� +𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩ +� += 𝜎𝐸 +� +𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ ∇𝐸(𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩) + d𝐵𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩ +� += 𝜎𝐸 +� +𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩𝜎−1 +𝐸 (∇𝐸(𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩)) +� ++ 𝜎𝐸 +� +d𝐵𝜎−1 +𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩ +� +. + +20 +BRANIMIR ĆAĆIĆ +We now show that ev𝐸 : 𝐸 ⊗𝐵 𝐸 → 𝐸 in Pic(𝐵) defines a corresponding arrow in DPic(𝐵). +Since ∇𝐵 = 𝜆−1 +Ω𝐵 ◦ d, it suffices to show that for all 𝑥, 𝑦 ∈ 𝐸, +d𝐵 ev𝐸(𝑥 ⊗ 𝑦) = 𝜆Ω𝐵 ◦ (ev𝐸 ⊗ idΩ𝐵) ◦ 𝛼−1 +𝐸,𝐸,Ω𝐵 +� +(id𝐸 ⊗ 𝜎𝐸) ◦ 𝛼𝐸,Ω𝐵,𝐸(∇𝐸𝑥 ⊗ 𝑦) + 𝑥 ⊗ ∇𝐸 𝑦 +� +. +Hence, let 𝑥, 𝑦 ∈ 𝐸 be given. By (2.30) applied to 𝜎𝐸 and (2.25) applied to ∇𝐸, +d𝐵 ev𝐸(𝑥 ⊗ 𝑦) = (∇𝐸𝑥, 𝑦 ⊗ 1) + (𝑥 ⊗ 1, ∇𝐸 𝑦) += +� +𝜎−1 +𝐸 (∇𝐸𝑥) ⟨0⟩ ⊗ 1, 𝜎−1 +𝐸 (∇𝐸𝑥) +∗ +⟨−1⟩(𝑦 ⊗ 1) +� ++ (𝑥 ⊗ 1, ∇𝐸 𝑦) += ev𝐸 +� +∇𝐸(𝑥) ⟨0⟩ ⊗ 𝜎𝐸(∇𝐸(𝑥) ⟨1⟩ ⊗ 𝑦) ⟨0⟩ +� +𝜎𝐸(∇𝐸(𝑥) ⟨1⟩ ⊗ 𝑦) ⟨−1⟩ ++ ev𝐸 +� +𝑥 ⊗ ∇𝐸(𝑦) ⟨0⟩ +� +∇𝐸(𝑦) ⟨1⟩ += 𝜆Ω𝐵 ◦ (ev𝐸 ⊗ idΩ𝐵) ◦ 𝛼−1 +𝐸,𝐸,Ω𝐵 +� +(id𝐸 ⊗ 𝜎𝐸) ◦ 𝛼𝐸,Ω𝐵,𝐸(∇𝐸𝑥 ⊗ 𝑦) + 𝑥 ⊗ ∇𝐸 𝑦 +� +. +□ +Example 2.31 (Connes [31, Thm 7], Polishchuk–Schwarz [84, Prop. 2.2]). We continue from +Example 2.28. The following defines a lift of the homomorphism 𝐸 : Γ𝜃 → Pic(𝐶∞ +𝜃 (T2)) of +Example 2.24 with respect to a homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ +𝜃 (T2)). +(1) Given 𝑔 ∈ Γ𝜃, let ˆ𝐸(𝑔) � (𝐸(𝑔), 𝜎𝑔, ∇𝑔), where (𝜎𝑔, ∇𝑔) is the Hermitian bimodule +connection on 𝐸(𝑔) of Example 2.28. +(2) Let ˆ𝐸(0) be given by id𝐶∞ +𝜃 (T2) � 𝐸(0). +(3) Given 𝑔, ℎ ∈ Γ𝜃, let ˆ𝐸(2) +𝑔,ℎ be given by 𝐸(2) +𝑔,ℎ . +In particular, that ˆ𝐸(0) and ˆ𝐸(2) satisfy the required commutative diagrams follows (with +superficial changes) from the result of Polishchuk–Schwarz. +2.4. Canonical actions of the differential Picard group. Again, let 𝐵 be a unital pre-𝐶∗- +algebra with ∗-exterior algebra (Ω𝐵, d𝐵). We show that the differential Picard group DPic(𝐵) +defines a generalised diffeomorphism group for (𝐵; Ω𝐵, d𝐵), whose action on the 𝐾0-monoid +V(𝐵) characterizes the fibres of the forgetful map DPic(𝐵) → V(𝐵) and whose action on +the graded centre Z(Ω𝐵) admits curvature as a canonical group 1-cocycle. +Let us first consider naïve diffeomorphisms of the manifold (𝐵; Ω𝐵, d𝐵). Gel’fand duality +initially suggests that a diffeomorphism of (𝐵; Ω𝐵, d𝐵) should be an automorphism of the +∗-exterior algebra (Ω𝐵, d𝐵) over 𝐵. However, as we shall see in Theorem 2.35, inner automor- +phisms of 𝐵, i.e., automorphisms of the form 𝑏 ↦→ 𝑢𝑏𝑢∗ for fixed 𝑢 ∈ U(𝐵), will generally only +satisfy the following conservative generalisation. +Definition 2.32. We define the extended diffeomorphism group of 𝐵 with respect to (Ω𝐵, d) to +be the subgroup � +Diff(𝐵) of (Ω1 +𝐵)sa ⋊ Aut(Ω𝐵) consisting of elements (𝜔, 𝜙) satisfying +∀𝛽 ∈ Ω𝐵, +d𝛽 − 𝜙 ◦ d ◦ 𝜙−1(𝛽) = i[𝜔, 𝛽], +(2.45) +where [·, ·] denotes the supercommutator in Ω𝐵 with respect to parity of degree. An extended +diffeomorphism of 𝐵 with respect to (Ω𝐵, d) is an element of � +Diff(𝐵). Moreover, we say that +(𝜔, 𝜙) ∈ � +Diff(𝐵) is topologically trivial whenever 𝜙↾𝐵= id𝐵, and we denote by � +Diff0(𝐵) the +normal subgroup of � +Diff(𝐵) consisting of topologically trivial elements. +Example 2.33. Continuing from Example 2.21, equip 𝐶∞(𝑋) with the de Rham ∗-exterior +calculus (Ω(𝑋, C), d). The isomorphism Diff(𝑋) → Aut(𝐶∞(𝑋)) yields an isomorphism +�(𝑓, 𝜔) ↦→ ((𝑓 −1)∗𝜔, (𝑓 −1)∗)� : Diff(𝑋) ⋉ Ω1(𝑋, R) → � +Diff(𝐶∞(𝑋)) + +NONCOMMUTATIVE U(1)-GAUGE THEORY +21 +where Diff(𝑋) acts on Ω1(𝑋, R) from the right by pullback. In particular, it follows that the +preimage of � +Diff0(𝑋) is {id𝑋} × Ω1(𝑋, R) � Ω1(𝑋, R). +Example 2.34. Recall the homomorphism 𝜏 : Aut(𝐵) → Pic(𝐵) of Example 2.14. The +following defines a lift of 𝜏 with respect to the forgetful homomorphism DPic(𝐵) → Pic(𝐵) +to a homomorphism ˆ𝜏 : � +Diff(𝐵) → DPic(𝐵). +(1) Given (𝜔, 𝜙) ∈ � +Diff(𝐵), let ˆ𝜏(𝜙,𝜔) � (𝐵𝜙, 𝜎𝜙, ∇(𝜔,𝜙)), where 𝐵𝜙 � 𝜏𝜙 and +∀𝛽 ∈ Ω𝐵, ∀𝑏 ∈ 𝐵, +𝜎𝜙(𝛽 ⊗ 𝑏𝜙) � 1𝜙 ⊗ 𝜙−1(𝛽𝑏), +(2.46) +∀𝑏 ∈ 𝐵, +∇(𝜙,𝜔) (𝑏𝜙) � 1𝜙 ⊗ 𝜙−1(d𝑏 + 𝑏 · i𝜔). +(2.47) +(2) Let ˆ𝜏 (0) be given by id𝐵 � 𝜏 (0). +(3) Given (𝜔1, 𝜙1), (𝜔2, 𝜙2) ∈ � +Diff(𝐵), let ˆ𝜏 (2) +(𝜔1,𝜙1),(𝜔2,𝜙2) be given by 𝜏 (2) +𝜙1,𝜙2. +Note that the homomorphism ˆ𝜏 : � +Diff(𝐵) → DPic(𝐵) is faithful on objects, so that it can +be viewed as embedding the group � +Diff(𝐵) in the coherent 2-group DPic(𝐵). +Now, let ΠPic(𝐵) : DPic(𝐵) → Pic(𝐵) be the group homomorphism induced by the +forgetful homomorphism DPic(𝐵) → Pic(𝐵), and recall that ΠV(𝐵) : Pic(𝐵) → V(𝐵) is +the set map induced by the forgetful functor Pic(𝐵) → Hilb(𝐵). Hence, note that the right +Pic(𝐵)-action on V(𝐵) of Proposition 2.19 pulls back via ΠPic(𝐵) to a right DPic(𝐵)-action on +V(𝐵); in turn, this right DPic(𝐵)-action correctly pulls back via 𝜋0(ˆ𝜏) to the usual pullback +action of isometric ∗-automorphisms on V(𝐵) Since this DPic(𝐵)-action is transitive on +the range of ΠV(𝐵) ◦ ΠPic(𝐵) : DPic(𝐵) → V(𝐵), we may use the resulting stabilizer group +DPic(𝐵)[𝐵] of [𝐵] ∈ ran(ΠV(𝐵) ◦ ΠPic(𝐵)) to characterize the fibres of the forgetful map from +the differential Picard group DPic(𝐵) to the 𝐾0-monoid V(𝐵). Moreover, since ΠPic(𝐵) is a +group homomorphism, its kernel yields the fibres of the forgetful map from DPic(𝐵) to the +(topological) Picard group Pic(𝐵). As it turns out, the subgroups DPic(𝐵)[𝐵] and ker ΠPic(𝐵) +are completely determined by the group homomorphism 𝜋0(ˆ𝜏) : � +Diff(𝐵) → DPic(𝐵). +Theorem 2.35. Let DPic(𝐵)[𝐵] denote the stabilizer subgroup of DPic(𝐵) with respect to the +isomorphism class [𝐵] ∈ ran(ΠK(𝐵) ◦ ΠPic(𝐵)), and let � +Ad : U(𝐵) → � +Diff(𝐵) be given by +∀𝑢 ∈ U(𝐵), +� +Ad𝑢 � (−i d𝐵(𝑢)𝑢∗, Ad𝑢) . +(2.48) +Then the homomorphisms 𝜋0(ˆ𝜏) and � +Ad fit into the exact sequences of groups +1 → U(Z(𝐵) ∩ ker d𝐵) → U(𝐵) +� +Ad +−−→ � +Diff(𝐵) +𝜋0(ˆ𝜏) +−−−−→ DPic(𝐵)[𝐵] → 1, +(2.49) +1 → U(Z(𝐵) ∩ ker d𝐵) → U(Z(Ω𝐵)0) +� +Ad +−−→ � +Diff0(𝐵) +𝜋0(ˆ𝜏) +−−−−→ DPic(𝐵) +ΠPic(𝐵) +−−−−−→ Pic(𝐵). (2.50) +Proof. Before continuing, we must show that (2.49) is a well-defined diagram of groups. A +straightforward calculation show that � +Ad : U(𝐵) → � +Diff(𝐵) is well-defined, so it remains to +check that ran 𝜋0(ˆ𝜏) ≤ DPic(𝐵)[𝐵]. However, given (𝜔, 𝜙) ∈ � +Diff(𝐵), the required isomor- +phism 𝑈 : 𝐵 ⊗𝐵 𝐵𝜙 → 𝐵 in Hilb(𝐵) is given by 𝑈 � �𝑏 ⊗ 𝑐𝜙 ↦→ 𝜙−1(𝑏𝑐)�. +We now show that (2.49) is an exact sequence. Exactness at U(𝐵) is immediate, so we +proceed to checking exactness at � +Diff(𝐵). On the one hand, let (𝜔, 𝜙) ∈ ker 𝜋0(ˆ𝜏) be given. +Thus, there exists an arrow 𝑈 : (𝐵𝜙, 𝜎𝜙, ∇𝜔,𝜙) → (𝐵, 𝜎𝐵, ∇𝐵) in DPic(𝐵). Set 𝑢 � 𝑈(1𝜙); we +claim that (𝜔, 𝜙) = � +Ad𝑢. First, observe that 𝑢 ∈ U(𝐵) since the singleton {1𝜙} defines both a +basis and strict cobasis for 𝐵𝜙. Next, observe that 𝜙 = Ad𝑢 since for all 𝛽 ∈ Ω𝐵, +𝛽𝑢 = 𝜆Ω𝐵 ◦ 𝜎0 ◦ (idΩ𝐵 ⊗ 𝑈)(𝛽 ⊗ 1𝜙) = 𝜆Ω𝐵 ◦ 𝜎0 ◦ (idΩ𝐵 ⊗ 𝑈) ◦ 𝜎−1 +𝜙 (1𝜙 ⊗ 𝜙−1(𝛽)) = 𝑢𝜙−1(𝛽). + +22 +BRANIMIR ĆAĆIĆ +Finally, given that 𝜙 = Ad𝑢, observe that 𝜔 = −i d(𝑢)𝑢∗ since +0 = 𝜆Ω𝐵 +�(𝑈 ⊗ idΩ𝐵)(∇𝜔,𝜙1𝜙) − d𝐵𝑢𝑈(1𝜙)� = 𝜆Ω𝐵 ◦ (𝑈 ⊗ idΩ𝐵) �1𝜙 ⊗ i𝜙−1(𝜔)� − d𝐵𝑢 += i(𝜔 + id𝐵(𝑢)𝑢∗)𝑢. +On the other hand, given 𝑢 ∈ U(𝐵), similar calculations show that (𝑏Ad𝑢 ↦→ 𝑏𝑢) : 𝐵Ad𝑢 → 𝐵 +defines an arrow ˆ𝜏� +Ad𝑢 → (𝐵, 𝜎0, ∇0) in DPic(𝐵). +Finally, we check exactness at DPic(𝐵)[𝐵]. Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule +with connection, and suppose that [(𝐸, 𝜎𝐸, ∇𝐸)] ∈ DPic(𝐵)[𝐵]. Using 𝜆𝐵 : 𝐵 ⊗𝐵 𝐵 → 𝐵, we +conclude that there exists an arrow 𝑈 : 𝐵 → 𝐸 in Hilb(𝐵). Set 𝑒0 � 𝑈(1); since the singleton +{1} defines both a basis and strict cobasis for 𝐵, it follows that {𝑒0} defines both a basis and +strict cobasis for 𝐸. We shall use 𝑒0 together with (𝜎𝐸, ∇𝐸) to construct (𝜔, 𝜙) ∈ � +Diff(𝐵) and +an arrow 𝑉 : ˆ𝜏(𝜙,𝜔) → (𝐸, 𝜎𝐸, ∇𝐸) in DPic(𝐵). +First, define a C-linear map Φ : Ω𝐵 → Ω𝐵 by Φ � (𝛽 ↦→ (𝑒0 ⊗ 1, 𝜎𝐸(𝛽 ⊗ 𝑒0))); once we +know that the degree-preserving map Φ is an element of Aut(Ω𝐵), we shall set 𝜙 � Φ−1. First, +note that Φ is unit-preserving since Φ(1) = (𝑒0, 𝑒0) = 1. Next, note that Φ is multiplicative +by (2.26) applied to 𝜎𝐸 and ∗-preserving by (2.30) applied to 𝜎𝐸. Next, note that Φ is bijective +since, for all 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, +𝛽 ⊗ 𝑥 = 𝛽 ⊗ 𝑒0(𝑒0, 𝑥) = 𝜎−1 +𝐸 (𝑒0 ⊗ (𝑒0 ⊗ 1, 𝜎−1 +𝐸 (𝛽 ⊗ 𝑒0)))(𝑒0, 𝑥) = 𝜎−1 +𝐸 (𝑒0 ⊗ Φ(𝛽))(𝑒0, 𝑒), +so that, in terms of the arrow ΥΩ𝐵,𝐸 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 in Bimod(𝐵), +∀𝛽 ∈ Ω𝐵, +Φ−1(𝛽) = +� +ΥΩ𝐵,𝐸(𝜎−1 +𝐸 (𝑒0 ⊗ 𝛽)), 𝑒0 ⊗ 1 +� +. +Finally, note that Φ is isometric on 𝐵 since, for all 𝑏 ∈ 𝐵, +∥Φ(𝑏)∥ = ∥(𝑒0, 𝑏𝑒0)∥ ≤ ∥𝑏∥ · ∥𝑒0∥2 = ∥𝑏∥ · ∥(𝑒0, 𝑒0)∥ = ∥𝑏∥, +∥Φ−1(𝑏)∥ = ∥(𝑒0𝑏, 𝑒0)∥ ≤ ∥𝑏∥ · ∥𝑒0∥2 = ∥𝑏∥ · ∥(𝑒0, 𝑒0)∥ = ∥𝑏∥. +Having constructed Φ, we now claim that (𝜔, 𝜙, ) � (−iΦ−1((𝑒0 ⊗ 1, ∇𝐸𝑒0)), Φ−1) defines +an element of � +Diff(𝐵). Note that 𝜔 ∈ Ω1 +𝐵 is self-adjoint since 𝜙 ∈ Aut(Ω𝐵) and since +0 = d𝐵(𝑒0, 𝑒0) = (∇𝐸𝑒0, 𝑒0 ⊗ 1) + (𝑒0 ⊗ 1, ∇𝐸𝑒0) = (𝑒0, ∇𝐸𝑒0)∗ + (𝑒0, ∇𝐸𝑒0). +Thus, it remains to show that (𝜙, 𝜔) satisfies (2.45). Let 𝛽 ∈ Ω𝐵 be given. Then +𝜎𝐸(d𝐵𝜙(𝛽) ⊗ 𝑒0) = ∇𝐸(𝜎𝐸(𝜙(𝛽) ⊗ 𝑒0)) − (−1) |𝛽|𝜎𝐸 +� +𝜙(𝛽) ⊗ ∇𝐸(𝑒0) ⟨0⟩ +� +∇𝐸(𝑒0) ⟨1⟩ += ∇𝐸(𝑒0 ⊗ 𝛽) − (−1) |𝛽|𝜎𝐸(𝜙(𝛽) ⊗ 𝑒0)(𝑒0, ∇𝐸𝑒0) += ∇𝐸(𝑒0)𝛽 + 𝑒0 ⊗ d𝛽 − (−1) |𝛽|𝑒0 ⊗ 𝜔(𝑒0, ∇𝐸𝑒0) += 𝑒0 ⊗ (d𝐵𝛽 + [(𝑒0, ∇𝐸𝑒0), 𝛽]) += 𝜎𝐸((𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)]) ⊗ 𝑒0), +so that, indeed, for every 𝑥 ∈ 𝐸, +d𝐵𝜙(𝛽)⊗𝑥 = d𝐵𝜙(𝛽)⊗𝑒0(𝑒0, 𝑥) = (𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)])⊗𝑒0(𝑒0, 𝑥) = (𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)])⊗𝑥. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +23 +Finally, define an arrow 𝑉 : 𝐵𝜙 → 𝐸 in Pic(𝐵) by 𝑉 (1𝜙) � 𝑒0; we claim that it yields an +arrow 𝑉 : ˆ𝜏𝜔,𝜙 → (𝐸, 𝜎𝐸, ∇𝐸) in DPic(𝐵). Indeed, for all 𝑏 ∈ 𝐵, +∇𝐸 +�𝑉 (𝑏𝜙)� = 𝜎𝐸(d𝑏 ⊗ 𝑒0) + 𝑏∇𝐸𝑒0 += 𝑒0 ⊗ ((𝑒0 ⊗ 1, 𝜎𝐸(d𝑏 ⊗ 𝑒0))) + 𝑏𝑒0 ⊗ (𝑒0 ⊗ 1, ∇𝐸𝑒0) += 𝑒0 ⊗ 𝜙−1(d𝑏) + 𝑏𝑒0 ⊗ i𝜙−1(𝜔) += (𝑉 ⊗ id)�∇(𝜔,𝜙)𝑏𝜙 +�. +□ +We have just seen that the generalised diffeomorphism group � +Diff(𝐵) embeds via ˆ𝜏 in the +differential Picard 2-group DPic(𝐵). The following refinement of Proposition-Definition +2.20 yields a surprising ‘moral converse’: the entire differential Picard group DPic(𝐵) acts +canonically as automorphisms on the graded centre of Ω𝐵 in a manner that will turn out to be +explicitly compatible with the embedding of � +Diff(𝐵) in DPic(𝐵) by Example 2.40. +Proposition-Definition 2.36 (Beggs–Majid [13, Prop. 5.9]). The Fröhlich homomorphism of 𝐵 +with respect to (Ω𝐵, d) is the unique group homomorphism ˆΦ : DPic(𝐵) → Aut(Z(Ω𝐵), d), +such that, for every Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸), +∀𝛽 ∈ Z𝐵(Ω𝐵), ∀𝑥 ∈ 𝐸, +ˆΦ[𝐸,∇𝐸](𝛽) ⊗ 𝑥 = 𝜎−1 +𝐸 (𝑥 ⊗ 𝛽); +(2.51) +in this case, we call ˆΦ[𝐸,∇𝐸] the Fröhlich automorphism of (𝐸, 𝜎𝐸, ∇𝐸). +Proof. Relative to [13, Prop. 5.9], it remains to check that for each (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)), +the automorphism ˆΦ[𝐸,∇𝐸] of the graded algebra Z𝐵(Ω𝐵) is ∗-preserving and commutes with +d𝐵 on Z(Ω𝐵); observe that the restriction ˆΦ[𝐸,∇𝐸]↾Z(𝐵)= Φ[𝐸] is isometric by Proposition- +Definition 2.20. Let (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)) be given, and fix a basis (𝑒𝑖)𝑛 +𝑖=1 and a strict +cobasis (𝜖𝑗)𝑚 +𝑗=1 for 𝐸. On the one hand, by the proof of Theorem 2.35, mutatis mutandis, +∀𝛽 ∈ Z(Ω𝐵), +ˆΦ[𝐸,∇𝐸](𝛽) = +∑︁𝑛 +𝑖=1 +� +ΥΩ𝐵,𝐸(𝜎−1 +𝐸 (𝑒𝑖 ⊗ 𝛽)), 𝑒𝑖 ⊗ 1 +� +, +(2.52) +∀𝛽 ∈ Z(Ω𝐵), +ˆΦ−1 +[𝐸,∇𝐸](𝛽) = +∑︁𝑚 +𝑗=1 +�𝜖𝑗 ⊗ 1, 𝜎𝐸(𝛽 ⊗ 𝜖𝑗)�; +(2.53) +hence, by (2.53) and (2.30) applied to 𝜎𝐸, the map ˆΦ[𝐸,∇𝐸] is ∗-preserving. On the other hand, +let 𝛽 ∈ Z(Ω𝐵) be given. Then, for all 𝑥 ∈ 𝐸, +𝑥 ⊗ d𝐵 ˆΦ−1 +[𝐸,∇𝐸](𝛽) = ∇𝐸 +�𝑥 ⊗ Φ[𝐸,∇𝐸](𝛽)� − ∇𝐸(𝑥)Φ−1 +[𝐸,∇𝐸](𝛽) = ∇𝐸(𝛽(𝑥 ⊗ 1)) − 𝛽∇𝐸𝑥 += 𝑥 ⊗ ˆΦ−1 +[𝐸,∇𝐸](d𝐵𝛽). +□ +Corollary 2.37. The canonical left action of 𝜋0(DPic(𝐵)) � DPic(𝐵) on the Abelian group +𝜋1(DPic(𝐵)) = U(Z(𝐵) ∩ ker d𝐵) is the left action induced by ˆΦ. +We can now introduce curvature as a 1-cocycle for this action of DPic(𝐵) on Z(Ω𝐵). For +convenience, let us define a pre-symplectic form on 𝐵 to be self-adjoint 𝛽 ∈ Z(Ω𝐵)2 satisfying +d𝛽 = 0. Hence, we denote by S(𝐵) the real subspace of all pre-symplectic forms on 𝐵, which +we endow with the right action of DPic(𝐵) defined by +∀[𝐸, ∇𝐸] ∈ DPic(𝐵), ∀𝜔 ∈ S(𝐵), +𝜔 ⊳ [𝐸, ∇𝐸] � ˆΦ−1 +[𝐸,∇𝐸](𝜔). +(2.54) +Moreover, recall that if Γ is a group and 𝑀 is a right Γ-module (written additively), then a map +𝑐 : Γ → 𝑀 is a right 1-cocycle whenever 𝑐(𝛾𝜂) = 𝑐(𝛾) ⊳ 𝜂 + 𝑐(𝜂) for all 𝛾, 𝜂 ∈ Γ. + +24 +BRANIMIR ĆAĆIĆ +Proposition-Definition 2.38 (Beggs–Majid [13, Prop. 5.9]). The curvature 1-cocycle of 𝐵 with +respect to (Ω𝐵, d) is the unique right 1-cocycle F : DPic(𝐵) → S(𝐵), such that, for every +Hermitian 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸), +∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, +∇2 +𝐸𝜉 = 𝜉 · iF[𝐸,∇𝐸]; +(2.55) +in this case, we call F[𝐸,∇𝐸] ∈ S(𝐵) the curvature 2-form of [𝐸, ∇𝐸]. +Proof. First, let (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)) be given; fix a basis (𝑒𝑖)𝑚 +𝑖=1 and a cobasis (𝜖𝑗)𝑛 +𝑗=1 +for 𝐸. The map ∇2 +𝐸 : 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 is right 𝐵-linear by repeated applications of (2.24) and left +𝐵-linear by (2.32). Thus, ∇2 +𝐸𝑥 = ∇2 +𝐸 +��𝑛 +𝑗=1(𝑥, 𝜖𝑗)𝜖𝑗 +� += �𝑚 +𝑗=1(𝑥, 𝜖𝑗)∇2 +𝐸𝜖𝑗 = 𝑥⊗�𝑚 +𝑗=1(𝜖𝑗 ⊗1, ∇2 +𝐸𝜖𝑗) +for every 𝑥 ∈ 𝐸, so that the 2-form F[𝐸,∇𝐸] � −i �𝑚 +𝑗=1(𝜖𝑗 ⊗ 1, ∇2 +𝐸𝜖𝑗) ∈ Ω2 +𝐵 satisfies +∀𝑥 ∈ 𝐸, +∇2 +𝐸𝑥 = 𝑥 ⊗ iF[𝐸,∇𝐸]. +(2.56) +On the one hand, F[𝐸,∇𝐸] is the unique 2-form satisfying (2.56) since, for any such 2-form 𝜛, +𝜛 = +∑︁𝑛 +𝑗=1(𝜖𝑗, 𝜖𝑗)𝜛 = +∑︁𝑛 +𝑗=1(𝜖𝑗 ⊗ 1, 𝜖𝑗 ⊗ 𝜛) = −i +∑︁𝑛 +𝑗=1(𝜖𝑗 ⊗ 1, ∇2 +𝐸𝜖𝑗) = F[𝐸,∇𝐸]; +in fact, this uniqueness implies that F[𝐸,∇𝐸] depends only on [𝐸, ∇𝐸] ∈ DPic(𝐵). On the other, +F[𝐸,∇𝐸] is self-adjoint by construction from (𝜖𝑗)𝑛 +𝑗=1 and repeated applications of (2.29). +We now show that F[𝐸,∇𝐸] is central, is closed, and satisfies (2.55). First, by repeated applica- +tions of (2.27), it follows that ∇2 +𝐸𝜎𝐸(𝛽 ⊗ 𝑥) = 𝜎𝐸(𝛽 ⊗ 𝑥) · iF[𝐸,∇𝐸] for all 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, so +that by invertibility of the map 𝜎𝐸, the 2-form F[𝐸,∇𝐸] satisfies (2.55). Next, by (2.55) together +with repeated applications of (2.24), it follows that for every 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, +𝑥 ⊗ F[𝐸,∇𝐸]𝛽 = −i∇2 +𝐸(𝑥 ⊗ 𝛽) = 𝑥 ⊗ 𝛽F[𝐸,∇𝐸], +so that F[𝐸,∇𝐸] is indeed central. Finally, F[𝐸,∇𝐸] is closed since for every 𝑥 ∈ 𝐸, by (2.55), +𝑥 ⊗ i d𝐵F[𝐸,∇𝐸] = ∇𝐸(𝑥 ⊗ iF[𝐸,∇𝐸]) − ∇𝐸(𝑥) · iF[𝐸,∇𝐸] = ∇𝐸(∇2 +𝐸𝑥) − ∇2 +𝐸(∇𝐸𝑥) = 0. +Finally, by [13, Prop. 5.9], mutatis mutandis, the map [𝐸, ∇𝐸] ↦→ F[𝐸,∇𝐸] satisfies +∀(𝐸, 𝜎𝐸, ∇𝐸), (𝐹, 𝜎𝐹, ∇𝐹) ∈ Obj(DPic(𝐵)), +F[𝐸⊗𝐵𝐹,∇𝐸⊗𝐵𝐹 ] = ˆΦ−1 +[𝐹,∇𝐹 ](F[𝐸,∇𝐸]) + F[𝐹,∇𝐹 ], +which is precisely the required right 1-cocycle identity. +□ +Example 2.39. We continue from Example 2.33. Let Ω2(𝑋, R)cl be the real vector space of +closed real 2-forms on 𝑋, which admits a right action of Diff(𝑋) by pullbacks. On the one +hand, let Ψ : DPic(𝐶∞(𝑋)) → Diff(𝑋) be the homomorphism induced by the Fröhlich +homomorphism of 𝐶∞(𝑋) with respect to the de Rham calculus. On the other, recall that +the ordinary differential cohomology group ˇ𝐻2(𝑋) is the group of isomorphism classes of +Hermitian line bundles on 𝑋 with unitary connection [54, Ex. 2.7]. Then, by Serre–Swan, +1 → ˇ𝐻2(𝑋) +[E,∇E]↦→[Γ(E),∇E] +−−−−−−−−−−−−−−−→ DPic(𝐶∞(𝑋)) +Ψ−→ Diff(𝑋) → 1 +defines a split exact sequence with canonical right splitting 𝜙 ↦→ [ˆ𝜏(0, (𝜙−1)∗)]. Given the +resulting isomorphism Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → DPic(𝐶∞(𝑋)) defined by +(𝜙, [E, ∇E]) ↦→ [Γ((𝜙−1)∗E), (𝜙−1)∗∇E][ˆ𝜏(0, (𝜙−1)∗)] +we may identify the Fröhlich homomorphism Φ with the quotient map +((𝜙, [E, ∇E]) ↦→ 𝜙) : Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → Diff(𝑋) +and the curvature 1-cocycle F : DPic(𝐶∞(𝑋)) → Ω2(𝑋, R)cl with the familiar-looking map +�(𝜙, [E, ∇E]) ↦→ 𝜙∗ tr(∇2 +E)� : Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → Ω2(𝑋, R)cl. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +25 +Example 2.40. The homomorphism ˆ𝜏 : � +Diff(𝐵) → DPic(𝐵) of Example 2.34 satisfies +∀(𝜔, 𝜙) ∈ � +Diff(𝐵), +ˆΦ ◦ 𝜋0(ˆ𝜏)(𝜔, 𝜙) = 𝜙↾Z(Ω𝐵), +F ◦ 𝜋0(ˆ𝜏)(𝜔, 𝜙) = 𝜙−1�d𝜔 − i𝜔2�. +Example 2.41 (Connes [31, Thm 7]). Continuing from Example 2.31, observe that Z(Ω𝜃(T2)) +is the complex Graßmann algebra in the self-adjoint generators 𝑒1 and 𝑒2 of degree 1. Hence, +the homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ +𝜃 (T2)) satisfies +∀𝑔 ∈ Γ𝜃, +ˆΦ ◦ 𝜋0( ˆ𝐸)(𝑔) = +2 +� +𝑘=0 +(𝑔21𝜃 + 𝑔22)𝑘 idZ(Ω𝜃)𝑘, +F ◦ 𝜋0( ˆ𝐸)(𝑔) = +2𝜋𝑔21 +𝑔21𝜃 + 𝑔22 +𝑒1𝑒2. +3. Reconstruction of noncommutative principal U(1)-bundles with connection +In this section, we generalise the familiar correspondence between Hermitian line bundles +with unitary connection and principal U(1)-bundles with principal connection to the nc +setting. This generalisation takes the form of an explicit equivalence of categories that can be +viewed as an adaptation of Pimsner’s construction [81] from the 𝐶∗-algebraic literature to the +setting of nc differential geometry. +In what follows, we say that a representation 𝑈 : U(1) → GL(𝑉) is of finite type whenever +𝑉 = �alg +𝑘∈Z 𝑉𝑘, where 𝑉𝑘 � {𝑣 ∈ 𝑉 | ∀𝑧 ∈ U(1), 𝑈𝑧𝑣 = 𝑧𝑘𝑣} for all 𝑘 ∈ Z. +3.1. Monoidal inversion and homomorphisms of coherent 2-groups. First we leverage +the coherence theorem for coherent 2-groups of Ulbrich [95] and Laplaza [66] to show that +every homomorphism of coherent 2-groups canonically defines a bar functor or involutive +monoidal functor in the sense of Beggs–Majid and Egger [44], respectively. This will obviate any +difficulties related to reconstructing ∗-structures on nc principal U(1)-bundle with principal +connection. +We first recall the additional categorical structure that will fully capture the behaviour of +monoidal inversion in a coherent 2-group. +Definition 3.1 (Beggs–Majid [12], Egger [44]). A strong bar category is a monoidal category G +together with a functor · : G → G, an isomorphism ★ : 1 → 1, and natural isomorphisms +bb = +� +bb𝑔 : 𝑔 → 𝑔 +� +𝑔∈Obj(G) and Υ = +� +Υ𝑔,ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 +� +(𝑔,ℎ) ∈Obj(G)2, such that bb𝑔 = bb𝑔 +for every 𝑔 ∈ Obj(G) and the following coherence diagrams commute for all 𝑔, ℎ, 𝑘 ∈ Obj(G): +1 +1 +1 +★ +bb1 +★ +𝑔 ⊗ 1 +1 ⊗ 𝑔 +𝑔 +1 ⊗ 𝑔 +Υ𝑔,1 +𝜌−1 +𝑔 +𝜌𝑔 +★⊗id𝑔 +1 ⊗ 𝑔 +𝑔 ⊗ 1 +𝑔 +𝑔 ⊗ 1 +Υ1,𝑔 +𝜆−1 +𝑔 +𝜆𝑔 +id𝑔 ⊗★ +(𝑔 ⊗ ℎ) ⊗ 𝑘 +𝑘 ⊗ 𝑔 ⊗ ℎ +𝑘 ⊗ (ℎ ⊗ 𝑔) +𝑔 ⊗ (ℎ ⊗ 𝑘) +ℎ ⊗ 𝑘 ⊗ 𝑔 +(𝑘 ⊗ ℎ) ⊗ 𝑔 +Υ𝑔⊗ℎ,𝑘 +id ⊗Υ𝑔,ℎ +Υ𝑔,ℎ⊗𝑘 +Υℎ,𝑘 ⊗id𝑔 +𝛼𝑔,ℎ,𝑘 +𝛼𝑘,ℎ,𝑔 +𝑔 ⊗ ℎ +𝑔 ⊗ ℎ +𝑔 ⊗ ℎ +ℎ ⊗ 𝑔 +bb𝑔 ⊗ bbℎ +Υ𝑔,ℎ +bb𝑔⊗ℎ +Υℎ,𝑔 +For example, given a unital pre-𝐶∗-algebra 𝐵, the monoidal category Bimod(𝐵) defines a +strong bar category with ★ : 𝐵 → 𝐵 and natural isomorphisms bb and Υ defined as follows: +∀𝑏 ∈ 𝐵, +★(𝑏) � 𝑏∗, +∀𝐸 ∈ Obj(Bimod(𝐵)), ∀𝑥 ∈ 𝐸, +bb𝐸(𝑥) � 𝑥, +∀𝐸, 𝐹 ∈ Obj(Bimod(𝐵)), ∀𝑥 ∈ 𝐸, ∀𝑦 ∈ 𝐹, +Υ𝐸,𝐹(𝑥 ⊗ 𝑦) � 𝑦 ⊗ 𝑥. + +26 +BRANIMIR ĆAĆIĆ +Next, let us recall the coherence theorem for coherent 2-groups on which everything will +hinge. Define a structural arrow of a coherent 2-group G to be an element of the smallest +subclass Str(G) of Hom(G) that: +(1) contains the identity arrows, associators, left unitors, and right unitors of G as a monoidal +category and the evaluation arrows of G as a coherent 2-group; +(2) is closed under composition and inversion in G as a category, the monoidal product in G +as a monoidal category, and monoidal inversion in G as a coherent 2-group. +Hence, given endofunctors 𝑃, 𝑄 : G → G of a coherent 2-group G, we say that a natural +transformation 𝜂 : 𝑃 ⇒ 𝑄 is structural whenever, for every 𝑔 ∈ Obj(G), the arrow 𝜂𝑔 is +structural. For example, given a coherent 2-group G, the natural isomorphisms coev and bb +of Theorem 2.4 are both structural [66, Lemm. 4.4 & 4.5]. +Theorem 3.2 (Ulbrich [95], Laplaza [66, §2]). Let G be a coherent 2-group. For every pair of +objects (𝑔, ℎ) in G, there exists at most one structural arrow 𝑔 → ℎ in G. +Our first application of the coherence theorem is that a coherent 2-group canonically +defines a strong bar category with respect to monoidal inversion. +Corollary 3.3 (cf. Laplaza [66, p. 310]). Let G be a coherent 2-group. There exist a unique +structural isomorphism ★ : 1 → 1 and a unique structural natural isomorphism +Υ = +� +Υ𝑔,ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 +� +𝑔,ℎ∈Obj(G) +making G into a strong bar category with respect to monoidal inversion and the natural isomorphism +bb of Theorem 2.4. +Proof. First, construct a structural arrow ★ : 1 → 1 by setting ★ � 𝜆1 ◦ coev1. Next, given +objects 𝑔 and ℎ of G, construct a structural arrow Υ𝑔,ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 as follows: first, +construct a structural arrow � +coev𝑔⊗ℎ : 1 → (𝑔 ⊗ ℎ) ⊗ (ℎ ⊗ 𝑔) by setting +� +coev𝑔⊗ℎ � 𝛼𝑔⊗ℎ,ℎ,𝑔 ◦ +� +𝛼−1 +𝑔,ℎ,ℎ ⊗ id𝑔 +� +◦ �(id𝑔 ⊗ coevℎ) ⊗ id𝑔 +� ◦ +� +𝜌−1 +𝑔 ⊗ id𝑔 +� +◦ coev𝑔, +and then set Υ𝑔,ℎ � 𝜆ℎ⊗𝑔 ◦ (ev𝑔⊗ℎ ⊗ idℎ⊗𝑔) ◦ 𝛼−1 +𝑔⊗ℎ,𝑔⊗ℎ,ℎ⊗𝑔 ◦ (id𝑔⊗ℎ ⊗ � +coev𝑔⊗ℎ) ◦ 𝜌−1 +𝑔⊗ℎ. The +claim now follows by Theorem 3.2. +□ +Remark 3.4. Let G be a coherent 2-group. The structural isomorphism ★ : 1 → 1 is the +unique isomorphism of the inverses (1, 𝜆1, 𝜆−1 +1 ) and (1, ev1, coev1) of 1. Likewise, given +𝑔, ℎ ∈ Obj(G), the structural isomorphism Υ𝑔,ℎ is the unique isomorphism of the inverses +(ℎ⊗𝑔, �ev𝑔⊗ℎ, � +coev𝑔⊗ℎ) and (𝑔 ⊗ ℎ, ev𝑔⊗ℎ, coev𝑔⊗ℎ) of 𝑔⊗ℎ, where �ev𝑔⊗ℎ : (ℎ⊗𝑔)⊗(𝑔⊗ℎ) → 1 +and � +coev𝑔⊗ℎ : 1 → (𝑔 ⊗ ℎ) ⊗ (𝑔 ⊗ ℎ) are the unique such structural arrows. +For example, let 𝐵 be a unital pre-𝐶∗-algebra. Then the canonical strong bar category +structure on Pic(𝐵) of Corollary 3.3 is that induced by the aforementioned strong bar category +structure on Bimod(𝐵). +We now come to the main definition of this subsection. +Definition 3.5 (Beggs–Majid [12], Egger [44]). Let G and G′ be strong bar categories. +(1) A bar functor 𝐹 : G → G′ consists of a monoidal functor 𝐹 : G → G′ together with +a natural isomorphism 𝐹 (−1) = +� +𝐹 (−1) +𝑔 +: 𝐹(𝑔) → 𝐹(𝑔) +� +𝑔∈Obj(G) making the following +diagrams commute for all 𝑔, ℎ ∈ Obj(G): + +NONCOMMUTATIVE U(1)-GAUGE THEORY +27 +𝐹(1) +𝐹(1) +𝐹(1) +1 +1 +𝐹 (−1) +1 +𝐹(★−1) +★−1 +𝐹 (0) +𝐹 (0) +(3.1) +𝐹(𝑔) +𝐹(𝑔) +𝐹(𝑔) +𝐹(𝑔) +𝐹(bb𝑔) +𝐹 (−1) +𝑔 +bb𝐹(𝑔) +𝐹 (−1) +𝑔 +(3.2) +𝐹(𝑔) ⊗ 𝐹(ℎ) +𝐹(ℎ) ⊗ 𝐹(𝑔) +𝐹(ℎ) ⊗ 𝐹(𝑔) +𝐹(𝑔 ⊗ ℎ) +𝐹(𝑔 ⊗ ℎ) +𝐹(ℎ ⊗ 𝑔) +Υ𝐹(𝑔),𝐹(ℎ) +𝐹 (−1) +𝑔 +⊗𝐹 (−1) +ℎ +𝐹 (−1) +𝑔⊗ℎ +𝐹(Υ𝑔,ℎ) +𝐹 (2) +𝑔,ℎ +𝐹 (2) +ℎ,𝑔 +(3.3) +(2) Given bar functors 𝑃, 𝑄 : G → G′, a monoidal natural transformation 𝜙 : 𝑃 ⇒ 𝑄 is a bar +natural transformation whenever 𝑄(−1) +𝑔 +◦ 𝜙𝑔 = 𝜙𝑔 ◦ 𝑃 (−1) +𝑔 +for all 𝑔 ∈ Obj(G). +Given a homomorphism of coherent 2-groups 𝐹, the following will supply the missing +natural isomorphism 𝐹 (−1). +Proposition 3.6 (Baez–Lauda [7, Thm 6.1]). Let 𝐹 : G → G′ be a homomorphism of coherent +2-groups. There exists a unique natural transformation 𝐹 (−1) = +� +𝐹 (−1) +𝑔 +: 𝐹(𝑔) → 𝐹(𝑔) +� +𝑔∈Obj(G), +such that the following diagrams commute for every object 𝑔 of G: +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +1 +𝐹(1) +𝐹(𝑔 ⊗ 𝑔) +𝐹 (−1) +𝑔 +⊗id𝑔 +𝐹 (0) +𝐹(ev𝑔) +ev𝑔 +𝐹 (2) +𝑔,𝑔 +(3.4) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +1 +𝐹(1) +𝐹(𝑔 ⊗ 𝑔) +id𝑔 ⊗𝐹 (−1) +𝑔 +𝐹 (0) +𝐹(coev𝑔) +coev𝑔 +𝐹 (2) +𝑔,𝑔 +(3.5) +We now prove that the natural isomorphism of Proposition 3.6 makes every homomorphism +of coherent 2-groups into a bar functor and every 2-isomorphism of coherent 2-groups into +a bar natural transformation. +Theorem 3.7. Let G and G′ be coherent 2-groups. +(1) Let 𝐹 : G → G′ be a monoidal functor. Then 𝐹 defines a bar functor with respect to the +canonical natural isomorphism 𝐹 (−1) of Proposition 3.6. +(2) Let 𝑃, 𝑄 : G → G′ be monoidal functors, so that 𝑃 and 𝑄 uniquely define bar functors +satisfying (3.4) and (3.5). Then every monoidal natural transformation 𝜂 : 𝑃 ⇒ 𝑄 is a bar +natural transformation. +Lemma 3.8 (Baez–Lauda [7, Proof of Thm 6.1]). Let G be a coherent 2-group. For every object 𝑔 of +𝐺 and every inverse (ℎ, e, i) of 𝑔, there exists a unique isomorphism of the inverses (𝑔, ev𝑔, coev𝑔) +and (ℎ, e, i) of 𝑔. Moreover, for every inverse (ℎ, e, i) of 𝑔 and every arrow 𝑢 : 𝑔 → ℎ, the arrow +𝑢 satisfies (2.3) with respect to the inverses (𝑔, ev𝑔, coev𝑔) and (ℎ, e, i) of 𝑔 if and only if 𝑢 is the +unique isomorphism of the inverses (𝑔, ev𝑔, coev𝑔) and (ℎ, e, i) of 𝑔. +Lemma 3.9. Let G and G′ be coherent 2-groups. Let 𝐹 : G → G′ be a monoidal functor, +and let 𝐹 (−1) be the natural isomorphism of Proposition 3.6. Let 𝑔 and ℎ be objects of G, and let +e𝑔,ℎ : (ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ) → 1 and e𝐹(𝑔),𝐹(ℎ) : (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) → 1 be the +unique such structural arrows in G and G′, respectively. The following diagram commutes: +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +𝐹(ℎ ⊗ 𝑔) ⊗ 𝐹(𝑔 ⊗ ℎ) +1 +𝐹(1) +𝐹((ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ)) +e𝐹(𝑔),𝐹(ℎ) +(𝐹 (−1) +ℎ +⊗𝐹 (−1) +𝑔 +) ⊗id +𝐹 (2) +ℎ,𝑔 ⊗𝐹 (2) +𝑔,ℎ +𝐹 (2) +ℎ⊗𝑔,𝑔⊗ℎ +𝐹(e𝑔,ℎ) +𝐹 (0) + +28 +BRANIMIR ĆAĆIĆ +Proof. Let 𝑔, ℎ ∈ Obj(G) be given. Observe that we can construct e𝑔,ℎ and e𝐹(𝑔),𝐹(ℎ) as +e𝑔⊗ℎ = evℎ ◦ 𝜌ℎ ◦ (id ⊗ ev𝑔) ⊗ id ◦ 𝛼−1 +ℎ,𝑔,𝑔, +e𝐹(𝑔) ⊗𝐹(ℎ) = ev𝐹(ℎ) ◦ 𝜌𝐹(ℎ) ◦ (id ⊗ ev𝐹(𝑔)) ⊗ id ◦ 𝛼−1 +𝐹(ℎ),𝐹(𝑔),𝐹(𝑔). +Thus, our claim follows from commutativity of the following diagram, where, for visual clarity, +we replace the symbol ⊗ with ·. +(𝐹(ℎ) · 𝐹(𝑔)) · (𝐹(𝑔) · 𝐹(ℎ)) +(𝐹(ℎ) · 𝐹(𝑔)) · (𝐹(𝑔) · 𝐹(ℎ)) +𝐹(ℎ · 𝑔) · (𝐹(𝑔) · 𝐹(ℎ)) +𝐹(ℎ · 𝑔) · 𝐹(𝑔 · ℎ) +𝐹((ℎ · 𝑔) · (𝑔 · ℎ)) +((𝐹(ℎ) · 𝐹(𝑔)) · 𝐹(𝑔)) · 𝐹(ℎ) +((𝐹(ℎ) · 𝐹(𝑔)) · 𝐹(𝑔)) · 𝐹(ℎ) +(𝐹(ℎ · 𝑔) · 𝐹(𝑔)) · 𝐹(ℎ) +(𝐹(ℎ) · (𝐹(𝑔) · 𝐹(𝑔))) · 𝐹(ℎ) +(𝐹(ℎ) · (𝐹(𝑔) · 𝐹(𝑔))) · 𝐹(ℎ) +(𝐹((ℎ · 𝑔) · 𝑔) · 𝐹(ℎ) +𝐹(((ℎ · 𝑔) · 𝑔) · ℎ) +(𝐹(ℎ) · 𝐹(𝑔 · 𝑔)) · 𝐹(ℎ) +𝐹(ℎ · (𝑔 · 𝑔)) · 𝐹(𝑔) +𝐹((ℎ · (𝑔 · 𝑔)) · ℎ) +(𝐹(ℎ) · 𝐹(1)) · 𝐹(ℎ) +𝐹(ℎ · 1) · 𝐹(𝑔) +𝐹((ℎ · 1) · ℎ) +(𝐹(ℎ) · 1) · 𝐹(ℎ) +(𝐹(ℎ) · 1) · 𝐹(ℎ) +𝐹(ℎ) · 𝐹(ℎ) +𝐹(ℎ) · 𝐹(ℎ) +𝐹(ℎ · ℎ) +1 +𝐹(1) +(𝐹 (−1) +ℎ +·𝐹 (−1) +𝑔 +) ·id +𝐹 (2) +ℎ,𝑔 ·id +id ·𝐹 (2) +𝑔,ℎ +𝐹 (2) +ℎ·𝑔,𝑔·ℎ +(𝐹 (2) +ℎ,𝑔 ·id)·id +(𝐹 (−1) +ℎ +·(𝐹 (−1) +𝑔 +·id)) ·id +𝐹 (2) +(ℎ·𝑔)·𝑔,ℎ +𝐹 (2) +ℎ,𝑔·𝑔 ·id +𝐹 (2) +ℎ·(𝑔·𝑔),ℎ·id +𝐹 (2) +ℎ,1 ·id +𝐹 (2) +ℎ·1,ℎ +(𝐹 (−1) +ℎ +·id) ·id +𝐹 (−1) +ℎ +·id +𝐹 (2) +ℎ,ℎ +𝐹 (0) +𝛼−1 +𝐹(ℎ)·𝐹(𝑔),𝐹(𝑔),𝐹(ℎ) +𝛼−1 +𝐹(ℎ)·𝐹(𝑔),𝐹(𝑔),𝐹(ℎ) +𝛼−1 +𝐹(ℎ·𝑔),𝐹(𝑔),𝐹(ℎ) +𝐹(𝛼−1 +ℎ·𝑔,𝑔,ℎ) +𝛼𝐹(ℎ),𝐹(𝑔),𝐹(𝑔) ·id +𝛼𝐹(ℎ),𝐹(𝑔),𝐹(𝑔) ·id +𝐹 (2) +ℎ·𝑔,𝑔 ·id +(id ·ev𝐹(𝑔))·id +(id ·𝐹 (2) +𝑔,𝑔 )·id +𝐹(𝛼ℎ,𝑔,𝑔)·id +𝐹(𝛼ℎ,𝑔,𝑔 ·id) +(id ·𝐹(ev𝑔)) ·id +𝐹(id · ev𝑔) ·id +𝐹((id · ev𝑔)·id) +(id ·𝐹 (0)) ·id +𝐹(𝜌ℎ) ·id +𝐹(𝜌ℎ·id) +𝜌𝐹(ℎ) +𝜌𝐹(ℎ) ·id +ev𝐹(ℎ) +𝐹(evℎ) +However, this now follows from applying naturality of 𝛼, monoidality of 𝐹, naturality of 𝐹 (2), +and commutativity of (3.4) as appropriate to each sub-diagram. +□ +Proof of Theorem 3.7. First, let 𝐹 : G → G′ be a monoidal functor. Before continuing, let us +recall the construction of the natural isomorphism 𝐹 (−1) of Proposition 3.6. Given 𝑔 ∈ Obj(G), +and ∗�ev𝐹(𝑔) � (𝐹 (0))−1 ◦ 𝐹(ev𝑔) ◦ 𝐹 (2) +𝑔,𝑔 and � +coev𝐹(𝑔) � (𝐹 (2) +𝑔,𝑔 )−1 ◦ 𝐹(coev𝑔) ◦ 𝐹 (0), so that +(𝐹(𝑔), �ev𝐹(𝑔), � +coev𝐹(𝑔)) is an inverse for 𝐹(𝑔). Hence, by Lemma 3.8, we may define 𝐹 (−1) as +follows: for each 𝑔 ∈ Obj(G), let 𝐹 (−1) +𝑔 +: 𝐹(𝑔) → 𝐹(𝑔) be the unique isomorphism of the +inverses (𝐹(𝑔), ev𝐹(𝑔), coev𝐹(𝑔)) and (𝐹(𝑔), �ev𝐹(𝑔), � +coev𝐹(𝑔)) of 𝐹(𝑔). +Let us first show that (3.1) commutes. By Lemma 3.8, it suffices to show that the composite ar- +row 𝐹(★)◦(𝐹 (0))−1◦★−1◦𝐹 (0) satisfies (2.3) with respect to the inverses (𝐹(1), ev𝐹(1), coev𝐹(1)) +and (𝐹(1), �ev𝐹(1), � +coev𝐹(1)) of 𝐹(1). This, in turn, is proved by the commutativity of the fol- +lowing diagram, which follows by applying bifunctoriality of ⊗, coherence in G, coherence in +G′, monoidality of 𝐹, or naturality of 𝐹 (2) as appropriate to each sub-diagram: +𝐹(1) ⊗ 𝐹(1) +1 ⊗ 𝐹(1) +1 ⊗ 𝐹(1) +𝐹(1) ⊗ 𝐹(1) +𝐹(1) ⊗ 𝐹(1) +1 ⊗ 1 +1 ⊗ 1 +𝐹(1 ⊗ 1) +1 +𝐹(1) +𝐹(1 ⊗ 1) +𝐹 (0) ⊗id +★−1 ⊗id +𝐹 (0) ⊗id +𝐹(★) ⊗id +★−1 ⊗id +𝐹 (0) +𝐹(ev1) +ev𝐹(1) +𝜆𝐹(1) +𝐹 (2) +1,1 +𝐹 (2) +1,1 +𝐹 (0) ⊗𝐹 (0) +id ⊗𝐹 (0) +ev1 +𝜆1 +𝐹(𝜆1) +𝐹(★⊗id) +Next, let us show that (3.2) commutes. Let 𝑔 ∈ Obj(G) be given. By Lemma 3.8, it suf- +fices to show that the arrow 𝐹(bb𝑔) ◦ bb−1 +𝐹(𝑔) ◦(𝐹 (−1) +𝑔 +)−1 satisfies (2.3) with respect to the +inverses (𝐹(𝑔), ev𝐹(𝑔), coev𝐹(𝑔)) and (𝐹(𝑔), �ev𝐹(𝑔), � +coev𝐹(𝑔)) of 𝐹(𝑔). This is now proved by +the commutativity of the following diagram, which follows by applying bifunctoriality of ⊗, + +NONCOMMUTATIVE U(1)-GAUGE THEORY +29 +coherence in G, coherence in G′, commutativity of (3.5), naturality of ev, and naturality of +𝐹 (2) as appropriate to each sub-diagram: +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔) ⊗ 𝐹(𝑔) +𝐹(𝑔 ⊗ 𝑔) +1 +𝐹(1) +𝐹(𝑔 ⊗ 𝑔) +(𝐹 (−1) +𝑔 +)−1 ⊗id +bb−1 +𝐹(𝑔) ⊗ id +𝐹(bb𝑔) ⊗id +bb𝐹(𝑔) ⊗ id +𝐹 (0) +𝐹(ev𝑔) +ev𝐹(𝑔) +𝐹 (−1) +𝑔 +⊗𝐹 (−1) +𝑔 +id ⊗𝐹 (−1) +𝑔 +𝐹 (2) +𝑔,𝑔 +𝐹 (2) +𝑔,𝑔 +𝐹(coev𝑔) +𝐹(bb𝑔 ⊗ id) +ev𝐹(𝑔) +coev𝐹(𝑔) +Finally, let us show that (3.3) commutes. Let 𝑔, ℎ ∈ Obj(G) be given; for convenience, let +e𝑔,ℎ and e𝐹(𝑔),𝐹(ℎ) be defined as in Lemma 3.9. By Lemma 3.8, it suffices to show that the +arrow 𝐹(Υ−1 +𝑔,ℎ) ◦ 𝐹 (2) +ℎ,𝑔 ◦ 𝐹 (−1) +ℎ +⊗ 𝐹 (−1) +𝑔 +◦ Υ𝐹(𝑔),𝐹(ℎ) ◦ (𝐹 (2) +𝑔,ℎ )−1 satisfies (2.3) with respect to the +inverses (𝐹(𝑔 ⊗ ℎ), ev𝐹(𝑔⊗ℎ), coev𝐹(𝑔⊗ℎ)) and (𝐹(𝑔 ⊗ ℎ), �ev𝐹(𝑔⊗ℎ), � +coev𝐹(𝑔⊗ℎ)) of 𝐹(𝑔 ⊗ ℎ). +This, in turn, is proved by commutativity of the following diagram, which follows by applying +coherence in G, coherence in G′, bifunctoriality of ⊗, naturality of ev, naturality of 𝐹 (2) and +Lemma 3.9 as appropriate to each sub-diagram: +𝐹(𝑔 ⊗ ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) +1 +𝐹(𝑔) ⊗ 𝐹(ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) +𝐹(𝑔) ⊗ 𝐹(ℎ) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ 𝐹(𝑔 ⊗ ℎ) +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ 𝐹(𝑔 ⊗ ℎ) +(𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +𝐹(ℎ ⊗ 𝑔) ⊗ 𝐹(𝑔 ⊗ ℎ) +𝐹(ℎ ⊗ 𝑔) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) +𝐹((ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ)) +𝐹(1) +𝐹(𝑔 ⊗ ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) +𝐹(𝑔 ⊗ ℎ ⊗ (𝑔 ⊗ ℎ)) +ev𝐹(𝑔⊗ℎ) +id ⊗𝐹 (2) +𝑔,ℎ +ev𝐹(𝑔)⊗𝐹(ℎ) ⊗ id +e𝐹(𝑔),𝐹(ℎ) +id ⊗𝐹 (2) +𝑔,ℎ +𝐹(e𝑔,ℎ) +𝐹 (2) +𝑔⊗ℎ,𝑔⊗ℎ +(𝐹 (2) +𝑔,ℎ )−1 ⊗id +𝐹 (0) +Υ𝐹(𝑔),𝐹(ℎ) ⊗id +Υ𝐹(𝑔),𝐹(ℎ) ⊗id +(𝐹 (−1) +ℎ +⊗𝐹 (−1) +𝑔 +) ⊗id +(𝐹 (−1) +ℎ +⊗𝐹 (−1) +𝑔 +) ⊗id +𝐹 (2) +ℎ,𝑔 ⊗id +𝐹 (2) +ℎ,𝑔 ⊗id +𝐹(Υ−1 +𝑔,ℎ ) ⊗id +𝐹(ev𝑔⊗ℎ) +𝐹(Υ−1 +𝑔,ℎ ⊗id) +𝐹 (2) +ℎ⊗𝑔,𝑔⊗ℎ +Now, let 𝑃, 𝑄 : G → G′ be monoidal functors, and let 𝜂 : 𝑃 ⇒ 𝑄 be a monoidal natural +transformation. Let 𝑃 (−1) and 𝑄(−1) be constructed as above, so that 𝑃 and 𝑄 define bar func- +tors satisfying (3.4) and (3.5). Let 𝑔 ∈ Obj(G) be given. To show that (3.3) commutes, it suffices +to show that 𝜙−1 +𝑔 ◦𝑄(−1) +𝑔 +◦ 𝜙𝑔 satisfies (2.3) with respect to the inverses (𝑃(𝑔), ev𝑃(𝑔), coev𝑃(𝑔)) +and (𝑃(𝑔), �ev𝑃(𝑔), � +coev𝑃(𝑔)) of 𝑃(𝑔). In turn, it suffices to show that the following diagram +commutes: +𝑃(𝑔) ⊗ 𝑃(𝑔) +𝑄(𝑔) ⊗ 𝑄(𝑔) +𝑄(𝑔) ⊗ 𝑄(𝑔) +𝑃(𝑔) ⊗ 𝑃(𝑔) +𝑄(1) +𝑄(𝑔 ⊗ 𝑔) +1 +𝑃(1) +𝑃(𝑔 ⊗ 𝑔) +𝜙𝑔 ⊗𝜙𝑔 +𝑄(−1) +𝑔 +⊗id +𝜙𝑔 ⊗𝜙𝑔 +𝑄(ev𝑔) +𝑃(ev𝑔) +𝑃 (0) +ev𝑃(𝑔) +ev𝑄(𝑔) +𝑄(2) +𝑔,𝑔 +𝑃 (2) +𝑔,𝑔 +𝑄(0) +𝜙1 +𝜙𝑔⊗𝑔 +However, this diagram commutes by applying naturality of ev, commutativity of (3.4) for 𝑄, +naturality of 𝜙, and monoidality of 𝜙 as appropriate to each sub-diagram. +□ +Example 3.10 (Buss–Meyer–Zhu [23, Thm 3.3]). Let 𝐵 be a unital pre-𝐶∗-algebra, let Γ be a +group, and let 𝐹 : Γ → Pic(𝐵) be a homomorphism. The disjoint union F � � +𝛾∈Γ 𝐹(𝛾) + +30 +BRANIMIR ĆAĆIĆ +defines a pre-Fell bundle over Γ in the sense of Exel [47, Def. 24.2] with respect to the fibrewise +multiplication F× F→ Fand the fibrewise ∗-operation F→ Fdefined, respectively, by +∀𝛾, 𝜂 ∈ Γ, ∀𝑝 ∈ 𝐹(𝛾), ∀𝑞 ∈ 𝐹(𝜂), +𝑝𝑞 � 𝐹 (2) +𝛾,𝜂 (𝛾 ⊗ 𝜂), +∀𝛾 ∈ Γ, ∀𝑝 ∈ 𝐹(𝛾), +𝑝∗ � 𝐹 (−1) +𝛾 +(𝑝), +where 𝐹 (−1) is the natural transformation of Theorem 3.7. Note that Theorem 3.7 as applied +to Hom(Γ, Pic(𝐵)) recovers Buss–Meyer–Zhu’s construction [23, Proof of Thm 3.3] of the +fibrewise ∗-operation on F. +3.2. Generalised crossed products via homomorphisms of coherent 2-groups. In this +section, we revisit the well-known theory of nc topological principal U(1)-bundles [1, 9, 4] +from the perspective of coherent 2-groups. This will permit us to generalise Abadie–Eilers– +Exel’s framework [1] of generalized crossed products via Pimsner’s construction [81] to the +setting of nc differential geometry by replacing the Picard 2-group with the differentiable +Picard 2-group. In what follows, let 𝐵 be a unital pre-𝐶∗-algebra. +Let us define a U(1)-pre-𝐶∗-algebra of finite type is a unital pre-𝐶∗-algebra 𝑃 equipped +with a strongly continuous U(1)-action of finite type by isometric ∗-automorphisms. In this +case, the spectral subspace 𝑃U(1) = 𝑃0 is a unital ∗-subalgebra of 𝑃, and the decomposition +of complex vector spaces 𝑃 = ⊕𝑘∈Z𝑃𝑘 defines a Z-grading of the unital ∗-algebra 𝑃 in the +sense that 𝑃𝑚 · 𝑃𝑛 ⊆ 𝑃𝑚+𝑛 for all 𝑚, 𝑛 ∈ Z and ∗(𝑃𝑚) ⊆ 𝑃−𝑚 for all 𝑚 ∈ Z. This permits the +following minimalistic definition of topological quantum principal U(1)-bundle. +Definition 3.11 (cf. Arici–Kaad–Landi [4, §4.2]). A topological quantum principal U(1)-bundle +is a pre-𝐶∗-algebra (𝑃, 𝛼) of finite type, such that there exist finite families (𝑒𝑖)𝑚 +𝑖=1 and (𝜖𝑗)𝑛 +𝑗=1 +in 𝑃1 satisfying �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 and �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1, respectively. +This definition is slightly unconventional but may be related to more familiar definitions +as follows. Suppose that (𝑃, 𝛼) is a topological quantum principal U(1)-bundle. On the one +hand, by an observation of Năstăsescu–Van Ostaeyen [78, Lemma i.3.2], the U(1)-action 𝛼 +is principal in the sense that SpanC{𝑧𝑘 ⊗ 𝑝𝑞 | 𝑘 ∈ Z, 𝑝 ∈ 𝑃𝑘, 𝑞 ∈ 𝑃} = O(U(1)) ⊗C 𝑃. On +the other hand, by an observation of Ulbrich [96, Lemma 2.1], it follows that the Z-grading +𝑃 = � +𝑘∈Z 𝑃𝑘 of 𝑃 is strong in the sense that 𝑃𝑚 · 𝑃𝑛 = 𝑃𝑚+𝑛 for all 𝑚, 𝑛 ∈ Z. The familiar +fact that 𝛼 is principal if and only if the Z-grading of 𝑃 is strong [78, Lemma i.3.2] yields the +familiar algebraic definition of (topological) quantum principal U(1)-bundle in the literature. +Example 3.12. Let 𝜋 : 𝑋 → 𝑌 be a compact differentiable principal U(1)-bundle with +principal right U(1)-action 𝜎 : U(1) → Diff(𝑋). Hence, let +𝐶∞ +alg(𝑋) � +alg +� +𝑘∈Z +{𝜔 ∈ 𝐶∞(𝑋) | ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧𝑘𝜔}, +which is norm-dense in 𝐶(𝑋) since it is Fréchet-dense in 𝐶∞(𝑋). Then 𝐶∞ +alg(𝑋) defines a +topological quantum principal U(1)-bundle with respect to 𝛼 � (𝑧 ↦→ (𝜎𝑧−1)∗); moreover, +the pullback homomorphism 𝜋∗ : 𝐶∞(𝑌) → 𝐶∞(𝑋)U(1) is an isometric ∗-isomorphism. In +particular, one can use an atlas of local principal U(1)-bundle trivialisations for 𝜋 : 𝑋 → 𝑌 +together with a subordinate smooth partition of unity on 𝑌 to construct a finite family (𝑒𝑖)𝑚 +𝑖=1 +in 𝐶∞ +alg(𝑋)1 satisfying �𝑛 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = �𝑛 +𝑖=1 𝑒∗ +𝑖 𝑒𝑖 = 1. +The following introduces our second main running example, the first genuinely nc example +of a topological quantum principal U(1)-bundle in the literature. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +31 +Example 3.13 (Brzeziński–Majid [22, §5.2]). Let 𝑞 ∈ (0, ∞) \ {1}, so that the corresponding +quantum special unitary group á la Woronowicz [98] is the universal 𝐶∗-algebra 𝐶𝑞(SU(2)) +generated by elements 𝑎 and 𝑐 satisfying +𝑎𝑐 = 𝑞𝑐𝑎, +𝑎𝑐∗ = 𝑞𝑐∗𝑎, +𝑐∗𝑐 = 𝑐𝑐∗, +𝑎∗𝑎 + 𝑐∗𝑐 = 1, +𝑎𝑎∗ + 𝑞2𝑐𝑐∗ = 1; +the corresponding unital pre-𝐶∗-algebra O𝑞(SU2) is the dense unital ∗-subalgebra of 𝐶𝑞(SU2) +consisting of complex polynomials in 𝑎, 𝑎∗, 𝑐, and 𝑐∗. Then O𝑞(SU(2)) defines a topological +quantum principal U(1)-bundle with respect to the unique U(1)-action of finite type 𝛼 +satisfying 𝛼𝑧(𝑎) = 𝑧𝑎 and 𝛼𝑧(𝑐) = 𝑧𝑐 for all 𝑧 ∈ U(1); in particular, the families (𝑎, 𝑞𝑐) and +(𝑎, 𝑐) in O𝑞(SU(2))1 respectively satisfy 𝑎𝑎∗ + (𝑞𝑐)(𝑞𝑐)∗ = 1 and 𝑎∗𝑎 + 𝑐∗𝑐 = 1. Moreover, +the U(1)-action 𝛼 satisfies O𝑞(SU(2))U(1) = O𝑞(CP1), where O𝑞(CP1), the algebraic standard +Podleś sphere [82], is the unital ∗-subalgebra of O𝑞(SU2) consisting of complex polynomials in +the elements 𝑐∗𝑐, 𝑎𝑐∗, and 𝑐𝑎∗. +We note that our rather strict definition of topological quantum principal U(1)-bundle +reduces to a simpler definition whenever the ∗-subalgebra of U(1)-invariant elements is +sufficiently like a 𝐶∗-algebra. +Proposition 3.14. Let 𝑃 be a unital pre-𝐶∗-algebra with U(1)-action of finite type 𝛼. Suppose +that the fixed-point subalgebra 𝑃U(1) admits polar decompositions. Then (𝑃, 𝛼) is a topological +quantum principal U(1)-bundle if and only if 𝑃1 · 𝑃−1 = 𝑃U(1) and 𝑃−1 · 𝑃1 = 𝑃U(1). +Proof. For each 𝑘 ∈ Z, the spectral subspace 𝑃𝑘 defines a 𝑃U(1)-bimodule with positive definite +𝑃U(1)-valued inner product (·, ·)𝑘 � ((𝑝, 𝑞) ↦→ 𝑝∗𝑞). Moreover, for each 𝑘 ∈ Z, the 𝑃U(1)- +valued inner products (·, ·)𝑘 and (·, ·)−𝑘 satisfy 𝑝 · (𝑞, 𝑟)𝑘 = (𝑝∗, 𝑞∗)−𝑘 · 𝑟 for all elements +𝑝, 𝑞, 𝑟 ∈ 𝑃𝑘. Hence, we may apply the proof of Proposition 2.23, mutatis mutandis, to 𝑃1 and +𝑃−1, where 𝑃−1 admits the isomorphism of 𝐵-bimodules (𝑝 ↦→ 𝑝∗) : 𝑃−1 → 𝑃1. +□ +Recall that 𝐵 is a given unital pre-𝐶∗-algebra. Let us define the concrete category Circ(𝐵) +of topological quantum principal U(1)-bundles over 𝐵 and their isomorphisms as follows: +(1) an object of Circ(𝐵) is a topological quantum principal U(1)-bundle together with an +isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1); +(2) an arrow 𝑓 : 𝑃 → 𝑄 in Circ(𝐵) is a U(1)-equivariant isometric ∗-isomorphism, such +that 𝑓 ◦ 𝜄𝑃 = 𝜄𝑄. +One can now make precise sense of associated line bundles in the nc setting. +Proposition 3.15 (Exel [46, §2], Schwieger–Wagner [93, §4.1]). The following defines a functor +L : Circ(𝐵) → Hom(Z, Pic(𝐵)). +(1) Let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵. Define a homomorphism +L(𝑃) : Z → Pic(𝐵) as follows: +(a) given 𝑘 ∈ Z, let L(𝑃)(𝑘) � 𝑃𝑘 as a complex vector space with the 𝐵-bimodule structure +∀𝑎, 𝑏 ∈ 𝐵, ∀𝑝 ∈ 𝑃𝑘, +𝑎𝑝𝑏 � 𝜄𝑃(𝑎)𝑝𝜄𝑃(𝑏) +(3.6) +and the 𝐵-valued inner products on 𝑃𝑘 and 𝑃𝑘 defined, respectively, by +∀𝑝, 𝑞 ∈ 𝑃𝑘, +(𝑝, 𝑞) � 𝜄−1 +𝑃 (𝑝∗𝑞), +(𝑝, 𝑞) � 𝜄−1 +𝑃 (𝑝𝑞∗); +(3.7) +(b) set L(𝑃)(0) � 𝜄−1 +𝑃 ; +(c) given 𝑚, 𝑛 ∈ Z, let L(𝑃)(2) +𝑚,𝑛 : L(𝑃)(𝑚) ⊗ L(𝑃)(𝑛) → L(𝑃)(𝑚 + 𝑛) be induced by +multiplication in 𝑃. + +32 +BRANIMIR ĆAĆIĆ +(2) Let 𝑓 : 𝑃 → 𝑄 be an isomorphism of topological quantum principal U(1)-bundles over 𝐵. +Define the corresponding 2-isomorphism L(𝑓) : L(𝑃) ⇒ L(𝑄) by +∀𝑘 ∈ Z, +L(𝑓)𝑘 � 𝑓↾𝑃𝑘 . +(3.8) +Proof. This is mostly a straightforward exercise in checking definitions. Let 𝑃 ∈ Obj(Pic(𝐵)) +be given. When checking that the functor L(𝑃) : Z → Pic(𝐵) is well defined, the only non- +trivial point is strict fullness of all 𝐵-valued inner product. However, let (𝑒𝑖)𝑚 +𝑖=1 and (𝜖𝑗)𝑛 +𝑗=1 be +families in 𝑃1 respectively satisfying �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 and �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1, and define 𝑒𝐼 � 𝑒𝑖1 . . . 𝑒𝑖𝑘 +for all 𝑘 ∈ N and 𝐼 = (𝑖1, . . . , 𝑖𝑘) ∈ {1, . . . , 𝑚}𝑘 and 𝜖𝐽 � 𝜖𝑗1 . . . 𝜖𝑗𝑘 for all 𝑘 ∈ N and +𝐽 = (𝑗1, . . . , 𝑗𝑘) ⊂ {1, . . . , 𝑛}𝑘. Then, for each 𝑘 ∈ N, it follows that (𝑒∗ +𝐼)𝐼 ∈{1,...,𝑚}𝑘 is a cobasis +for L(𝑃)(−𝑘), that (𝜖∗ +𝐽)𝐽∈{1,...,𝑛}𝑘 is a cobasis for L(𝑃)(−𝑘), that (𝜖𝐽)𝐽∈{1,...,𝑛}𝑘 is a cobasis for +L(𝑃)(𝑘), and that (𝑒𝐼)𝐼 ∈{1,...,𝑚}𝑘 is a cobasis for L(𝑃)(𝑘). From here, monoidality of L(𝑃) +follows from elementary algebraic properties of 𝑃: coherence with respect to unitors follows +from multiplicativity of the isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1), while coherence with +respect to associators follows from associativity of multiplication in 𝑃. Similarly, if 𝑓 : 𝑃 → 𝑄 +is an arrow in Pic(𝐵), then L(𝑓) intertwines L(𝑃)(0) and L(𝑄)(0) since 𝑓 intertwines the +given isometric ∗-isomorphisms 𝜄𝑃 : 𝐵 → 𝑃U(1) and 𝜄𝑄 : 𝐵 → 𝑄U(1), while coherence of +L(𝑓) with respect to L(𝑃)(2) and L(𝑄)(2) follows from multiplicativity of 𝑓. +□ +For example, let 𝜋 : 𝑋 → 𝑌 be a compact differentiable principal U(1)-bundle with +principal U(1)-action 𝜎 : U(1) → Diff(𝑋), so that 𝐶∞ +alg(𝑋) defines an object of Circ(𝐶∞(𝑌)) +with respect to 𝜋∗ : 𝐶∞(𝑌) → 𝐶∞ +alg(𝑋)U(1). By Serre–Swan duality for smooth Hermitian +vector bundles on 𝑌, for each 𝑘 ∈ Z, the Hermitian line 𝐵-bimodule L𝑘(𝐶∞ +alg(𝑋)) recovers +the associated Hermitian line bundle to 𝑌 of winding number −𝑘. +Likewise, in the setting of Example 3.13, the homomorphism E : Z → Pic(O𝑞(CP1)) +given by E � L(O𝑞(SU(2))) recovers (up to a sign convention) the canonical line bundles +on O𝑞(CP1) as studied by Landi–Reina–Zampini [65]. In fact, it follows from a result of +Carotenuto–Ó Buachalla [29, Prop. 4.4] that the homomorphism Eexhausts the left O𝑞(SU(2))- +covariant Hermitian line O𝑞(CP1)-bimodules up to O𝑞(SU(2))-covariant isomorphism. +We now recover the known result that the functor Lextracting associated line bundles +is an equivalence of categories. As a preliminary, recall that a conditional expectation of a +unital pre-𝐶∗-algebra 𝐴2 onto a unital pre-𝐶∗-algebra 𝐴1 with respect to an isometric ∗- +homomorphism 𝜄 : 𝐴1 → 𝐴2 is a contractive unit-preserving and ∗-preserving 𝐴1-bimodule +map E : 𝐴2 → 𝐴1, such that E((𝐴2)+) ⊆ (𝐴1)+ and E ◦ 𝜄 = id𝐴1. In this case, we say that E is +faithful whenever it satisfies {𝑎 ∈ (𝐴2)+ | E(𝑎) = 0} = {0}. +Proposition 3.16. Let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵. Define a +complex-linear map E𝑃 : 𝑃 → 𝐵 by setting E𝑃↾𝑃𝑗� �𝑝 ↦→ 𝜄−1 +𝑃 +�𝛿 𝑗,0𝑝�� for all 𝑗 ∈ Z. Then E is a +U(1)-invariant faithful conditional expectation of 𝑃 onto 𝐵 with respect to 𝜄𝑃. +Proof. Let 𝜎 denote the U(1)-action on 𝑃, and let 𝑚 denote the normalised Haar measure on +U(1). Note that E𝑃 is manifestly U(1)-invariant, unit-preserving, ∗-preserving, and 𝐵-bilinear +and that it satisfies E𝑃 ◦ 𝜄𝑃 = id𝐵. Since 𝜎 is of finite type, we may use Bochner integration on +U(1) to write E𝑃 = +� +𝑝 ↦→ 𝜄𝑃 +�∫ +U(1) 𝜎𝑧(𝑝) d𝑚(𝑧) +�� +. Since 𝜎 acts isometrically on 𝑃, it follows +that E is contractive; since 𝜎 acts by unital ∗-automorphisms and by our convention for +positive cones, it follows that the E𝑃 maps positive elements to positive elements. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +33 +Let us now show that E is faithful.1 Let 𝑝 ∈ 𝑃+ \ {0}, so that there exists a bounded state +𝜙 : 𝑃 → C, such that 𝜙(𝑝) > 0. Since (𝑧 ↦→ 𝜙(𝜎𝑧(𝑝))) : U(1) → [0, ∞) is continuous, there +exists an open neighbourhood 𝐼 of 1, such that 𝜙(𝜎𝑧(𝑝)) > 1 +2𝜙(𝑝) for all 𝑧 ∈ 𝐼. Hence, by +norm-continuity of E𝑃, (𝜙 ◦ 𝜄𝑃)(E𝑃(𝑝)) = +∫ +U(1) (𝜙 ◦ 𝜎𝑧)(𝑝) d𝑚(𝑧) ≥ 1 +2𝜙(𝑝)𝑚(𝐼) > 0. +□ +Theorem 3.17 (Buss–Meyer–Zhu [19, Thm 3.3], Schwieger–Wagner [93, Thmm 4.21 & 5.2]). +The following defines a a weak inverse Σ : Hom(Z, Pic(𝐵)) → Circ(𝐵) of the functor L. +(1) Given a homomorphism 𝐹 : Z → Pic(𝐵), construct a topological quantum principal U(1)- +bundle Σ(𝐹) over 𝐵 as follows: +(a) define the unital ∗-algebra Σ(𝐹) by equipping the complex vector space � +𝑘∈Z 𝐹(𝑘) with +the multiplication and ∗-operation defined, respectively, by +∀𝑚, 𝑛 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), ∀𝑞 ∈ 𝐹(𝑛), +𝑝𝑞 � 𝐹 (2) +𝑚,𝑛(𝑝 ⊗ 𝑞), +(3.9) +∀𝑚 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), +𝑝∗ � 𝐹 (−1) +𝑚 +(𝑝); +(3.10) +(b) equip Σ(𝐹) with the unique 𝐶∗-norm ∥ · ∥Σ(𝐹), such that +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), +∥𝑝∥2 +Σ(𝐹) = ∥(𝑝, 𝑝)∥; +(3.11) +(c) define a U(1)-action of finite type 𝛼 on Σ(𝐹) by +∀𝑧 ∈ U(1), ∀𝑚 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), +𝛼𝑧(𝑝) � 𝑧𝑚𝑝; +(3.12) +(d) set 𝜄Σ(𝐹) � (𝐹 (0))−1. +(2) Given a 2-isomorphism 𝜂 : 𝑅 → 𝑆, construct an isomorphism Σ(𝜂) : Σ(𝑅) → Σ(𝑆) of +topological quantum principal U(1)-bundles over 𝐵 by +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑅(𝑘), +Σ(𝜂)𝑘(𝑝) � 𝜂𝑘(𝑝). +(3.13) +Hence, in particular, the category Circ(𝐵) is essentially small. +Proof. We supply a proof that we can (and shall) adapt to other contexts. Let us first show that +Σ is well-defined on objects. Let 𝐹 : Z → Pic(𝐵) be a given homomorphism, which is a bar +functor by Theorem 3.7. This now implies that Σ(𝐹) is a unital ∗-algebra and that 𝜄Σ(𝐹) is a +∗-isomorphism. Indeed, coherence of 𝐹 with respect to associators implies associativity of +Σ(𝐹), while coherence of 𝐹 with respect to unitors implies that Σ(𝐹) is unital and that 𝜄Σ(𝐹) +is a unital homomorphism. From there, commutativity of (3.3), (3.2), and (3.1) imply that the +∗-operation is antimultiplicative, involutive, and unital, respectively, while commutativity of +(3.1) also implies that 𝜄Σ(𝐹) is a ∗-homomorphism. +Now, recall from Example 3.10 that 𝐹 canonically defines a a pre-Fell bundle Fover Z in +the sense of Exel [47, Def. 24.2]; it follows immediately that Σ(𝐹) is precisely the ∗-algebra of +compactly supported cross-sections of F. Thus, by [47, Propp. 17.9.(iv) & 19.8], the 𝐶∗-norm +on the reduced cross-sectional 𝐶∗-algebra [47, Def. 17.6] of the Fell bundle completion [47, Def. +24.7] of Fyields the unique 𝐶∗-norm on Σ(𝐹) satisfying (3.11); since 𝐹 (0) satisfies (2.11), this +implies that 𝜄Σ(𝐹) is isometric. Finally, by [85, Thm 3], it follows (3.12) correctly defines a U(1)- +action of finite type on the unital pre-𝐶∗-algebra Σ(𝐹); that (Σ(𝐹); 𝛼) defines a topological +quantum principal U(1)-bundle over 𝐵 now follows from the existence of strict cobases for +𝐹(1) and 𝐹(1), respectively. +Next, let us show that Σ is well-defined on arrows. Let 𝜂 : 𝑅 → 𝑆 be a 2-isomorphism +between homomorphisms 𝑅, 𝑆 : Z → Pic(𝐵), so that 𝜂 is automatically a bar natural transfor- +mation by Theorem 3.7. This now implies that the U(1)-equivariant vector space isomorphism +1This elementary argument, which is surely folkloric, was found in an anonymous answer to a MathOverflow +question (https://mathoverflow.net/q/72624). + +34 +BRANIMIR ĆAĆIĆ +Σ(𝜂) : Σ(𝑅) → Σ(𝑆) is a unital ∗-isomorphism intertwining 𝜄Σ(𝑅) and 𝜄Σ(𝑆). Indeed, coherence +of 𝜂 with respect to 𝑅(2) and 𝑆(2) implies that Σ(𝜂) is multiplicative, that 𝜂1 intertwines 𝑅(0) +and 𝑆(0) implies that Σ(𝜂) is unital and intertwines 𝜄Σ(𝑅) and 𝜄Σ(𝑆), and the fact that 𝜂 is a bar +natural transformation implies that Σ(𝜂) is ∗-preserving. Since, for each 𝑘 ∈ Z, the arrows 𝜂𝑘 +and 𝜂−1 +𝑘 both satisfy (2.11), it follows that the bar natural transformation 𝜂 induces a isomor- +phism of the Fell bundle completions of the pre-Fell bundles induced by 𝑅 and 𝑆 respectively, +so that Σ(𝜂) is isometric by [47, Prop. 21.3]. +Now, functoriality of Σ is easily checked, so it remains to construct natural isomorphisms +𝜇 : idCirc(𝐵) ⇒ Σ ◦ L and 𝜈 : idHom(Z,𝑃𝑖𝑐(𝐵)) ⇒ L ◦ Σ. On the one hand, let 𝑃 be a +topological quantum principal U(1)-bundle over 𝐵. Since the Z-grading 𝑃 = � +𝑘∈Z 𝑃𝑘 is +strong, the spectral subspaces of 𝑃 define a pre-Fell bundle over Z fibrewise-isometrically +isomorphic (mutatis mutandis) over 𝜄−1 +𝑃 to the pre-Fell bundle over Z induced by L(𝑃); note, +moreover, by Proposition 3.16, that this Z-grading is topological in the sense of Exel [46, Def. +19.2] and that averaging over the U(1)-action yields a faithful conditional expectation of the +𝐶∗-completion of 𝑃 onto the 𝐶∗-completion of 𝐵, cf. [3, §4]. Hence, by [47, Prop. 21.3], there +exists unique U(1)-equivariant isometric ∗-isomorphism 𝜇𝑃 : 𝑃 → Σ ◦ L(𝑃) that satisfies +𝜇𝑃 ◦ 𝜄𝑃 = 𝜄Σ◦L(𝑃), namely, set 𝜇𝑃 ↾𝑃𝑘� id for 𝑘 ∈ Z \ {0} and 𝜇𝑃 ↾𝑃0� 𝜄−1 +𝑃 . Naturality of +𝜇 � (𝜇𝑃 : 𝑃 → Σ ◦ L(𝑃))𝑃∈Obj(Circ(𝐵)) now follows by uniqueness. On the other hand, +given monoidal 𝐹 : Z → Pic(𝐵), define 𝜈𝐹 : 𝐹 ⇒ L ◦ Σ(𝐹) as follows: for each 𝑘 ∈ Z, +let (𝜈𝐹)𝑘 : 𝐹(𝑘) → (L◦ Σ(𝐹))(𝑘) be the inclusion of 𝐹(𝑘) in Σ(𝐹) as a direct sum of 𝐵- +bimodules. Naturality of 𝜈 � (𝜈𝐹 : 𝐹 ⇒ L◦ Σ(𝐹))𝐹∈Obj(Hom(Z,Pic(𝐵))) follows from the fact +that direct sums in Bimod(𝐵) are coproducts. +□ +Remark 3.18. Let 𝑇 be a compact Abelian group with Pontrjagin dual ˆ𝑇; suppose that 𝐵 admits +polar decompositions. The results above generalise to yield an equivalence of categories +between Hom( ˆ𝑇, Pic(𝐵)) and an analogous category of topological quantum principal 𝑇- +bundles over 𝐵, thereby recovering the relevant classification results of Schwieger–Wagner +[93] in a manner that is adaptable to nc differential geometry. +The construction of the natural isomorphism 𝜇 : idCirc(𝐵) ⇒ Σ◦Lin the proof of Theorem +3.17 implies the following useful characterisation of relevant 𝐶∗-norms. +Corollary 3.19 (Arici–Kaad–Landi [4, Thm. 3.10]). Let 𝑃 be a topological quantum principal +U(1)-bundle on 𝐵; let ∥ · ∥ denote its 𝐶∗-norm. Let ∥ · ∥′ be a U(1)-invariant 𝐶∗-norm on 𝑃. +Then ∥ · ∥′ = ∥ · ∥ if and only if ∥ · ∥′↾𝑃U(1) = ∥ · ∥↾𝑃U(1) . +Combining Theorem 3.17 with Corollary 2.9 recovers Arici–Kaad–Landi’s characterisation +of topological quantum principal U(1)-bundles in terms of Pimsner’s construction [81]. +Corollary 3.20 (Arici–Kaad–Landi [4, §3]; cf. Abadie–Eilers–Exel [1, Thm 3.1], Beggs–Brzez- +iński [9, Thm 7.3]). The functor 𝜖1 ◦ L : Circ(𝐵) → Pic(𝐵) is an equivalence of categories. +Definition 3.21 (Abadie–Eilers–Exel [1]). Let 𝐸 be a Hermitian line 𝐵-bimodule. The crossed +product of 𝐵 by 𝐸 is the essentially unique topological quantum principal U(1)-bundle 𝐵 ⋊𝐸 Z +over 𝐵, such that L(𝐵 ⋊𝐸 Z)(1) � 𝐸. +One may justify this terminology as follows. Let 𝜙 ∈ Aut(𝐵), so that its algebraic crossed +product 𝐵 ⋊alg +𝜙 Z is the unital ∗-algebra obtained from 𝐵 by adjoining a unitary 𝑈 satisfying +𝑈𝑏𝑈∗ = 𝜙(𝑏) for all 𝑏 ∈ 𝐵. Then 𝐵 ⋊alg +𝜙 Z defines topological quantum U(1)-bundle over 𝐵 +when equipped with the reduced crossed product 𝐶∗-norm and the unique U(1)-action of +finite type 𝛼, such that 𝛼𝑧↾𝐵= id and 𝛼𝑧(𝑈) = 𝑧𝑈 for all 𝑧 ∈ U(1). Since +�𝑏𝜙 ↦→ 𝑈𝜙−1(𝑏)� : 𝐵𝜙 → L(𝐵 ⋊𝜙 Z)(1) + +NONCOMMUTATIVE U(1)-GAUGE THEORY +35 +is an isomorphism of Hermitian line 𝐵-bimodules, we may therefore take 𝐵⋊𝜏(𝜙) Z � 𝐵⋊alg +𝜙 Z. +3.3. Horizontal calculi as generalised crossed products. As promised, we now adapt the +considerations of the last subsection to the setting of nc differential geometry by replacing +the Picard 2-group with the differential Picard 2-group. However, in the absence of additional +constraints, we can only reconstruct the horizontal calculus of a quantum principal U(1)-bundle. +In what follows, let 𝐵 be a given unital pre-𝐶∗-algebra with ∗-exterior algebra (Ω𝐵, d𝐵). +Let 𝑃 be a U(1)-pre-𝐶∗-algebra of finite type with U(1)-action 𝛼. We define a U(1)-∗- +quasi-dga of finite type over 𝑃 to be a ∗-quasi-dga (Ω, d) over 𝑃 together with a pointwise +extension of 𝛼 to a group homomorphism ˆ𝛼 : U(1) → Aut(Ω, d), such that, for each 𝑘 ∈ N0, +the restriction of ˆ𝛼 to a 𝑈-action on the complex vector space Ω𝑘 is of finite type. In this +case, we call (Ω, d) a U(1)-∗-exterior algebra of finite type over 𝑃 whenever the underlying +∗-quasi-dga is a ∗-exterior algebra. At last, we denote by CDGAU(1) the concrete category +whose objects (𝑃; Ω, d) consist of a U(1)-pre-𝐶∗-algebra of finite type 𝑃 together with a U(1)- +∗-quasi-dga of finite type (Ω, d) over 𝑃 and whose arrows 𝑓 : (𝑃1; Ω1, d1) → (𝑃2; Ω2, d2) are +U(1)-equivariant morphisms of ∗-quasi-dga. +The following definition characterises the differentiable structure that a Hermitian line +𝐵-bimodule with connection can generally induce on the corresponding topological quantum +principial U(1)-bundle over 𝐵. +Definition 3.22 (Ðurđević [38, §2], cf. Ćaćić [24]). Let 𝑃 be a topological quantum principal +U(1)-bundle over 𝐵 with isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1). A horizontal calculus for +𝑃 is a U(1)-∗-quasi-dga (Ω𝑃,hor, d𝑃,hor) of finite type over 𝑃 together with an isomorphism of +quasi-∗-dga ˆ𝜄𝑃 : (𝐵; Ω𝐵, d) → (𝑃U(1), (Ω𝑃,hor)U(1), d𝑃,hor↾(Ω𝑃,hor)U(1) ) extending the isometric +∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1), such that Ω𝑃,hor = 𝑃 · (Ω𝑃,hor)U(1) · 𝑃. +Example 3.23 (Majid [67, §3]). We continue from Example 3.13. Let Ω𝑞,hor(SU(2)) be the +graded ∗-algebra over O𝑞(SU(2)) generated by 𝑒+ ∈ Ω1 +𝑞,hor(SU(2)) and 𝑒− � −(𝑒+)∗ satisfying +𝑒±𝑎 = 𝑞−1𝑎𝑒±, +𝑒±𝑎∗ = 𝑞𝑎∗𝑒±, +𝑒±𝑐 = 𝑞−1𝑐𝑒±, +𝑒±𝑐∗ = 𝑞𝑐∗𝑒±, +(𝑒±)2 = 0, +𝑒−𝑒+ + 𝑞−2𝑒+𝑒− = 0. +Define complex-linear maps 𝜕± : O𝑞(SU(2)) → O𝑞(SU(2) by +𝜕+(𝑎) � −𝑞𝑐∗, +𝜕+(𝑎∗) � 0, +𝜕+(𝑐) � 𝑎∗, +𝜕+(𝑐∗) � 0, +𝜕−(𝑎) � 0, +𝜕−(𝑎∗) � 𝑐, +𝜕−(𝑐) � 0, +𝜕−(𝑐∗) � −𝑞−1𝑎, +together with the twisted Leibniz rule +∀𝑥 ∈ O𝑞(SU(2)), ∀𝑗 ∈ Z, ∀𝑦 ∈ O𝑞(SU(2))𝑗, +𝜕±(𝑥𝑦) = 𝜕±𝑥𝑦𝑞−𝑗 + 𝑥𝜕±(𝑦); +hence, define d𝑞,hor : O𝑞(SU(2)) → Ω1 +𝑞,hor(SU(2)) by setting +∀𝑝 ∈ O𝑞(SU(2)), +d𝑞,hor(𝑝) � 𝜕+(𝑝)𝑒+ + 𝜕−(𝑝)𝑒−, +and extend d𝑞,hor to Ω𝑞,hor(SU(2)) by setting d𝑞,hor(𝑒±) � 0. Finally, extend the U(1)- +action from O𝑞(SU(2)) to Ω𝑞,hor(SU(2)) by setting 𝛼𝑧(𝑒±) = 𝑧±2𝑒± for all 𝑧 ∈ U(1). Then +(Ω𝑞,hor(SU(2)), d𝑃,hor) defines a horizontal calculus for the topological quantum principal +U(1)-bundle O𝑞(SU(2)) over O𝑞(CP1) with respect to the ∗-exterior algebra +(Ω𝑞(CP1), d) � +� +Ω𝑞,hor(SU(2))U(1), d𝑞,hor↾Ω𝑞,hor(SU(2))U(1) +� +on O𝑞(CP1), which, by Majid’s result, recovers the 2-dimensional calculus on O𝑞(CP1) first +constructed by Podleś [83]. + +36 +BRANIMIR ĆAĆIĆ +Let us now define the concrete category DCirchor(𝐵) of horizontally differentiable quantum +principal U(1)-bundles over 𝐵 and their isomorphisms as follows: +(1) an object (𝑃; Ω𝑃,hor, d𝑃,hor) consists of a topological quantum principal U(1)-bundle 𝑃 +over 𝐵 together with a horizontal calculus (Ω𝑃,hor, d𝑃,hor) on 𝑃; +(2) an arrow 𝑓 : (𝑃; Ω𝑃,hor, d𝑃,hor) → (𝑄; Ω𝑄,hor, d𝑄,hor) is an isomorphism of U(1)-∗-quasi- +dga, such that ˆ𝜄𝑄 ◦ 𝑓 = 𝑓 ◦ ˆ𝜄𝑃. +It is useful to observe that the forgetful functor DCirchor(𝐵) → Circ(𝐵) is faithful: an +arrow 𝑓 : (𝑃; Ω𝑃,hor, d𝑃,hor) → (𝑄; Ω𝑄,hor, d𝑄,hor) in DCirc(𝐵) is uniquely determined by the +corresponding arrow 𝑓 ↾𝑃: 𝑃 → 𝑄 in Circ(𝐵) precisely because Ω𝑃,hor is generated as an +algebra by 𝑃 and ˆ𝜄𝑃(d(𝐵)) ⊂ d𝑃,hor(𝑃). We can now make precise sense of associated line +bundles with connection in the nc setting. +Proposition 3.24 (cf. Ćaćić–Mesland [26, Appx B]). Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally +differentiable quantum principal U(1)-bundle over 𝐵. +(1) Observe that Ω𝑃,hor defines a 𝐵-bimodule with respect to 𝜄𝑃 : 𝐵 → 𝑃U(1). There exists a +unique U(1)-equivariant isomorphism of 𝐵-bimodules ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵, such that +∀𝑝 ∈ 𝑃, ∀𝛽 ∈ Ω𝐵, +ˆℓ−1 +𝑃 (𝑝 ⊗ 𝛽) = 𝑝ˆ𝜄𝑃(𝛽). +(3.14) +(2) Let 𝑘 ∈ Z be given. Define functions 𝜎𝑃;𝑘 : Ω𝐵 ⊗𝐵 L(𝑃)(𝑘) → L(𝑃)(𝑘) ⊗𝐵 Ω𝐵 and +∇𝑃;𝑘 : L(𝑃)(𝑘) → L(𝑃) ⊗𝐵 Ω1 +𝐵 by +∀𝛽 ∈ Ω𝐵, ∀𝑝 ∈ 𝑃𝑘, +𝜎𝑃;𝑘(𝛽 ⊗ 𝑝) � ˆℓ𝑃(ˆ𝜄𝑃(𝛽)𝑝), +(3.15) +∀𝑝 ∈ 𝑃𝑘, +∇𝑃;𝑘(𝑝) � ˆℓ𝑃 +�d𝑃,hor(𝑝)�, +(3.16) +respectively. Then (𝜎𝑃;𝑘, ∇𝑃;𝑘) defines a Hermitian bimodule connection on the Hermitian line +𝐵-bimodule L(𝑃)(𝑘). +Proof. Let us first show that ˆℓ𝑃 is well-defined; uniqueness and U(1)-equivariance will then +follow from the explicit form of ˆℓ−1 +𝑃 . Given 𝑘 ∈ Z, let (𝑒𝑖)𝑚 +𝑖=1 be a basis for L(𝑃)(𝑘), and +define ˆℓ𝑃;𝑘 : (Ω𝑃,hor)𝑘 → L(𝑃)(𝑘) ⊗𝐵 Ω𝐵 by ˆℓ𝑃;𝑘 � �𝜔 ↦→ �𝑚 +𝑖=1 𝑒𝑖 ⊗ ˆ𝜄−1 +𝑃 (𝑒∗ +𝑖 𝜔)�; that ˆℓ𝑃;𝑘 is an +isomorphism of 𝐵-bimodules with inverse given by (3.14) now follows from observing that +(𝑒𝑖)𝑚 +𝑖=1 satisfies 1 = 𝜄𝑃 +��𝑚 +𝑖=1(𝑒𝑖, 𝑒𝑖)� = �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 . We may now set ˆℓ𝑃 � � +𝑘∈Z ˆℓ𝑃;𝑘. +We now fix 𝑘 ∈ Z and show that (𝜎𝑃;𝑘, ∇𝑃;𝑘) defines a Hermitian bimodule connection on +the Hermitian line 𝐵-bimodule L(𝑃)(𝑘). Let (𝑒𝑖)𝑚 +𝑖=1 be a basis and let (𝜖𝑗)𝑛 +𝑗=1 be a strict cobasis +for L(𝑃)(𝑘). Recall that �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 and observe that �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 𝜄𝑃 +��𝑛 +𝑗=1(𝜖𝑗, 𝜖𝑗) +� += 1. On +the one hand, the fact that �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1 implies that 𝜎𝑃;𝑘 is indeed an isomorphism of graded +𝐵-bimodules with inverse 𝜎−1 +𝑃;𝑘 = +� +𝑝 ⊗ 𝛽 ↦→ �𝑛 +𝑗=1 ˆ𝜄−1 +𝑃 +� +𝑝𝛽𝜖∗ +𝑗 +� +⊗ 𝜖𝑗 +� +. On the other hand, the fact +that � +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 implies, that for all 𝛼, 𝛽 ∈ Ω𝐵 and 𝑝 ∈ 𝑃𝑘, +ˆℓ−1 +𝑃 ◦ 𝜎𝑃;𝑘(𝛼𝛽 ⊗ 𝑝) = ˆ𝜄𝑃(𝛼𝛽)𝑝 = ˆℓ−1 +𝑃 +� +𝜎𝑃;𝑘(𝛼 ⊗ 𝜎𝑃;𝑘(𝛽 ⊗ 𝑝) ⟨0⟩)𝜎𝑃;𝑘(𝛽 ⊗ 𝑝) ⟨1⟩ +� +, +which yields (2.26). Thus, 𝜎𝑃;𝑘 is a well-defined Hermitian generalised braiding; it remains to +show that ∇𝑃;𝑘 is a right Hermitian connection satisfying (2.27) with respect to 𝜎𝑃;𝑘. However, +we may again use the maps ˆℓ𝑃 and ˆ𝜄𝑃 together with the equality �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 to derive (2.24), +(2.25), and (2.27) from the Leibniz rule for d𝑃,hor. +□ +Proposition 3.25 (cf. Beggs–Majid [13, Prop. 5.56], Saldaña [72, §3]). The functor L of Propo- +sition 3.15 lifts with respect to the obvious forgetful functors DCirchor(𝐵) → Circ(𝐵) and +Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)) to the functor ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)) +defined as follows. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +37 +(1) Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)-bundle over 𝐵. +Define a homomorphism ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) : Z → DPic(𝐵) as follows: +(a) given 𝑘 ∈ Z, let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(𝑘) � (L(𝑃)(𝑘), 𝜎𝑃,𝑘, ∇𝑃,𝑘), where (𝜎𝑃;𝑘, ∇𝑃;𝑘) is +the Hermitian bimodule connection of Proposition 3.24; +(b) let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(0) be the unique lift of id𝑃0 � L(𝑃)(0); +(c) given 𝑚, 𝑛 ∈ Z, let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(2) +𝑚,𝑛 be the unique lift of L(𝑃)(2) +𝑚,𝑛. +(2) Given an isomorphism 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) of horizontally differen- +tiable quantum principal U(1)-bundles over 𝐵, let +ˆL(𝑓) : ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) ⇒ ˆL(𝑄, Ω𝑄,hor, d𝑄,hor) +be the unique lift of the 2-isomorphism L(𝑓) : L(𝑃) ⇒ L(𝑄). +Proof. First, let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)- +bundle over 𝐵. For notational simplicity, we set 𝐹 � L(𝑃) and denote our would-be +homomorphism ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) by ˆ𝐹. The functor ˆ𝐹 : Z → DPic(𝐵) is well defined by +Proposition 3.24; that the arrow 𝐹 (0) : 𝐹(0) → 𝐵 satisfies (2.33) follows from the fact that +ˆ𝜄𝑃 ◦ d = d𝑃,hor. Given 𝑚, 𝑛 ∈ Z, the arrow 𝐹 (2) +𝑚,𝑛 : 𝐹(𝑚) ⊗𝐵 𝐹(𝑛) → 𝐹(𝑚 + 𝑛) satisfies (2.33) by +applying the isomorphism ˆℓ−1 +𝑃 of Proposition 3.24 to both sides of the desired equality and +then applying the Leibniz rule for d𝑃,hor in Ω𝑃,hor; thus, the natural isomorphism ˆ𝐹 (2) is well +defined. Commutativity of the relevant commutative diagrams now follows from observing +that the forgetful functor DPic(𝐵) → Pic(𝐵) is faithful. +Now, let 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) be an isomorphism of horizontally +differentiable quantum principal U(1)-bundle over 𝐵. Again, for notational simplicity, set +𝑅 � L(𝑃), ˆ𝑅 � ˆL(𝑃, Ω𝑃,hor, d𝑃,hor), 𝑆 � L(𝑄), and ˆ𝑆 � ˆL(𝑄, Ω𝑄,hor, d𝑄,hor). Observe +that 𝑓 ⊗ idΩ𝐵 necessarily intertwines the isomorphisms ˆℓ𝑃 and ˆℓ𝑄 of Proposition 3.24, so that +for each 𝑘 ∈ Z, the arrow L(𝑓)𝑘 : 𝑅(𝑘) → 𝑆(𝑘) in Pic(𝐵) satisfies (2.33) precisely since +d𝑄,hor ◦ 𝑓 = 𝑓 ◦ d𝑃,hor; it follows that ˆL(𝑓) : ˆ𝑅 → ˆ𝑆 is well defined as a natural transformation. +Once more, commutativity of the relevant commutative diagrams now follows from observing +that the forgetful functor DPic(𝐵) → Pic(𝐵) is faithful. +□ +Example 3.26 (Landi–Reina–Zampini [65], Khalkhali–Landi–Van Suijlekom [61]). We continue +from Example 3.23; in particular, we now equip O𝑞(CP1) with Podleś’s 2-dimensional calculus +(Ω𝑞(CP1), d). The homomorphism E � L(O𝑞(SU(2))) lifts to the corresponding homo- +morphism ˆE : Z → DPic(O𝑞(CP2)) by setting ˆE � ˆL(O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor). In +particular, given 𝑘 ∈ Z, it follows that ˆE(𝑘) = (E(𝑘), 𝜎𝑘, ∇𝑘), where ∇𝑘 and 𝜎𝑘 respectively +recover the canonical connection [65, §4.1] and ‘twisted flip’ [61, §§3.5-6] on E(𝑘). +At last, we show that the functor ˆLis, indeed, an equivalence of categories. +Proposition 3.27. Let ˆ𝐹 : Z → DPic(𝐵) be a homomorphism with image 𝐹 : Z → Pic(𝐵) +under the forgetful functor Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)), and let 𝑃 � Σ(𝐹) be the +topological quantum principal U(1)-bundle over 𝐵 induced by 𝐹. The following defines a horizontal +calculus (Ω𝑃,hor, d𝑃,hor) on 𝑃: +(1) define the graded ∗-algebra Ω𝑃,hor by equipping the complex vector space 𝑃 ⊗𝐵 Ω𝐵 with the +multiplication and ∗-operation defined, respectively, by +∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝑝 ∈ Z, ∀𝑘 ∈ Z, ∀𝑞 ∈ 𝐹(𝑛), +(𝑝 ⊗ 𝛼)(𝑞 ⊗ 𝛽) � 𝑝𝜎𝐹(𝑘) (𝛼 ⊗ 𝑞)𝛽, +(3.17) +∀𝛼 ∈ Ω𝐵, ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), +(𝑝 ⊗ 𝛼)∗ � 𝜎𝐹(−𝑘) (𝛼∗ ⊗ 𝑝∗), +(3.18) +and with the grading induced by the grading on Ω𝐵; + +38 +BRANIMIR ĆAĆIĆ +(2) define d𝑃,hor : Ω𝑃,hor → Ω𝑃,hor by +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), ∀𝛽 ∈ Ω𝐵, +d𝑃,hor(𝑝 ⊗ 𝛽) � ∇𝐹(𝑘) (𝑝) ⊗ 𝛽 + 𝑝 ⊗ d𝛽; +(3.19) +(3) extend the canonical U(1)-action 𝛼 on 𝑃 pointwise to ˆ𝛼 : U(1) → Aut(Ω𝑃,hor) by +∀𝑧 ∈ U(1), ∀𝑝 ∈ 𝑃, ∀𝛽 ∈ Ω𝐵, +ˆ𝛼𝑧(𝑝 ⊗ 𝛽) � 𝛼𝑧(𝑝) ⊗ 𝛽; +(3.20) +(4) let ˆ𝜄𝑃 : (Ω𝐵, d) → (ΩU(1) +𝑃,hor, d𝑃,hor↾ΩU(1) +𝑃,hor) be induced by (𝐹 (0) ⊗ id) ◦ 𝜆Ω𝐵. +Proof. The construction of Ω𝑃,hor, ˆ𝛼, and ˆ𝜄𝑃 from ˆ𝐹 follows, mutatis mutandis, from the explicit +construction of 𝑃 � Σ(𝐹), 𝛼, and 𝜄𝑃 from 𝐹 in the proof of Theorem 3.17. Indeed, recall that +ˆ𝐹 canonically defines a bar functor by Theorem 3.7. Hence, each definitional commutative +diagram satisfied by the bar functor ˆ𝐹 yields a corresponding commutative diagram satis- +fied by the family of Hermitian generalised braidings (𝜎𝐹(𝑘))𝑘∈Z, which, in turn, yields the +corresponding properties of Ω𝑃,hor. +Let us now turn to d𝑃,hor, which is U(1)-equivariant and complex-linear by construction; it +also clearly satisfies d𝑃,hor ◦ ˆ𝜄𝑃 = ˆ𝜄𝑃 ◦d. Given 𝑚, 𝑛 ∈ Z, the fact that 𝐹 (2) +𝑚,𝑛 satisfies (2.33) implies +that d𝑃,hor satisfies the Leibniz rule on (Ω𝑃,hor)𝑚 · (Ω𝑃,hor)𝑛 = (Ω𝑃,hor)𝑚+𝑛. Finally, given +𝑘 ∈ Z, the fact that 𝐹 (−1) +𝑘 +satisfies (2.33) implies that d𝑃,hor is ∗-preserving on the subspace +∗�(Ω𝑃,hor)𝑘 +� = (Ω𝑃,hor)−𝑘 of Ω𝑝,hor. +□ +Theorem 3.28. The functor Σ : Hom(Z, Pic(𝐵)) → Circ(𝐵) of Theorem 3.17 lifts with respect +to the forgetful functors DCirchor(𝐵) → Circ(𝐵) and Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)) +to the weak inverse ˆΣ : Hom(Z, DPic(𝐵)) → DCirchor(𝐵) of ˆL defined as follows. +(1) Given a homomorphism ˆ𝐹 : Z → DPic(𝐵) descending to 𝐹 : Z → Pic(𝐵), let +ˆΣ( ˆ𝐹) � (Σ(𝐹), ΩΣ(𝐹),hor, dΣ(𝐹),hor), +where (ΩΣ(𝐹),hor, dΣ(𝐹),hor) is the horizontal calculus of Proposition 3.27. +(2) Given a 2-isomorphism ˆ𝜂 : ˆ𝑅 → ˆ𝑆 between homomorphisms ˆ𝑅, ˆ𝑆 : Z → DPic(𝐵) that +descends to a 2-isomorphism 𝜂 : 𝑅 → 𝑆 between homomorphisms 𝑅, 𝑆 : Z → Pic(𝐵), let +ˆΣ( ˆ𝜂) : ˆΣ( ˆ𝑅) → ˆΣ( ˆ𝑆) be the unique lift of Σ(𝜂) : Σ(𝑅) → Σ(𝑆). +Hence, in particular, the category DCirchor(𝐵) is essentially small. +Proof. We have seen that Σ is well defined on objects, so let us check that it is well defined +on arrows. Let ˆ𝜂 : ˆ𝑅 → ˆ𝑆 be an arrow in Hom(Z, DPic(𝐵)) descending to 𝜂 : 𝑅 → 𝑆 in +Hom(Z, Pic(𝐵)), so that ˆ𝜂 and 𝜂 define bar natural transformations by Theorem 3.7. We can +extend Σ(𝜂) : Σ(𝑅) → Σ(𝑆) to a U(1)-equivariant isomorphism of graded 𝐵-bimodules +ˆΣ( ˆ𝜂) : ΩΣ(𝑅),hor → ΩΣ(𝑆),hor by setting ˆΣ � Σ(𝜂) ⊗ idΩ𝐵. Coherence of 𝜂 with respect to 𝑅(2) +and 𝑆(2) implies that ˆΣ( ˆ𝜂) is multiplicative, that 𝜂1 intertwines 𝑅(0) and 𝑆(0) implies that ˆΣ( ˆ𝜂) is +unital, and the fact that ˆ𝜂 is a bar functor implies that ˆΣ(𝜂) is ∗-preserving. Finally, given 𝑘 ∈ Z, +the fact that 𝜂(𝑘) satisfies (2.33) implies that ˆΣ(𝜂) satisfies ˆΣ(𝜂) ◦ dΣ(𝑅),hor = dΣ(𝑆),hor ◦ ˆΣ(𝜂) +on (ΩΣ(𝑅),hor)𝑘. The rest now follows from Theorem 3.17, mutatis mutandis. +□ +Remark 3.29. Building on a proposal of Ðurđević [40, §4.4], Saldaña proves analogues of +Proposition 3.27 [72, Thm 3.11] and Theorem 3.28 [72, Thm 3.12] for quantum principal bundles +with structure quantum group given by a Hopf ∗-algebra in terms of certain heavily structured +functors that resemble bar functors à la Beggs–Majid [12]. By contrast, in the special case of +quantum principal U(1)-bundles, Theorem 3.7 allows us to use monoidal functors simpliciter. +Indeed, after suitable generalisation, the same will still be true in the somewhat more general +case where the structure quantum group is a group ring. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +39 +By combining Theorem 3.28 with Corollary 2.9, we obtain the promised generalisation of +Abadie–Eilers–Exel’s generalised crossed product construction to the setting of nc differential +geometry in the absence of any further constraints. +Corollary 3.30. The functor 𝜖1 ◦ ˆL : DCirchor(𝐵) → DPic(𝐵) is an equivalence. +Definition 3.31. The horizontal crossed product of (𝐵; Ω𝐵, d) by a Hermitian line 𝐵-bimodule +with connection (𝐸, 𝜎𝐸, ∇𝐸) is the essentially unique horizontally differentiable quantum +principal U(1)-bundle (𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z over (𝐵; Ω𝐵, d), such that +ˆL +� +(𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z +� +(1) � (𝐸, 𝜎𝐸, ∇𝐸). +One may justify this terminology as follows. Let (𝜔, 𝜙) be an extended diffeomorphism +of 𝐵, so that 𝐵 ⋊alg +𝜙 Z admits the horizontal calculus (Ω𝐵 ⋊alg +𝜙 Z, d(𝜔,𝜙)), where the graded +∗-algebra Ω𝐵 ⋊𝜙 Z is obtained from Ω𝐵 by adjoining a unitary 𝑈 ∈ (Ω𝐵 ⋊𝜙 Z)0 that satisfies +𝑈𝜙𝛽𝑈−1 +𝜙 += 𝜙(𝛽) for all 𝛽 ∈ Ω𝐵, the ∗-derivation d(𝜔,𝜙) is uniquely determined by requiring +that d(𝜔,𝜙)↾Ω𝐵� d𝐵 and d(𝜔,𝜙) (𝑈𝜙) � i𝜔𝑈𝜙, and the U(1)-action ˆ𝛼 on Ω𝐵 ⋊alg +𝜙 Z is uniquely +determined by setting ˆ𝛼𝑧↾Ω𝐵= idΩ𝐵 and 𝛼𝑧(𝑈𝜙) � 𝑧𝑈𝜙 for all 𝑧 ∈ U(1). Since +(𝑏𝜙 ↦→ 𝑈𝜙−1(𝑏)) : ˆ𝜏(𝜔, 𝜙) → ˆL(𝐵 ⋊alg +𝜙 Z; Ω𝐵 ⋊alg +𝜙 Z, d(𝜔,𝜙))(1) +is an isomorphism of Hermitian line 𝐵-bimodules with connection, we may therefore take +(𝐵; Ω𝐵, d) ⋊hor +ˆ𝜏(𝜔,𝜙) Z � (𝐵 ⋊alg +𝜙 Z; Ω𝐵 ⋊alg +𝜙 Z, d(𝜔,𝜙)). +We conclude this subsection by discussing curvature. In general, the curvature of a ∗-quasi- +dga (Ω, d) is the map d2, which vanishes for a ∗-exterior algebra. Thus, the curvature (in this +sense) of a horizontally differentiable quantum principal U(1)-bundle (𝑃, Ω𝑃,hor, d𝑃,hor) over +𝐵 is the map d2 +𝑃,hor, which is U(1)-equivariant ∗-derivation that vanishes on Ω𝐵 and hence, +in particular, is left and right Ω𝐵-linear. Passing this notion of curvature through the lens of +Proposition 3.25 and Theorem 3.28 yields the following more refined definition. +Proposition-Definition 3.32 (cf. Ðurđević [38, Lemma 2.2]). Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a hori- +zontally differentiable quantum principal U(1)-bundle over 𝐵. +(1) Its Fröhlich automorphism is the unique U(1)-equivariant automorphism ˆΦ𝑃 of the U(1)- +∗-quasi-dga of finite type (Z(Ω𝐵), d↾Z(Ω𝐵)), such that +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, ∀𝛽 ∈ Z(Ω𝐵), +ˆ𝜄𝑃 +� +ˆΦ𝑘 +𝑃(𝛽) +� +𝑝 = 𝑝ˆ𝜄𝑃(𝛽). +(3.21) +(2) Its curvature 1-cocycle is the unique group 1-cocycle F𝑃 : Z → S(𝐵) for the right Z-action +generated by ˆΦ−1 +𝑃 , such that +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, +d2 +𝑃,hor(𝑝) = 𝑝 · ˆ𝜄𝑃(iF𝑃(𝑘)). +(3.22) +Hence, its curvature data is the pair (Φ𝑃, F𝑃). +Proof. By Proposition 3.25 together with Proposition-Definition 2.38, we can and must take +ˆΦ𝑃 � Φ +� ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) +� +(1), +F𝑃 � F +� ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) +� +. +□ +Suppose that (𝑃, Ω𝑃,hor, d𝑃,hor) is a horizontally differentiable quantum principal U(1)- +bundle over 𝐵 with curvature data (Φ𝑃, F𝑃). On the one hand, by Theorem 3.28, every homo- +morphism ˆ𝐹 : Z → DPic(𝐵) that is 2-isomorphic to ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) satisfies +ˆΦ ◦ 𝜋0( ˆ𝐹)(1) = ˆΦ𝑃, +F ◦ 𝜋0( ˆ𝐹(1)) = F𝑃. + +40 +BRANIMIR ĆAĆIĆ +On the other hand, by Corollary 3.30, every Hermitian line 𝐵-bimodule (𝐸, 𝜎𝐸, ∇𝐸) that is +isomorphic to ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(1) satisfies +ˆΦ[𝐸,∇𝐸] = ˆΦ𝑃, +F[𝐸,∇𝐸] = F𝑃(1); +in other words, for every Hermitian line 𝐵-bimodule (𝐸, 𝜎𝐸, ∇𝐸), the resulting horizontal +crossed product (𝐵, Ω𝐵, d) ⋊(𝐸,𝜎𝐸,∇𝐸) Z has curvature data (Φ[𝐸,∇𝐸], F[𝐸,∇𝐸]). +Example 3.33 (Landi–Reina–Zampini [65, Prop. 4.2]). We continue from Example 3.26. Recall +(O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) from Example 3.26; let (ΦO𝑞(SU(2)), FO𝑞(SU(2))) be its curva- +ture data. Using the pbw basis for O𝑞(SU(2)), one shows that Z(Ω𝑞(CP1)) = C[i𝑒+𝑒−]. Since +the generators 𝑎, 𝑐 ∈ O𝑞(SU(2))1 satisfy 𝑎𝑎∗ + (𝑞𝑐)(𝑞𝑐)∗ = 1 and 𝑎∗𝑎 + 𝑐∗𝑐 = 1, we may use +them to compute +ˆΦO𝑞(SU(2)) (i𝑒+𝑒−) = 𝑞2i𝑒+𝑒−, +FO𝑞(SU(2)) (1) = 𝑞−2i𝑒+𝑒−. +(3.23) +3.4. Reconstruction of total calculi. At last, we leverage structural results of Ðurđević [39] +and Beggs–Majid [14] to obtain the promised nc generalisation of the classical correspondence +between Hermitian line bundles with unitary connection and principal U(1)-bundles with +principal connection. Once more, let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior algebra +(Ω𝐵, d𝐵), which we view as a fixed nc base manifold. In what follows, given 𝑞 ∈ (0, ∞), +we define the corresponding 𝑞-integers by setting [𝑘]𝑞 � 1−𝑞𝑘 +1−𝑞 for 𝑘 ∈ Z when 𝑞 ≠ 1 and +[𝑘]𝑞 � 𝑘 for 𝑘 ∈ Z when 𝑞 = 1. +We begin by noting that U(1) does not always appear in nc differential geometry with its +usual smooth structure as a Lie group. Instead, we must allow for all possible 1-dimensional +bi-invariant ∗-exterior algebras on the unital pre-𝐶∗-algebra O(U(1)) of trigonometric polyno- +mials. These, in turn, are exhausted up to isomorphism by the family ((Ω𝜅(U(1)), d𝜅))𝜅∈(0,∞) +of ∗-exterior algebras on O(U(1)) whose construction is conveniently generalised as follows. +Definition 3.34. Let 𝜅 ∈ (0, ∞). We define 𝜅-deformed Chevalley–Eilenberg extension to be +the faithful functor CE𝜅 : QDGAU(1) → QDGAU(1) constructed as follows. +(1) Given a U(1)-∗-quasi-dga of finite type (𝑃; Ω, d), let +CE𝜅(𝑃; Ω, d) � (𝑃; CE𝜅(Ω), CE𝜅(d)) , +where CE𝜅(Ω) is the graded ∗-algebra obtained from Ω by adjoining a self-adjoint element +𝑒𝜅 of degree 1 satisfying the relations 𝑒2 +𝜅 = 0 and +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 +𝑘, +𝑒𝜅𝜔 = (−1)𝑛𝜅−𝑘𝜔𝑒𝜅, +(3.24) +where CE𝜅(d) is defined by setting CE𝜅(d)(𝑒𝜅) � 0 and +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 +𝑘, +CE𝜅(d)(𝜔) � (−1)𝑛𝜅−𝑘2𝜋i[𝑘]𝜅𝜔𝑒𝜅 + d𝜔. +(3.25) +and where the U(1)-action on CE𝜅(Ω) is the unique extension of the U(1)-action on Ω +leaving 𝑒𝜅 invariant. +(2) Given a morphism 𝑓 : (𝑃, Ω𝑃, d𝑃) → (𝑄, Ω𝑄, d𝑄) of U(1)-∗-quasi-dga, let +CE𝜅(𝑓) : CE𝜅(𝑃, Ω𝑃, d𝑃) → CE𝜅(𝑄, Ω𝑄, d𝑄) +be the unique extension of 𝑓 : Ω𝑃 → Ω𝑄 satisfying CE𝜅(𝑓)(𝑒𝜅) = 𝑒𝜅. +Thus, given 𝜅 > 0, the ∗-exterior algebra (Ω𝜅(U(1)), d𝜅) � (CE𝜅(O(U(1))), CE𝜅(0)) on +O(U(1)) is the essentially unique ∗-exterior algebra on O(U(1)) of dimension 1 that satisfies +the relation d𝜅(𝑧) · 𝑧 = 𝜅𝑧 · d𝜅(𝑧), where d𝜅(𝑧) = 2𝜋i𝑒𝜅 · 𝑧. Note that 𝜅 = 1 recovers the usual +de Rham calculus on U(1) as a Lie group. In general, differentiability of a U(1)-action with +respect to the ∗-exterior algebra (Ω𝜅(U(1)), d𝜅) can now be characterised as follows. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +41 +Definition 3.35 (cf. Ðurđević [39, §3], Beggs–Brzeziński [10, §7]). Let 𝑃 be a U(1)-pre-𝐶∗- +algebra of finite type and let (Ω, d) be a U(1)-∗-exterior algebra over 𝑃. We say that (Ω, d) +is 𝜅-vertical whenever there exists a (necessarily unique) lift of id𝑃 to a morphism of U(1)- +∗-quasi-dga ver : (𝑃, Ω, d) → CE𝜅(𝑃, Ω, d), the vertical coevaluation on (Ω, d). In this case, +we define horizontal form in Ω to be an element of the U(1)-invariant graded ∗-subalgebra +Ωhor � {𝜔 ∈ Ω | ver(𝜔) = 𝜔} of Ω, and a basic form to be an element of the U(1)-invariant +and d-invariant graded ∗-subalgebra Ωbas � (Ωhor)U(1) of Ω. +At last, given 𝜅 > 0, we can make precise sense of nc differentiable principal U(1)-bundles, +where U(1) carries the bi-invariant ∗-exterior algebra (Ω𝜅(U(1)), d𝜅)). +Definition 3.36 (Brzeziński–Majid [22, §4], Hajac [52], Ðurđević [39, §3], Beggs–Brzeziński [10, +§7], Beggs–Majid [14, §5.5]; cf. Ćaćić [24]). Let 𝜅 ∈ (0, ∞). A 𝜅-differentiable quantum principal +U(1)-bundle over 𝐵 is a triple (𝑃, Ω𝑃, d𝑃), where 𝑃 is a topological quantum principal U(1)- +bundle over 𝐵 and (Ω𝑃, d𝑃) is a 𝜅-vertical U(1)-∗-exterior algebra over 𝑃 together with an +isomorphism ˆ𝜄𝑃 : (Ω𝐵, d𝐵) → (Ω𝑃,bas, d𝑃↾Ω𝑃,bas) of ∗-quasi-dga extending 𝜄𝑃, such that +Ω𝑃,hor = 𝑃 · Ω𝑃,bas · 𝑃. +Example 3.37. Continuing from Example 3.12, let 𝜕 +𝜕𝑡 be the fundamental vector field of the +U(1)-action on 𝑋, and let Ωalg(𝑋) � �alg +𝑘∈Z{𝜔 ∈ Ω(𝑋) | ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧−𝑘𝜔}, +which we equip with the U(1)-action 𝑧 ↦→ (𝜎𝑧−1)∗ and the de Rham exterior derivative. +Then (𝐶∞ +alg(𝑋), Ωalg(𝑋), d) defines a 1-differentiable quantum principal U(1)-bundle over +(𝐶∞(𝑌), Ω(𝑌), d) with respect to 𝜋∗ : Ω(𝑌) → Ωalg(𝑋)U(1). Note, in particular, that the +vertical coevaluation reduces to the map Ωalg(𝑋) → Ω(U(1))U(1) �⊗C Ωalg(𝑋) that dualises +contraction of differential forms with the fundamental vector field 𝜕 +𝜕𝑡. +The following necessary and sufficient conditions are of both theoretical and practical +importance. Note that they involve the strong connection condition first identified by Hajac [52]. +Proposition 3.38 (Beggs–Majid [14, Cor. 5.53 & Lemma 5.60]). Let 𝜅 ∈ (0, ∞), let 𝑃 be a +topological quantum principal U(1)-bundle over 𝐵, let (Ω𝑃, d𝑃) be a 𝜅-vertical U(1)-∗-exterior +algebra over 𝑃, and let ˆ𝜄𝑃 : (Ω𝐵, d𝐵) → (Ω𝑃,bas, d𝑃↾Ω𝑃,bas) be an injective morphism of ∗-quasi- +dga extending 𝜄𝑃. Then (𝑃, Ω𝑃, d𝑃) defines a 𝜅-differentiable quantum principal U(1)-bundle +over 𝐵 with respect to ˆ𝜄𝑃 if and only if +Ω𝑃,hor = 𝑃 · ˆ𝜄𝑃(Ω𝐵). +(3.26) +Moreover, if, for each 𝑛 ∈ N0, the left 𝐵-module Ω𝑛 +𝐵 is flat, then (𝑃, Ω𝑃, d𝑃) defines a 𝜅- +differentiable quantum principal U(1)-bundle over 𝐵 with respect to ˆ𝜄𝑃 if and only if +Ω1 +𝑃,hor = 𝑃 · ˆ𝜄𝑃(Ω1 +𝐵). +(3.27) +We now recall the notions of principal Ehresmann connection and connection 1-form +appropriate to our nc setting. +Definition 3.39 (Brzeziński–Majid [22, §4.2 & Appx. a], Hajac [52, §4], Ðurđević [39, §4], +Beggs–Majid [14, §5.5]). Let 𝜅 ∈ (0, ∞), and let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum +principal U(1)-bundle over 𝐵 with respect to (Ω𝐵, d). +(1) A connection on (𝑃, Ω𝑃, d𝑃) is a surjective U(1)-equivariant grading- and ∗-preserving +algebra homomorphism Π : Ω𝑃 → Ω𝑃,hor, such that Π2 = Π and +∀𝜔 ∈ Ω1 +𝑃, +(id −Π)(𝜔)2 = 0. +(3.28) + +42 +BRANIMIR ĆAĆIĆ +(2) A connection 1-form on (𝑃, Ω𝑃, d𝑃) is U(1)-invariant self-adjoint 𝜗 ∈ Ω1 +𝑃, such that +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑛 +𝑃)𝑘, +𝜗𝜔 = (−1)𝑛𝜅−𝑘𝜔𝜗, +(3.29) +ver(𝜗) = 𝑒𝜅 + 𝜗. +(3.30) +Remark 3.40. In the terminology of Brzeziński–Majid [22, §4.2 & Appx. a], Hajac [52, §4], and +Beggs–Majid [14, §5.5], the restriction of a connection Π to 1-forms is a ∗-preserving strong +bimodule connection. In the terminology of Ðurđević [39, §4], the datum of a connection 1-form +is equivalent to the datum of a multiplicative regular connection. +The bijection between principal connections and connection 1-forms persists in the nc +context. +Proposition 3.41 (cf. Brzeziński–Majid [22, Propp. 4.4 & 5.10], Ðurđević [39, Proof of Thm +4.12]). Let 𝜅 ∈ (0, ∞); let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum principal U(1)-bundle over +𝐵. For every connection Π on (𝑃, Ω𝑃, d𝑃), there exists a unique connection 1-form 𝜗, such that +∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, +(id −Π) ◦ d𝑃(𝑝) = 2𝜋i[𝑘]𝜅𝜅−𝑘𝑝𝜗. +(3.31) +Conversely, for every connection 1-form 𝜗 on (𝑃, Ω𝑃, d𝑃), there exists a unique connection Π that +satisfies (3.31). +Proof of Proposition 3.41. We begin with preliminary observations. By a lemma of Beggs–Majid +[14, Lemma 5.59], the vertical coevaluation of (𝑃, Ω𝑃, d𝑃) satisfies +∀𝑛 ∈ N, +(ver − id)(Ω𝑛 +𝑃) ⊆ Ω𝑛−1 +𝑃,hor · 𝑒𝜅. +(3.32) +Together with (3.26), this yields a short exact sequence +0 → Ω𝑃,hor → Ω𝑃 +ver−id +−−−−→ Ω𝑃,hor · 𝑒𝜅 → 0 +(3.33) +of ∗-closed U(1)-invariant Ω𝑃,hor-sub-bimodules of Ω𝑃 and U(1)-equivariant left and right +Ω𝑃,hor-linear maps preserving both the ambient ∗-operation and N0-grading on CE𝜅(Ω𝑃). +First, suppose that Π is a connection on 𝑃, Ω𝑃, d𝑃). Then Π is a U(1)-equivariant left +and right Ω𝑃,hor-linear left splitting of (3.33) preserving both the ambient ∗-operation and +N0-grading in CE𝜅(Ω𝑃), so that (ver − id) ↾ran(id −Π): ran(id −Π) → Ω𝑃,hor · 𝑒𝜅 is a U(1)- +equivariant isomorphism of Ω𝑃,hor-bimodules preserving both the ambient ∗-operation and +the ambient N0-grading in CE𝜅(Ω𝑃). Hence, let 𝜗 � �(ver − id)↾ran(id −Π) +�−1 (𝑒𝜅), which is +thus a U(1)-invariant self-adjoint element of Ω1 +𝑃 satisfying (3.30) by construction and (3.31) +by (3.25) applied to d𝑃(𝑃). It remains to show that 𝜗 satisfies (3.29). Since 𝜗2 = 0 by (3.28), it +suffices to show that (3.29) holds for horizontal 𝜔, but this now follows from the fact that +(ver − id) ↾ran(id −Π) is an isomorphism of Ω𝑃,hor-bimodules. Finally, let us show that 𝜗 is +uniquely determined by Π. Let (𝜖𝑗)𝑛 +𝑗=1 be a finite family in 𝑃1 satisfying �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1. Then +𝜗 = +∑︁𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗𝜗 = +𝜅 +2𝜋i +∑︁𝑛 +𝑗=1 𝜖∗ +𝑗 (2𝜋i[1]𝜅𝜅−1𝜖𝑗𝜗) = (id −Π) +� +𝜅 +∑︁𝑛 +𝑗=1 𝜖∗ +𝑗 d𝑃(𝜖𝑗) +� +. +Now, suppose that 𝜗 is a connection 1-form on (𝑃, Ω𝑃, d𝑃). On the one hand, by construc- +tion of CE𝜅(Ω𝑃), the element 𝑒𝜅 freely generates the left Ω𝑃,hor-submodule Ω𝑃 ·𝑒𝜅 ⊆ CE𝜅(Ω𝑃). +On the other hand, by (3.29) and (3.30), the element 𝜗 satisfies the same relations in Ω𝑃 that 𝑒𝜅 +satisfies in CE𝜅(Ω𝑃). Hence, the identity map idΩ𝑃 extends to a surjective U(1)-equivariant +algebra homomorphism 𝜓𝜗 : CE𝜅(Ω𝑃) → Ω𝑃 intertwining ∗-operations and N0-gradings by +setting 𝜓𝜗(𝑒𝜅) � 𝜗. We now show that Π � idΩ𝑃 −𝜓𝜗 ◦ (ver − idΩ𝑃) defines a connection +satisfying (3.31) with respect to 𝜗. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +43 +First, by construction, the map Π is U(1)-equivariant and unital, is left and right Ω𝑃,hor- +linear, and intertwines ∗-operations and N0-gradings; moreover, by definition of Ω𝑃,hor, it +follows that Π↾Ω𝑃,hor= idΩ𝑃,hor. Next, by (3.32) together with (3.30), it follows that +(ver − id) ◦ Π = (ver − id) − (ver − id) ◦ 𝜓𝜗 ◦ (ver − id) = 0, +so that ran Π ⊂ Ω𝑃,hor; from this, it follows that Π2 = Π, and hence, in particular, that +ran(id −Π) = Ω𝑃,ℎ𝑜𝑟 · 𝜗, so that (3.28) follows since 𝜗2 = 0. Multiplicativity now follows from +left and right Ω𝑃,hor-linearity of Π together with the decomposition Ω𝑃 = Ω𝑃,hor ⊕ Ω𝑃,hor · 𝜗 +of Ω𝑃,hor-bimodules. Finally, that Π is uniquely determined by 𝜗 follows from multiplicativity +of Π together with the fact that 𝑃 and d𝑃(𝑃) generate Ω𝑃. +□ +Hence, just as in the classical case, one may now use the connection 1-form to define the +curvature 2-form of a principal connection. +Definition 3.42. Let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum principal U(1)-bundle over +𝐵. Let Π be a connection on (𝑃, Ω𝑃, 𝜄𝑃) with connection 1-form 𝜗. The curvature of Π is the +closed self-adjoint 2-form FΠ � −ˆ𝜄−1 +𝑃 (d𝑃(𝜗)) ∈ Z(Ω𝐵)2. +Example 3.43. We continue from Example 3.37. Let 𝐻∗𝑋 → 𝑋 be the horizontal cotangent +bundle of the total space 𝑋, whose fibre at 𝑥 ∈ 𝑋 is the annihilator of 𝜕 +𝜕𝑡 at 𝑥, so that +Ωalg(𝑋)hor = +� +𝑘∈Z +� +𝜔 ∈ Γ +�� +𝐻∗𝑋 ⊗ C +� ��� ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧−𝑘𝜔 +� +. +Hence, let Π be a principal connection on 𝜋 : 𝑋 → 𝑌, which we view as a U(1)-equivariant +real vector bundle endomorphism Π : 𝑇∗𝑋 → 𝑇∗𝑋 satisfying Π2 = Π and ran Π = 𝐻∗𝑋. Then +Π induces a connection on (𝐶∞ +alg(𝑋), Ωalg(𝑋), d), whose connection 1-form and curvature +2-form respectively recover the usual connection 1-form and curvature 2-form of Π. +We now leverage structural results of Ðurđević [39] to obtain the promised correspondence +between nc Hermitian line bundles with connection and nc principal U(1)-bundles with +principal connection. +Let 𝜅 ∈ (0, ∞). We may define the concrete category Gauge𝜅(𝐵) of 𝜅-differentiable quantum +principal U(1)-bundle with connection over 𝐵 and their isomorphisms as follows: +(1) an object is a triple (𝑃, Ω𝑃, d𝑃; Π) consisting of a 𝜅-differentiable quantum principal U(1)- +bundle (𝑃, Ω𝑃, d𝑃) over 𝐵 and a connection Π𝑃 on (𝑃, Ω𝑃, d𝑃); +(2) an arrow 𝑓 : (𝑃, Ω𝑃, d𝑃; Π𝑃) → (𝑄, Ω𝑄, d𝑄; Π𝑄) is an isomorphism of curved U(1)-∗-dga +𝑓 : (𝑃, Ω𝑃, d𝑃) → (𝑄, Ω𝑄, d𝑄) that satisfies 𝑓 ◦ ˆ𝜄𝑃 = ˆ𝜄𝑄 and 𝑓 ◦ Π𝑃 = Π𝑄 ◦ 𝑓. +Hence, we may define a functor Hor𝜅 : Gauge𝜅(𝐵) → DCirchor(𝐵) as follows: +(1) given an object (𝑃, Ω, d; Π), let Hor𝜅(𝑃, Ω, d; Π) � �𝑃, Ωhor, Π ◦ d↾Ωhor +�; +(2) given an arrow 𝑓 : (𝑃, Ω𝑃, d𝑃; Π𝑃) → (𝑄, Ω𝑄, d𝑄; Π𝑄), let +Hor𝜅(𝑓) : Hor𝜅(𝑃, Ω𝑃, d𝑃, Π𝑃) → Hor𝜅(𝑄, Ω𝑄, d𝑄, Π𝑄) +be given by the map 𝑓↾Ω𝑃,hor: Ω𝑃,hor → Ω𝑄,hor. +Thus, the functor Hor𝜅 takes a 𝜅-differentiable quantum principal U(1)-bundle with connec- +tion and extracts the horizontal calculus induced by the choice of connection. A straight- +forward calculation shows that the essential range of this functor satisfies a simple algebraic +constraint: the curvature 2-form solves the eigenvector equation for the Fröhlich automor- +phism with respect to 𝜅. + +44 +BRANIMIR ĆAĆIĆ +Proposition 3.44 (cf. Ðurđević [39, §6.6]). Let 𝜅 ∈ (0, ∞), and let (𝑃, Ω𝑃, d𝑃; Π) be a 𝜅- +differentiable quantum principal U(1)-bundle with connection over 𝐵. Let FΠ be the curvature +2-form of Π, and let ( ˆΦ𝑃,Π, F𝑃,Π) be the curvature data of Hor𝜅(𝑃, Ω𝑃, d𝑃; Π), so that +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝛽 ∈ (Ω𝑛 +𝑃,hor)𝑘, +(Π ◦ d𝑃)2(𝛽) = 𝛽 · ˆ𝜄𝑃 +�i F𝑃,Π(𝑘)�. +Then F𝑃,Π : Z → S(𝐵) is given by F𝑃,Π = �𝑘 ↦→ 2𝜋[𝑘]𝜅𝜅−𝑘FΠ +�, so that, in particular, +ˆΦ𝑃,Π +�F𝑃,Π(1)� = 𝜅F𝑃,Π(1). +Definition 3.45. Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal +U(1)-bundle over 𝐵 with curvature data ( ˆΦ𝑃, F𝑃). We say that (𝑃, Ω𝑃,hor, d𝑃,hor) is flat when- +ever F𝑃 = 0. When F𝑃(1) is an eigenvector of ˆΦ𝑃, the vertical deformation parameter 𝜅𝑃 ∈ R× +of (𝑃, Ω𝑃,hor, d𝑃,hor) is defined to be the corresponding eigenvalue of ˆΦ𝑃. +Remarkably, this single algebraic constraint suffices to characterize the essential range of +the functor Hor𝜅, which therefore yields an equivalence of categories. +Theorem 3.46 (Ðurđević [39, Thm 4.12 & §6.5]). Let 𝜅 ∈ (0, ∞), and let DCirchor,𝜅(𝐵) denote +the strictly full subcategory of DCirchor(𝐵) whose objects are either flat or have vertical deformation +parameter 𝜅. Then Hor𝜅 defines an equivalence Gauge𝜅(𝐵) → DCirchor,𝜅(𝐵) with weak inverse +Tot𝜅 : DCirchor,𝜅(𝐵) → Gauge𝜅(𝐵) defined as follows. +(1) Given a horizontally differentiable quantum principal U(1)-bundle (𝑃, Ω𝑃,hor, d𝑃,hor) over 𝐵 +with curvature 1-cocycle FΠ, let +Tot𝜅(𝑃, Ω𝑃,hor, d𝑃,hor) � (𝑃, CE𝜅(Ω𝑃,hor), CE𝜅(d𝑃,hor) + iΠ, Π𝜅), +where iΠ : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑃,hor) is the complex-linear map defined by +∀𝜔1, 𝜔2 ∈ Ω𝑃,hor, +iΠ(𝜔1 + 𝜔2𝑒𝜅) � − 𝜅 +2𝜋 𝜔2FΠ(1), +and where Π𝜅 : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑃,hor) is the unique algebra homomorphism satisfying +Π𝜅↾Ω𝑃,hor= idΩ𝑃,hor and Π𝜅(𝑒𝜅) = 0. +(2) Given an isomorphism 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) of horizontally differen- +tiable quantum principal U(1)-bundles over 𝐵, let +Tot𝜅(𝑓) : Tot𝜅(𝑃, Ω𝑃,hor, d𝑃,hor) → Tot𝜅(𝑄, Ω𝑄,hor, d𝑄,hor) +be given by the map CE𝜅(𝑓) : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑄,hor). +In particular, a canonical natural isomorphism Ð : idGauge𝜅 (𝐵) ⇒ Tot𝜅 ◦ Hor𝜅 is defined as +follows: given a 𝜅-differentiable quantum principal U(1)-bundle with connection (𝑃; Ω𝑃, d𝑃, Π) +over 𝐵, define Ð(𝑃;Ω𝑃,d𝑃;Π) : (𝑃; Ω𝑃, d𝑃; Π) → Tot𝜅 ◦ Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) by +∀𝜔 ∈ Ω𝑃, +Ð(𝑃;Ω𝑃,d𝑃;Π) (𝜔) � (ver − id) ◦ (id −Π)(𝜔) + Π(𝜔). +(3.34) +By combining this theorem with Theorem 3.28, Proposition-Definition 3.32, and Corollary +2.9, we generalise of Abadie–Eilers–Exel’s generalised crossed product construction to a +precise nc generalisation of the classical correspondence between Hermitian line bundles +with unitary connection and principal U(1)-bundles with principal connection. +Definition 3.47. Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connection. We say that +(𝐸, 𝜎𝐸, ∇𝐸) is flat whenever F[𝐸,∇𝐸] = 0. When F[𝐸,∇𝐸] is an eigenvector of the automorphism +ˆΦ[𝐸,∇𝐸], the vertical deformation parameter 𝜅[𝐸,∇𝐸] ∈ R× of (𝐸, 𝜎𝐸, ∇𝐸) is defined to be the +corresponding eigenvalue of ˆΦ[𝐸,∇𝐸]. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +45 +Corollary 3.48. Let 𝜅 ∈ (0, ∞), and let DPic𝜅(𝐵) be the strictly full subcategory of DPic(𝐵) +whose objects are either full or have vertical deformation parameter 𝜅. Then DPic𝜅(𝐵) is the +essential image of DCirchor,𝜅(𝐵) under the equivalence 𝜖1 ◦ ˆL : DCirchor(𝐵) → DPic(𝐵), so +that 𝜖1 ◦ ˆL◦ Hor𝜅 : DCirc𝜅,tot(𝐵) → DPic𝜅(𝐵) is an equivalence of categories. +Definition 3.49. Let 𝜅 ∈ (0, ∞), and let (𝐸, 𝜎, ∇) be a Hermitian line 𝐵-bimodule with +connection that is flat or has vertical deformation parameter 𝜅. We define the 𝜅-total crossed +product of (𝐵; Ω𝐵, d) by (𝐸, 𝜎𝐸, ∇𝐸) to be the essentially unique 𝜅-differentiable quantum +principal U(1)-bundle with connection (𝐵; Ω𝐵, d) ⋊𝜅,tot +(𝐸,𝜎𝐸,∇𝐸) Z on 𝐵, such that +( ˆL◦ Hor𝜅) +� +(𝐵; Ω𝐵, d) ⋊𝜅,tot +(𝐸,𝜎𝐸,∇𝐸) Z +� +� (𝐸, 𝜎𝐸, ∇𝐸); +in this case, we define a ∗-exterior algebra (Ω𝐵, d𝐵) ⋊𝜅,tot +(𝐸,𝜎𝐸) Z and connection Π(𝐸,𝜎𝐸,∇𝐸) by +� +𝐵 ⋊𝐸 Z; (Ω𝐵, d𝐵) ⋊𝜅,tot +(𝐸,𝜎𝐸) Z; Π(𝐸,𝜎𝐸,∇𝐸) +� +� (𝐵; Ω𝐵, d) ⋊𝜅,tot +(𝐸,𝜎𝐸,∇𝐸) Z. +Thus, given 𝜅 ∈ (0, ∞) and (𝐸, 𝜎𝐸, ∇𝐸) that is either flat or has vertical deformation +parameter 𝜅, we may always take (𝐵; Ω𝐵, d) ⋊𝜅,tot +(𝐸,𝜎𝐸,∇𝐸) Z � Tot𝜅((𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z), +where (𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z is any horizontal crossed product of (𝐵; Ω𝐵, d) by (𝐸, 𝜎𝐸, ∇𝐸). +Note that (𝐸, 𝜎𝐸, ∇𝐸) is flat if and only if (𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z is flat and that (𝐸, 𝜎𝐸, ∇𝐸) has +vertical deformation parameter 𝜅 if and only if (𝐵; Ω𝐵, d) ⋊hor +(𝐸,𝜎𝐸,∇𝐸) Z has vertical deformation +parameter 𝜅. +There are certain examples that are naturally described in terms of homomorphisms from +Z to DPic(𝐵) or that give rise to homomorphisms of particular interest. In such cases, it +convenient to have a straightforward algebraic characterization of the essential range of the +composite functor ˆL◦ Hor𝜅. +Corollary 3.50. Let 𝜅 ∈ (0, ∞), and let Hom𝜅(Z, DPic(𝐵)) be the essential image of the +subcategory DCirchor,𝜅(𝐵) under the equivalence ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)). +Then a homomorphism ˆ𝐹 : Z → DPic(𝐵) defines an object of Hor𝜅(Z, DPic(𝐵)) if and only if +ˆ𝐹(1) is flat or has vertical deformation parameter 𝜅, so that, in the latter case, +∀𝑚 ∈ Z, +F ◦ 𝜋0( ˆ𝐹)(𝑚) = 𝜅−𝑚+1[𝑚]𝜅F[ ˆ𝐹(1)]. +(3.35) +Proof. Given the discussion after Proposition-Definition 3.32, it remains to check (3.35). Sup- +pose that ˆ𝐹 : Z → DPic(𝐵) is a homomorphism, such that ˆ𝐹(1) has vertical deformation +parameter 𝜅. The right 1-cocycle identity for F : DPic(𝐵) → S(𝐵) specialises to +∀𝑚, 𝑛 ∈ Z, +F ◦ 𝜋0( ˆ𝐹)(𝑚 + 𝑛) = ˆΦ−𝑛 +[ ˆ𝐹(1)] +� +F ◦ 𝜋0( ˆ𝐹)(𝑚) +� ++ F ◦ 𝜋0( ˆ𝐹)(𝑛). +By induction together with the equation ˆΦ[ ˆ𝐹(1)](F[ ˆ𝐹(1)]) = 𝜅F[ ˆ𝐹(1)], it follows that F ◦ 𝜋0( ˆ𝐹) +satisfies F ◦ 𝜋0( ˆ𝐹) = +� +𝑚 ↦→ [𝑚]𝜅−1F[ ˆ𝐹(1)] +� +, which, in turn, yields (3.35). +□ +Example 3.51 (Ðurđević [39, §4]). We continue from Example 3.33. By (3.23), it follows that +(O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) has deformation parameter 𝑞2; hence, by (3.23) and (3.35), +the homomorphism ˆE : Z → DPic(O𝑞(CP1)) of Example 3.26 satisfies +∀𝑚 ∈ Z, +F ◦ 𝜋0( ˆE)(𝑚) = [𝑚]𝑞2𝑞−2𝑚i𝑒+𝑒−. +At last, the results of Ðurđević show that +(O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞; Π𝑞) � Tot𝑞2 (O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor)) + +46 +BRANIMIR ĆAĆIĆ +recovers the 3-dimensional calculus (Ω𝑞(SU(2)), d𝑞) on O𝑞(SU(2)) of Woronowicz [98] and +the non-universal 𝑞-monopole connection Π𝑞 of Brzeziński–Majid [22]. In other words, we +may obtain Ω𝑞(SU(2)) from Ω𝑞,hor(SU(2)) by adjoining the skew-adjoint U(1)-invariant +1-form 𝑒0 = 2𝜋i𝑞−2𝑒𝑞2 subject to the relations (𝑒0)2 = 0 and +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 +𝑞,hor(SU(2))𝑘, +𝑒0𝜔 = (−1)𝑛𝑞−2𝑘𝜔𝑒0, +and we may obtain d𝑞 from d𝑞,hor by setting d𝑞(𝑒0) � 𝑞−2𝑒+𝑒− and +∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 +𝑞,hor(SU(2))𝑘, +d𝑞(𝜔) � (−1)𝑛[𝑘]𝑞−2𝜔𝑒0 + d𝑞,hor(𝜔). +From now on, we shall refer to (O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞; Π𝑞) as the 𝑞-monopole. +Example 3.52. We continue from Example 2.41. Since the map (𝑔 ↦→ 𝑔21𝜃 + 𝑔22) : Γ𝜃 → R× +is an injective group homomorphism [53, Thm 5.2.10], there exists a unique generator 𝛾 of the +infinite cyclic group Γ𝜃 satisfying 𝛾21𝜃 + 𝑔22 > 1; hence, let 𝜖𝜃 � 𝛾21𝜃 + 𝛾22, which recovers +the norm-positive fundamental unit of the real quadratic field generated by 𝜃. Next, since +ˆΦ[ ˆ𝐸(𝛾)](𝑒1𝑒2) = (𝛾21𝜃 + 𝛾22)2𝑒1𝑒2 = 𝜖2 +𝜃 𝑒1𝑒2, +it follows that ˆ𝐸(𝛾) has vertical deformation parameter 𝜖2 +𝜃. Thus, the composite homomor- +phism ˆ𝐸◦(𝑘 ↦→ 𝛾𝑘) is an object of Hom𝜖2 +𝜃 (Z, DPic(𝐶∞ +𝜃 (T2))), so that ˆΣ +� +ˆ𝐸 ◦ (𝑘 ↦→ 𝛾𝑘) +� +defines +an object of DCirchor,𝜖2 +𝜃 (𝐶∞ +𝜃 (T2)). Hence, at last, we define the real multiplication instanton to +be the 𝜖2 +𝜃-differentiable quantum principal U(1)-bundle over 𝐶∞ +𝜃 (T2) given by +(𝑃𝜃; Ω𝑃𝜃, d𝑃𝜃; Π𝑃𝜃) � Tot𝜖2 +𝜃 ◦ˆΣ +� +ˆ𝐸 ◦ (𝑘 ↦→ 𝛾𝑘) +� +which recovers a construction of Ćaćić [24]. Note that 𝐶∗-algebraic completion of 𝑃𝜃 is part +of a family of Cuntz–Pimsner algebras first considered by Nawata [76]. +4. Lifting problems for noncommutative Riemannian structures +In the commutative case, a Riemannian metric on the base space of a principal U(1)-bundle +with principal connection lifts canonically to the total space. In this section, we use study the +analogous lifting problems for two closely interrelated notions of Riemannian structure on +nc manifolds, which are based, respectively, on generalised Hodge star operators and formal +spectral triples. In particular, we show that these lifted Riemannian structures inexorably +involve modular phenomena in both vertical and horizontal directions that are generally non- +trivial and distinct. Along the way, we construct moduli spaces of U(1)-instantons, show that +quantum SU(2) qua total space of the 𝑞-monopole does not admit a non-pathological U(1)- +equivariant twisted spectral triple, and obtain a geometric formal derivation of Kaad–Kyed’s +compact quantum metric space [58] on quantum SU(2) for a canonical choice of parameters. +For the entirety of this section, let 𝐵 be a unital separable pre-𝐶∗-algebra with ∗-differential +calculus (Ω𝐵, d𝐵), which we assume has dimension 𝑁 ∈ N; let 𝛾𝐵 : Ω𝐵 → Ω𝐵 denote the +Z/2Z-grading on Ω𝐵 by parity of degree. Moreover, given a horizontal quantum principal +U(1)-bundle (𝑃; Ω𝑃,hor; d𝑃,hor) over 𝐵, we suppress the isomorphism ˆ𝜄𝑃 : Ω𝐵 → ΩU(1) +𝑃,hor. +4.1. Basic noncommutative Hodge theory and moduli spaces of U(1)-instantons. We +begin by considering the bare minimum of Riemannian geometry required for classical U(1)- +gauge theory on nc manifolds: the Hodge star operator and integration against the Riemannian +volume form. Such an approach was first pursued by Kustermans–Murphy–Tuset for quantum +groups [63] and then by Majid [67] and Zampini [99] for quantum CP1 and quantum SU(2); +it has attained its fullest expression in the context of nc Kähler geometry in the sense of Ó + +NONCOMMUTATIVE U(1)-GAUGE THEORY +47 +Buachalla [79]. We combine the relevant nc Hodge decomposition theorem with the results of +§2.4 to construct moduli spaces of solutions to Maxwell’s equations, and we obtain a robust nc +generalisation of the notion of conformal orientation-preserving diffeomorphism to the entire +differential Picard group that makes conformal factors into a multiplicative group 1-cocycle +on the resulting conformal subgroup. +We begin with a straightforward generalisation of the Hodge star operator. +Definition 4.1 (Kustermans–Murphy–Tuset [63], Majid [67], Zampini [99], Ó Buachalla [79]). +A Hodge operator on (Ω𝐵, d𝐵) is a ∗-preserving 𝐵-bimodule morphism ★ : Ω𝐵 → Ω𝐵, such +that, for every 𝑘 ∈ {0, . . . , 𝑁}, the restriction of ★ to Ω𝑘 +𝐵 satisfies +★(Ω𝑘 +𝐵) ⊆ Ω𝑁−𝑘 +𝐵 +, +★2↾Ω𝑘 +𝐵= (−1)𝑘(𝑁−𝑘) idΩ𝑘 +𝐵, +(4.1) +∀𝜔, 𝜂 ∈ Ω𝑘 +𝐵, +𝜔 · ★(𝜂) = ★−1(𝜔) · 𝜂. +(4.2) +Hence, the inverse metric induced by ★ is the right 𝐵-valued inner product 𝑔 on Ω𝐵 given by +∀𝜔, 𝜂 ∈ Ω𝐵, +𝑔(𝜔, 𝜂) � ★(𝜔∗ · ★(𝜂)). +(4.3) +By combining a generalised Hodge star operator with a suitable generalisation of integra- +tion against the corresponding Riemannian volume form, we obtain our first notion of nc +Riemannian structure; in particular, following Connes [33] and Kustermans–Murphy–Tuset +[62], we impose Stokes’s theorem for divergence as a requirement. +Definition 4.2 (cf. Kustermans–Murphy–Tuset [63], Ó Buachalla [79], Saldaña [73]). A Rie- +mannian geometry on (𝐵; Ω𝐵, d𝐵) is a pair (★, 𝜏), where ★ is a Hodge operator on (Ω𝐵, d𝐵) +whose inverse metric 𝑔 admits a basis as a right 𝐵-valued inner product on Ω𝐵 and satisfies +∀𝑏 ∈ 𝐵, ∀𝜔 ∈ Ω𝐵, +𝑔(𝑏𝜔, 𝑏𝜔) ≤ ∥𝑏∥2𝑔(𝜔, 𝜔), +(4.4) +and where 𝜏 is a bounded state on 𝐵 that satisfies +∀𝜔 ∈ Ω𝑁−1 +𝐵 +, +(𝜏 ◦ ★ ◦ d𝐵)(𝜔) = 0, +(4.5) +∀𝑏 ∈ 𝐵, +sup{𝜏(𝑎∗𝑏∗𝑏𝑎) | 𝑎 ∈ 𝐴, 𝜏(𝑎∗𝑎) ≤ 1} = ∥𝑏∥2. +(4.6) +Example 4.3. We continue from Example 2.39. Suppose that 𝑋 is orientable. Equip 𝑋 +with an orientation and a Riemannian metric 𝑔; let ★𝑔 and vol𝑔 respectively denote the +resulting Hodge star operator and Riemannian volume form. Then (★𝑔, +∫ +𝑋 (·) vol𝑔) defines +a Riemannian geometry on (𝐶∞(𝑋); Ω(𝑋, C), d), whose inverse metric is the usual inverse +Riemannian metric; in particular, the inner product of (4.7) is the usual Riemannian 𝐿2 inner +product on differential forms. Note that a basis for Ω(𝑋, C) with respect to the inverse metric +can be constructed from local orthonormal frames using a smooth partition of unity. +Example 4.4. We continue from Example 3.51. Let ℎ𝑞 denote Woronowicz’s Haar state on +O𝑞(SU(2)). Since (O𝑞(CP1), Ω𝑞(CP1), d𝑞) is a nc Kähler manifold à la Ó Buachalla [79, §§4.4, +5.4], it admits a canonical Riemannian geometry (★𝑞, ℎ𝑞↾O𝑞(CP1)), where ★𝑞(1) � i𝑒+𝑒− and +★𝑞 restricts to ±i id on O𝑞(SU(2))∓2 · 𝑒±. note that ★𝑞 recovers Zampini’s modification [99, +Eq. 5.14] of Majid’s Hodge operator [67, §4] for the choice of parameter 𝛼′′ = −𝑞2. Note that +(4.6) is satisfied by a theorem of Nagy [75], which shows that the Haar state ℎ𝑞 remains faithful +on 𝐶𝑞(SU(2)). +Just as in the classical case, we may now equip Ω𝐵 with an 𝐿2-inner product and compute +the (formal) adjoint of the exterior derivative d𝐵 in terms of the Hodge star operator. + +48 +BRANIMIR ĆAĆIĆ +Proposition 4.5 (Ó Buachalla [79, §§5.2–3]). Let (★, 𝜏) be a Riemannian geometry on (𝐵; Ω𝐵, d𝐵); +let 𝑔 be the resulting inverse metric. Then Ω𝐵 defines a 𝐵-self-correspondence of finite type with +respect to 𝑔 that decomposes as an orthogonal direct sum Ω𝐵 = �𝑁 +𝑘=0 Ω𝑘 +𝐵 of sub-𝐵-bimodules. +Hence, the C-vector space Ω𝐵 defines a separable pre-Hilbert space with respect to the inner product +⟨·, ·⟩𝜏 defined by +∀𝜔, 𝜂 ∈ Ω𝐵, +⟨𝜔, 𝜂⟩𝜏 � 𝜏(𝑔(𝜔, 𝜂)), +(4.7) +with respect to which the left 𝐵-module structure on Ω𝐵 defines an isometric ∗-representation of 𝐵, +the direct sum decomposition Ω𝐵 = �𝑁 +𝑘=0 Ω𝑘 +𝐵 is orthogonal, the Hodge operator ★ is unitary, and +the operator d𝐵 is adjointable with adjoint d∗ +𝐵 = ★−1 ◦ d𝐵 ◦ ★ ◦ 𝛾𝐵. +Proof. Relative to the references (compare the proof of Proposition 4.29 below), it remains +to show that Ω𝐵 is separable as a pre-Hilbert space and that the left 𝐵-module structure is +isometric as a ∗-homomorphism. Let 𝔅 be the 𝐶∗-algebraic completion of 𝐵, so that 𝜏 extends +to a state on 𝔅. Let 𝑚 ∈ {0, . . . , 𝑁}, and let (𝑒𝑖)𝑛 +𝑖=1 be a basis for Ω𝑚 +𝐵 with respect to 𝑔, so +that the matrix 𝑋 � �𝑔(𝑒𝑖, 𝑒𝑗)�𝑛 +𝑖,𝑗=1 ∈ 𝑀𝑛(𝐵) is positive with unique positive square root +√ +𝑋 ∈ 𝑀𝑛(𝔅). Let 𝑎 � (𝑎1, . . . , 𝑎𝑛) ∈ 𝐵𝑛 ⊂ 𝔅𝑛 and set 𝜔 � �𝑛 +𝑖=1 𝑒𝑖𝑎𝑖. Then +⟨𝜔, 𝜔⟩𝜏 = 𝜏 +�∑︁𝑛 +𝑖,𝑗=1 𝑎∗ +𝑖 𝑔(𝑒𝑖, 𝑒𝑗)𝑎𝑗 +� += 𝜏((𝑎, 𝑋𝑎)𝐵𝑛) ≤ 𝜏 +���� +√ +𝑋 +��� +2 +(𝑎, 𝑎)𝔅𝑛 +� +≤ ∥𝑋∥ +∑︁𝑛 +𝑖=1∥𝑎𝑖∥2. +Since 𝐵 is separable as a normed vector space and since (𝑒𝑖)𝑚 +𝑖=1 generates Ω𝑚 +𝐵 as a right 𝐵- +module, it follows that Ω𝑚 +𝐵 is separable as a pre-Hilbert space. Hence, the finite orthogonal +direct sum Ω𝐵 = �𝑁 +𝑚=0 Ω𝑚 +𝐵 is also separable as a pre-Hilbert space. +Let us now show that the left 𝐵-module structure 𝜋 : 𝐵 → L(Ω𝐵) is isometric. Let 𝑏 ∈ 𝐵 +be given. Since the 𝜋 is bounded, it necessarily contractive, so that, by (4.6), it follows that +∥𝑏∥2 ≥ ∥𝜋(𝑏)2∥ ≥ sup{⟨𝜋(𝑏)𝑎, 𝜋(𝑏)𝑎⟩𝜏 | 𝑎 ∈ 𝐵, ⟨𝑎, 𝑎⟩𝜏 ≤ 1} = ∥𝑏∥2. +□ +Given a Riemannian geometry (★𝐵, 𝜏) on our nc manifold (𝐵; Ω𝐵, d𝐵), we may now +consider the Yang–Mills equation d∗ +𝐵F[𝐸,∇𝐸] = 0 for unknown [𝐸, ∇𝐸] ∈ DPic(𝐵), where +d𝐵F[𝐸,∇𝐸] = 0 is automatically satisfied; in fact, for any closed self-adjoint j ∈ Z(Ω𝐵)3, we +may consider the (Euclidean) Maxwell’s equation d∗ +𝐵F[𝐸,∇𝐸] = j. The following nc Hodge +decomposition theorem will allow us to apply Theorem 2.35 to the construction of moduli +spaces of solutions in a fixed topological sector. +Definition 4.6. We say that a symmetric operator 𝑆 on a pre-Hilbert space His Fréchet- +diagonalisable with spectral gap whenever the following all hold: +(1) there is a countable maximal orthonormal subset of Hconsisting of eigenvectors of 𝑆; +(2) the vector space Hdefines a Fréchet space with respect to the countable family of pre- +Hilbert space norms (∥ · ∥𝑘)𝑘∈N0 defined by +∀𝑘 ∈ N0, ∀𝜉 ∈ H, +∥𝜉∥𝑘 � +�∑︁𝑘 +𝑚=0⟨𝑆𝑚𝜉, 𝑆𝑚𝜉⟩ +�1/2 +; +(4.8) +(3) the non-zero eigenvalues of 𝑆 are bounded away from 0. +Theorem 4.7 (Ó Buachalla [79, Thm. 6.2], Ó Buachalla–Šťoviček–Van Roosmalen [80, Prop. +6.3]; cf. Kustermans–Murphy–Tuset [63, Thm. 4.1]). Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on +𝐵 with respect to (Ω𝐵, d𝐵); suppose that d𝐵 + d∗ +𝐵 is diagonalisable or Fréchet-diagonalisable with +spectral gap. For each 𝑚 ∈ {0, . . . , 𝑁}, the pre-Hilbert space Ω𝑚 +𝐵 decomposes orthogonally as +Ω𝑚 +𝐵 = d𝐵(Ω𝑚−1 +𝐵 +) ⊕ �Ω𝑚 +𝐵 ∩ ker(d𝐵 + d∗ +𝐵)2� ⊕ d∗ +𝐵(Ω𝑚+1 +𝐵 +). + +NONCOMMUTATIVE U(1)-GAUGE THEORY +49 +The case where d𝐵 + d∗ +𝐵 is diagonalisable (i.e., admits an Hamel basis for Ω𝐵 consisting of +eigenvectors) is given in the literature. The case where d𝐵 + d∗ +𝐵 is Fréchet-diagonalisable with +spectral gap is a consequence of the following lemma. +Lemma 4.8. Let 𝑉 be a pre-Hilbert space, let 𝑠 : 𝑉 → 𝑉 be an adjointable complex-linear map +satisfying 𝑠2 = 0. Suppose that 𝑆 � 𝑠 + 𝑠∗ is Fréchet-diagonalisable with spectral gap. Then 𝑉 +decomposes orthogonally as 𝑉 = ran 𝑠 ⊕ ker 𝑆 ⊕ ran 𝑠∗, where ker 𝑠 = ran 𝑠 ⊕ ker 𝑆. +Proof. Let 𝔙 denote the Hilbert space completion of 𝑉, let (𝜉𝑖)𝑖∈N be an orthonormal basis +for 𝔙 consisting of eigenvectors of 𝑆 in 𝑉, and let Vdenote the algebraic span of (𝜉𝑖)𝑖∈N, so +that 𝑆 is essentially self-adjoint on V; since 𝑆 is assumed to be Fréchet-diagonalisable with +spectral gap, it follows that 𝑉 is the Fréchet space of smooth vectors for the unique self-adjoint +extension 𝑆 of 𝑆. In particular, if we set 𝔙 � 𝔙0 and, for 𝑘 ∈ N, set 𝔙𝑘 to be the Hilbert space +completion of 𝑉 with respect to the inner product ⟨·, ·⟩𝑘 � +� +(𝑣, 𝑤) ↦→ �𝑘 +𝑚=0⟨𝑆𝑚𝑣, 𝑆𝑚𝑤⟩ +� +, +then the Fréchet space 𝑉 is the projective limit of the decreasing countable family of Hilbert +spaces (𝔙𝑘)𝑘∈N0 with contractive inclusions. Thus, a subspace 𝑋 of 𝑉 is Fréchet-closed if and +only if 𝑋 = �∞ +𝑘=0 𝑋 +𝔙𝑘, where, for each 𝑘 ∈ N0, we denote by 𝑋 +𝔙𝑘 the closure of 𝑋 in 𝔙𝑘. +First, let 𝑃 : 𝔙 → 𝔙be the orthogonal projection onto ker 𝑆, so that id −𝑃 is the orthogonal +projection onto the closure in 𝔙 of ran 𝑆 For each 𝑖 ∈ N, let 𝜆𝑖 denote the eigenvalue of 𝑆 +corresponding to 𝜉𝑖; hence, let I � {𝑖 ∈ N | 𝜆𝑖 ≠ 0}, so that 𝐶 � sup𝑖∈I 𝜆−1 +𝑖 +< +∞. Hence, +we may define a Fréchet-continuous map 𝑅 : 𝑉 → 𝑉 by 𝑅 � �𝑣 ↦→ � +𝑖∈I 𝜆−1 +𝑖 ⟨𝜉𝑖, 𝑣⟩𝜉𝑖 +�. Since +id −𝑅𝑆 = id −𝑆𝑅 = 𝑃↾𝑉, it follows, in particular, that ran 𝑆 is Fréchet-closed. +Next, let us look at the kernel of 𝑠 and the range of 𝑠∗; recall that 𝑠2 = 0. On the one hand, +one can show that ⟨𝑆𝑘𝑠𝑣, 𝑆𝑘𝑠𝑣⟩ + ⟨𝑆𝑘𝑠∗𝑣, 𝑆𝑘𝑠∗𝑣⟩ = ⟨𝑆𝑘+1𝑣, 𝑆𝑘+1𝑣⟩ for all 𝑘 ∈ N and 𝑣 ∈ 𝑉, so +that 𝑠 is Fréchet continuous, and hence ker 𝑠 is Fréchet-closed. On the other hand, one can +also show that ⟨𝑆𝑘𝑠𝑣, 𝑆𝑘𝑠∗𝑤⟩ = 0 for all 𝑘 ∈ N and 𝑣, 𝑤 ∈ 𝑉, so that ran 𝑆 = ran 𝑠 ⊕ ran 𝑠∗, +where, for each 𝑘 ∈ N the direct sum is orthogonal with respect to the inner product ⟨·, ·⟩𝑘; it +now follows that ran 𝑠 = �∞ +𝑘=0 ran 𝑠𝔙𝑘 is Fréchet-closed since +ran 𝑠 ⊕ ran 𝑠∗ = ran 𝑆 = +∞ +� +𝑘=0 +ran 𝑆 +𝔙𝑘 = +∞ +� +𝑘=0 +ran 𝑠𝔙𝑘 ⊕ ran 𝑠∗𝔙𝑘 = +∞ +� +𝑘=0 +ran 𝑠𝔙𝑘 ⊕ +∞ +� +𝑘=0 +ran 𝑠∗𝔙𝑘. +Finally, by the proof of [80, Prop. 6.3], we know that V ⊆ ker 𝑠 ⊕ran 𝑠∗. Since Vis Fréchet– +dense in 𝑉 and ker 𝑠 and ran 𝑠∗ are both Fréchet–closed, it follows that 𝑉 = ker 𝑠 ⊕ ran 𝑠∗, +which suffices by the proof of [80, Prop. 6.3]. +□ +Example 4.9. We continue from Example 3.52. The canonical Riemannian geometry (★, 𝜏) +on 𝐶∞ +𝜃 (T2) with respect to the canonical ∗-exterior algebra (Ω𝜃(T2), d) is given by +★(1) � 𝑒1𝑒2, +★(𝑒1) � 𝑒2, +★(𝑒2) � −𝑒1; +∀(𝑚, 𝑛) ∈ Z2, +𝜏(𝑈𝑚𝑉 𝑛) � 𝛿𝑚,0𝛿 𝑛,0; +so that 𝜏 is the canonical U(1)-invariant trace on 𝐶∞ +𝜃 (T2). Since Ω𝜃(T2) = 𝐶∞ +𝜃 (T2) ·C[𝑒1, 𝑒2], +where 𝐶∞ +𝜃 (T2) is the Fréchet completion of � +𝑚,𝑛∈Z C · 𝑢𝑚𝑣𝑛 with respect to the family of +seminorms induced by −(𝛿2 +1 + 𝛿2 +2) = (d + d∗)2↾𝐶∞ +𝜃 (T2), an elementary calculation shows that +d + d∗ is Fréchet-diagonalisable with spectral gap. +At last, we construct moduli spaces of U(1)-instantons—more generally, solutions to +Maxwell’s equations—with fixed topological sector. In the classical case, the Picard group of +line bundles up to isomorphism is canonically isomorphic to integral singular cohomology in +degree 2, so that a choice of topological sector can be equivalently specified with respect to +either group. Hence, in the absence of well-defined singular cohomology for nc topological +spaces, we define a choice of topological sector to be a choice of 𝑐 ∈ Pic(𝐵). + +50 +BRANIMIR ĆAĆIĆ +Corollary 4.10. Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), and +suppose that d𝐵 + d∗ +𝐵 is diagonalisable or Fréchet-diagonalisable with spectral gap. Let +� +Diff00(𝐵) � +� +(𝜔, 𝜙) ∈ � +Diff0(𝐵) +��� d𝐵𝜔 − i𝜔2 = 0 +� +≤ � +Diff0(𝐵), +and suppose that � +Diff0(𝐵) itself satisfies +∀(𝜔, 𝜙) ∈ � +Diff0(𝐵), +d𝐵𝜔 − i𝜔2 ∈ d𝐵 +�Z(Ω𝐵)1�. +(4.9) +Finally, let 𝑐 ∈ Pic(𝐵) and j ∈ Ω1 +𝐵 be given, let +A(𝑐, j) � +� +[𝐸, ∇𝐸] ∈ DPic(𝐵) +�� [𝐸] = 𝑐, d∗ +𝐵F[𝐸,∇𝐸] = j +� +, +and suppose that A(𝑐, j) ≠ ∅. Then the curvature 1-cocycle F is constant on A(𝑐, j) and the group +� +Diff00(𝐵)/� +Ad�U(Z(𝐵)0)� acts freely and transitively on A(𝑐, j) by +∀(𝜔, 𝜙) ∈ � +Diff00(𝐵), ∀[𝐸, ∇𝐸] ∈ A(𝑐, j), +[𝜔, 𝜙] ⊲ [𝐸, ∇𝐸] � [ˆ𝜏(𝜔, 𝜙)][𝐸, ∇𝐸]. +Proof. First, we show that the curvature 1-cocycle F : DPic(𝐵) → Ω2 +𝐵 ∩ ker d𝐵 descends +to a map c : ran ΠPic(𝐵) → 𝐻2 +dR(𝐵), where ΠPic(𝐵) : DPic(𝐵) → Pic(𝐵) is the forgetful +homomorphism and 𝐻2 +dR(𝐵) � (Ω2 +𝐵 ∩ ker d𝐵)/d𝐵(Ω1 +𝐵). Let [𝐸, ∇𝐸], [𝐹, ∇𝐹] ∈ DPic(𝐵), +and suppose that [𝐸] = [𝐹]. By Theorem 2.35, there exists (𝜔, 𝜙) ∈ � +Diff0(𝐵), such that +[𝐸, ∇𝐸][𝐹, ∇𝐹]−1 = [ˆ𝜏(𝜔, 𝜙)]. But now, by (4.9), d𝐵𝜔 − i𝜔2 = d𝐵𝛽 for some 𝛽 ∈ Z(Ω𝐵)1, so +that F[𝐸,∇𝐸] − F[𝐹,∇𝐹 ] = ˆΦ−1 +[𝐹,∇𝐹 ] +�F[ˆ𝜏(𝜔,𝜙)] +� = ˆΦ−1 +[𝐹,∇𝐹 ] +�𝜙−1(d𝐵𝛽)� = d𝐵 +� +ˆΦ−1 +[𝐹,∇𝐹 ] ◦ 𝜙−1(𝛽) +� +. +Next, by Theorem 4.7 together with our hypothesis on d𝐵 + d∗ +𝐵, we obtain a vector space +isomorphism �𝜔 ↦→ ([𝜔], d∗ +𝐵𝜔)� : ker +� +d𝐵↾Ω2 +𝐵 +� +→ 𝐻2 +dR(𝐵) ⊕ d∗ +𝐵(Ω1 +𝐵). Thus, the map F is +constant on A(𝑐, j) with value F𝑐,j uniquely determined by ([F𝑐,j], d∗ +𝐵F𝑐,j) = (c([𝐸]), j), so +that, in particular, A(𝑐, j) = +� +[𝐸, ∇𝐸] ∈ DPic(𝐵) +�� [𝐸] = 𝑐, F[𝐸,∇𝐸] = F𝑐,j +� +. +Now, let 𝑐 ∈ Pic(𝐵), j ∈ Ω1 +𝐵, and [𝐸, ∇𝐸] ∈ DPic(𝐵) with F[𝐸,∇𝐸] = 0 be given; we show +that A([𝐸] · 𝑐, j) = [𝐸] · A(𝑐, j). Suppose that [𝐹, ∇𝐹] ∈ A(𝑐, j). Then [𝐸, ∇𝐸][𝐹, ∇𝐹] satisfies +both ΠPic(𝐵) ([𝐸, ∇𝐸][𝐹, ∇𝐹]) = [𝐸]𝑐 and d∗ +𝐵 +�F[𝐸,∇𝐸] [𝐹,∇𝐹 ] +� = d∗ +𝐵 +� ˆΦ[𝐹,∇𝐹 ](0) + F[𝐹,∇𝐹 ] +� = j, +so that [𝐸, ∇𝐸][𝐹, ∇𝐹] ∈ A([𝐸]𝑐, j). Since F[𝐸,∇𝐸]−1 = − ˆΦ[𝐸,∇𝐸] +�F[𝐸,∇𝐸] +� = 0, we similarly +find that [𝐸, ∇𝐸]−1[𝐺, ∇𝐺] ∈ A(𝑐, j) for every [𝐺, ∇𝐺] ∈ A([𝐸] · 𝑐, j). +Finally, let 𝑐 ∈ Pic(𝐵) and j ∈ Ω1 +𝐵 be given, and suppose that A(𝑐, j) ≠ ∅. On the one +hand, let (𝜔, 𝜙) ∈ � +Diff00(𝐵). Since [ˆ𝜏(𝜔, 𝜙)] = [𝐵] = 1 and F[ˆ𝜏(𝜔,𝜙)] = 0, it now follows +that [ˆ𝜏(𝜔, 𝜙)] · A(𝑐, j) = A(𝑐, j). On the other hand, let [𝐸, ∇𝐸], [𝐹, ∇𝐹] ∈ A(𝑐, j). Then +ΠPic(𝐵) ([𝐸, ∇𝐸][𝐹, ∇𝐹]−1) = 𝑐𝑐−1 = 1 and +F[𝐸,∇𝐸] [𝐹,∇𝐹 ]−1 = ˆΦ[𝐹,∇𝐹 ] +�F[𝐸,∇𝐸] − F[𝐹,∇𝐹 ] +� = ˆΦ[𝐹,∇𝐹 ] +�F𝑐,j − F𝑐,j +� = 0, +hence [𝐸, ∇𝐸][𝐹, ∇𝐹]−1 ∈ 𝜋0(ˆ𝜏) +� +� +Diff00(𝐵) +� +by Theorem 2.35. +□ +Example 4.11. We continue from Example 4.3. Let 𝐻1(𝑋, R)Z ≤ 𝐻1(𝑋, R) be the lattice of +integral classes, so that 𝐻1(𝑋, R)Z = {[−i d𝑓 · 𝑓 −1] | 𝑓 ∈ 𝐶∞(𝑋, U(1))} by the isomorphism +([𝑓] ↦→ 𝑓∗) : [𝑋, U(1)] → Hom(𝜋1(𝑋), 𝜋1(U(1))) � 𝐻1(𝑋, Z). +Note that d + d∗ is the usual Hodge–de Rham operator, which is Fréchet–diagonalisable with +spectral gap by the theory of elliptic regularity. Fix c ∈ 𝐻2(𝑋, Z) and j ∈ Ω1(𝑋, R) and let +𝔄(c, j) � +� +[E, ∇E] ∈ ˇ𝐻2(𝑋) +�� 𝑐1(E) = c, d∗�tr ∇2 +E +� = j +� +, +where 𝑐1 : Pic(𝑋) → 𝐻2(𝑋, Z) denotes the integral first Chern class on line bundles; thus, +the set 𝔄(c, j) is the moduli space of gauge equivalence classes of solutions in the instanton + +NONCOMMUTATIVE U(1)-GAUGE THEORY +51 +sector c to Maxwell’s equations with current 1-form j. Let 𝑐 ∈ Pic(𝐶∞(𝑋)) be the preimage +of (id, c) under the isomorphism Pic(𝑋) → Diff(𝑋) ⋉ 𝐻2(𝑋, Z) induced by the integral +first Chern class and the isomorphism Diff(𝑋) ⋉ Pic(𝑋) → Pic(𝐶∞(𝑋)) of Example 2.21. +Then 𝔄(c, j) is non-empty if and only if A(𝑐, j) is non-empty, in which case the bijection +([E, ∇E] ↦→ [Γ(E), ∇E]) : 𝔄(c, j) → A(𝑐, j) and the group isomorphism +�[𝛽] + 𝐻1(𝑋, R)Z ↦→ [(𝛽, id)]� : 𝐻1(𝑋, R)/𝐻1(𝑋, R)Z → � +Diff00(𝐶∞(𝑋))/� +Ad(U(𝐶∞(𝑋))) +combine to recover 𝔄(c, j) as a torsor for the torus 𝐻1(𝑋, R)/𝐻1(𝑋, R)Z � T dim 𝐻1(𝑋,R). +Example 4.12. We continue from Example 4.9. Recall from Examples 2.24 and 2.31 the +homomorphism 𝐸 : Γ𝜃 → Pic(𝐶∞ +𝜃 (T2)) and its lift ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ +𝜃 (T2)). On the one +hand, note that � +Diff0(𝐶∞(T2)) = SpanR{𝑒1, 𝑒2} × {id} � R2 by a computation of Ćaćić– +Karthik [25], so that � +Diff0(𝐶∞ +𝜃 (T2)) satisfies (4.9). On the other hand, note that Z(𝐶∞ +𝜃 (T2)) = C +since 𝜃 is irrational. Hence, by Corollary 4.10, for each 𝑔 ∈ Γ𝜃, the set A([𝐸(𝑔)], 0) is a 2- +dimensional real affine space with basepoint [ ˆ𝐸(𝑔)], on which the curvature 1-cocycle F takes +the constant value +2𝜋𝑔21 +𝑔21𝜃+𝑔22 𝑒1𝑒2. Note the contrast with Example 4.11 as applied to T2, where a +moduli space A(𝑐, j) � 𝔄(c, j), if non-empty, is a T2-torsor instead. +We now generalise the notion of conformal orientation-preserving diffeomorphism to +our nc setting. For convenience, let Z>0(𝐵) denote the multiplicative group of all positive +invertible elements of Z(Ω𝐵)0, so that Z>0(𝐵) admits a canonical right action of the differential +Picard group DPic(𝐵) defined by +∀𝜇 ∈ Z>0(𝐵), ∀[𝐸, ∇𝐸] ∈ DPic(𝐵), +𝜇 ⊳ [𝐸, ∇𝐸] � ˆΦ−1 +[𝐸,∇𝐸](𝜇). +(4.10) +Note from Examples 2.34 and 2.39 and the proof of Theorem 2.35 that the dynamical content +of Hermitian line 𝐵-bimodule with connection is encoded by its generalised braiding. Hence, +we promote the behaviour of the usual Hodge star operator under orientation-preserving +conformal diffeomorphisms [17, Thm. 1.159.h] into the following definition. +Definition 4.13. Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵). A Hermitian line 𝐵-bimodule with +connection (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal when there exists (necessarily unique) 𝜇 ∈ Z>0(𝐵), +the conformal factor of (𝐸, 𝜎𝐸, ∇𝐸), such that +∀𝑥 ∈ 𝐸, ∀𝑘 ∈ {0, . . . , 𝑁}, ∀𝛼 ∈ Ω𝑘 +𝐵, +𝜎𝐸(★𝐵(𝛼) ⊗ 𝑥) = 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ +� +𝜇𝑁−2𝑘. +(4.11) +We denote by DPic(𝐵; ★𝐵) the strictly full subcategory of DPic(𝐵) whose objects are ★𝐵- +conformal, we denote by DPic(𝐵; ★𝐵) the corresponding subset of DPic(𝐵), and we define +𝜇 : DPic(𝐵; ★𝐵) → Z>0(𝐵) to be the function that maps [𝐸, ∇𝐸] ∈ DPic(𝐵; ★𝐵) to the +conformal factor 𝜇[𝐸,∇𝐸] of any (and hence every) representative. +In the classical case, orientation-preserving conformal diffeomorphisms form a group and +their conformal factors define a multiplicative 1-cocycle on this group. The same is true in +the noncommutaitve setting. +Proposition 4.14. Suppose that ★𝐵 is a Hodge operator on (Ω𝐵, d𝐵). Then DPic(𝐵; ★𝐵) defines +a sub-2-group of DPic(𝐵), the subset DPic(𝐵; ★𝐵) defines a subgroup of DPic(𝐵), and the +function 𝜇 : DPic(𝐵; ★𝐵) → Z>0(𝐵) defines a group 1-cocycle with respect to the restriction to +DPic(𝐵; ★𝐵) of the right DPic(𝐵)-action on Z>0(𝐵) defined by (4.10). + +52 +BRANIMIR ĆAĆIĆ +Proof. First, note that the monoidal unit (𝐵, 𝜎𝐵, ∇𝐵) is trivially ★𝐵-conformal with confor- +mal factor 𝜇[𝐵,∇𝐵] = 1. On the one hand, suppose that (𝐸, 𝜎𝐸, ∇𝐸) and (𝐹, 𝜎𝐹, ∇𝐹) are ★𝐵- +conformal. Then, given 𝑘 ∈ {0, . . . , 𝑁}, 𝛼 ∈ Ω𝑘 +𝐵, 𝑥 ∈ 𝐸, and 𝑦 ∈ 𝐹, +𝜎𝐸⊗𝐵𝐹 (★𝐵(𝛼) ⊗ (𝑥 ⊗ 𝑦)) += +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜎𝐹 +� +★𝐵(𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩)𝜇𝑁−2𝑘 +[𝐸,∇𝐸] ⊗ 𝑦 +� +⟨0⟩ +� +⊗ 𝜎𝐹 +� +★𝐵(𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩)𝜇𝑁−2𝑘 +[𝐸,∇𝐸] ⊗ 𝑦 +� +⟨1⟩ += +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜎𝐹 +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ ⊗ 𝑦 +� +⟨0⟩ +� +⊗ ★𝐵 +� +𝜎𝐹 (𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ ⊗ 𝑦) ⟨1⟩ +� +ˆΦ−1 +[𝐹,∇𝐹 ] (𝜇𝑁−2𝑘 +[𝐸,∇𝐸])𝜇𝑁−2𝑘 +[𝐹,∇𝐹 ] += 𝜎𝐸⊗𝐵𝐹 (𝛼 ⊗ (𝑥 ⊗ 𝑦)) ⟨0⟩ ⊗ ★𝐵 +� +𝜎𝐸⊗𝐵𝐹 (𝛼 ⊗ (𝑥 ⊗ 𝑦)) ⟨1⟩ +� � +ˆΦ−1 +[𝐹,∇𝐹 ] (𝜇[𝐸,∇𝐸])𝜇[𝐹,∇𝐹 ] +�𝑁−2𝑘 +. +Hence, the subcategory DPic(𝐵; ★𝐵) is closed under the monoidal product and the map +𝜇 satisfies the required 1-cocycle identity. On the other hand, suppose that (𝐸, 𝜎𝐸, ∇𝐸) is +★𝐵-conformal. Then, given 𝑘 ∈ {0, . . . , 𝑁}, 𝛼 ∈ Ω𝑘 +𝐵, and 𝑥 ∈ 𝐸, +𝜎𝐸(★𝐵(𝛼) ⊗ 𝑥) = 𝜎−1 +𝐸 (𝑥 ⊗ ★𝐵(𝛼)∗) ⟨0⟩ ⊗ 𝜎−1 +𝐸 (𝑥 ⊗ ★𝐵(𝛼)∗) +∗ +⟨−1⟩ += 𝜎−1 +𝐸 (𝑥 ⊗ 𝛼∗) ⟨0⟩𝜇−𝑁+2𝑘 +[𝐸,∇𝐸] ⊗ ★𝐵 +� +𝜎−1 +𝐸 (𝑥 ⊗ 𝛼∗) +∗ +⟨−1⟩ +� += 𝜇−𝑁+2𝑘 +[𝐸,∇𝐸]𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ +� += 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 +� +𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ +� +ˆΦ[𝐸,∇𝐸](𝜇−1 +[𝐸,∇𝐸])𝑁−2𝑘. +Hence, the the subcategory DPic(𝐵; ★𝐵) is also closed under monoidal inversion. +□ +Given a Hodge operator ★𝐵 on (Ω𝐵, d𝐵), we may therefore call DPic(𝐵; ★𝐵) the conformal +sub-2-group of DPic(𝐵). +Example 4.15. We continue from Example 4.3. On the one hand, for every Hermitian line +bundle with unitary connection (E, ∇E) on 𝑋, (Γ(E), flip, ∇E) is ★𝑔-conformal with confor- +mal factor 1. On the other, for every 𝜙 ∈ Diff(𝑋), ˆ𝜏(0, (𝜙−1)∗) is ★𝑔-conformal if and only +if 𝜙 ∈ Conf+(𝑋, 𝑔), in which case 𝜇 ◦ 𝜋0(ˆ𝜏)(0, (𝜙−1)∗) = +√︃ +𝜙∗𝑔 +𝑔 . Hence, the isomorphism of +Example 2.39 restricts to an isomorphism Conf+(𝑋, 𝑔) ⋉ ˇ𝐻2(𝑋) → DPic(𝐶∞(𝑋); ★𝑔), with +respect to which 𝜇 : DPic(𝐶∞(𝑋); ★𝑔) → 𝐶∞(𝑋, (0, ∞)) reduces to the map +� +(𝜙, [E, ∇E]) ↦→ +√︃ +𝜙∗𝑔 +𝑔 +� +: Conf+(𝑋, 𝑔) ⋉ ˇ𝐻2(𝑋) → 𝐶∞(𝑋, (0, ∞)). +In light of Theorem 3.28 and Proposition-Definition 3.32, we may equivalently consider +conformality of horizontally differentiable quantum principal U(1)-bundles over 𝐵 with +respect to a given Hodge operator on (Ω𝐵, d𝐵). +Definition 4.16. Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵). Let (𝑃; Ω𝑃,hor, d𝑃,hor) be a hori- +zontally differentiable quantum principal U(1)-bundle over 𝐵 with Fröhlich automorphism +ˆΦ𝑃; define a right Z-action on Z>0(𝐵) by +∀𝜇 ∈ Z>0(𝐵), ∀𝑘 ∈ Z, +𝜇 ⊳ 𝑘 � ˆΦ−𝑘 +𝑃 (𝜇). +(4.12) +We say that (𝑃; Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal if there exists a (necessarily unique) group +1-cocycle 𝜇𝑃 : Z → Z>0(𝐵), the conformal factor of (𝑃; Ω𝑃,hor, d𝑃,hor), such that +∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝛼 ∈ Ω𝑚 +𝐵, ∀𝑝 ∈ 𝑃𝑗, +★𝐵(𝛼)𝑝 = ˆℓ𝑃(𝛼𝑝) ⟨0⟩★𝐵 +� +ˆℓ𝑃(𝛼𝑝) ⟨1⟩ +� +𝜇𝑃(𝑗)𝑁−2𝑘, +(4.13) + +NONCOMMUTATIVE U(1)-GAUGE THEORY +53 +where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is the 𝐵-bimodule isomorphism of Proposition 3.24. We denote +by DCirchor(𝐵; ★𝐵) the strictly full subcategory of DCirchor(𝐵) with ★𝐵-conformal objects. +Proposition 4.17. Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵). Then Hom(Z, DPic(𝐵; ★𝐵)) is the +essential image of DCirchor(𝐵; ★𝐵) under the functor ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)), +so that the composite functor 𝜖1 ◦ ˆL : DCirchor(𝐵; ★𝐵) → DPic(𝐵; ★𝐵) is an equivalence +of categories. In particular, if (𝑃; Ω𝑃,hor, d𝑃,hor) is a ★𝐵-conformal horizontally differentiable +quantum principal U(1)-bundle over 𝐵, then its conformal factor 𝜇𝑃 satisfies +𝜇𝑃 = 𝜇 ◦ 𝜋0 +� +ˆL(𝑃; Ω𝑃,hor, d𝑃,hor) +� +. +(4.14) +Thus, a Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal if and only +if (𝑃; Ω𝑃,hor, d𝑃,hor) � (𝐵; Ω𝐵, d𝐵) ⋊(𝐸,𝜎𝐸,∇𝐸) Z is ★𝐵-conformal, in which case, the conformal +factor 𝜇𝑃 of (𝑃; Ω𝑃,hor, d𝑃,hor) is uniquely determined by 𝜇𝑃(1) = 𝜇[𝐸,∇𝐸]. +4.2. The lifting problem for Riemannian structures via Hodge operators. We now at- +tack the problem of lifting Riemannian geometries in terms of Hodge operators to the total +spaces of nc principal U(1)-bundles with connection. The existence of such lifts will be +entirely governed by conformality in our nc sense. As a result, the resulting lifted Riemannian +geometries necessarily involve modular phenomena in both vertical and horizontal directions +that are generally non-trivial and distinct. +In what follows, let 𝜅 > 0, let (𝑃; Ω𝑃, d𝑃; Π) be a 𝜅-differentiable quantum principal U(1)- +bundle with connection over 𝐵, let 𝜗 be the connection 1-form of Π, and let ˆΦ𝑃 be the Fröhlich +automorphism of Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) = (𝑃, Ω𝑃,hor, d𝑃,hor). +We begin with a general definition of U(1)-equivariant Hodge operator on a total space that +draws on standard requirements imposed in the classical case: that the canonical surjection +onto the base be a Riemannian submersion, that the principal Ehresmann connection be +fibrewise orthogonal, that the fibres all have unit length, and that the total space have the +’fibre-first’ orientation. We go beyond the definitions proposed by Kustermans–Murphy–Tuset +[63] by carefully controlling failure of the Hodge operator to be right linear and ∗-preserving in +terms of (possibly distinct) modular automorphisms in the vertical and horizontal directions. +On the one hand, we define a modular automorphism of Ω𝑃 is a U(1)-equivariant automor- +phism Δ of Ω𝑃 as a unital graded C-algebra satisfying Δ↾ΩU(1) +𝑃 += id and +∀𝑗 ∈ Z, ∀𝑝 ∈ 𝑃𝑗, +𝑝∗Δ(𝑝) ≥ 0; +(4.15) +for example, given 𝑡 ∈ (0, ∞), we may define a modular automorphism Λ𝑡 of Ω𝑃 by +∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑚 +𝑗 , +Λ𝑡(𝜔) � 𝑡−𝑗𝜔. +(4.16) +On the other hand, we use the connection Π to define a convenient bigrading (Ω𝑗,𝑘 +𝑃 )(𝑗,𝑘) ∈N2 +0 of +Ω𝑃 as follows. For each 𝑘 ∈ {0, . . . , 𝑁}, let +Ω0,𝑘 +𝑃 +� Π(Ω𝑘 +𝑃) = Ω𝑘 +𝑃,hor, +Ω1,𝑘 +𝑃 +� (id −Π)(Ω𝑘+1 +𝑃 ) = 𝜗 · Ω𝑘 +𝑃,hor, +(4.17) +and for (𝑗, 𝑘) ∉ {0, 1} × {0, . . . , 𝑁}, set Ω𝑗,𝑘 +𝑃 � 0. Then the family (Ω𝑗,𝑘 +𝑃 )(𝑗,𝑘) ∈N2 +0 satisfies: +∀𝑚 ∈ {0, . . . , 𝑁 + 1}, +1 +� +𝑗=0 +𝑚−1 +� +𝑘=0 +Ω𝑗,𝑘 +𝑃 = Ω𝑚 +𝑃 , +∀(𝑗1, 𝑘1), (𝑗2, 𝑘2) ∈ N2 +0, +Ω𝑗1,𝑘1 +𝑃 +· Ω𝑗2,𝑘2 +𝑃 += Ω𝑗1+𝑗2,𝑘1+𝑘2 +𝑃 +, +∀(𝑗, 𝑘) ∈ N2 +0, +� +Ω𝑗,𝑘 +𝑃 +�∗ += Ω𝑗,𝑘 +𝑃 . + +54 +BRANIMIR ĆAĆIĆ +Definition 4.18. Let (Δver, Δhor) be a commuting pair of modular automorphisms of Ω𝑃 that +commute with Π. A (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to Π is a +U(1)-equivariant left 𝑃-linear map that commutes with both Δver and Δhor, satisfies +∀(𝑗, 𝑘) ∈ {0, 1} × {0, . . . , 𝑁}, +★ +� +Ω𝑗,𝑘 +𝑃 +� +⊆ Ω1−𝑗,𝑁−𝑘 +𝑃 +, +(4.18) +∀𝑚 ∈ {0, . . . , 𝑁 + 1}, +★2↾Ω𝑚 +𝑃 = (−1)𝑚(𝑁+1−𝑚) idΩ𝑚 +𝑃 , +(4.19) +and satisfies, for every (𝑗, 𝑘) ∈ {0, 1} × {0, . . . , 𝑁}, +∀𝑝 ∈ 𝑃, ∀𝜔 ∈ Ω𝑗,𝑘 +𝑃 , +★(𝜔𝑝) = ★(𝜔) · (Δ2𝑘−𝑁 +hor +◦ Δ2𝑗−1 +ver )(𝑝), +(4.20) +∀𝜔 ∈ Ω𝑗,𝑘 +𝑃 , +★(𝜔)∗ = ★ +� +(Δ2𝑘−𝑁 +hor +◦ Δ2𝑗−1 +ver )(𝜔)∗� +, +(4.21) +∀𝜔 ∈ Ω𝑗,𝑘 +𝑃 , +★ +� +𝛿 𝑗,0𝜔 +� += (−1)𝑁−𝑘★(𝜔𝜗)𝜗, +(4.22) +∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 +𝑃 , +𝜔 · ★(𝜂) = ★−1(𝜔) · (Δ2𝑘−𝑁 +hor +◦ Δ2𝑗−1 +ver )(𝜂). +(4.23) +Hence, in this case, the inverse metric induced by the (Δver, Δhor)-modular Hodge operator ★ +is the R-bilinear map 𝑔 : Ω𝑃 × Ω𝑃 → 𝑃 defined by +∀𝜔, 𝜂 ∈ Ω𝑃, +𝑔(𝜔, 𝜂) � ★(𝜔∗ · ★(𝜂)). +(4.24) +Notwithstanding the appearance of modular automorphisms, the following properties of +inverse metrics will suffice for our purposes. +Proposition 4.19. Let Δver and Δhor be a commuting pair of modular automorphisms of Ω𝑃 that +commute with Π, and let ★ be a (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to +Π. Then the inverse metric 𝑔 : Ω𝑃 × Ω𝑃 → 𝑃 is U(1)-equivariant in the sense that +∀(𝑚, 𝑗), (𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 +𝑃 )𝑗, ∀𝜂 ∈ (Ω𝑛 +𝑃)𝑘, +𝑔(𝜔, 𝜂) ∈ 𝑃−𝑗+𝑘, +(4.25) +makes Π into an orthogonal projection in the sense that +∀𝜔, 𝜂 ∈ Ω𝑃, +𝑔(𝜔, 𝜂) = 𝑔(Π(𝜔), Π(𝜂)) + 𝑔((id −Π)(𝜔), (id −Π)(𝜂)), +(4.26) +and satisfies, for each (𝑗, 𝑘) ∈ {0, 1} × {0, . . . , 𝑁}, +∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 +𝑃 , ∀𝑝 ∈ 𝑃, +𝑔(𝜔, 𝜂 · 𝑝) = 𝑔(𝜔, 𝜂) · (Δ2𝑗 +ver ◦ Δ2𝑘 +hor)(𝑝), +(4.27) +∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 +𝑃 , +𝑔(𝜔, 𝜂)∗ = (Δ2𝑗 +ver ◦ Δ2𝑘 +hor)(𝑔(𝜂, 𝜔)). +(4.28) +Proof. The non-trivial claims are equations 4.26 and 4.28. On the one hand, let 𝜔, 𝜂 ∈ Ω𝑃 be +given; since (id −Π)(Ω𝑃)2 = 0, it now follows by (4.18) that +★(𝑔(𝜔, 𝜂)) = (Π(𝜔∗) + (id −Π)(𝜔∗)) ★ (Π(𝜂) + (id −Π)(𝜂)) += (Π(𝜔∗)) ★ (Π(𝜂)) + (id −Π)(𝜔∗)) ★ ((id −Π)(𝜂)) += ★(𝑔(Π(𝜔), Π(𝜂)) + 𝑔((id −Π)(𝜔), (id −Π)(𝜂))). +On the other hand, let (𝑗, 𝑘) ∈ {0, 1} × {0, . . . , 𝑁} and 𝜔, 𝛽 ∈ Ω𝑗,𝑘 +𝑃 be given; by (4.23), +★(𝑔(𝜔, 𝜂)∗) = Δver ◦ Δ𝑁 +hor(★(𝑔(𝜔, 𝜂))∗) += Δver ◦ Δ𝑁 +hor +� +(−1)(𝑗+𝑘) (𝑁+1−𝑗−𝑘)★(𝜂)∗𝜔 +� += Δver ◦ Δ𝑁 +hor +� +(−1)(𝑗+𝑘) (𝑁+1−𝑗−𝑘) (★ ◦ Δ2𝑗−1 +ver ◦ Δ2𝑘−𝑁 +hor +)(𝜂∗) · 𝜔 +� += Δ2𝑗 +ver ◦ Δ2𝑘 +hor(𝜂∗ · ★(𝜔)) += ★ +� +Δ2𝑗 +ver ◦ Δ2𝑘 +hor(𝑔(𝜂, 𝜔)) +� +. +□ + +NONCOMMUTATIVE U(1)-GAUGE THEORY +55 +At last, the following definitions give our proposed notion of lifted Riemannian geometry. +Definition 4.20. We define a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π) to be a quadruple +(Δver, Δhor, ★, 𝜏), where (Δver, Δhor) is a commuting pair of modular automorphisms of Ω𝑃 +that commute with Π, where ★ is a (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with +respect to Π whose inverse metric restricts, for each (𝑚, 𝑗) ∈ N0 × Z, to a 𝐵-valued inner +product on (Ω𝑚 +𝑃 )𝑗 admits a basis and satisfies +∀𝑏 ∈ 𝐵, ∀𝜔 ∈ (Ω𝑚 +𝑃 )𝑗, +𝑔(𝑏𝜔, 𝑏𝜔) ≤ ∥𝑏∥2𝑔(𝜔, 𝜔), +(4.29) +and where 𝜏 is a U(1)-equivariant bounded state on 𝑃 that satisfies +∀𝜔 ∈ Ω𝑁 +𝑃 , +(𝜏 ◦ ★ ◦ d)(𝜔) = 0; +(4.30) +∀𝑝 ∈ 𝑃, +sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1} = ∥𝑝∥2. +(4.31) +Definition 4.21. Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), let +(Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π), and suppose that +∀𝛽 ∈ Ω𝐵, +★(𝜗𝛽) = ★𝐵(𝛽), +(4.32) +∀𝑏 ∈ 𝐵, +𝜏(𝑏) = 𝜏𝐵(𝑏). +(4.33) +We call (★𝐵, 𝜏𝐵) a restriction of (Δver, Δhor, ★, 𝜏) to (𝐵; Ω𝐵, d𝐵) and we call (Δver, Δhor, ★, 𝜏) a +lift of (★𝐵, 𝜏𝐵) to (𝑃; Ω𝑃, d𝑃; Π). +Our definitions are justified by the following existence and uniqueness theorem, which +characterizes existence of lifts in terms of conformality and demonstrates the inexorability of +non-trivial modular phenomena outside of an narrow range. +Theorem 4.22. Let Λ𝜅 denote the modular automorphism of Ω𝑃 defined by (4.16). +(1) Suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π). There exists +a unique restriction (★𝐵, 𝜏𝐵) of (Δver, Δhor, ★, 𝜏) to (𝐵; Ω𝐵, d𝐵). Moreover, it follows that +Δver = Λ𝜅 and that (𝑃; Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor 𝜇𝑃 satisfying +∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 +𝑃 )𝑗, +Δhor(𝜔) = 𝜔𝜇𝑃(𝑗) +(4.34) +(2) Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵; Ω𝐵, d𝐵), and suppose that (𝑃; Ω𝑃,hor, d𝑃,hor) +is ★𝐵-conformal with conformal factor 𝜇𝑃. Hence, define a modular automorphism Δhor of +Ω𝑃 by (4.34). There exists a unique (Λ𝜅, Δhor)-modular Hodge operator ★ on (Ω𝑃, d𝑃) with +respect to Π and faithful U(1)-equivariant bounded state 𝜏 on 𝑃 making (Λ𝜅, Δhor, ★, 𝜏) into +a lift of (★𝐵, 𝜏𝐵) to (𝑃; Ω𝑃, d𝑃; Π), namely +∀𝑝 ∈ 𝑃, ∀𝑘 ∈ {0, . . . , 𝑁}, ∀𝛽 ∈ Ω𝑘 +𝐵, +★𝑃(𝑝𝛽) � (−1)𝑘𝑝𝜗★𝐵(𝛽), +(4.35) +∀𝑝 ∈ 𝑃, ∀𝑘 ∈ {0, . . . , 𝑁}, ∀𝛽 ∈ Ω𝑘 +𝐵, +★𝑃(𝑝𝜗𝛽) � 𝑝★𝐵(𝛽), +(4.36) +∀𝑗 ∈ Z, ∀𝑝 ∈ 𝑃𝑗, +𝜏𝑃(𝑝) � 𝜏𝐵 +� +𝛿 𝑗,0𝑝 +� +. +(4.37) +Lemma 4.23. For every modular automorphism Δ of Ω𝑃, there exists a unique group 1-cocycle +𝜇 : Z → Z>0(𝐵) for the right Z-action defined by (4.12), such that +∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 +𝑃 )𝑗, +Δ(𝜔) = 𝜔𝜇(𝑗). +(4.38) +Conversely, for every group 1-cocycle 𝜇 : Z → Z>0(𝐵), Equation 4.38 defines a modular +automorphism Δ of Ω𝑃. + +56 +BRANIMIR ĆAĆIĆ +Proof. Let Δ be a modular automorphism. For each 𝑗 ∈ Z, the map Δ restricts to a 𝐵-bimodule +morphism L(𝑃)(𝑗) → L(𝑃)(𝑗), so that, by Proposition 2.16, there exists unique 𝜇(𝑗) ∈ +Z(𝐵) that satisfies (4.38) for 𝑚 = 0; given any cobasis (𝑒𝑖)𝑁 +𝑖=1 for L(𝑃)𝑗, it follows that 0 ≤ +�𝑁 +𝑖=1 𝑒∗ +𝑖 Δ(𝑒𝑖) = �𝑁 +𝑖=1 𝑒∗ +𝑖 𝑒𝑖𝜇(𝑗) = 𝜇(𝑗) and 𝜇(𝑗)𝛼 = �𝑁 +𝑖=1 𝑒∗ +𝑖 Δ(𝑒𝑖)𝛼 = �𝑁 +𝑖=1 𝑒∗ +𝑖 Δ(𝑒𝑖𝛼) = 𝛼𝜇𝑃(𝑗) +for all 𝛼 ∈ Ω𝐵, so that 𝜇𝑃(𝑗) ∈ Z(Ω𝐵)0. Given 𝑗, 𝑘 ∈ Z, for all 𝑥 ∈ 𝑃𝑗, and 𝑦 ∈ 𝑃𝑘, we find +that 𝑥𝑦𝜇(𝑗 + 𝑘) = Δ(𝑥𝑦) = Δ(𝑥)Δ(𝑦) = 𝑥𝜇𝑃(𝑗) · 𝑦 · 𝜇𝑃(𝑘) = 𝑥𝑦Φ−𝑘 +𝑃 (𝜇(𝑗))𝜇(𝑘), so that +𝜇𝑃(𝑗 + 𝑘) = ˆΦ−𝑘 +𝑃 (𝜇𝑃(𝑗))𝜇𝑃(𝑘) by the equality 𝑃𝑗+𝑘 = 𝑃𝑗 · 𝑃𝑘 together with uniqueness of +𝜇𝑃(𝑗 + 𝑘). Since Δ acts as the identity on 𝑃0, it follows that 𝜇(0) = 1; hence, for each 𝑗 ∈ Z, +it follows that 𝜇(𝑗) ∈ Z>0(𝐵) with inverse Φ−𝑗 +𝑃 (𝜇(−𝑗))−1. In sum, we obtain a unique group +1-cocycle 𝜇 : Z → Z(𝐵)× +≥0 satisfying (4.38) for 𝑚 = 0. Finally, since Ω𝑃 = 𝑃 · Ω𝐵 ⊕ 𝑃 · 𝜗 · Ω𝐵 +and since Δ acts as the identity on Ω𝐵 and on 𝜗, it follows that 𝜇 : Z → Z>0(𝐵) is the unique +group 1-cocycle satisfying (4.38) in general. Reversing this argument almost suffices to show +that a group 1-cocycle 𝜇 : Z → Z>0(𝐵) defines a modular automorphism Δ by (4.38); all that +is left is that 𝑝∗Δ(𝑝) = 𝑝∗𝑝 · 𝜄𝑃(𝜇𝑃(𝑗)) = 𝑝∗Φ𝑗 +𝑃(𝜇𝑃(𝑗))𝑝 ≥ 0 for all 𝑗 ∈ Z and 𝑝 ∈ 𝑃𝑗. +□ +Proof of Theorem 4.22. First, suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on +(𝑃; Ω𝑃, d𝑃; Π). We begin by showing that Δver = Λ𝜅. By Lemma 4.23, let 𝜇 : Z → Z>0(𝐵) be +the unique group 1-cocycle satisfying (4.38) with respect to Δ = Δver. Then Δ2 +ver = Λ2 +𝜅 since +★(𝑝) = (−1)𝑁★(𝑝𝜗)𝜗 += (−1)𝑁★(𝜗) · (Δ−𝑁 +hor ◦ Δver ◦ Λ−1 +𝜅 )(𝑝) · 𝜗 += ★(1) · (Δ−𝑁 +hor ◦ Δver ◦ Λ−2 +𝜅 )(𝑝) += ★�Δ2 +ver ◦ Λ−2 +𝜅 (𝑝)� +for every 𝑝 ∈ 𝑃. Let 𝑗 ∈ Z and let (𝑒𝑖)𝑁 +𝑖=1 be a cobasis for L(𝑃)𝑗. Then 𝜇(𝑗) = 𝜅−𝑗 by positivity +of 𝜇(𝑗) together with the calculation 𝜅−2𝑗 = �𝑁 +𝑖=1 𝑒∗ +𝑖 Λ2 +𝜅(𝑒𝑖) = �𝑁 +𝑖=1 𝑒∗ +𝑖 Δ2 +ver(𝑒𝑖) = 𝜇(𝑗)2. +Next, we show that there is a unique Hodge operator ★𝐵 on 𝐵 with respect to (Ω𝐵, d𝐵) +satisfying (4.32). By (4.22) and (4.19), there exists a unique C-linear map ★𝐵 : Ω𝐵 → Ω𝐵 +satisfying (4.32), which is given by ★𝐵(𝛽) � ★(𝜗𝛽) for all 𝑘 ∈ {0, . . . , 𝑁} and 𝛽 ∈ Ω𝑘 +𝐵. The +map ★𝐵 is left 𝐵-linear by construction and U(1)-equivariant by U(1)-equivariance of ★ and +U(1)-invariance of 𝜗. Moreover, since both Λ𝜅 and Δhor act as the identity on Ω𝐵 and on 𝜗 and +since 𝜗 supercommutes with Ω𝐵, it follows that ★𝐵 is right 𝐵-linear by (4.20), is ∗-preserving +by (4.21), satisfies (4.1) by (4.18) and (4.19), and satisfies (4.2) by (4.23). +Next, we show that the pair (★𝐵, 𝜏↾𝐵) defines a Riemannian geometry on 𝐵 with respect +to (Ω𝐵, d𝐵). On the one hand, since both Λ𝜅 and Δhor act trivially on Ω𝐵, Proposition 4.19 +together with the 𝑗 = 0 case of (4.29) shows that the inverse metric induced by ★𝐵 satisfies +(4.4); indeed, for each 𝑚 ∈ {0, . . . , 𝑀}, one can obtain a basis for Ω𝑚 +𝐵 from a basis for (Ω𝑚 +𝑃 )U(1) +by applying Π retaining any non-zero vectors. On the other hand, for every 𝛽 ∈ Ω𝑁−1 +𝐵 +, since +d𝑃(𝜗𝛽) = d𝑃(𝜗)𝛽 − 𝜗d𝐵(𝛽) = −FΠ𝛽 − 𝜗d𝐵(𝛽) = −𝜗d𝐵(𝛽) for FΠ the curvature 2-form of +the connection Π, it follows that 𝜏 ◦ ★𝐵 ◦ d𝐵(𝛽) = 𝜏(−★(𝜗d𝐵(𝛽))) = 𝜏 ◦ ★ ◦ d(𝜗𝛽) = 0. +Finally, by Lemma 4.23, let 𝜇𝑃 : Z → Z>0(𝐵) be the unique group 1-cocycle satisfying +(4.34). Given (4.34) and the fact that Δhor(𝜗) = 𝜗, that (𝑃; Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with +conformal factor 𝜇𝑃 is now equivalent to (4.20). Uniqueness of 𝜏𝐵 is trivial. +Now, let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), and suppose +that (𝑃; Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor ★𝐵. By Lemma 4.38, the modular +automorphism Δhor of Ω𝑃 constructed from 𝜇𝑃 by (4.34) is well-defined. +Let us first show that there is a unique (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) +with respect to Π satisfying (4.32). First, by Proposition 3.24, (4.35) and (4.36) define the unique + +NONCOMMUTATIVE U(1)-GAUGE THEORY +57 +U(1)-equivariant left 𝑃-linear map ★ : Ω𝑃 → Ω𝑃 satisfying (4.32) and (4.22). Next, the map ★ +satisfies (4.18) by construction, satisfies (4.20) by (4.34) and (4.13), satisfies (4.19) by (4.1) and left +𝑃-linearity, and satisfies (4.21) by the fact that ★𝐵 is ∗-preserving together with left 𝑃-linearity +of ★ and (4.20). Finally, the map ★ satisfies (4.23) by a case-by-case application of (4.2) together +with (4.20) and left 𝑃-linearity of ★. +Next, let us show that the inverse metric 𝑔 induced by ★ satisfies the requirements in the +definition of total Riemannian geometry. Let 𝑔𝐵 denote the inverse metric induced by ★𝐵. Let +(𝑚, 𝑗) ∈ {0, . . . , 𝑁} × Z be given, and let (·, ·)𝑗 denote the 𝐵-valued inner product on L(𝑃)(𝑗). +Let 𝑝1, 𝑝2 ∈ 𝑃𝑗 and 𝛼1, 𝛼2 ∈ Ω𝑚 +𝐵 be given. On the one hand, +★(𝑔(𝑝1𝛼1, 𝑝2𝛼2)) = 𝛼∗ +1𝑝∗ +1★(𝑝2𝛼2) = 𝛼∗ +1𝑝∗ +1𝑝2(−1)𝑁★𝐵(𝛼2)𝜗 = ★�𝑔𝐵(𝛼1, (𝑝1, 𝑝2)𝑗𝛼2)�, +while on the other, a similar calculation shows that 𝑔(𝑝1𝛼1𝜗, 𝑝2𝛼2𝜗) = 𝑔𝐵(𝛼1, (𝑝1, 𝑝2)𝑗𝛼2). +Thus, the 𝐵-bimodule isomorphism ˆℓ𝑃 of Proposition 3.24 induces 𝐵-bimodule isomorphisms +�𝜔 ↦→ ˆℓ𝑃(𝜔)� : (Ω𝑚 +𝑃,hor)𝑗 → L(𝑃)(𝑗)⊗𝐵Ω𝑚 +𝐵, +�𝜔𝜗 ↦→ ˆℓ𝑃(𝜔)� : 𝜗(Ω𝑚 +𝑃,hor)𝑗 → L(𝑃)(𝑗)⊗𝐵Ω𝑚 +𝐵 +that respectively realise the restrictions of 𝑔 to (Ω𝑚 +𝑃,hor)𝑗 and 𝜗(Ω𝑚 +𝑃,hor)𝑗 as pullbacks of the +𝐵-valued inner product on the tensor product L(𝑃)(𝑗) ⊗𝐵 Ω𝑚 +𝐵 of 𝐵-self-correspondences +of finite type. Hence, both (Ω𝑚 +𝑃,hor)𝑗 = Π((Ω𝑚 +𝑃 )𝑗) and 𝜗(Ω𝑚 +𝑃,hor)𝑗 = (id −Π)((Ω𝑚+1 +𝑃 +)𝑗) defines +𝐵-self-correspondences of finite type with respect to 𝑔, which suffices for our purposes. +Finally, let us show that (4.37) defines the unique U(1)-equivariant bounded state 𝜏 on 𝑃 +satisfying 𝜏↾𝐵= 𝜏𝐵 and satisfying (4.30) and (4.31). Recall the bounded faithful conditional +expectation E𝑃 : 𝑃 → 𝐵 of Proposition 3.16. First, the map 𝜏 : 𝑃 → C defined by (4.37) can now +be rewritten as 𝜏 = 𝜏𝐵 ◦ E𝑃, which therefore defines a faithful bounded U(1)-equivariant state +on 𝑃 restricting to 𝜏𝐵 on 𝐵. Next, by continuity and U(1)-invariance, any faithful bounded +U(1)-equivariant state 𝜏′ on 𝑃 satisfying 𝜏′↾𝐵= 𝜏𝐵 must satisfy 𝜏′ = 𝜏′ ◦ E𝑃 = 𝜏𝐵 ◦ E𝑃 = 𝜏. +Finally, we show that 𝜏 satisfies (4.30) with respect to ★. On the one hand, let 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗, +and 𝛽 ∈ Ω𝑁 +𝐵 . Since 𝛿 𝑗,0d𝑃(𝑝𝛽) = 𝛿 𝑗,0 �2𝜋i[𝑗]𝜅𝜗𝑝𝛽 + d𝑃,hor(𝑝)𝛽 + 𝑝d𝐵𝛽� = 0, it follows by +(4.35) that 𝜏 ◦ ★ ◦ d(𝑝𝛽) = 𝜏𝐵 ◦ ★�𝛿 𝑗,0d(𝑝𝛽)� = 0. On the other hand, let 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗, and +𝛼 ∈ Ω𝑁−1 +𝐵 +. Since +𝛿 𝑗,0d𝑃(𝑝𝛼𝜗) = d𝑃 +� +(𝛿 𝑗,0𝑝)𝛼𝜗 +� += d𝐵 +� +(𝛿 𝑗,0𝑝)𝛼 +� +· 𝜗 + (−1)𝑁 (𝛿 𝑗,0𝑝)𝛼FΠ = d𝐵 +� +(𝛿 𝑗,0𝑝)𝛼 +� +𝜗, +it follows by (4.36) and (4.37) that +𝜏 ◦ ★ ◦ d(𝑝𝛼𝜗) = 𝜏𝐵 ◦ ★ +� +d𝑃(𝛿 𝑗,0𝑝𝛼𝜗) +� += (−1)𝑁𝜏𝐵 ◦ ★𝐵 ◦ d𝐵 +� +(𝛿 𝑗,0𝑝)𝛼 +� += 0. +Thus, either way, the composition 𝜏 ◦ ★ ◦ d↾Ω𝑁 +𝑃 vanishes. +Let us now show that 𝜏 satisfies (4.31). Define ∥ · ∥′ : 𝑃 → [0, ∞) by +∀𝑝 ∈ 𝑃, +(∥𝑝∥′)2 � sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1}. +Since ∥ · ∥′ is the operator norm on 𝑃 with respect to the gns representation of 𝑃 induced by +the faithful bounded state 𝜏, it follows that ∥ · ∥′ is a 𝐶∗-norm bounded from above by ∥ · ∥; +since 𝜏 is U(1)-invariant, it follows that ∥ · ∥′ is a U(1)-invariant 𝐶∗-norm on 𝑃. Hence, by +Corollary 3.19, it suffices to show that 𝜏 satisfies (4.31) on 𝑃U(1) = 𝐵. But now, given 𝑏 ∈ 𝐵, it +follows from (4.6) applied to 𝜏𝐵 that +(∥𝑏∥′)2 = sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1} ≥ sup{𝜏(𝑐∗𝑝∗𝑝𝑐) | 𝑞 ∈ 𝐵, 𝜏(𝑐∗𝑐) ≤ 1} = ∥𝑏∥2. +□ +The construction of Lemma 4.23 will be used frequently enough to warrant the following. + +58 +BRANIMIR ĆAĆIĆ +Definition 4.24. Equip Z>0(𝐵) with the right Z-action constructed from ˆΦ𝑃 by (4.12). The +symbol of a modular automorphism Δ is the unique group 1-cocycle 𝜇 : Z → Z>0(𝐵) that +satisfies (4.38) with respect to Δ. +In particular, we may use Proposition 4.17 to rewrite Theorem 4.22 as follows. +Corollary 4.25. Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (Ω𝐵, d𝐵). Let (𝐸, 𝜎𝐸, ∇𝐸) be a +Hermitian line 𝐵-bimodule with connection that is flat or has vertical deformation parameter 𝜅. +Then (★𝐵, 𝜏𝐵) admits a lift (Δver, Δhor, ★, 𝜏) to (𝐵; Ω𝐵, Σ𝐵)⋊𝜅,tot +(𝐸,𝜎𝐸,∇𝐸)Z if and only if (𝐸, 𝜎𝐸, ∇𝐸) is +★𝐵-conformal, in which case the lift is unique, Δver = Λ𝜅, and Δhor has symbol 𝜇◦(𝑚 ↦→ [𝐸, ∇𝐸]𝑚). +Example 4.26. We continue from Examples 3.51 and 4.4. On the one hand, observe that +(O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) = Hor𝜅(O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞, Π𝑞) +is ★𝑞-conformal with conformal factor 𝜇O𝑞(SU(2)) = (𝑘 ↦→ 𝑞−𝑘); compare [99, Lemma 5.6]. On +the other hand, recall that the usual basis for the free left O𝑞(SU(2))-module Ω𝑞(SU(2)) is +{𝑒0, 𝑒+, 𝑒−}, where 𝑒0 � 2𝜋i𝑞−2𝑒𝑞2. Hence, by Theorem 4.22, the unique lift of (★𝑞, ℎ𝑞↾O𝑞(CP1)) +to (O𝑞(SU(2)); Ω𝑞(SU(2)), d𝑞; Π𝑞) is (Λ𝑞2, Λ𝑞, �★𝑞, ℎ𝑞), where �★𝑞 is uniquely determined by +�★𝑞(1) � 𝑞2 +2𝜋 𝑒0𝑒+𝑒−, +�★𝑞(𝑒0) � − 2𝜋 +𝑞2 𝑒+𝑒−, +�★𝑞(𝑒+) � − 𝑞6 +2𝜋 𝑒0𝑒+, +�★𝑞(𝑒−) � +1 +2𝜋𝑞2 𝑒0𝑒−, +and ℎ𝑞 is Woronowicz’s Haar state on O𝑞(SU(2)). In fact, the operator �★𝑞 recovers the Hodge +operator of Zampini [99, Eq. 4.20] for the choice of parameters (𝛼′, 𝛽, 𝜈, 𝛾) = ( −2𝜋 +𝑞4 , 1, 𝑞−2, 4𝜋2 +𝑞4 ), +which satisfies his canonical constraints [99, Remark 5.7] with respect to the choice of parameter +𝛼′′ = −𝑞2 from Example 4.26; it also necessarily recovers the Hodge operator of Kustermans– +Murphy–Tuset [63, Thm. 8.1 et seq.] up to suitable rescaling in each respective degree. Note +that Λ𝑞2 ≠ Λ𝑞 since 𝑞 ≠ 1. +Example 4.27. We continue from Examples 3.52 and 4.9. By Example 2.41, the homomorphism +ˆ𝐸 of Example 2.31 defines a homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ +𝜃 (T2); ★) satisfying +∀𝑔 ∈ Γ𝜃, +𝜇 ◦ 𝜋0( ˆ𝐸)(𝑔) = (𝑔21𝜃 + 𝑔22)−1. +Hence, by Proposition 4.17 and Theorem 4.22, the unique lift of (★, 𝜏) from Example 4.9 to the +real multiplication instanton (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃) is (Λ𝜖2 +𝜃, Λ𝜖𝜃, �★,�𝜏), where �★ is determined by +�★(1) � 𝑒0𝑒1𝑒2, +�★(𝑒0) � 𝑒1𝑒2, +�★(𝑒1) � −𝑒0𝑒2, +�★(𝑒2) � 𝑒0𝑒1, +and ˜𝜏 is determined by ˜𝜏↾𝐶∞ +𝜃 (T2)= 𝜏. Note that Λ𝜖2 +𝜃 ≠ Λ𝜖𝜃 since 𝜖𝜃 > 1. +In fact, these examples typify the important special case where the base manifold is even- +dimensional, the curvature 2-form is a symplectic form, and the Riemannian volume form is +a constant multiple of the appropriate power of the curvature 2-form. +Corollary 4.28. Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵; Ω𝐵, d𝐵). Suppose that 𝑁 is +even and that there exists 𝑐 ∈ R \ {0}, such that ★𝐵(1) = 𝑐F𝑁/2 +Π +. If (★𝐵, 𝜏𝐵) admits a lift +(Δver, Δhor, ★, 𝜏) to (𝑃; Ω𝑃, d𝑃; Π), then (Δver, Δhor) = (Λ𝜅, Λ𝜅1/2). +Proof. Let 𝜗 be the connection 1-form of the connection Π, let 𝜇𝑃 be the conformal factor of +Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) with respect to the Hodge operator ★𝐵, and let (𝜖𝑖)𝑁 +𝑖=1 be a finite family in +𝑃1 satisfying �𝑁 +𝑖=1 𝜖∗ +𝑖 𝜖𝑖 = 1. Then 𝜇𝑃(1) = 𝜅−1/2 since +★𝑃(1) = +𝑁 +∑︁ +𝑖=1 +𝜖∗ +𝑖 ★𝑃 (1)(Δ−𝑁 +hor ◦ Δ−1 +ver)(𝜖𝑖) = +𝑁 +∑︁ +𝑖=1 +𝜖∗ +𝑖 𝜗𝑐F𝑁/2 +Π +𝜖𝑖𝜅𝜇𝑃(1)−𝑁 = ★𝑃(1)𝜅−𝑁/2𝜇𝑃(1)−𝑁. +□ + +NONCOMMUTATIVE U(1)-GAUGE THEORY +59 +We conclude this section by showing that modular phenomena are no obstacle to equipping +Ω𝑃 with an 𝐿2-inner product and computing the formal adjoint of the exterior derivative d𝑃 +in terms of the Hodge star operator. In fact, one can straighforwardly generalise the nc Hodge +decomposition of Theorem 4.7 to total Riemannian geometries by combining the abstract +Hodge decomposition of Ó Buachalla–Sťoviček–Van Roosmalen [80, Prop. 6.3] with Lemma +4.8, but we shall not need it in the sequel. +Proposition 4.29. Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π) with +inverse metric 𝑔. Then Ω𝑃 defines a pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 +defined by +∀𝜔, 𝜂 ∈ Ω𝑃, +⟨𝜔, 𝜂⟩ � 𝜏(𝑔(𝜔, 𝜂)). +(4.39) +Moreover, with respect to this pre-Hilbert space structure, the U(1)-action on Ω𝑃 defines a unitary +representation of finite type, the direct sum decomposition Ω𝑃 = �1 +𝑗=0 +�𝑁 +𝑘=0 Ω𝑗,𝑘 +𝑃 is orthogonal, +the left 𝑃-module structure on Ω𝑃 defines a U(1)-equivariant isometric ∗-representation on 𝑃, the +connection Π defines an orthogonal projection, the operator ★𝑃 is unitary, and the operator d𝑃 is +adjointable with adjoint d∗ +𝑃 = ★−1 ◦ d𝑃 ◦ ★ ◦ 𝜒𝑃. +Proof. Recall the faithful conditional expectation E𝑃 : 𝑃 → 𝐵 of Proposition 3.16, so that the +state 𝜏 satisfies 𝜏 = 𝜏 ◦ E𝑃 = 𝜏𝐵 ◦ E𝑃. +First, let (𝑚, 𝑗) ∈ {0, . . . , 𝑁} × Z be given, so that 𝑔 makes (Ω𝑚 +𝑃 )𝑗 and hence its direct +summands (Ω0,𝑚 +𝑃 )𝑗 = Π((Ω𝑚 +𝑃 )𝑗) and (Ω1,𝑚−1 +𝑃 +)𝑗 = (id −Π)((Ω𝑚 +𝑃 )𝑗) into 𝐵-self-correspondences +of finite type. Since the state 𝜏 is faithful and positive, ⟨·, ·⟩𝜏 restricts to U(1)-invariant positive- +definite inner products on both (Ω0,𝑚 +𝑃 )𝑗 and (Ω1,𝑚−1 +𝑃 +)𝑗; moreover, the proof of Proposition 4.5 +shows that both (Ω0,𝑚 +𝑃 )𝑗 and (Ω1,𝑚−1 +𝑃 +)𝑗 are separable as pre-Hilbert spaces by separability of 𝐵 +as a pre-𝐶∗-algebra. But now, by (4.37), for every (𝑚, 𝑗), (𝑛, 𝑘) ∈ {0, . . . , 𝑁} × Z, 𝜔 ∈ (Ω0,𝑚 +𝑃,hor)𝑗, +and 𝜂 ∈ (Ω0,𝑛 +𝑃,hor)𝑘, we find that ⟨𝜔, 𝜂⟩𝜏 = 𝜏(𝑔(𝜔, 𝜂)) = 𝜏(𝛿𝑚,𝑛𝑔(𝜔, 𝜂)) = 𝜏�𝛿𝑚,𝑛𝛿 𝑗,𝑘𝑔(𝜔, 𝜂)� +and similarly that ⟨𝜔𝜗, 𝜂𝜗⟩𝜏 = 𝜏(𝛿𝑚,𝑛𝑔(𝜔𝜗, 𝜂𝜗)), while ⟨𝜔, 𝜂𝜗⟩𝜏 = ⟨𝜔𝜗, 𝜂⟩𝜏 = 0 by (4.26). This +shows that Ω𝑃 = �1 +𝑗=0 +�𝑁 +𝑘=0 +�∞ +ℓ=−∞(Ω𝑗,𝑘 +𝑃 )ℓ is an orthogonal decomposition with respect +to ⟨·, ·⟩𝜏, so that ⟨·, ·⟩𝜏 defines a U(1)-invariant positive-definite inner product on Ω𝑃, with +respect to which Ω𝑃 is separable and the U(1)-action is unitary and of finite type. +Next, we show that the left 𝑃-module structure on Ω𝑃 yields a U(1)-equivariant isometric +∗-representation of 𝑃; note that U(1)-equivariance is automatic. First, let us show that each +𝑝 ∈ 𝑃 acts as an adjointable operator on Ω𝑃,hor with formal adjoint given by 𝑝∗. Indeed, let +𝑝 ∈ 𝑃 be given. Then, ★(𝑔(𝑝𝜔, 𝜂)) = (𝑝𝜔)∗ · ★(𝜂) = 𝜔∗𝑝∗★(𝜂) = 𝜔∗★(𝑝∗𝜂) = ★(𝑔(𝜔, 𝑝∗𝜂)) +for all 𝜔, 𝜂 ∈ Ω𝑃, so that ⟨𝑝𝜔, 𝜂⟩𝜏 = ⟨𝜔, 𝑝∗𝜂⟩. Now, let us show that each 𝑝 ∈ 𝑃 acts as +a bounded operator on Ω𝑃. Indeed, let 𝑝 ∈ 𝑃 be given, and write 𝑝 = � +𝑘∈Z ˆ𝑝(𝑘), where +ˆ𝑝(𝑘) ∈ 𝑃𝑘 for each 𝑘 ∈ Z, so that 𝐸(𝑝∗𝑝) = 𝐸�� +𝑘,ℓ ∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(ℓ)� = � +𝑘∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(𝑘). Let +(𝑚, 𝑗) ∈ {0, . . . , 𝑁} × Z and let 𝜔 ∈ (Ω𝑚 +𝑃 )𝑗. Then, by adjointability of 𝑝, Equation 4.29, the +proof of Proposition 4.5, and contractivity of 𝐸, +(E𝑃 ◦ 𝑔)(𝑝𝜔, 𝑝𝜔)) = (E𝑃 ◦ 𝑔) +� +𝜔, +∑︁ +𝑘,ℓ ∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(ℓ)𝜔 +� += 𝑔(𝜔, E𝑃(𝑝∗𝑝)𝜔) ≤ ∥𝑝∥2𝑔(𝜔, 𝜔). +Thus, the left 𝑃-module structure defines a bounded ∗-representation of 𝑃. That this ∗- +representation is isometric follows, mutatis mutandis, from the proof of Proposition 4.5. +Next, we show that d𝑃 is adjointable with adjoint ★−1 ◦d𝑃 ◦★◦ 𝜒𝑃. Let 𝑚 ∈ {0, . . . , 𝑁 +1}, +let 𝜔 ∈ Ω𝑚−1 +𝑃 +, and let 𝜂 ∈ Ω𝑚 +𝑃 . Then, since 𝜏𝑃 ◦ ★ ◦ d𝑃(𝜔∗𝜂) = 0 by (4.30), it follows that +d𝑃(𝜔)∗★(𝜂) = d𝑃(𝜔∗𝜂) + (−1)𝑚𝜔∗(d𝑃 ◦ ★)(𝜂) = d𝑃(𝜔∗𝜂) + 𝜔∗★�(★−1 ◦ d𝑃 ◦ ★ ◦ 𝛾𝑃)(𝜂)�. + +60 +BRANIMIR ĆAĆIĆ +Finally, we show that ★𝑃 is unitary. Let (𝑗, 𝑘) ∈ {0, 1} × {0, . . . , 𝑁}; let 𝜔, 𝜂 ∈ Ω𝑗,𝑘 +𝑃 . Then +★(𝜔)∗ · ★(★(𝜂)) = ★ +� +(Δ1−2𝑗 +ver ◦ Δ𝑁−2𝑘 +hor +)(𝜔∗) +� +· (−1)𝑚(𝑁+1−𝑚)𝜂 += (Δ1−2𝑗 +ver ◦ Δ𝑁−2𝑘 +hor +) +� +★−1(𝜔∗) · (Δ2𝑘−𝑁 +hor +◦ Δ2𝑗−1 +ver )(𝜂) +� += (Δ1−2𝑗 +ver ◦ Δ𝑁−2𝑘 +hor +)(𝜔∗ · ★(𝜂)), +which suffices to show that ⟨★(𝜔), ★(𝜂)⟩𝜏 = ⟨𝜔, 𝜂⟩𝜏. +□ +4.3. Unbounded lifts of commutator representations. We now consider the analogous +lifting problem for Connes’s nc Riemannian geometry in terms of spectral triples [32]. In this +approach, analogues of Dirac-type operators—for example, the Hodge–de Rham operator +d + d∗ on a compact oriented Riemannian manifold—simultaneously encode differential +calculus (to first order), index theory, Riemannian geometry, and metric geometry. Follow- +ing Schmüdgen [90], we restrict our attention to the first aspect and consider commutator +representations of ∗-exterior algebras through degree 1. However, the resulting lifted commu- +tator representations also generally involve non-trivial modular phenomena in the form of +unboundedness of represented 1-forms. +Just as before, let 𝜅 > 0, let (𝑃; Ω𝑃, d𝑃; Π) be a 𝜅-differentiable quantum principal U(1)- +bundle with connection over 𝐵, let 𝜗 be the connection 1-form of Π, and let ˆΦ𝑃 be the Fröhlich +automorphism of Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) = (𝑃, Ω𝑃,hor, d𝑃,hor). Moreover, given a pre-Hilbert +space 𝐻, let L(𝐻) denote the unital pre-𝐶∗-algebra of bounded adjointable operators on 𝐻, +which is Z/2Z-graded as a ∗-algebra whenever the 𝐻 is as a pre-Hilbert space. +We begin with a simplified version of the notion of spectral triple, which we shall apply to +the nc base manifold (Ω𝐵, d𝐵). In short, it generalises Clifford actions of 1-forms in terms of +bounded commutators with an abstract Dirac-type operator. For a detailed introduction, see +the survey article of Carey–Phillips–Rennie [28]. +Definition 4.30 (Baaj–Julg [6], Connes [32], Schmüdgen [90]). Let 𝐻 be a separable Z/2Z- +graded pre-Hilbert space equipped with a bounded ∗-homomorphism 𝜋 : 𝐵 → L(𝐻)odd and +an odd formally self-adjoint C-linear map 𝐷 : 𝐻 → 𝐻, so that L(𝐻) defines a 𝐵-bimodule with +respect to 𝜋. We call (𝐻, 𝜋, 𝐷) a bounded commutator representation of (𝐵; Ω𝐵, d𝐵) whenever +there exists a (necessarily unique) 𝐵-bimodule homomorphism 𝜋𝐷 : Ω1 +𝐵 → L(𝐻), such that +∀𝑏 ∈ 𝐵, +𝜋𝐷 ◦ d𝐵(𝑏) = i[𝐷, 𝜋(𝑏)]. +(4.40) +In this case, we say that (𝐻, 𝜋, 𝐷) is faithful whenever 𝜋 is isometric and 𝜋𝐷 is injective. +Remark 4.31. Let 𝔅 denote the 𝐶∗-algebra completion of 𝐵. A bounded commutator rep- +resentation (𝐻, 𝜋, 𝐷) of (𝐵; Ω𝐵, d𝐵) defines an even spectral triple for 𝔅 if and only if 𝐷 is +essentially self-adjoint and has compact resolvent. +Example 4.32 (D˛abrowski–Sitarz [41], Majid [67]). We continue from Example 4.4 and con- +sider the best-known spectral triple on O𝑞(CP1). Let /𝑆𝑞,±(CP1) � O𝑞(SU(2))∓1 with the +inner product ⟨·, ·⟩ defined by ⟨𝑠1, 𝑠2⟩ � ℎ𝑞(𝑠∗ +1𝑠2), for all 𝑠1, 𝑠2 ∈ /𝑆𝑞,±(CP1), where ℎ𝑞 : +O𝑞(SU(2)) → C, as usual, is Woronowicz’s Haar state; hence, by the proof of Proposition 4.5 +together with Nagy’s result [75] on faithfulness of ℎ𝑞 on 𝐶𝑞(SU(2)), each of /𝑆𝑞,±(CP1) defines a +separable pre-Hilbert space admitting isometric 𝜋± : O𝑞(CP1) → L(/𝑆𝑞,±(CP1)), respectively, +given by left multiplication in O𝑞(SU(2)). Hence, let /𝑆𝑞(CP1) � /𝑆𝑞,+(CP1) ⊕ /𝑆𝑞,−(CP1) as an +orthogonal direct sum with Z/2Z-grading id ⊕(− id) and define 𝜋 : O𝑞(CP1) → L(/𝑆𝑞(CP1)) +by setting 𝜋 � (𝑏 ↦→ 𝜋+(𝑏) ⊕ 𝜋−(𝑏)). Finally, let /𝐷𝑞 : /𝑆𝑞(CP1) → /𝑆𝑞(CP1) be Majid’s spin +Dirac operator [67, Prop. 5.5], which is constructed from the maps 𝜕+ and 𝜕− of Example 3.26 + +NONCOMMUTATIVE U(1)-GAUGE THEORY +61 +by /𝐷𝑞 � +� +0 +𝑞−1𝜕+ +𝑞𝜕− +0 +� +. Then (/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) is a faithful bounded commutator representation +of (Ω𝑞(CP1), d𝑞) that recovers Majid’s 𝑞-deformed Clifford action [67, §5] as the induced map +𝜋 /𝐷𝑞 : Ω1 +𝑞(CP1) → L(/𝑆𝑞(CP1)) is. Moreover, a straightforward calculation now shows that +(/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) recovers the spectral triple on O𝑞(CP1) constructed by Dąbrowski–Sitarz +[41] as reformulated by Neshveyev–Tuset [77, §3]. +The following familiar proposition shows that nc Riemannian geometry in terms of abstract +Dirac-type operators generalises nc Riemannian geometry in terms of abstract Hodge star +operators. In particular, it shows that the Hodge–de Rham operator of a compact oriented +Riemannian manifold still makes perfect sense in the nc setting. +Proposition 4.33 (cf. Das–Ó Buachalla–Somberg [36, §3.2]). Let (★, 𝜏) be a Riemannian geom- +etry on (𝐵; Ω𝐵, d𝐵), so that Ω𝐵 defines a Z/2Z-graded separable pre-Hilbert space with respect to +the inner product ⟨·, ·⟩𝜏 induced by (★, 𝜏) and the Z/2Z-grading 𝛾𝐵. Let 𝜋 : 𝐵 → L(Ω𝐵) denote +the bounded ∗-representation of 𝐵 on Ω𝐵 defined by left multiplication. The triple (Ω𝐵, 𝜋, d𝐵 + d∗ +𝐵) +defines a faithful bounded commutator representation of (Ω𝐵, d𝐵) that satisfies +∀𝛼 ∈ Ω1 +𝐵, ∀𝛽 ∈ Ω𝐵 +𝜋d𝐵+d∗ +𝐵 (𝛼)𝛽 = i𝛼 · 𝛽 + i★−1(𝛼 · (★ ◦ 𝛾𝐵)(𝛽)). +(4.41) +Lemma 4.34. Under the hypotheses of Proposition 4.33, given 𝑘 ∈ {0, . . . , 𝑁} and 𝜔 ∈ Ω𝑘 +𝐵, define +e(𝜔) : Ω𝐵 → Ω𝐵 to be left multiplication by 𝜔 in Ω𝐵. Then, for all 𝑘 ∈ {0, . . . , 𝑁} and 𝜔 ∈ Ω𝑘 +𝐵, +the map e(𝜔) defines a bounded operator on the pre-Hilbert space Ω𝐵 with adjoint +e(𝜔)∗ = (−1)𝑘★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑘 +𝐵. +Proof. Let 𝑘 ∈ {0, . . . , 𝑁} and 𝜔 ∈ Ω𝑘 +𝐵 be given. Let 𝑔 be the inverse metric induced by (★, 𝜏), +so that Ω𝐵 defines a 𝐵-self-correspondence of finite type with respect to 𝑔. Hence, the right +𝐵-linear map e(𝜔) is adjointable and bounded as an operator on (Ω𝐵, 𝑔) with operator norm +∥e(𝜔)∥ < +∞. Moreover, given 𝑗 ∈ {0, . . . , 𝑁}, 𝛼 ∈ Ω𝑗 +𝐵, and 𝛽 ∈ Ω𝑘+𝑗 +𝐵 , we see that +(e(𝜔)𝛼)∗★(𝛽) = (−1)𝑗𝑘𝛼∗𝜔∗★(𝛽) = 𝛼∗ · ((−1)𝑘★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑘 +𝐵)(𝛽), +so that e(𝜔)∗ = (−1)𝑘★−1◦e(𝜔∗)◦★◦𝛾𝐵 for e(𝜔) as operators on the 𝐵-self-correspondence +of finite type Ω𝐵. But now, recall that ⟨·, ·⟩𝜏 = 𝜏 ◦ 𝑔, which immediately implies that e(𝜔)∗ +remains the adjoint of e(𝜔) as an operator on the pre-Hilbert space Ω𝐵. Then e(𝜔) is also +bounded with operator norm at most ∥e(𝜔)∥, since, for all 𝛼 ∈ Ω𝐵, +⟨e(𝜔)𝛼, e(𝜔)𝛼⟩𝜏 = 𝜏(𝑔(e(𝜔)𝛼, e(𝜔)𝛼)) ≤ 𝜏�∥e(𝜔)∥2𝑔(𝛼, 𝛼)� = ∥e(𝜔)∥2⟨𝛼, 𝛼⟩𝜏. +□ +Proof of Proposition 4.33. In light of Proposition 4.5 and Lemma 4.34, it suffices to show that +∀𝑏 ∈ 𝐵, ∀𝛽 ∈ Ω𝐵, +[d𝐵 + d∗ +𝐵, 𝜋(𝑏)]𝛽 = d𝐵(𝑏) · 𝛽 + ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)). +Hence, let 𝑏 ∈ 𝐵, 𝑘 ∈ {0, . . . , 𝑁}, and 𝛽 ∈ Ω𝑘 +𝐵 be given. On the one hand, the Leibniz rule for +d𝐵 immediately implies that [d𝐵, 𝜋(𝑏)]𝛽 = d𝐵(𝑏) · 𝛽. On the other hand, together with left +𝐵-linearity of 𝛾𝐵 and ★, it also implies that [d∗ +𝐵, 𝜋(𝑏)]𝛽 = ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)), since +d∗ +𝐵(𝑏 · 𝛽) = ★−1 ◦ d𝐵 ◦ ★ ◦ 𝛾𝐵(𝑏 · 𝛽) = ★−1 ◦ d𝐵(𝑏 · (★ ◦ 𝛾𝐵)(𝛽)) += ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)) + 𝑏 · d∗ +𝐵𝛽. +Hence, in the notation of Lemma 4.34, we find that +i[d𝐵 + d∗ +𝐵, 𝜋(𝑏)] = ie(d𝐵𝑏) + i(★−1 ◦ e(d𝐵𝑏) ◦ ★ ◦ 𝛾𝐵) = ie(d𝐵𝑏) + (ie(d𝐵𝑏∗))∗. +□ + +62 +BRANIMIR ĆAĆIĆ +Definition 4.35. Let (★, 𝜏) be a Riemannian geometry on (𝐵; Ω𝐵, d𝐵). The Hodge–de Rham +commutator representation induced by (★, 𝜏) is the faithful bounded commutator representation +(Ω𝐵, 𝜋𝐵, d𝐵 + d∗ +𝐵) of (𝐵; Ω𝐵, d𝐵) constructed from (★, 𝜏) by Proposition 4.33. +We now turn to the construction of commutator representations for (𝑃; Ω𝑃, d𝑃; Π). In +particular, we seek a canonical construction for lifting faithful bounded commutator repre- +sentations of (𝐵; Ω𝐵, d𝐵). The following generalisation of Schmüdgen’s no-go theorem for +quantum SU(2) with Woronowicz’s 3-dimensional calculus [90] shows that faithful bounded +commutator representations of (𝑃; Ω𝑃, d𝑃) do not exist when 𝜅 ≠ 1. This forces us to consider +commutator representations where 1-forms may be represented by unbounded operators. +Proposition 4.36 (cf. Schmüdgen [90, Lemma 6]). Suppose that (𝐻, 𝜋, 𝐷) is a bounded com- +mutator representation of (𝑃, Ω𝑃, d𝑃). If 𝜅 ≠ 1, then (id −Π)(Ω1 +𝑃) ⊆ ker 𝜋𝐷. +Proof. Let (𝑒𝑖)𝑚 +𝑖=1 and (𝜖𝑗)𝑛 +𝑗=1 be finite families in 𝑃1 satsifying �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 and �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1, +respectively, and define bounded completely positive maps 𝜙± : LU(1) (𝐻) → LU(1) (𝐻) by +∀𝑇 ∈ LU(1) (𝐻), +𝜙+(𝑇) � +∑︁𝑚 +𝑖=1 𝜋(𝑒𝑖)𝑇𝜋(𝑒∗ +𝑖 ), +𝜙−(𝑇) � +∑︁𝑛 +𝑗=1 𝜋(𝜖∗ +𝑗 )𝑇𝜋(𝜖𝑗), +which are unit preserving and hence contractive. Since 𝜅−1 �𝑚 +𝑖=1 𝑒𝑖𝜗𝑒∗ +𝑖 = 𝜗 = 𝜅 �𝑚 +𝑗=1 𝜖∗ +𝑗 𝜗𝜖𝑗, it +follows that ∥𝜋𝐷(𝜗)∥ = 𝜅∓1∥𝜙± ◦ 𝜋𝐷(𝜗)∥ ≤ 𝜅∓1∥𝜋𝐷(𝜗)∥. Thus, if 𝜅 ≠ 1, then 𝜋𝐷(𝜗) = 0, so +that 𝜋𝐷 vanishes on (id −Π)(Ω1 +𝑃) = 𝑃 · 𝜗. +□ +Example 4.37 (Schmüdgen [90, Thm. 3]). Continuing from Example 3.51, let us suppose that +(𝐻, 𝜋, 𝐷) is a bounded commutator representation of (O𝑞(SU(2)); Ω𝑞(SU(2)), d𝑞). On the +one hand, since 𝑞2 ≠ 1, Proposition 4.36 shows that 𝜋𝐷(𝑒0) = 0. On the other hand, since +𝑞 ≠ 1, the proof of Proposition 4.36, mutatis mutandis, shows that 𝜋𝐷(𝑒±) = 0. Hence, it +follows that 𝜋𝐷 = 0. Note that this recasts Schmüdgen’s original argument [90, Lemma 6 et +seq.] in way that does not use unitarity of +� +𝑎 −𝑞𝑐∗ +𝑐 +𝑎∗ +� +∈ 𝑀2(O𝑞(SU(2))). +Example 4.38. Continuing from Example 3.52, let us suppose that (𝐻, 𝜋, 𝐷) is a bounded +commutator representation of (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃). On the one hand, since 𝜖2 +𝜃 ≠ 1, Proposition 4.36 +shows that 𝜋𝐷(𝑒0) = 0. On the other hand, since 𝜖𝜃 ≠ 1, the proof of Proposition 4.36, mutatis +mutandis, shows that 𝜋𝐷(𝑒1) = 0 and 𝜋𝐷(𝑒2) = 0. Hence, it follows that 𝜋𝐷 = 0. +This catastrophic failure of bounded commutator representations to accommodate im- +portant examples of nc differentiable principal U(1)-bundles forces us to consider a more +general notion of commutator representation where elements of Ω1 +𝑃 may be represented by +unbounded operators. The type of unboundedness that arises can be characterized as follows. +Definition 4.39. Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped with a +unitary representation 𝑉 : U(1) → L(𝐻)even of finite type. We say that an operator 𝑇 : 𝐻 → +𝐻 is locally bounded whenever it satisfies both of the following conditions: +(1) for all 𝑗, 𝑘 ∈ Z, the map 𝑃𝑗𝑇𝑃𝑘↾𝐻𝑘: 𝐻𝑘 → 𝐻𝑗 is bounded and adjointable; +(2) the set {𝑐 ∈ Z | ∃𝑘 ∈ Z, 𝑃𝑘+𝑐𝑇𝑃𝑘 ≠ 0} is bounded. +Hence, we denote by LU(1) +loc (𝐻) the Z/2Z-graded unital ∗-algebra of locally bounded operators +on 𝐻, where the ∗-operation is given by taking operator adjoints, and where the Z/2Z-grading +is induced by the Z/2Z-grading on 𝐻. At last, set LU(1) (𝐻) � L(𝐻) ∩ LU(1) +loc (𝐻). +Example 4.40. Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped with a unitary +representation 𝑉 : U(1) → U(1)(L(𝐻))even of finite type. Then each (𝑇𝑗)𝑗∈Z ∈ � +𝑗∈Z L(𝐻𝑗) + +NONCOMMUTATIVE U(1)-GAUGE THEORY +63 +yields a U(1)-equivariant operator � +𝑗∈Z 𝑇𝑗 ∈ LU(1) +loc (𝐻)U(1). In particular, given 𝜅 > 0, we +may define even U(1)-equivariant operators Λ𝜅, 𝜕𝜅 ∈ LU(1) +loc (𝐻), respectively, by +Λ𝜅 � +� +𝑗∈Z +𝜅−𝑗 id𝐻𝑗, +𝜕𝜅 � +� +𝑗∈Z +2𝜋i[𝑗]𝜅 id𝐻𝑗, +(4.42) +so that Λ𝜅 is formally self-adjoint while 𝜕𝜅 is formally skew-adjoint. +We now weaken the definition of bounded commutator representation accordingly. +Definition 4.41. Let 𝐴 be a U(1)-pre-𝐶∗-algebra of finite type, and let (Ω, d) be a U(1)-∗- +quasi-dga of finite type over 𝐴. Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped +with a unitary representation𝑉 : U(1) → L(𝐻)even of finite type, a U(1)-equivariant bounded +∗-automorphism 𝜋 : 𝐴 → LU(1) (𝐻)even, and a U(1)-invariant odd formally self-adjoint C- +linear map 𝐷 : 𝐻 → 𝐻, so that LU(1) +loc (𝐻)odd defines a 𝐴-bimodule with respect to 𝜋. We +call (𝐻, 𝜋, 𝐷) a locally bounded commutator representation of (𝐴; Ω, d) whenever there exists a +(necessarily unique) 𝐴-bimodule homomorphism 𝜋𝐷 : Ω1 → LU(1) +loc (𝐻)odd, such that +∀𝑎 ∈ 𝐴, +𝜋𝐷 ◦ d(𝑎) = i[𝐷, 𝜋(𝑎)]. +(4.43) +In this case, we say that (𝐻, 𝜋, 𝐷) is faithful whenever 𝜋 is isometric and 𝜋𝐷 is injective. +At last, we propose a refined notion of locally bounded commutator representation for +𝜅-differentiable quantum principal U(1)-bundles over 𝐵. In the case where 𝜅 = 1, it reduces to +a multigraded variation on a Dąbrowski–Sitarz’s definition of principal U(1)-spectral triples +[42] in the spirit of Ćaćić–Mesland [26]. +Definition 4.42. A projectable commutator representation of (𝑃; Ω𝑃, d𝑃; Π) is a quadruple of +the form (𝐻, 𝜋, 𝐷,Γ), where: +(1) the datum (𝐻, 𝜋, 𝐷) is a locally bounded commutator representation of (𝑃, Ω𝑃, d𝑃), such +that (𝑝 ⊗ 𝜉 ↦→ 𝜋(𝑝)𝜉) : 𝑃 ⊗𝐵 𝐻U(1) → 𝐻 is bijective and 𝜋𝐷(𝜗)2 = Λ2 +𝜅; +(2) the datum Γ ∈ LU(1) (𝐻) is an even U(1)-invariant self-adjoint unitary commuting with +ran 𝜋 and anticommuting with 𝜋𝐷(𝜗), such that the horizontal Dirac operator +𝐷hor � 1 +2 (𝐷 + Γ𝐷Γ) +(4.44) +supercommutes with 𝜋𝐷(𝜗) and the remainder +𝑍 � 1 +2 (𝐷 − Γ𝐷Γ) + i𝜋𝐷(𝜗)𝜕𝜅 +(4.45) +is bounded and supercommutes with ran 𝜋. +In this case, we say that (𝐻, 𝜋, 𝐷,Γ) is faithful whenever (𝐻, 𝜋, 𝐷) is faithful and the maps +�𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : 𝐵 → L(𝐻U(1)), +�𝛽 ↦→ 𝜋𝐷(𝛽)↾𝐻U(1) � : Ω1 +𝐵 → L(𝐻U(1)) +are isometric and injective, respectively. +Remark 4.43. Let 𝔓 denote the 𝐶∗-algebra completions of 𝑃. A projectable commutator +representation (𝐻, 𝜋, 𝐷,Γ) of 𝑃 can be viewed as defining a formal U(1)-equivariant un- +bounded 𝐾𝐾1-cycle (𝑃, 𝐻, 𝐷) for (𝔓, C), where the U(1)-invariant odd self-adjoint unitary +−iΓ𝜋𝐷(𝜗)Λ−1 +𝜅 generates the requisite 1-multigrading. If 𝜅 = 1, the horizontal Dirac operator +𝐷hor has bounded commutators with 𝜋(𝑃), and the operator 𝐷 is essentially self-adjoint +with compact resolvent, then (𝑃, 𝐻, 𝐷) gives rise to a genuine U(1)-equivariant odd spectral +triple for 𝔓. Otherwise, the formal unbounded 𝐾𝐾1-cycle (𝑃, 𝐻, 𝐷) generally lies outside the +current scope of unbounded 𝐾𝐾-theory. + +64 +BRANIMIR ĆAĆIĆ +The following proposition shows that a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃; Π) +induces a canonical projectable commutator representation just as a Riemannian geometry +on (𝐵; Ω𝐵, d𝐵) induces a bounded commutator representation. This demonstrates the non- +triviality of our definitions. +Proposition 4.44. Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃, Π). +Hence, view Ω𝑃 as a Z/2Z-graded separable pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 +induced by (Δver, Δhor, ★, 𝜏) and the Z/2Z-grading 𝛾𝑃, so that the U(1)-action ˆ𝜎 on Ω𝑃 defines +a unitary U(1)-representation of finite type by even operators. Let 𝜋 : 𝑃 → L(Ω𝑃) denote the +isometric ∗-representation of 𝑃 on Ω𝑃 defined by left multiplication. Then (Ω𝑃, 𝜋, d𝑃 +d∗ +𝑃, 2Π−id) +defines a faithful projectable commutator representation of (𝑃, Ω𝑃, d𝑃, Π) that satisfies +∀𝜔 ∈ Ω1 +𝑃, ∀𝜂 ∈ Ω𝑃, +𝜋d𝑃+d∗ +𝑃 (𝜔)𝜂 = i𝜔 · 𝜂 + ★−1(i𝜔 · (★ ◦ 𝛾𝑃)(𝜂)). +(4.46) +Moreover, the remainder 𝑍 of (Ω𝑃, 𝜋, d𝑃 + d∗ +𝑃, 2Π − id) is given by +∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝛼1, 𝛼2 ∈ Ω𝐵, +𝑍(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −𝑝2FΠ𝛼2 − 𝑝1𝜗★−1 +𝐵 (FΠ★𝐵(𝛼1)), +(4.47) +where FΠ is the curvature 2-form of the connection Π and where (★𝐵, 𝜏𝐵) is the restriction of +(Δver, Δhor, ★, 𝜏) to (𝐵; Ω𝐵, d𝐵). +Lemma 4.45. Under the hypotheses of Proposition 4.44, let 𝑚 ∈ {0, . . . , 𝑁 + 1} and 𝜔 ∈ Ω𝑚 +𝑃 be +given, and let e(𝜔) : Ω𝑃 → Ω𝑃 be left multiplication by 𝜔 in Ω𝑃. Then e(𝜔) ∈ LU(1) +loc (𝐻) and +e(𝜔)∗ = (−1)𝑚★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑚 +𝑃 . +Proof. The proof of Lemma 4.34 applies verbatim except for showing that e(𝜔) ∈ LU(1) +loc (𝐻). +First, suppose that 𝑚 = 1 and 𝜔 = 𝜗. Let (𝑚, 𝑗) ∈ {0, . . . , 𝑁 + 1} × Z and let 𝜂 ∈ (Ω𝑚 +𝑃 )𝑗, +so that 𝜂 = 𝜂1 + 𝜗𝜂2 for 𝜂1 ∈ (Ω𝑚 +𝑃,hor)𝑗 and 𝜂2 ∈ (Ω𝑚−1 +𝑃,hor)𝑗. Then e(𝜔)(𝜂) = 𝜗𝜂1 ∈ (Ω𝑚+1 +𝑃 +)𝑗, +so that ⟨e(𝜔)(𝜂), e(𝜔)(𝜂)⟩𝜏 = ⟨𝜗𝜂1, 𝜗𝜂1⟩𝜏 = 𝜅−2𝑗⟨𝜂1, 𝜂1⟩𝜏 ≤ 𝜅−2𝑗⟨𝜂, 𝜂⟩𝜏. by the proof of +Proposition 4.29. Thus, given 𝑗, 𝑘 ∈ Z, we see that E𝑗e(𝜔)E𝑘 ≠ 0 only if 𝑗 = 𝑘, in which case +∥E𝑗e(𝜔)E𝑗∥ ≤ 𝜅−𝑗. Hence, the operator e(𝜔) is locally bounded when 𝜔 = 𝜗. +Next, suppose that 𝜔 ∈ Ω𝑚 +𝐵. Let (𝑟, 𝑠, 𝑗) ∈ {0, 1} × {0, . . . , 𝑁} × Z be given, and recall +from Proposition 4.29 that both (Ω𝑟,𝑠 +𝑃 )𝑗 and (Ω𝑟,𝑠+𝑚 +𝑃 +)𝑗 are 𝐵-self-correspondences of finite type +with respect to the inverse metric 𝑔 induced by ★. Let 𝑆𝑟,𝑠 +𝑗 denote the restriction of e(𝜔) to +(Ω𝑟,𝑠 +𝑃 )𝑗, whose range is therefore contained in (Ω𝑟,𝑠+𝑚 +𝑃 +)𝑗. Since 𝑆𝑟,𝑠 +𝑗 +: (Ω𝑟,𝑠 +𝑃 )𝑗 → (Ω𝑟,𝑠+𝑚 +𝑃 +)𝑗 is right +𝐵-linear, it is bounded as a map of right pre-Hilbert 𝐵-modules, and hence, since ⟨·, ·⟩𝜏 = 𝜏 ◦ 𝑔, +as a map of pre-Hilbert spaces. Thus, given 𝑗, 𝑘 ∈ Z, it follows that E𝑗e(𝜔)E𝑘 ≠ 0 only if 𝑗 = 𝑘, +in which case ∥E𝑗e(𝜔)E𝑗∥ ≤ sup{∥𝑆𝑟,𝑠 +𝑗 ∥ | (𝑟, 𝑠) ∈ {0, 1} × {0, . . . , 𝑁}}. Hence, the operator +e(𝜔) is locally bounded when 𝜔 ∈ Ω𝑚 +𝐵. +Let us finally consider the general case. Without loss of generality, there exist 𝑝1, 𝑝2 ∈ 𝑃, +𝛼1 ∈ Ω𝑚 +𝐵, and 𝛼2 ∈ Ω𝑚−1 +𝐵 +, such that 𝜔 = 𝑝1𝛼1 + 𝑝2𝜗𝛼2. Recall that 𝜋(𝑝1), 𝜋(𝑝2) ∈ LU(1) (Ω𝑃) +by Proposition 4.29. Hence, e(𝜔) = 𝜋(𝑝1)e(𝛼1) + 𝜋(𝑝2)e(𝜗)e(𝛼2) ∈ LU(1) +loc (Ω𝑃). +□ +Proof of Proposition 4.44. In what follows, let e : Ω𝑃 → LU(1) +loc (Ω𝑃) be the U(1)-equivariant +C-linear map defined by Lemma 4.45 together with linearity, let i : Ω𝑃 → LU(1) +loc (Ω𝑃) be the +U(1)-equivariant C-linear map defined by i(𝜔) � (−1)𝑚★−1 ◦ e(𝜔) ◦ ★ ◦ 𝛾𝑚 +𝑃 = e(𝜔∗)∗ for +all 𝑚 ∈ {0, . . . , 𝑁 + 1} and 𝜔 ∈ Ω1 +𝑃, let 𝑐 � i(e− i), and set 𝐷 � d𝑃 + d∗ +𝑃, By analogy, define +maps e𝐵, i𝐵, 𝑐𝐵 : Ω𝐵 → L(Ω𝐵) and set 𝐷𝐵 � d𝐵 + d∗ +𝐵, so that 𝜋𝐷𝐵 = 𝑐𝐵↾Ω1 +𝐵. Finally, let 𝜗 +denote the connection 1-form of Π, let ∇ � ˆℓ𝑃 ◦ Π ◦ d𝑃↾𝑃, where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is + +NONCOMMUTATIVE U(1)-GAUGE THEORY +65 +the U(1)-equivariant isomorphism of 𝐵-bimodules of Proposition 3.24, let 𝐷ver � −i𝜋𝐷(𝜗)𝜕𝜅, +and let Γ � 2Π − id. +First, after substituting Proposition 4.29 for Proposition 4.5 and Lemma 4.45 for Lemma +4.34, the proof of Proposition 4.33 shows that (Ω𝑃, 𝜋, 𝐷) defines a faithful locally bounded +commutator representation satisfying (4.46). Moreover, Proposition 3.24 combined with +Theorem 3.46 yields bijectivity of the multiplication map (𝑝 ⊗ 𝜔 ↦→ 𝑝𝜔) : 𝑃 ⊗𝐵 ΩU(1) +𝑃 +→ Ω𝑃. +Let us now consider the operator 𝜋𝐷(𝜗), the would-be horizontal Dirac operator 𝐷hor, and +the would-be remainder 𝑍. Before continuing, note that e(𝜗) maps Ω𝑃,hor to 𝜗 · Ω𝑃,hor and +vanishes on 𝜗 · Ω𝑃,hor = Ω⊥ +𝑃,hor, so that its adjoint i(𝜗) maps 𝜗 · Ω𝑃,hor to Ω𝑃,hor and vanishes +on Ω𝑃,hor; since Γ acts as id on Ω𝑃,hor and as − id on 𝜗 · Ω𝑃,hor, this already suffices to show +that Γ anticommutes with 𝜋𝐷(𝜗). Now, let 𝑝 ∈ 𝑃, 𝑚 ∈ {0, . . . , 𝑁}, and 𝛼 ∈ Ω𝑚 +𝐵 be given. On +the one hand, we find that e(𝜗)(𝑝𝛼) = Λ𝜅(𝑝)𝜗𝛼 and +𝐷(𝑝𝛼) = d𝑃(𝑝𝛼) + (−1)𝑚★−1 ◦ d𝑃(𝑝𝜗 ★𝐵 (𝛼)) += (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 + ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝛼 + 𝑝d𝐵(𝛼) ++ ★−1� +−∇(𝑝) ⟨0⟩𝜗∇(𝑝) ⟨1⟩ ★𝐵 (𝛼) − 𝑝FΠ★𝐵(𝛼) + 𝑝𝜗(d𝐵 ◦ ★𝐵)(𝛼) +� += (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 + ∇(𝑝) ⟨0⟩e𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝d𝐵(𝛼) +− ∇(𝑝) ⟨0⟩i𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝜗i𝐵(FΠ)(𝛼) + 𝑝d∗ +𝐵(𝛼) += +� +−i∇(𝑝) ⟨0⟩𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝𝐷𝐵(𝛼) +� ++ ((Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 − 𝑝𝜗i𝐵(FΠ)(𝛼)) , +so that 𝐷hor(𝑝𝛼) = −i∇(𝑝) ⟨0⟩𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝𝐷𝐵(𝛼), and hence +𝑍(𝑝𝛼) = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 − 𝑝𝜗i𝐵(FΠ)(𝛼) − i𝑐(𝜗)𝜕𝜅(𝑝𝛼) = −𝑝𝜗i𝐵(FΠ)(𝛼). +On the other hand, +i(𝜗)(𝑝𝜗𝛼) = (−1)𝑚★−1(𝜗 · 𝑝★𝐵(𝛼)) = (−1)𝑚★−1(Λ𝜅(𝑝)𝜗★𝐵(𝛼)) = Λ𝜅(𝑝)𝛼, +𝐷(𝑝𝜗𝛼) = d𝑃(𝑝𝜗𝛼) + (−1)𝑚+1★−1 ◦ d𝑃(𝑝★𝐵(𝛼)) += ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝜗𝛼 − 𝑝FΠ𝛼 − 𝑝𝜗d𝐵(𝛼) ++ (−1)𝑚+1★−1� +(Λ𝜅 ◦ 𝜕𝜅)(𝑝)★𝐵(𝛼) + ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩★𝐵(𝛼) + 𝑝(d𝐵 ◦ ★𝐵)(𝛼) +� += ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝜗𝛼 − 𝑝FΠ𝛼 − 𝑝𝜗d𝐵(𝛼) +− (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 + ∇(𝑝) ⟨0⟩𝜗i𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝜗d∗ +𝐵(𝛼) += (−(Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 − 𝑝FΠ𝛼) + +� +i∇(𝑝) ⟨0⟩𝜗𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝐷𝐵(𝛼) +� +, +so that 𝐷hor(𝑝𝜗𝛼) = i∇(𝑝) ⟨0⟩𝜗𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝐷𝐵(𝛼), and hence +𝑍(𝑝𝜗𝛼) = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 − 𝑝FΠ𝛼 − i𝑐(𝜗)𝜕𝜅(𝑝𝜗𝛼) = −𝑝e𝐵(FΠ)(𝛼). +Thus, for all 𝑝1, 𝑝2 ∈ 𝑃 and 𝛼1, 𝛼2 ∈ Ω𝐵, +𝜋𝐷(𝜗)(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = Λ𝜅(𝑝2)𝛼2 + Λ𝜅(𝑝1)𝜗𝛼1, +𝐷hor(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −i∇(𝑝1) ⟨0⟩𝑐𝐵(∇(𝑝1) ⟨1⟩)(𝛼1) + 𝑝1𝐷𝐵(𝛼1) ++ i∇(𝑝2) ⟨0⟩𝜗𝑐𝐵(∇(𝑝2) ⟨1⟩)(𝛼2) − 𝑝2𝜗𝐷𝐵(𝛼2) +𝑍(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −𝑝2e𝐵(FΠ)(𝛼2) − 𝑝1𝜗i𝐵(FΠ)(𝛼1). +These expressions for 𝜋𝐷(𝜗), 𝐷hor, and 𝑍 now make clear that 𝜋𝐷(𝜗)2 = Λ2 +𝜅, that 𝐷hor super- +commutes with 𝜋𝐷(𝜗), and that 𝑍 supercommutes with ran 𝜋. + +66 +BRANIMIR ĆAĆIĆ +Finally, we show that 𝑍 is bounded. Recall the faithful conditional expectation E𝑃 : 𝑃 → 𝐵 +of Proposition 3.16, so that 𝑃 ⊗ C2 defines a countably generated right pre-Hilbert 𝐵-module +with respect to the 𝐵-valued inner product (·, ·) given by +∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝑣1, 𝑣2 ∈ C2, +(𝑝1 ⊗ 𝑣1, 𝑝2 ⊗ 𝑣2) � ⟨𝑣1, 𝑣2⟩E𝑃(𝑝∗ +1𝑝2). +Thus, (𝑃 ⊗ C2) ⊗𝐵 Ω𝐵 defines a pre-Hilbert space with respect to the inner product defined, +mutatis mutandis, by (2.12). Moreover, by Proposition 3.24, Theorem 3.46, and the proof of +Proposition 4.29, we may define unitary 𝑀 : (𝑃 ⊗ C2) ⊗𝐵 Ω𝐵 → Ω𝑃 by +∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝛼 ∈ Ω𝐵, +𝑀 +��𝑝1 +𝑝2 +� +⊗ 𝛼 +� +� (𝑝1 + 𝑝2𝜗)𝛼. +Since the left 𝐵-linear maps e(FΠ) and i(FΠ) are both bounded as operators on the pre- +Hilbert space Ω𝐵, we may now apply standard Hilbert 𝐶∗-module lore to conclude that +𝑍 = −𝑀 +��0 +1 +0 +0 +� +⊗ e(FΠ) + +�0 +0 +1 +0 +� +⊗ i(FΠ) +� +𝑀∗ +is bounded as an operator on the pre-Hilbert space Ω𝑃. +We conclude by showing that the maps +� +𝑏 ↦→ 𝜋(𝑝)↾ΩU(1) +𝑃 +� +: 𝐵 → L(ΩU(1) +𝑃 +), +� +𝛽 ↦→ 𝜋d𝑃+d∗ +𝑃 (𝛽)↾ΩU(1) +𝑃 +� +: Ω1 +𝐵 → L(ΩU(1) +𝑃 +) +are isometric and injective, respectively. First, let 𝑏 ∈ 𝐵 be given. On the one hand, the operator +𝜋(𝑏)↾ΩU(1) +𝑃 +block-diagonal with respect to orthogonal decomposition ΩU(1) +𝑃 += Ω𝐵 ⊕ 𝜗Ω𝐵, +where 𝜋(𝑏)↾Ω𝐵 is simply left multiplication by 𝑏 on Ω𝐵. On the other hand, by Proposition +4.33, left multiplication of 𝐵 on Ω𝐵 defines an isometric ∗-homorphism 𝐵 → L(Ω𝐵). Hence, +it follows that ∥𝑏∥ = ∥𝜋(𝑝)↾Ω𝐵 ∥ ≤ ∥𝜋(𝑏)↾ΩU(1) +𝑃 +∥ ≤ ∥𝜋(𝑏)∥ ≤ ∥𝑏∥. Now, let 𝛽 ∈ Ω1 +𝐵 be +given. On the one hand, both e(𝛽) ↾ΩU(1) +𝑃 +and e(𝛽∗) ↾ΩU(1) +𝑃 +are both block-diagonal with +respect to the orthogonal decomposition ΩU(1) +𝑃 += Ω𝐵 ⊕ 𝜗Ω𝐵, where e(𝛽)↾Ω𝐵= e𝐵(𝛽) and +e(𝛽∗)↾Ω𝐵= e𝐵(𝛽∗), so that 𝑐(𝛽)↾ΩU(1) +𝑃 +is similarly block-diagonal with 𝑐(𝛽)↾Ω𝐵= 𝑐𝐵(𝛽). On +the other hand, by Proposition 4.33, the map 𝑐𝐵 : Ω1 +𝐵 → L(Ω𝐵) is injective. Hence, it follows +that 𝑐(𝛽)↾ΩU(1) +𝑃 += 0 only if 𝛽 = 0. +□ +Definition 4.46. Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π). +The total Hodge–de Rham commutator representation induced by (Δver, Δhor, ★, 𝜏) is the faithful +projectable commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 +d∗ +𝑃, 2Π−id) of (𝑃; Ω𝑃, d𝑃; Π) constructed +from (Δver, Δhor, ★, 𝜏) by Proposition 4.44. +We now show that a faithful projectable commutator representation of (𝑃; Ω𝑃, d𝑃; Π) +lives up to our terminology by canonically projecting to a faithful bounded commutator +representation of (𝐵; Ω𝐵, d𝐵). This, in turn, will make precise the notion of lifting a faithful +bounded commutator representation of (𝐵; Ω𝐵, d𝐵) to (𝑃; Ω𝑃, d𝑃; Π). +On the one hand, define the concrete category BCRep(𝐵) of faithful bounded commutator +representations of (𝐵; Ω𝐵, d𝐵) and their isomorphisms as follows: +(1) an object is a faithful bounded commutator represention (𝐻, 𝜋, 𝐷) of (Ω𝐵, d𝐵); +(2) an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1) → (𝐻2, 𝜋2, 𝐷2) is a unitary 𝑈 : 𝐻1 → 𝐻2 that satisfies +𝑈𝜋1(·)𝑈∗ = 𝜋2, +𝑈𝐷1𝑈∗ = 𝐷2. +On the other hand, given 𝜅 > 0 and (𝑃; Ω𝑃, d𝑃; Π) a 𝜅-differentiable quantum principal +U(1)-bundle with connection over 𝐵, define the concrete category PCRep(𝑃; Π) of faithful +projectable commutator representations of (𝑃; Ω𝑃, d𝑃; Π) and their isomorphisms as follows: + +NONCOMMUTATIVE U(1)-GAUGE THEORY +67 +(1) an object is a faithful projectable commutator representation (𝐻, 𝜋, 𝐷,Γ) of (𝑃; Ω𝑃, d𝑃; Π); +(2) an arrow (𝑈, 𝑍) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2) consists of an even U(1)-equivariant +unitary 𝑈 : 𝐻1 → 𝐻2 and odd U(1)-invariant self-adjoint 𝑍 ∈ LU(1) (𝐻1) supercommut- +ing with ran 𝜋 and Γ, such that +𝑈𝜋1(·)𝑈∗ = 𝜋1, +𝑈(𝐷1 − 𝑍)𝑈∗ = 𝐷2, +𝑈Γ1𝑈∗ = Γ2; +(3) given objects (𝐻1, 𝜋1, 𝐷1,Γ1), (𝐻2, 𝜋2, 𝐷2,Γ2), (𝐻3, 𝜋3, 𝐷3,Γ3𝑥), and arrows +(𝑈1, 𝑍1) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2), +(𝑈2, 𝑍2) : (𝐻2, 𝜋2, 𝐷2,Γ2) → (𝐻3, 𝜋3, 𝐷3,Γ3), +the composition (𝑈2, 𝑍2) ◦ (𝑈1, 𝑍1) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻3, 𝜋3, 𝐷3,Γ3) is given by +(𝑈2, 𝑍2) ◦ (𝑈1, 𝑍1) � �𝑈2𝑈1,𝑈∗ +1𝑍2𝑈1 + 𝑍1 +� ; +(4) the identity arrow of an object (𝐻, 𝜋, 𝐷,Γ) is given by (id, 0). +Note that an arrow (𝑈, 𝑍) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2) in PCRep(𝑃; Π) encodes U(1)- +equivariant unitary equivalence of (𝐻1, 𝜋1, 𝐷1,Γ1) and (𝐻2, 𝜋2, 𝐷2,Γ2) after perturbation by +the relative remainder 𝑍. +Proposition 4.47. The following defines a functor 𝜄∗ +𝑃 : PCRep(𝑃; Π) → BCRep(𝐵). +(1) Given an object (𝐻, 𝜋, 𝐷,Γ), let 𝜄∗ +𝑃(𝐻, 𝜋, 𝐷,Γ) � �𝑃𝐻U(1), 𝑃𝜋(·)𝑃, 𝑃𝐷hor𝑃�, where we set +𝑃 � 1 +2 (id +Γ)↾𝐻U(1) and 𝐷hor is the horizontal Dirac operator of (𝐻, 𝜋, 𝐷,Γ). +(2) Given an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2), let 𝜄∗ +𝑃𝑈 be given by 𝑃2𝑈𝑃1, where +𝑃1 � 1 +2 (id +Γ1)↾𝐻U(1) +1 +and 𝑃2 � 1 +2 (id +Γ2)↾𝐻U(1) +2 +. +Proof. This is a routine verification except for one subtlety. Let (𝐻, 𝜋, 𝐷,Γ) be a faithful +projectable commutator representation of (𝑃; Ω𝑃, d𝑃; Π). It remains to show that the bounded +commutator representation (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ +𝑃(𝐻, 𝜋, 𝐷,Γ) of (𝐵; Ω𝐵, d𝐵) is faithful. Observe +that 𝐻U(1) admits the orthogonal decomposition 𝐻U(1) = 𝐻𝐵 ⊕ Γ𝐻𝐵, where Γ restricts to a +unitary 𝑉 : 𝐻𝐵 → Γ𝐻𝐵. Hence, it follows that 𝜋(𝑏)↾𝐻U(1) = 𝜋𝐵(𝑏) ⊕ (𝑉𝜋𝐵(𝑏)𝑉∗) for all 𝑏 ∈ 𝐵, +so that 𝜋𝐵 is isometric since �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : 𝐵 → L(𝐻U(1)) is isometric. A qualitatively +identical argument shows that 𝜋𝐷 is injective. +□ +Definition 4.48. Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation +of (𝐵; Ω𝐵, d𝐵), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a faithful projectable commutator representation of +(𝑃; Ω𝑃, d𝑃; Π). We say that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) is a lift of (𝐵; Ω𝐵, d𝐵) to (𝑃; Ω𝑃, d𝑃; Π) whenever +𝜄∗ +𝑃( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) and (𝐻, 𝜋, 𝐷) are isomorphic in BCRep(𝐵). +Example 4.49. Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃, Π), and +let (★𝐵, 𝜏𝐵) be its restriction to a Riemannian geometry on (𝐵; Ω𝐵, d𝐵). Then the total Hodge– +de Rham commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ +𝑃, 2Π − id) induced by (Δver, Δhor, ★, 𝜏) is +a lift of the Hodge–de Rham commutator representation (Ω𝐵, 𝜋𝐵, d𝐵+d∗ +𝐵) induced by (★𝐵, 𝜏𝐵). +Indeed, the inclusion map ˆ𝜄𝑃 : Ω𝐵 +∼−→ ΩU(1) +𝑃,hor = Π(ΩU(1) +𝑃 +) defines an isomorphism +ˆ𝜄𝑃 : (Ω𝐵, 𝜋𝐵, d𝐵 + d∗ +𝐵) → 𝜄∗ +𝑃(Ω𝑃, 𝜋𝑃, d𝑃 + d∗ +𝑃, 2Π − id). +At last, we show that every faithful bounded commutator representation of (𝐵; Ω𝐵, d𝐵) has +an essentially unique lift to (𝑃; Ω𝑃, d𝑃; Π), i.e., up to U(1)-equivariant unitary equivalence after +perturbation by a relative remainder. Note that this excludes the use of Schwieger–Wagner’s +lifting construction [92], even after modification to permit 𝜅 ≠ 1, since it requires unnatural +choices of representation-theoretic data that need not even yield locally bounded commutator +representations of (𝑃; Ω𝑃, d𝑃). + +68 +BRANIMIR ĆAĆIĆ +We first show that lifts always exist. When 𝜅 = 1, the right 𝐵-module Ω1 +𝐵 is free with +basis consisting of self-adjoint elements of Z(Ω𝐵)1, the Fröhlich automorphism ˆΦ𝑃 is the +identity map, and the faithful bounded commutator representation of (𝐵; Ω𝐵, d𝐵) takes a +certain restrictive form, our construction recovers a lifting construction for spectral triples +first proposed by Gabriel–Grensing [51]. +In what follows, recall the self-adjoint Pauli matrices +𝜎1 � +�0 +1 +1 +0 +� +, +𝜎2 � +�0 +−i +i +0 +� +, +𝜎3 � +�1 +0 +0 +−1 +� += −i𝜎1𝜎2. +Proposition 4.50. Let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐵; Ω𝐵, d𝐵). +Define a map ∇ : 𝑃 → 𝑃 ⊗𝐵 Ω𝐵 by ∇ � ˆℓ𝑃 ◦ Π ◦ d𝑃↾𝑃, where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is +the 𝐵-bimodule isomorphism of Proposition 3.24, and let E𝑃 : 𝑃 → 𝐵 be the faithful conditional +expectation of Proposition 3.16. Equip the left 𝑃-module 𝑃 ⊗ C2 with the right 𝐵-module structure +∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀𝑏 ∈ 𝐵, +(𝑝 ⊗ 𝑥) · 𝑏 � 𝑝𝑏 ⊗ 𝑥, +equip 𝐻 with the left 𝐵-module structure defined by 𝜋, and equip (𝑃 ⊗ C2) ⊗𝐵 𝐻 with the inner +product ⟨·, ·⟩ defined by +∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝑥1, 𝑥2 ∈ C2, ∀ℎ1, ℎ2 ∈ 𝐻, +⟨𝑝1 ⊗ 𝑥1 ⊗ ℎ1, 𝑝2 ⊗ 𝑥2 ⊗ ℎ2⟩ � ⟨𝑥1, 𝑥2⟩⟨ℎ1, 𝜋(E𝑃(𝑝∗ +1𝑝2))ℎ2⟩, +the Z/2Z-grading id ⊗𝜎3 ⊗ 𝜒𝐻 and the linear U(1)-representation induced by the U(1)-action +on 𝑃. Finally, define an operator (id ⊗𝜎3) ⊗∇ 𝐷 on (𝑃 ⊗ C2) ⊗𝐵 𝐻 by +∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀ℎ ∈ 𝐻, +(id ⊗𝜎3) ⊗∇ 𝐷(𝑝 ⊗ 𝑥 ⊗ ℎ) � −i∇(𝑝) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋𝐷 +� +∇(𝑝) ⟨1⟩ +� +ℎ + 𝑝 ⊗ 𝜎3𝑥 ⊗ 𝐷ℎ. +Then the data +�(𝑃 ⊗ C2) ⊗𝐵 𝐻, id ⊗ id ⊗𝜋(·), i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2 ⊗ id +(id ⊗𝜎3) ⊗∇ 𝐷, id ⊗𝜎3 ⊗ id� +define a lift of (𝐻, 𝜋, 𝐷) to (𝑃; Ω𝑃, d𝑃; Π) with horizontal Dirac operator (id ⊗𝜎3) ⊗∇ 𝐷 and +remainder 0. +Lemma 4.51. Let (𝐻, 𝜋, 𝐷) be a bounded commutator representation of (𝐵; Ω𝐵, d𝐵), and let +(𝐸, 𝜎, ∇) be a Hermitian line 𝐵-bimodule with connection. Equip 𝐸 ⊗𝐵 𝐻 with the positive definite +inner product defined, mutatis mutandis, by (2.12). +(1) For every 𝑥 ∈ 𝐸, we obtain contractive 𝜙𝐸[𝑥] : 𝐸 ⊗𝐵 𝐻 → 𝐻 by setting +∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, +𝜙𝐸[𝑥](𝑦 ⊗ ℎ) = 𝜋((𝑥, 𝑦)𝐸)ℎ +Hence, in particular, the pre-Hilbert space 𝐸 ⊗𝐵 𝐻 is separable. +(2) We obtain formally self-adjoint id ⊗∇𝐷 : 𝐸 ⊗𝐵 𝐻 → 𝐸 ⊗𝐵 𝐻 by setting +∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, +(id ⊗∇𝐷)(𝑦 ⊗ ℎ) = −i∇(𝑦) ⟨0⟩ ⊗ 𝜋𝐷(∇(𝑦) ⟨1⟩)ℎ + 𝑦 ⊗ 𝐷ℎ. +(3) For every 𝛼 ∈ Ω1 +𝐵, we obtain bounded 𝜌𝐸[𝛼] : 𝐸 ⊗𝐵 𝐻 → 𝐸 ⊗𝐵 𝐻 by setting +∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, +𝜌𝛼(𝑦 ⊗ ℎ) = 𝜎 (𝛼 ⊗ 𝑦) ⟨0⟩ ⊗ 𝜋(𝜎 (𝛼 ⊗ 𝑦) ⟨1⟩)ℎ. +Proof. Before continuing, let us fix a basis (𝑒𝑖)𝑁 +𝑖=1 for 𝐸. Recall, moreover, that by the proof of +Proposition 4.5, mutatis mutandis, every positive 𝑋 ∈ 𝑀𝑛(𝐵) satisfies +∀ℎ = (ℎ𝑖)𝑁 +𝑖=1 ∈ 𝐻𝑁, +⟨ℎ, 𝜋𝑛(𝑋)ℎ⟩ ≤ ∥𝑋∥ +∑︁𝑁 +𝑖=1∥ℎ𝑖∥2, +(4.48) + +NONCOMMUTATIVE U(1)-GAUGE THEORY +69 +where 𝜋𝑛 : 𝑀𝑛(𝐵) → L(𝐻𝑛) is the bounded ∗-homomorphism canonically induced by +𝜋 : 𝐵 → L(𝐻). Note that this applies, in particular, to the matrix 𝑋 � ((𝑒𝑖, 𝑒𝑗))𝑁 +𝑖,𝑗=1. +First, let 𝑥 ∈ 𝐸 be given. Define 𝜓𝐸[𝑥] : 𝐻 → 𝐸 ⊗𝐵 𝐻 by 𝜓𝐸[𝑥] � (ℎ ↦→ 𝑥 ⊗ ℎ). A +standard calculation show that 𝜙𝐸[𝑥] = 𝜓𝐸[𝑥]∗ and that 𝜓𝐸[𝑥] is bounded with operator +norm ∥𝜓𝐸[𝑥]∥ = ∥𝜋(⟨𝑥, 𝑥⟩)∥1/2 ≤ 1, so that 𝜙𝐸[𝑥] is contractive. Since (𝑒𝑖)𝑁 +𝑖=1 is a basis for +𝐸, it now follows that +∀𝜉 ∈ 𝐸 ⊗𝐵 𝐻, +𝜉 = +∑︁𝑁 +𝑖=1 𝑒𝑖 ⊗ 𝜙𝐸[𝑒𝑖]𝜉. +(4.49) +Next, let 𝑉 be a countable dense subset of 𝐻; we claim that {�𝑁 +𝑖=1 𝑒𝑖 ⊗ 𝑣𝑖 | 𝑣1, . . . , 𝑣𝑁 ∈ 𝑉} +is dense in 𝐸 ⊗𝐵 𝐻. Let 𝜉 ∈ 𝐸 ⊗𝐵 𝐻 and 𝜖 > 0 be given. Let 𝑋 � ((𝑒𝑖, 𝑒𝑗))𝑁 +𝑖,𝑗=1, and choose +𝑣1, . . . , 𝑣𝑁 ∈ 𝑉, such that ∥𝜙𝐸[𝑒𝑖]𝜉 − 𝑣𝑖∥2 < +𝜖2 +𝐶𝑁+1. Then, by (4.49) and (4.48), +�����𝜉 − +𝑁 +∑︁ +𝑖=1 +𝑒𝑖 ⊗ 𝑣𝑖 +����� +2 += +𝑁 +∑︁ +𝑖,𝑗=1 +⟨𝜙𝑒𝑖𝜉 − 𝑣𝑖, 𝜋((𝑒𝑖, 𝑒𝑗))(𝜙𝐸[𝑒𝑗]𝜉 − 𝑣𝑗⟩ ≤ ∥𝑋∥ +𝑁 +∑︁ +𝑖=1 +∥𝜙𝐸[𝑒𝑖]𝜉 − 𝑣𝑖∥2 < 𝜖2. +Next, that id ⊗∇𝐷 is well-defined and formally self-adjoint is well-known in the literature +on unbounded 𝐾𝐾-theory—see, e.g., [19, Lemma 2.28]. +Finally, let 𝛼 ∈ Ω1 +𝐵 be given. Then right 𝐵-linearity of the generalised braiding 𝜎 guarantees +that 𝜌𝐸[𝛼] is a well-defined map. Hence, by (4.49), for every 𝜉 ∈ 𝐸 ⊗𝐵 𝐻, it follows that +∥𝜌𝐸[𝛼]𝜉∥ ≤ +𝑁 +∑︁ +𝑖,𝑗=1 +��𝑒𝑖 ⊗ 𝜋𝐷 +�(𝑒𝑖, 𝜎 (𝛼 ⊗ 𝑒𝑗))�𝜙𝐸(𝑒𝑗)𝜉 +�� ≤ �� +� +𝑁 +∑︁ +𝑖,𝑗=1 +∥𝑒𝑖∥ · ∥𝜋𝐷 +�(𝑒𝑖, 𝜎 (𝛼 ⊗ 𝑒𝑗))�∥�� +� +∥𝜉∥. +□ +Proof of Proposition 4.50. For convenience, we shall permit the following abuse of notation: +given 𝑗 ∈ Z, we conflate the isotypical subspace 𝑃𝑗 with the Hermitian line 𝐵-bimodule 𝑃𝑗. +Moreover, we shall also use the notation of Lemma 4.51 and its proof. Let us first show that +( ˜𝐻, ˜𝜋, ˜𝐷) � �(𝑃 ⊗ C2) ⊗𝐵 𝐻, id ⊗ id ⊗𝜋(·), i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2 ⊗ id +(id ⊗𝜎3) ⊗∇ 𝐷� +defines a faithful locally bounded commutator representation of (𝑃; Ω𝑃, d𝑃). Let 𝔅 be the +𝐶∗-algebraic completion of 𝐵, and let 𝜏 : ˜𝐻 → C2 ⊗ (𝑃 ⊗𝐵 𝐻) be the canonical unitary +defined by 𝜏 � (𝑝 ⊗ 𝑥 ⊗ ℎ ↦→ 𝑥 ⊗ 𝑝 ⊗ ℎ). +First, we show that ˜𝐻 is separable. It follows from Lemma 4.51 that the pre-Hilbert space +𝑃 ⊗𝐵 𝐻 = � +𝑗∈Z 𝑃𝑗 ⊗𝐵 𝐻 is separable, so that ˜𝐻 � (𝑃 ⊗𝐵 𝐻)2 is also separable. It is now +straightforward to check that 𝜒 ˜𝐻 � id⊗𝜎3⊗ 𝜒𝐻 defines a Z2-grading on ˜𝐻 and that 𝜎·⊗id ⊗ id +defines a unitary U(1)-representation of finite type on ˜𝐻 with ˜𝐻𝑗 = (𝑃𝑗 ⊗ C2) ⊗𝐵 𝐻 � +(𝑃 ⊗𝐵 𝐻)2 for 𝑗 ∈ Z. +Next, let us show that ˜𝜋 is well-defined. It suffices to show that the left 𝑃-module structure +on 𝑃 ⊗𝐵 𝐻 defines a bounded ∗-homomorphism 𝜆 : 𝑃 → L(𝑃 ⊗𝐵 𝐻), since this will imply +boundedness of ˜𝜋 = 𝜏∗ ◦ (id ⊗𝜆(·)) ◦ 𝜏; the other properties of ˜𝜋 will follow by routine checks. +In turn, the only non-trivial points are that 𝜆 is well-defined and bounded as a map of Banach +spaces. Let 𝑝 ∈ 𝑃 be given, so that there exists 𝑁 ∈ N, such that 𝑝 ∈ �𝑁 +𝑗=−𝑁 𝑃𝑗; hence, +we may unique write 𝑝 = �𝑁 +𝑗=−𝑁 ˆ𝑝(𝑗), where ˆ𝑝(𝑗) ∈ 𝑃𝑗 for each 𝑗 ∈ {−𝑁, . . . , 𝑁}, so that +E𝑃(𝑝∗𝑝) = �𝑁 +𝑗=−𝑁 ˆ𝑝(𝑗)∗ ˆ𝑝(𝑗). Let 𝑘 ∈ N and 𝜉 ∈ 𝑃𝑗 ⊗𝐵 𝐻 be given. Let (𝑒𝑖)𝑀 +𝑖=1 be a basis for 𝑃𝑗. +Then, in the notation of Lemma 4.51, +∥𝜆(𝑝)𝜉∥2 = +𝑀 +∑︁ +𝑚,𝑛=1 +⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(E𝑃(𝑒∗ +𝑚𝑝∗𝑝𝑒𝑛))𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ = +𝑀 +∑︁ +𝑚,𝑛=1 +⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(𝑒∗ +𝑚E𝑃(𝑝∗𝑝)𝑒𝑛)⟩. + +70 +BRANIMIR ĆAĆIĆ +Now, let 𝑏 � +√︁ +E𝑃(𝑝∗𝑝) ∈ 𝔅. After passing to Hilbert 𝐶∗-module and Hilbert space comple- +tions, we may apply [64, p. 42] to conclude that +∑︁𝑀 +𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(𝑒∗ +𝑚E𝑃(𝑝∗𝑝)𝑒𝑛)⟩ = +∑︁𝑁 +𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋((𝑏𝑒𝑚, 𝑏𝑒𝑛))𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ +≤ ∥𝑏∥2 ∑︁𝑁 +𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋((𝑒𝑚, 𝑒𝑛)𝑗)𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ +≤ ∥𝑝∥2∥𝜉∥2. +Next, let us show that ˜𝐷 is U(1)-invariant, odd, and formally self-adjoint. Define operators +𝑆 and 𝑇 satisfying ˜𝐷 = 𝑆 +𝑇 by 𝑆 � i(Λ𝜅 ◦𝜕𝜅) ⊗ 𝜎2 ⊗ id and 𝑇 � (id ⊗𝜎3) ⊗∇ 𝐷, respectively. +On the one hand, the block-diagonal operator 𝑆 = � +𝑗∈Z(−2𝜋[𝑗]𝜅𝜅−𝑗 id) ⊗ 𝜎2 ⊗ id is U(1)- +invariant, odd, and formally self-adjoint by construction. On the other hand,the operator +𝑇 is likewise U(1)-invariant and odd by construction; by U(1)-invariance, it follows that +𝑇 = � +𝑗∈Z 𝑇↾ ˜𝐻𝑗, where 𝑇↾ ˜𝐻𝑗= 𝜏∗ ◦ +� +𝜎3 ⊗ (id ⊗∇𝑃,𝑗𝐷) +� +◦ 𝜏↾ ˜𝐻𝑗 is formally self-adjoint for each +𝑗 ∈ Z by Lemma 4.51. +Next, we show that ˜𝜋 ˜𝐷 : Ω1 +𝑃 → LU(1) +loc ( ˜𝐻) is well-defined. On the one hand, recall that +(id −Π)(Ω1 +𝑃) is freely generated as a left 𝑃-module by the connection 1-form 𝜗 of Π; hence, +we may define a map ˜𝜋ver : (id −Π)(Ω1 +𝑃) → LU(1) +loc ( ˜𝐻) by +∀𝑝 ∈ 𝑃, +˜𝜋ver(𝑝𝜗) � ˜𝜋(𝑝) · (Λ𝜅 ⊗ 𝜎2 ⊗ id). +On the other hand, since multiplication (𝑝 ⊗ 𝛼 ↦→ 𝑝𝛼) : 𝑃 ⊗𝐵 Ω1 +𝐵 → Π(Ω1 +𝑃) is a 𝐵-bimodule +isomorphism, we may apply Lemma 4.51 to define a map ˜𝜋hor : Π(Ω1 +𝑃) → LU(1) +loc ( ˜𝐻) by +∀𝑝 ∈ 𝑃, ∀𝛼 ∈ Ω1 +𝐵, ∀𝑗 ∈ Z, ∀ +˜𝜋hor(𝑝𝛼)↾ ˜𝐻𝑗� ˜𝜋(𝑝) ◦ 𝜏∗ ◦ (𝜎3 ⊗ 𝜌𝑃𝑗 [𝛼]) ◦ 𝜏↾ ˜𝐻𝑗 . +Since ˜𝐷 = 𝑆 + 𝑇, it now suffices to show that +∀𝑝 ∈ 𝑃, +i[𝑆, ˜𝜋(𝑝)] = ˜𝜋ver ◦ (id −Π) ◦ d𝑃(𝑝), +i[𝑇, ˜𝜋(𝑝)] = ˜𝜋hor ◦ Π ◦ d𝑃(𝑝). +Finally, let 𝑗, 𝑘 ∈ Z, 𝑝 ∈ 𝑃𝑗, 𝑞 ∈ 𝑃𝑘, 𝑥 ∈ C2, and ℎ ∈ 𝐻 be given. On the one hand, +[𝑆, ˜𝜋(𝑝)](𝑞 ⊗ 𝑥 ⊗ ℎ) = −2𝜋[𝑗 + 𝑘]𝜅𝜅−𝑗−𝑘𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ + 2𝜋[𝑘]𝜅𝜅−𝑘𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ += 2𝜋[𝑗]𝜅𝜅−𝑗𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ +− i˜𝜋ver(2𝜋i[𝑗]𝜅𝜅−𝑗𝑝𝜗)(𝑞 ⊗ 𝑥 ⊗ ℎ) += −i( ˜𝜋ver ◦ (id −Π) ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ). +On the other hand, since Π ◦ d𝑃 is a derivation, it follows that +∇𝑃;𝑗+𝑘(𝑝𝑞) = ∇𝑃;𝑗(𝑝) ⟨0⟩𝜎𝑃;𝑘(∇𝑃;𝑗(𝑝)) ⟨1⟩ ⊗ 𝑞) ⟨0⟩ ⊗ 𝜎𝑃;𝑘(∇𝑃;𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨1⟩ ++ 𝑝∇𝑃;𝑘(𝑞) ⟨0⟩ ⊗ ∇𝑃;𝑘(𝑞) ⟨1⟩, +so that +i𝑇(𝑝𝑞 ⊗ 𝑥 ⊗ ℎ) = ∇𝑃;𝑗(𝑝) ⟨0⟩𝜎𝑃;𝑘(∇𝑃;𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋(𝜎𝑃;𝑘(∇𝑃;𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨1⟩)ℎ ++ 𝑝∇𝑃;𝑘(𝑞) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋(∇𝑃;𝑘(𝑞) ⟨1⟩)ℎ + 𝑝𝑞 ⊗ 𝜎3𝑥 ⊗ 𝐷ℎ += ( ˜𝜋hor ◦ Π ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ) + i˜𝜋(𝑝)𝑇(𝑞 ⊗ 𝑥 ⊗ ℎ), +and hence i[𝑇, ˜𝜋(𝑝)](𝑞 ⊗ 𝑥 ⊗ ℎ) = ( ˜𝜋hor ◦ Π ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ). +Finally, we show that ( ˜𝐻, ˜𝜋, ˜𝐷) is faithful. We first show that ˜𝜋 is isometric. Since ˜𝜋 is +bounded, faithful, and U(1)-equivariant, it suffices by Corollary 3.19 to show that ˜𝜋 ↾𝐵 is +isometric. Indeed, let 𝑏 ∈ 𝐵. Since 𝜏∗(𝐵 ⊗𝐵 𝐻) � 𝐻 is an orthogonal direct summand of ˜𝐻 + +NONCOMMUTATIVE U(1)-GAUGE THEORY +71 +and 𝜋 is isometric, it follows that ∥𝑏∥ ≥ ∥ ˜𝜋(𝑏)∥ ≥ ∥𝜏 ˜𝜋(𝑏)𝜏∗↾𝐵⊗𝐵𝐻∥ = ∥𝜋(𝑏)∥ = ∥𝑏∥. Now, let +us show that ˜𝜋 ˜𝐷 is injective; to do so, it suffices to show that ˜𝜋ver and ˜𝜋hor are both injective. On +the one hand, ˜𝜋ver is injective since ˜𝜋 is injective and ˜𝜋ver(𝜗) is invertible. On the other, to show +that ˜𝜋hor is injective, it suffices to show injectivity of 𝑓 : 𝑃 ⊗𝐵 Ω1 +𝐵 → EndC(𝐻, 𝑃 ⊗𝐵 𝐻) defined +by 𝑓 (𝑝 ⊗ 𝛽)ℎ � 𝜋(𝑝)𝜋𝐷(𝛽)ℎ for 𝑝 ∈ 𝑃, 𝛽 ∈ Ω1 +𝐵, and ℎ ∈ 𝐻. Indeed, fix 𝑗 ∈ Z, and note that +𝑓𝑗↾𝑃𝑗 ⊗𝐵Ω1 +𝐵= 𝑟𝑗◦𝑠𝑗, where 𝑠𝑗 : 𝑃𝑗⊗𝐵Ω1 +𝐵 → 𝑃𝑗⊗𝐵L(𝐻) and 𝑟𝑗 : 𝑃𝑗⊗𝐵L(𝐻) → EndC(𝐻, 𝑃𝑗⊗𝐵 𝐻) +are given by 𝑠𝑗 � id ⊗𝜋𝐷 and 𝑟𝑗 � (𝑝 ⊗ 𝑆 ↦→ 𝜓𝑃𝑗 [𝑝]𝑆), respectively. Then 𝑠𝑗 is injective +by flatness of the projective right 𝐵-module 𝑃𝑗, while 𝑟𝑗 is injective by existence of the left +inverse 𝑇 ↦→ �𝑁 +𝑖=1 𝑒𝑖 ⊗ 𝜙𝑃𝑗 [𝑒𝑖]𝑇, where (𝑒𝑖)𝑁 +𝑖=1 is any basis for the Hermitian line 𝐵-bimodule +𝑃𝑗. Hence, the map 𝑓𝑗↾𝑃𝑗 ⊗𝐵Ω1 +𝐵: 𝑃𝑗 ⊗𝐵 Ω1 +𝐵 → EndC(𝐻, 𝑃𝑗 ⊗𝐵 𝐻) is also injective. +Now, let ˜Γ � id ⊗𝜎3 ⊗ id, which, by construction, is an even U(1)-invariant self-adjoint +unitary commuting with ran ˜𝜋. Let us check that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) defines a lift of (𝐻, 𝜋, 𝐷). +First, let 𝑀 : 𝑃 ⊗𝐵 ˜𝐻U(1) → ˜𝐻 be given by 𝑀 � (𝑝 ⊗ 𝜉 ↦→ ˜𝜋𝜉). Define a left 𝐵-linear +unitary Φ : C2 ⊗ 𝐻 → ˜𝐻U(1) by Φ � (𝑥 ⊗ ℎ ↦→ 1 ⊗ 𝑥 ⊗ ℎ), and observe that +∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀ℎ ∈ 𝐻, +𝜏 ◦ 𝑀 ◦ (id ⊗Φ)(𝑝 ⊗ 𝑥 ⊗ ℎ) = 𝑥 ⊗ 𝑝 ⊗ ℎ, +so that 𝜏◦𝑀◦(id ⊗Φ) : 𝑃 ⊗𝐵 (C2 ⊗ 𝐻) → C2 ⊗ (𝑃 ⊗𝐵 𝐻) is manifestly bijective, which implies +that 𝑀 is bijective as well. Next, note that ˜𝑝𝑖 ˜𝐷(𝜗)2 = Λ2 +𝜅 since ˜𝜋 ˜𝐷(𝜗) = ˜𝜋ver(𝜗) = Λ𝜅 ⊗ 𝜎2 ⊗ id, +which also shows that ˜Γ anticommutes with ˜Γ. Next, observe that ˜Γ anticommutes with 𝑆 and +commutes with 𝑇, so that ˜𝐷hor = 𝑇 and 𝑍 = 𝑆 + i˜𝜋 ˜𝐷(𝜗)𝜕𝜅 = 0. Next, since 𝜏∗(𝐵 ⊗ 𝐻) � 𝐻 +is an orthogonal direct summand of 𝐻U(1), the proof that ˜𝜋 is isometric also shows that the +map (𝑏 ↦→ ˜𝜋(𝑏)↾ ˜𝐻U(1) ) : 𝐵 → L( ˜𝐻U(1)) is isometric. Likewise, the proof that ˜𝜋hor is injective, +specialised to 𝑗 = 0, shows that the map (𝛽 ↦→ ˜𝜋 ˜𝐷(𝛽)↾ ˜𝐻U(1) ) : Ω1 +𝐵 → LU(1) ( ˜𝐻U(1)) is injective. +Finally, an arrow 𝑉 : (𝐻, 𝜋, 𝐷) → 𝜄∗ +𝑃( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) is given by 𝑉 � �ℎ ↦→ 1 ⊗ � 1 +0 +� ⊗ ℎ�. +□ +Having proved existence of lifts, we now show that they are indeed unique up to U(1)- +equivariant unitary equivalence after perturbation by a relative remainder. +Theorem 4.52. The functor 𝜄∗ +𝑃 of Proposition 4.47 is an equivalence of categories with weak +inverse (𝜄𝑃)! : BCRep(𝐵) → PCRep(𝑃; Π) defined as follows. +(1) Given an object (𝐻, 𝜋, 𝐷), let (𝜄𝑃)!(𝐻, 𝜋, 𝐷) be the projectable commutator representation of +(𝑃; Ω𝑃, d𝑃; Π) constructed from (𝐻, 𝜋, 𝐷) by Proposition 4.50. +(2) Given an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1) → (𝐻2, 𝜋2, 𝐷2), let (𝜄𝑃)!(𝑈) � (id ⊗ id ⊗ 𝑈, 0). +Thus, in particular, every bounded commutator representation of (𝐵; Ω𝐵, d𝐵) has an essentially +unique lift to (𝑃; Ω𝑃, d𝑃; Π). +Proof. It remains to construct 𝑈 : idPCRep(𝑃;Π) ⇒ (𝜄𝑃)! ◦ 𝜄∗ +𝑃 and 𝑉 : idBCRep(𝐵) ⇒ 𝜄∗ +𝑃 ◦ (𝜄𝑃)!. +First, let (𝐻, 𝜋, 𝐷,Γ) be an object of PCRep(𝑃; Π); let 𝐷hor be its horizontal Dirac operator +and 𝑍 its remainder, and let (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ +𝑃(𝐻, 𝜋, 𝐷,Γ). Define an even U(1)-equivariant +unitary Υ : (𝑃 ⊗ C2) ⊗𝐵 𝐻𝐵 → 𝐻 by +∀𝑝 ∈ 𝑃, ∀ +�𝑣1 +𝑣2 +� +∈ C2, ∀ℎ ∈ 𝐻𝐵, +Υ +� +𝑝 ⊗ +�𝑣1 +𝑣2 +� +⊗ ℎ +� +� 𝜋(𝑝) �𝑣1 id −i𝑣2Γ𝜋𝐷(𝜗)Λ−1 +𝜅 +� ℎ. +A straightforward if tedious calculation generalising the proof of Proposition 4.44 now shows, +in the notation of Proposition 4.50, that +Υ∗(−i𝜋𝐷(𝜗)𝜕𝜅)Υ = i(Λ𝜅 ◦𝜕𝜅) ⊗𝜎2 ⊗id, +Υ∗𝐷horΥ = (id⊗𝜎3) ⊗∇ 𝐷𝐵, +Υ∗ΓΥ = id⊗𝜎3 ⊗id, +so that we may define 𝑈(𝐻,𝜋,𝐷,Γ) : (𝜄𝑃)! ◦ 𝜄∗ +𝑃(𝐻, 𝜋, 𝐷,Γ) → (𝐻, 𝜋, 𝐷,Γ) by 𝑈(𝐻,𝜋,𝐷,Γ) � (Υ∗, 𝑍). +Now, let (𝐻, 𝜋, 𝐷) be an object of BCRep(𝐵). Then, as in the proof of Proposition 4.50, we +may define 𝑉(𝐻,𝜋,𝐷) : (𝐻, 𝜋, 𝐷) → 𝜄∗ +𝑃 ◦ (𝜄𝑃)!(𝐻, 𝜋, 𝐷) by 𝑉(𝐻,𝜋,𝐷) � �ℎ ↦→ 1 ⊗ � 1 +0 +� ⊗ ℎ�. +□ + +72 +BRANIMIR ĆAĆIĆ +Note that Corollary 3.48 and Theorem 4.52 combine to yield a formalisation of the con- +structions of Bellissard–Marcolli–Reihani [16] and Gabriel–Grensing [51] for (generalised) +crossed product spectral triples. Indeed, let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator +representation of (𝐵; Ω𝐵, d𝐵). Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connec- +tion, such that 𝜖1 ◦ ˆL◦ Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) � (𝐸, 𝜎𝐸, ∇𝐸). Then the 𝜅-total crossed product of +(𝐻, 𝜋, 𝐷) by (𝐸, 𝜎𝐸, ∇𝐸) is the canonical lift (𝐻, 𝜋, 𝐷)⋊𝜅,tot +(𝐸,𝜎𝐸,Z) Z � (𝜄𝑃)!(𝐻, 𝜋, 𝐷) of (𝐻, 𝜋, 𝐷) +to (𝑃; Ω𝑃, d𝑃) of Proposition 4.50. +Remark 4.53. We continue from Remarks 4.31 and 4.43. The right pre-Hilbert 𝐵-module +𝑃 ⊗ C2 � � +𝑗∈Z L(𝑃)(𝑗)2 gives rise to a formal U(1)-equivariant unbounded 𝐾𝐾1-cycle +(𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2) for (𝔓, 𝔅), where id ⊗ 𝜎1 generates the 1-multigrading. This +defines a genuine U(1)-equivariant unbounded 𝐾𝐾1-cycle for (𝔓, 𝔅) if and only if 𝜅 = 1, +in which case, it recovers a well-known construction of Carey–Neshveyev–Nest–Rennie [27, +Cor. 2.10] up to 1-multigrading; in all cases, its formal bounded transform recovers, up to +1-multigrading, the canonical representative of the extension class [𝜕] ∈ 𝐾𝐾1(𝔓, 𝔅) of 𝔓 +as a Pimsner algebra [4, §2.2]. Moreover, we may now reinterpret Theorem 4.52 in terms of +formal unbounded Kasparov products [71, 59]: +(1) given an object (𝐻, 𝜋, 𝐷) of BCRep(𝐵), we may write +(𝑃, ˜𝐻, ˜𝐷) � (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2; ∇) ⊗𝐵 (𝐵, 𝐻, 𝐷); +(2) given an object (𝐻, 𝜋, 𝐷,Γ) of PCRep(𝑃; Π), the natural isomorphism 𝑈(𝐻,𝜋,𝐷,Γ) of the +proof of Theorem 4.52 yields +(𝑃, 𝐻, 𝐷 − 𝑍) � (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2; ∇) ⊗𝐵 (𝐵, 𝐻𝐵, 𝐷𝐵), +where 𝑍 is the remainder of (𝐻, 𝜋, 𝐷,Γ) and (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ +𝑃(𝐻, 𝜋, 𝐷,Γ). +In both cases, ∇ is the represented connection on (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦𝜕𝜅) ⊗ 𝜎2) constructed from +Π in Proposition 4.50. In the second case, if (𝑃, 𝐻, 𝐷) defines a genuine U(1)-equivariant +unbounded 𝐾𝐾1-cycle for (𝔓, C), then this formal unbounded Kasparov product defines a +genuine unbounded Kasparov product by results of Ćaćić–Mesland [26, Thm. 2.44]. Otherwise, +the 𝐾𝐾-theoretic significance of Theorem 4.52 is an open question. +Corollary 4.54. Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵; Ω𝐵, d𝐵), and suppose that +(★𝐵, 𝜏𝐵) lifts to a total Riemannian geometry (Δver, Δhor, ★, 𝜏) on (𝑃, Ω𝑃, d𝑃, Π). Then the total +Hodge–de Rham commutator representation (𝑃; 𝜋𝑃, d𝑃 +d∗ +𝑃; 2Π−id) induced by (Δver, Δhor, ★, 𝜏) +is the essentially unique lift of the Hodge–de Rham commutator representation (𝐵; 𝜋𝐵, d𝐵 + d∗ +𝐵) +induced by (★𝐵, 𝜏𝐵) to (𝑃; Ω𝑃, d𝑃; Π). +Example 4.55. Continuing from Examples 3.51 and 4.32, we shall use Proposition 4.50 to +construct 𝜄∗ +O𝑞(SU(2)) (/𝑆𝑞(CP1), 𝜋, /𝐷1). First, let /𝑆𝑞(SU(2)) � C2 ⊗ C2 ⊗ O𝑞(SU(2)) with the +inner product ⟨·, ·⟩ given by +∀𝑥1, 𝑥2, 𝑦1, 𝑦2 ∈ C2, ∀𝑝1, 𝑝2 ∈ O𝑞(SU(2)), +⟨𝑥1 ⊗ 𝑦1 ⊗ 𝑝1, 𝑥2 ⊗ 𝑦2 ⊗ 𝑝2⟩ � ⟨𝑥1, 𝑥2⟩⟨𝑦1, 𝑦2⟩ℎ𝑞(𝑝∗ +1𝑝2), +the Z2-grading 𝜎3 ⊗ 𝜎3 ⊗ id, and the unitary U(1)-representation of finite type ˜𝑈 defined +by setting ˜𝑈 � �𝑧 ↦→ id ⊗ � 𝑧 +0 +0 𝑧−1 +� ⊗ 𝛼𝑧 +�; hence, define Λ𝑞2 and 𝜕𝑞2 on /𝑆𝑞(SU(2)) in terms of +˜𝑈. Next, let ˜𝜋 : O𝑞(SU(2)) → LU(1) (/𝑆𝑞(SU(2)))even be induced by multiplication from the +left in O𝑞(SU(2)). At last, let �/𝐷𝑞 � �/𝐷𝑞,ver + �/𝐷𝑞,hor, where the operators �/𝐷𝑞,ver and �/𝐷𝑞,hor on +/𝑆𝑞(SU(2)) are given by +�/𝐷𝑞,ver � i(𝜎2 ⊗id ⊗ id)◦Λ𝑞2 ◦𝜕𝑞2, +�/𝐷𝑞,hor � 𝜎3 ⊗ � 1 +2 (𝜎1 − i𝜎2) ⊗ 𝑞𝜕− + 1 +2 (𝜎1 + i𝜎2) ⊗ 𝑞𝜕+ +� , + +NONCOMMUTATIVE U(1)-GAUGE THEORY +73 +and let Γ𝑞 � 𝜎3 ⊗ id ⊗ id. Since the maps (𝑝 ⊗ 𝑥 ↦→ 𝑝 · 𝑥) : O𝑞(SU(2)) ⊗O𝑞(CP1) /𝑆𝑞,±(CP1) +are left O𝑞(SU(2))-module isomorphisms by Proposition 3.15, we may construct an even +U(1)-equivariant unitary Φ : (O𝑞(SU(2)) ⊗ C2) ⊗O𝑞(CP1) /𝑆𝑞(CP1) → /𝑆𝑞(SU(2)) by +∀𝑝 ∈ O𝑞(SU(2)), ∀𝑥 ∈ C2, ∀ � 𝑠+𝑠− +� ∈ /𝑆𝑞(CP1), +Φ�𝑝 ⊗ 𝑥 ⊗ � 𝑠+𝑠− +�� � 𝑥 ⊗ �� 1 +0 +� ⊗ 𝑝 · 𝑠+ + � 0 +1 +� ⊗ 𝑝 · 𝑠− +� , +which yields the desired isomorphism of projective commutator representations +(Φ, 0) : 𝜄O𝑞(SU(2)) (/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) → +� +/𝑆𝑞(SU(2)), ˜𝜋, �/𝐷𝑞,Γ𝑞 +� +. +Note, in particular, that +� +/𝑆𝑞(SU(2)), ˜𝜋, �/𝐷𝑞,Γ𝑞 +� +is faithful since (/𝑆𝑞(CP1), 𝜋, /𝐷1) is. +4.4. Twisted boundedness of lifted commutator representations. We have solved the +lifting problem for faithful bounded commutator representations, but at a cost: the resulting +faithful projectable commutator representations generally involve unbounded represented +1-forms. Here, we control this unboundedness in the spirit of Connes–Moscovici’s twisted +spectral triples [34] by allowing for possibly distinct vertical and horizontal twists. One upshot +is that quantum SU(2) as the total space of the 𝑞-monopole does not admit a non-pathological +U(1)-equivariant twisted spectral triple. The other is that Kaad–Kyed’s compact quantum +metric space [58] on quantum SU(2) for a canonical choice of parameters can be geometrically +derived, up to equivalence of Lipschitz seminorms, from the spin Dirac spectral triple on +quantum CP1 using the 𝑞-monopole. +Once more, let 𝜅 > 0, let (𝑃, Ω𝑃, d𝑃, Π) be a 𝜅-differentiable quantum principal U(1)- +bundle over 𝐵, let 𝜗 be the connection 1-form of Π, let ˆΦ𝑃 be the Fröhlich automorphism +of Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) = (𝑃, Ω𝑃,hor, d𝑃,hor), and let Φ𝑃 be the Fröhlich automorphism of the +Hermitian line 𝐵-bimodule L(𝑃)(1), so that ˆΦ𝑃 and Φ𝑃 agree on Z(Ω𝐵)0. Hence, recall that +ˆΦ𝑃 induces the right Z-action on Z>0(𝐵) � (Z(Ω𝐵)0)× ++ defined by (4.12), which therefore +extends, mutatis mutandis to a right Z-action on Z(𝐵)× ++ in terms of Φ𝑃. +We begin with the analogue for locally bounded commutator representations of modular +automorphisms. +Definition 4.56. Suppose that (𝐻, 𝜋, 𝐷) is a locally bounded commutator representation +of (𝑃; Ω𝑃, d𝑃). A modular symmetry of (𝐻, 𝜋, 𝐷) is an even positive U(1)-invariant invertible +operator 𝑁 ∈ LU(1) +loc (𝐻) the restricts to the identity on 𝐻U(1), commutes with 𝜋(𝐵), and +satisfies 𝑁 ran(𝜋)𝑁−1 = ran(𝜋). +Remark 4.57. Let 𝔓 denote the 𝐶∗-completion for 𝑃. Suppose that (𝐻, 𝜋, 𝐷) is a locally +bounded commutator representation of (𝑃; Ω𝑃, d𝑃), that 𝜈 is a modular automorphism of +Ω𝑃, and that 𝑁 is a modular symmetry of (𝐻, 𝜋, 𝐷), such that 𝑁−1𝜋(·)𝑁 = 𝜋 ◦ 𝜈. Hence, let +𝐷𝑁 � 𝑁𝐷𝑁. Since, for all 𝑝 ∈ 𝑃, +𝑁[𝐷, 𝜋(𝑝)]𝑁 = 𝐷𝑁𝜋(𝜈(𝑝)) − 𝜋 �𝜈−1(𝑝)�𝐷𝑁 = 𝐷𝑁𝜋(𝑝) − 𝜋 �𝜈−2(𝑝)�𝐷𝑁, +it follows that (𝑃, 𝐻, 𝐷𝑁) defined a U(1)-equivariant even 𝜈−2-twisted spectral triple for 𝔓 +only if 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻). +In light of the above remark, the following theorem will exclude the existence of non- +pathological U(1)-equivariant twisted spectral triples that faithfully represent the total spaces +of the 𝑞-monopole or the real multiplication instanton of Example 3.52, respectively. In +particular, it will imply that faithful projectable commutator representations of these two +examples cannot naturally be accommodated by the theory of twisted spectral triples. + +74 +BRANIMIR ĆAĆIĆ +Theorem 4.58. Suppose that Z(𝐵) = C. Let (𝐻, 𝜋, 𝐷) be a locally bounded commutator rep- +resentation of (𝑃; Ω𝑃, d𝑃); suppose that 𝜋 is injective, the subspace 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻, +and there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷), such that 𝑁 · 𝜋𝐷(Ω1 +𝑃)𝑁 ⊆ LU(1) (𝐻). Let +𝜂 ∈ Ω1 +𝑃,hor \ {0} and 𝑡 ∈ (0, ∞) \ {𝜅}, and suppose that +∀𝑝 ∈ 𝑃, +𝜂 · 𝑝 = Λ𝑡(𝑝) · 𝜂. +(4.50) +Then (id −Π)(Ω1 +𝑃) ⊆ ker 𝜋𝐷 or 𝜋𝐷(𝜂) = 0. +Lemma 4.59. Let (𝐻, 𝜋, 𝐷) be a U(1)-equivariant commutator representation of (𝑃; Ω𝑃, d𝑃). If +𝜋 is injective, the map �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : Z(𝐵) → L(𝐻U(1)) is isometric, and 𝜋(𝑃) · 𝐻U(1) is +dense in 𝐻, then for every modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷), there exists a unique right 1-cocycle +𝜇 : Z → Z(𝐵)× ++, such that +𝑁 = +� +𝑗∈Z +𝜋(𝜇(−𝑗))↾𝐻𝑗 . +(4.51) +Conversely, for every 1-cocycle 𝜇 : Z → Z(𝐵)× ++, (4.51) defines a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷). +Proof. We prove the non-trivial direction. Suppose that 𝜋 is injective, the map 𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) +restricts to an isometry on Z(𝐵), and 𝜋(𝑃)·𝐻U(1) is dense in 𝐻. Let 𝑁 be a modular symmetry +of (𝐻, 𝜋, 𝐷). Since 𝜋 is injective, there exists a unique U(1)-equivariant algebra automorphism +Δ of 𝑃, such that 𝜋 ◦ Δ = 𝑁−1𝜋(·)𝑁; in particular, Δ↾𝐵= id𝐵 since 𝑁 commutes with 𝜋(𝐵). +Hence, by Lemma 4.23, mutatis mutandis, there exists a unique 1-cocycle 𝜇 : Z → Z(𝐵)×, +such that Δ(𝑝) = 𝑝 · 𝜇(𝑗) for all 𝑗 ∈ Z and 𝑝 ∈ 𝑃𝑗. Hence, for all 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗 and +ℎ ∈ 𝐻U(1), 𝑁𝜋(𝑝)ℎ = 𝑁𝜋(𝑝)𝑁−1ℎ = 𝜋(Δ−1(𝑝))ℎ = 𝜋(𝑝𝜇(𝑗)−1)ℎ = 𝜋(𝜇(−𝑗))𝜋(𝑝)ℎ since +ˆΦ𝑗 +𝑃(𝜇(𝑗)−1) = 𝜇(−𝑗), so that (𝑁, 𝜇) satisfies (4.51) since 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻. Finally, let +(𝜖𝑖)𝑁 +𝑖=1 be a finite family in 𝑃1 satisfying �𝑁 +𝑖=1 𝜖∗ +𝑖 𝜖𝑖 = 1. Then +0 ≤ +∑︁𝑁 +𝑖=1 𝜋(𝜖𝑖)∗𝑁𝜋(𝜖𝑖) = +∑︁𝑁 +𝑖=1 𝜋(𝜖𝑖)∗𝜋(𝜖𝑖𝜇(1)−1)𝑁 = 𝜋(𝜇(1)−1)𝑁, +so that 𝜋(𝜇(1)↾𝐻U(1))≥ 0. Hence, since �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : Z(𝐵) → L(𝐻U(1)) is isometric, it +follows that 𝜇(1) ≥ 0, so that 𝜇 takes its values in the subgroup Z(𝐵)× +≥0. +□ +Lemma 4.60. Suppose that Z(𝐵) = C. Let 𝜂 ∈ Ω1 +𝑃,hor \ {0} and 𝑡 ∈ (0, ∞) \ {𝜅}, and +suppose that 𝜂 and 𝑡 satisfy (4.50). Let (𝐻, 𝜋, 𝐷) be a locally bounded commutator representation of +(𝑃; Ω𝑃, d𝑃), such that 𝜋 is injective and 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻. For every modular symmetry +𝑁 of (𝐻, 𝜋, 𝐷), the operator 𝑁𝜋𝐷(𝜔)𝑁 is bounded only if 𝑁 = Λ𝑡−1/2 or 𝜋𝐷(𝜔) = 0. +Proof. Let 𝑁 be a modular symmetry of (𝐻, 𝜋, 𝐷); suppose that 𝑁𝜋𝐷(𝜔)𝑁 is bounded. Since +Z(𝐵) = C, it follows from Lemma 4.23 that there exists unique 𝑠 ∈ (0, ∞), such that 𝑁 = Λ𝑠. +Now, let (𝑒𝑖)𝑚 +𝑖=1 and (𝜖𝑗)𝑛 +𝑗=1 be finite families in 𝑃1, such that �𝑚 +𝑖=1 𝑒𝑖𝑒∗ +𝑖 = 1 and �𝑛 +𝑗=1 𝜖∗ +𝑗 𝜖𝑗 = 1; +hence, let 𝜙± : LU(1) (𝐻) → LU(1) (𝐻) be the unit-preserving contractions from the proof of +Proposition 4.36 induced by (𝑒𝑖)𝑚 +𝑖=1 and (𝜖𝑗)𝑛 +𝑗=1, respectively. Then +𝜙+(𝑁𝜋𝐷(𝜂)𝑁) = +∑︁𝑚 +𝑖=1 𝜋(𝑒𝑖)Λ𝑠𝜋𝐷(𝜂)Λ𝑠𝜋(𝑒∗ +𝑖 ) = 𝜋 +�∑︁𝑚 +𝑖=1 𝑒𝑖 · (Λ𝑠 ◦ Λ𝑡 ◦ Λ𝑠)(𝑒∗ +𝑖 ) +� +𝑁𝜋𝐷(𝜂)𝑁 += 𝑠2𝑡𝑁𝜋𝐷(𝜂)𝑁, +while a similar calculation shows that 𝜙−(𝑁𝜋𝐷(𝜂)𝑁) = (𝑠2𝑡)−1𝑁𝜋𝐷(𝜂)𝑁. Hence, +∥𝑁𝜋𝐷(𝜂)𝑁∥ = (𝑠2𝑡)∓1∥𝜙±(𝑁𝜋𝐷(𝜂)𝑁)∥ ≤ (𝑠2𝑡)∓1∥𝑁𝜋𝐷(𝜂)𝑁∥, +so that 𝑁𝜋𝐷(𝜂)𝑁 = 0 or 𝑠2𝑡 = 1. +□ + +NONCOMMUTATIVE U(1)-GAUGE THEORY +75 +Proof of Theorem 4.58. Let 𝑁 be a modular symmetry of the locally bounded commutator +representation (𝐻, 𝜋, 𝐷) that satisfies 𝑁 · 𝜋𝐷(Ω1 +𝑃) · 𝑁 ⊆ LU(1) (𝐻). Suppose that 𝜋𝐷(𝜂) ≠ ∅. +By Lemma 4.60 applied to 𝜂, it follows that 𝑁 = Λ𝑡−1/2 ≠ Λ𝜅−1/2; hence, by Lemma 4.60 applied +to 𝜗, it follows that 𝜋𝐷(𝜗) = 0. Since 𝜗 generates (id −Π)(Ω1 +𝑃) as a left 𝑃-module, it follows +that (id −Π)(Ω1 +𝑃) ⊆ ker 𝜋𝐷. +□ +Example 4.61. Continuing from Example 4.37, let (𝐻, 𝜋, 𝐷) be a locally bounded commuta- +tor representation of (O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞), such that 𝜋 is injective and the subspace +𝜋(O𝑞(SU(2))) · 𝐻U(1) is dense in 𝐻. If there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷) sat- +isfying 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻), then (id −Π𝑞)(Ω1 +𝑃) ⊆ ker 𝜋𝐷 or Ω1 +𝑞,hor(SU(2)) ⊆ ker 𝜋𝐷. +Suppose that 𝑁 is such a modular symmetry and (id −Π𝑞)(Ω1 +𝑞(SU(2)))\ker 𝜋𝐷 ≠ ∅. Note that +(𝜂, 𝑡) � (𝑒±, 𝑞) satisfy (4.50), where 𝑞 ≠ 𝑞2, so that 𝜋𝐷(𝑒±) = 0 by Theorem 4.58. Since {𝑒+, 𝑒−} +generates Ω1 +𝑞,hor(SU(2)) as a left O𝑞(SU(2))-module, it follows that Ω1 +𝑞,hor(SU(2)) ⊆ ker 𝜋𝐷. +Example 4.62. Continuing from Example 4.38, let (𝐻, 𝜋, 𝐷) be a U(1)-equivariant commu- +tator representation of (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃), such that 𝜋 is injective and 𝜋(𝑃𝜃) · 𝐻U(1) is dense in 𝐻, +If there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷) satisfying 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻), then +(id −Π𝑃𝜃)(Ω1 +𝑃𝜃) ⊆ ker 𝜋𝐷 or Ω1 +𝑃𝜃,hor ⊆ ker 𝜋𝐷. Indeed, note that the left 𝑃-module Ω1 +𝑃,hor is +freely generated by 𝑒1, 𝑒2 ∈ Z(Ω𝜃(T2))1, where (4.50) is satisfied by (𝜂, 𝑡) = (𝑒𝑖, 𝜖𝜃) for 𝑖 = 1, 2 +by Example 2.41. Since 𝜖𝜃 ≠ 𝜖2 +𝜃, we may apply Theorem 4.58 exactly as in Example 4.55. +Since a single modular symmetry cannot generally be used to control the unboundedness +of represented 1-forms, we are forced to allow for distinct modular symmetries in the vertical +and horizontal directions. Recall that (id −Π)(Ω1 +𝑃) = 𝑃 · 𝜗 and Π(Ω1 +𝑃) = 𝑃 · Ω1 +𝐵. +Definition 4.63. Let (𝐻, 𝜋, 𝐷,Γ) be a projectable commutator representation of (𝑃; Ω𝑃, d𝑃; Π). +(1) A vertical twist for (𝐻, 𝜋, 𝐷,Γ) is a pair (𝑁ver, 𝜈ver), where 𝑁ver is a modular symmetry of +(𝐻, 𝜋, 𝐷) commuting with both Γ and 𝜋𝐷(𝜗) and 𝜈ver is a modular automorphism of Ω𝑃, +such that 𝑁−1 +ver𝜋(·) · 𝑁ver = 𝜋 ◦ 𝜈ver↾𝑃 and 𝑁ver𝜋𝐷(𝜗)𝑁ver ∈ LU(1) (𝐻). +(2) A horizontal twist for (𝐻, 𝜋, 𝐷,Γ) is a pair (𝑁hor, 𝜈hor), where 𝑁hor is a modular symmetry +of (𝐻, 𝜋, 𝐷) commuting with both Γ and 𝜋𝐷(𝜗) and 𝜈hor is a modular automorphism of +Ω𝑃, such that 𝑁−1 +hor𝜋(·)𝑁hor = 𝜋 ◦ 𝜈hor↾𝑃 and 𝑁hor𝜋𝐷 +�Ω1 +𝐵 +�𝑁hor ⊆ LU(1) (𝐻). +Thus, if (𝐻, 𝜋, 𝐷,Γ) is a projectable commutator representation of (𝑃; Ω𝑃, d𝑃; Π), then any +vertical twist (𝑁ver, 𝜈ver) satisfies 𝑁ver𝜋𝐷 +�(id −Π)(Ω1 +𝑃)�𝑁ver ⊆ LU(1) (𝐻), and any horizontal +twist (𝑁hor, 𝜈hor) satisfies 𝑁hor𝜋𝐷 +�Π(Ω1 +𝑃)�𝑁hor ⊆ LU(1) (𝐻). +We now study the existence of vertical and horizontal twists for faithful projectable com- +mutator representations. Lemmata 4.23 and 4.59 justify the following convenient definition. +Definition 4.64. Suppose that (𝐻, 𝜋, 𝐷,Γ) is faithful projectable commutator representation +of (𝑃; Ω𝑃, d𝑃; Π). We define a modular pair for (𝐻, 𝜋, 𝐷,Γ) to be a pair (𝑁, 𝜈), where 𝑁 is +a modular symmetry of (𝐻, 𝜋, 𝐷) and 𝜈 is a modular automorphism of Ω𝑃 satisfying the +equation 𝑁−1𝜋(·)𝑁 = 𝜋 ◦ 𝜈↾𝑃. In this case, the symbol of (𝑁, 𝜈) is the unique right 1-cocycle +𝜆 : Z → Z>0(𝐵), such that +∀𝑗 ∈ Z, +𝑁↾𝐻𝑗= 𝜋(𝜆(−𝑗))↾𝐻𝑗, +𝜈↾(Ω𝑃)𝑗= (𝜔 ↦→ 𝜔𝜆(𝑗)) . +We first show that there is a canonical choice of vertical twist, which is unique whenever 𝐵 +satisfies Z(𝐵) = C, e.g., when 𝐵 is O𝑞(CP1) for 𝑞 ∈ (0, ∞) \ {1} or 𝐶∞ +𝜃 (T2) for 𝜃 irrational. +Proposition 4.65. Suppose that (𝐻, 𝜋, 𝐷,Γ) is a faithful projectable commutator representation +of (𝑃; Ω𝑃, d𝑃; Π). Then (Λ𝜅−1/2, Λ𝜅1/2) defines a vertical twist of (𝐻, 𝜋, 𝐷,Γ), which is unique +whenever Z(𝐵) = C. + +76 +BRANIMIR ĆAĆIĆ +Proof. Suppose that (𝑁, 𝜈) is a modular pair for (𝐻, 𝜋, 𝐷,Γ) with symbol 𝜆. Since the operator +𝜋𝐷(𝜃) satisfies 𝜋𝐷(𝜃)2 = Λ4 +𝜅, it follows that (𝑁𝜋𝐷(𝜃)𝑁)2 = Λ2 +𝜅𝑁4, so that (𝑁, 𝜈) is a vertical +twist for (𝐻, 𝜋, 𝐷,Γ) if and only if sup𝑗∈Z 𝜅−𝑗/2∥𝜋(𝜆(−𝑗)) ↾𝐻𝑗 ∥ < +∞, which is certainly +satisfied by 𝜆 � (𝑗 ↦→ 𝜅−𝑗/2). Moreover, if Z(𝐵) = C, then 𝜆 = (𝑗 ↦→ 𝑡𝑗) for unique real 𝑡 > 0, +so that (𝑁, 𝜈) is a vertical twist for (𝐻, 𝜋, 𝐷,Γ) if and only if sup𝑗∈Z(𝜅1/2𝑡)−𝑗 < +∞, if and +only if 𝑡 = 𝜅−1/2. +□ +To characterize existence of horizontal twists, we shall need the following broad gener- +alisation of a definition from the literature on spectral triples for crossed products due to +Bellissard–Marcolli–Reihani [16]. +Definition 4.66. Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation +of (𝐵; Ω𝐵, d𝐵). Let Γ be a group, and let ˆ𝐹 : Γ → DPic(𝐵) be a homomorphism, so that the +right DPic(𝐵)-action on Z>0(𝐵) of (4.10) pulls back via ˆΦ ◦ 𝜋0( ˆ𝐹) to a right Γ-action. For +each 𝛾 ∈ Z, let (𝐹(𝛾), 𝜎𝛾, ∇𝛾) � ˆ𝐹(𝛾), and equip 𝐹(𝛾) ⊗𝐵 𝐻 with the inner product defined +by (2.12); hence, for each 𝛽 ∈ Ω1 +𝐵, define 𝜌𝛾 [𝛽] : 𝐹(𝛾) ⊗𝐵 𝐻 → 𝐹(𝛾) ⊗𝐵 𝐻 by +∀𝑥 ∈ 𝐹(𝛾), ∀ℎ ∈ 𝐻, +𝜌𝛾 [𝛽](𝑥 ⊗ ℎ) � 𝜎𝛾(𝛽 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜋𝐷 +� +𝜎𝛾(𝛽 ⊗ 𝑥) ⟨1⟩ +� +ℎ, +(4.52) +and let ∥𝜌𝛾 [𝛽]∥ denote the resulting operator norm of 𝜌𝛾 [𝛽], which we set to equal +∞ +whenever 𝜌𝛾 [𝛽] is not bounded. Given a 1-cocycle 𝜆 : Γ → Z>0(𝐵), we say that ˆ𝐹 is 𝜆- +metrically equicontinuous with respect to (𝐻, 𝜋, 𝐷) whenever +∀𝛽 ∈ Ω1 +𝐵, +sup +𝛾∈Γ +��𝜌𝛾 +� +𝜆(𝛾−1)2𝛽 +��� < +∞. +(4.53) +Example 4.67. Recall the homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ +𝜃 (T2)) of Example 2.31; de- +fine a group homomorphism 𝜆 : Γ𝜃 → R>0 by 𝜆 � �𝑔 ↦→ (𝑔21𝜃 + 𝑔22)−1/2�. Then ˆ𝐸 is 𝜆- +metrically equicontinuous with respect to every faithful bounded commutator representation +of (𝐶∞ +𝜃 (T2), Ω𝜃(T2), d). Indeed, let (𝐻, 𝜋, 𝐷) be such a bounded commutator representation. +Recall that the left 𝐶∞ +𝜃 (T2)-module Ω1 +𝜃 (T2) is generated by {𝑒1, 𝑒2} ⊂ Z(Ω𝜃(T2))1. Given +𝑖 = 1, 2 and 𝑔 ∈ Γ𝜃, it follows by construction of ˆ𝐸 that 𝜌𝑔[𝑒𝑖] = +1 +𝑔21𝜃+𝑔22 id ⊗𝜋𝐷(𝑒𝑖), so that +∥𝜌𝑔[𝜆(𝑔−1)2𝑒𝑖]∥ = ∥id ⊗𝜋𝐷(𝑒𝑖)∥ ≤ ∥id∥∥𝜋𝐷(𝑒𝑖)∥ ≤ ∥𝜋𝐷(𝑒𝑖)∥. +Example 4.68. Recall from Example 3.26 the homomorphisms E : Z → Pic(O𝑞(CP1)) +and ˆE : Z → DPic(O𝑞(CP1)), and define a group homomorphism 𝜆 : Z → R>0 by +𝜆 � (𝑘 ↦→ 𝑞−𝑗). Then ˆE is 𝜆-metrically equicontinuous with respect to every faithful +bounded commutator representation of (O𝑞(CP1); Ω𝑞(CP1), d𝑞). Indeed, let (𝐻, 𝜋, 𝐷) be such +a bounded commutator representation. Recall that Ω𝑞(CP1) = E−2·𝑒+⊕ E2·𝑒−. Choose a coba- +sis (𝜂∓ +𝑖 )𝑁∓ +𝑖=1 for E∓2, and define 𝜏± : 𝐻 → E±2 ⊗O𝑞(CP1) 𝐻 by 𝜏±(ℎ) � �𝑁∓ +𝑖=1(𝜂∓ +𝑖 )∗ ⊗ 𝜋𝐷(𝜂∓ +𝑖 𝑒±)ℎ +for ℎ ∈ 𝐻; note that 𝜏± is bounded and left 𝐵-linear since, for all ℎ, 𝑘 ∈ 𝐻 and 𝑥 ∈ E±2, +⟨𝑥 ⊗ 𝑘, 𝜏±(ℎ)⟩ = +� +𝑘, 𝜋𝐷 +� +𝑥∗ �∑︁𝑁∓ +𝑖=1(𝜂∓ +𝑖 )∗𝜂∓ +𝑖 +� +𝑒±� +ℎ +� += ⟨𝑘, 𝜋𝐷(𝑥∗𝑒±)ℎ⟩. +For all 𝑖, 𝑗 ∈ Z, define a unitary 𝑉𝑖,𝑗 : E𝑖⊗O𝑞(CP1) (E𝑗⊗O𝑞(CP1) 𝐻) → (E𝑖⊗O𝑞(CP1) E𝑗)⊗O𝑞(CP1) 𝐻 +by 𝑉𝑖,𝑗 � (𝑥 ⊗ (𝑦 ⊗ ℎ) ↦→ (𝑥 ⊗ 𝑦) ⊗ ℎ) and, for each 𝑝 ∈ E𝑖, the bounded adjointable map +𝜋𝑖,𝑗(𝑝) : E𝑗 ⊗O𝑞(CP1) 𝐻 → E𝑖+𝑗 ⊗O𝑞(CP1) 𝐻 by 𝜋𝑖,𝑗(𝑝) � (𝑦 ⊗ ℎ ↦→ 𝑝 · 𝑦 ⊗ ℎ), which satisfies +∥𝜆𝑖,𝑗(𝑝)∥ ≤ ∥𝑝∥. At last, let 𝑗 ∈ Z and 𝑝 ∈ E∓2 be given. Since 𝑝𝑒±𝑥 = 𝑞−𝑗 �𝑁∓ +𝑖=1 𝑝𝑥(𝜂∓)∗ +𝑖 𝜂∓ +𝑖 𝑒± +for all 𝑥 ∈ E𝑗, it follows that 𝜌𝑗[𝜆(−𝑗)2𝑝𝑒±] = 𝜋∓2,𝑗±2(𝑝) ◦ (E(2) +𝑗,±2 ⊗ id) ◦ 𝑉𝑗,±2 ◦ (id ⊗ 𝜏±), and +hence ∥𝜌𝑗[𝜆(−𝑗)2𝑝𝑒±]∥ ≤ ∥𝜋∓2,𝑗±2(𝑝)∥∥E(2) +𝑗,±2 ⊗ id∥∥𝑉𝑗,±2∥∥id ⊗ 𝜏±∥ ≤ ∥𝑝∥∥𝜏±∥. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +77 +In light of Corollary 2.9, one may ask whether our generalised notion of metric equiconti- +nuity makes sense at the level of Hermitian line 𝐵-bimodules with connection. The following +proposition answer this question in the affirmative. +Proposition 4.69. Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation of +(𝐵; Ω𝐵, d𝐵). Let Γ be a group, let ˆ𝐹1, ˆ𝐹2 : Γ → DPic(𝐵) be homomorphisms, and suppose that +ˆ𝐹1 � ˆ𝐹2 in Hom(Γ, DPic(𝐵)). Hence, let 𝜆 : Γ → Z>0(𝐵) be a right 1-cocycle for the pullback +of the DPic(𝐵)-action of (4.10) by 𝜋0( ˆ𝐹1) = 𝜋0( ˆ𝐹2). Then ˆ𝐹1 is 𝜆-metrically equicontinuous if +and only if ˆ𝐹2 is. +Proof. Choose a natural isomorphism 𝜂 : ˆ𝐹1 → ˆ𝐹2. Let 𝛾 ∈ Γ be given. For 𝑖 = 1, 2, +let (𝐹𝑖(𝛾), 𝜎𝑖;𝛾, ∇𝑖;𝛾) � ˆ𝐹𝑖(𝛾)), and define 𝜌𝑖;𝛾 [𝛽] : 𝐹𝑖(𝛾) ⊗𝐵 𝐻 → 𝐹𝑖(𝛾) ⊗𝐵 𝐻 for each +𝛽 ∈ Ω1 +𝐵 by (4.52). Since 𝜂𝛾 : (𝐹1(𝛾), 𝜎1;𝛾, ∇1;𝛾) → (𝐹2(𝛾), 𝜎2;𝛾, ∇2;𝛾) is an isomorphism is +DPic(𝐵), the map 𝜂𝛾 ⊗ id : 𝐹1(𝛾) ⊗𝐵 𝐻 → 𝐹2(𝛾) ⊗𝐵 𝐻 is a well-defined unitary that satisfies +(𝜂𝛾 ⊗ id) ◦ 𝜌1;𝛾 [𝛽] = 𝜌2;𝛾 [𝛽] ◦ (𝜂𝛾 ⊗ id) for all 𝛽 ∈ Ω1 +𝐵. +□ +Hence, given a Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) and a group 1- +cocycle 𝜆 : Z → Z>0(𝐵) for the right Z-action generated by ˆΦ−1 +𝐸 , we define (𝐸, 𝜎𝐸, ∇𝐸) to be 𝜆- +metrically equicontinuous whenever some (and hence every) homomorphism ˆ𝐹 : Z → DPic(𝐵) +satisfying ˆ𝐹(1) � (𝐸, 𝜎𝐸, ∇𝐸) is 𝜆-metrically equicontinuous. The following characterisation +of metric equicontinuity in our general sense for crossed products by extended diffeomor- +phisms now shows that metric equicontinuity (in our sense) with respect to a trivial 1-cocycle +corresponds to the existing definition in the literature on crossed product spectral triples. +Proposition 4.70 (cf. Bellissard–Marcolli–Reihani [16]). Let (𝐻, 𝜋, 𝐷) be a bounded commu- +tator representation of (𝐵; Ω𝐵, d𝐵). Let (𝜔, 𝜙) ∈ � +Diff(𝐵) and let 𝜆 : Z → Z>0(𝐵) be a right +1-cocycle for the right Z-action generated by 𝜙−1. Then ˆ𝜏(𝜔, 𝜙) is 𝜆-metrically equicontinuous +with respect to (𝐻, 𝜋, 𝐷) if and only if +∀𝑏 ∈ 𝐵, +sup +𝑘∈Z +��𝜋(𝜆(𝑘)−1) · [𝐷, 𝜋(𝜙−𝑘(𝑏))] · 𝜋(𝜆(𝑘)−1) +�� < +∞. +(4.54) +Proof. By Proposition 4.70, it suffices to check that the homomorphism ˆ𝜏 ◦ (𝑘 ↦→ (𝜔, 𝜙)𝑘) is +𝜆-metrically equicontinuous. Let 𝑘 ∈ Z be given. Define a unitary 𝑉𝑘 : 𝐵𝑘 +𝜙 ⊗𝐵 𝐻 → 𝐻 by +𝑉 � (𝑏𝑘 +𝜙 ⊗ ℎ ↦→ 𝜋(𝜙−𝑘(𝑏))ℎ). By construction of ˆ𝜏((𝜔, 𝜙)𝑘) � (𝐵𝜙, 𝜎𝜙𝑘, ∇(𝜔,𝜙)𝑘), it follows +that 𝑉𝑘𝜌𝑘[𝛽]𝑉∗ +𝑘 = 𝜋𝐷 +�𝜙−𝑘(𝛽)� for all 𝛽 ∈ Ω1 +𝐵, so that +𝑉𝑘𝜌𝑘 +� ˆΦ[ˆ𝜏(𝜔,𝜙)](𝜆(−𝑘)2) · d𝐵(𝑏) +� +𝑉∗ +𝑘 = 𝜋𝐷 +� +𝜆(𝑘)−2d𝐵𝜙−𝑘(𝑏) +� += i𝜋(𝜆(𝑘)−1)[𝐷, 𝜋(𝜙−𝑘(𝑏))]𝜋(𝜆(𝑘)−1). +for every 𝑏 ∈ 𝐵. Since d𝐵(𝐵) generates Ω1 +𝐵 as a left 𝐵-module, this proves our claim. +□ +At last, we characterise horizontal twists among all modular pairs. +Proposition 4.71. Let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐵; Ω𝐵, d𝐵), +and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) to (𝑃; Ω𝑃, d𝑃, Π). Let (𝑁, 𝜈) be a modular pair for +( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) with symbol 𝜆. Then (𝑁, 𝜈) defines a horizontal twist for ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) if and only if +ˆL◦ Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) is 𝜆-metrically equicontinuous with respect to (𝐻, 𝜋, 𝐷). +Proof. By Theorem 4.52, we may assume that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) = (𝜄𝑃)!(𝐻, 𝜋, 𝐷) without any loss of +generality. Hence, we reprise the notation of the proof of Proposition 4.50. In particular, it +follows that 𝜏 ◦ 𝑁 ◦ 𝜏∗ = id ⊗𝜈 ⊗ id, which makes it clear that 𝑁 ˜𝜋(·)𝑁−1 = ˜𝜋 ◦ 𝜈. + +78 +BRANIMIR ĆAĆIĆ +Let 𝛽 ∈ Ω1 +𝐵 be given. Fix 𝑗 ∈ Z. By the proof of Proposition 4.50, it follows that +˜𝜋 ˜𝐷(𝛽)↾ ˜𝐻𝑗= 𝜏∗ ◦ (𝜎3 ⊗ 𝜌𝑗[𝛽]) ◦ 𝜏↾ ˜𝐻𝑗 . +Hence, for every 𝑥 ∈ C2, 𝑝 ∈ 𝑃𝑗, and ℎ ∈ 𝐻, we find that +𝜏𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁𝜏∗(𝑥 ⊗ 𝑝 ⊗ ℎ) = (id ⊗𝜈 ⊗ id) +� +𝜎3𝑥 ⊗ 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨0⟩ ⊗ 𝜋𝐷 +� +𝜎𝑗(𝛽 ⊗ 𝑝) ⟨1⟩𝜆(𝑗) +� +ℎ +� += 𝜎3 ⊗ 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨0⟩𝜆(𝑗) ⊗ 𝜋𝐷 +� +𝜎𝑗(𝛽 ⊗ 𝑝) ⟨1⟩𝜆(𝑗) +� +ℎ += 𝜎3 ⊗ 𝜌𝑗 +� +𝜆(−𝑗)2𝛽 +� +(𝑥 ⊗ 𝑝 ⊗ ℎ), +which implies that ∥𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁↾ ˜𝐻𝑗 ∥ = +��𝜎3 ⊗ 𝜌𝑗 +� +𝜆(−𝑗)2𝛽 +��� = +��𝜌𝑗 +� +𝜆(−𝑗)2𝛽 +��� by unitarity of +𝜎3 ∈ 𝑀2(C). Thus, the operator 𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁 ∈ LU(1) +loc (𝐻) is bounded if and only if the set of +operator norms +���𝜌𝑗 +� +𝜆(−𝑗)2𝛽 +��� �� 𝑗 ∈ Z +� +is bounded from above. +□ +Example 4.72. Continuing from Examples 3.52 and 4.67, let (𝐻, 𝜋, 𝐷) be a faithful bounded +commutator representation of (𝐶∞ +𝜃 (T2), Ω𝜃(T2), d), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) +to the real multiplication instanton (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃). On the one hand, by Proposition +4.65, the modular pair (Λ𝜖−1 +𝜃 , Λ𝜖𝜃) is the unique vertical twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ). On the other +hand, by Example 4.67, the homomorphism ˆL ◦ Hor𝜖2 +𝜃 (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃) is (𝑚 ↦→ 𝜖−𝑚/2 +𝜃 +)- +equicontinuous with respect to (𝐻, 𝜋, 𝐷), so that (Λ𝜖−1/2 +𝜃 +, Λ𝜖1/2 +𝜃 ) is the unique horizontal twist +of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) by Proposition 4.71 together with Lemma 4.60 applied to (𝜂, 𝑡) = (𝑒1, 𝜖𝜃). Note +that (Λ𝜖−1 +𝜃 , Λ𝜖𝜃) and (Λ𝜖−1/2 +𝜃 +, Λ𝜖1/2 +𝜃 ) are non-trivial and distinct since 𝜖𝜃 ≠ 1. +Example 4.73. Continuing from Example 4.55, let (𝐻, 𝜋, 𝐷) be a faithful bounded commu- +tator representation of (O𝑞(CP1), Ω𝑞(CP1), d𝑞), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) to +the 𝑞-monopole (O𝑞(SU(2)); Ω𝑞(SU(2)), d𝑞; Π𝑞). On the one hand, by 4.65, the modular pair +(Λ𝑞−1, Λ𝑞) is the unique vertical twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ). On the other hand, by Example 4.68, the +homomorphism ˆE : Z → DPic(O𝑞(CP1)) of Example 3.26 is (𝑚 ↦→ 𝑞−𝑚/2)-equicontinuous +with respect to (𝐻, 𝜋, 𝐷), so that (Λ𝑞−1/2, Λ𝑞1/2) is the unique horizontal twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) +by Proposition 4.71 together with Lemma 4.60 applied to (𝜂, 𝑡) = (𝑒±, 𝑞). Note that (Λ𝑞−1, Λ𝑞) +and (Λ𝑞−1/2, Λ𝑞1/2) are non-trivial and distinct since 𝑞 ≠ 1. +We now show that total Hodge–de Rham commutator representations admit canonical +horizontal twists under a mild hypothesis that is vacuous when Z(𝐵) = C or when 𝐵 is +commutative and admits polar decompositions. +Theorem 4.74. Suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on (𝑃; Ω𝑃, d𝑃; Π). +Let 𝜇𝑃 be the symbol of Δhor, and suppose that 𝜇𝑃(1) has a square root in Z>0(𝐵); hence, let +𝜇1/2 +𝑃 +: Z → Z>0(𝐵) be the unique right 1-cocycle for the right Z-action of (4.12) that satisfies +𝜇1/2 +𝑃 (·)2 = 𝜇𝑃, and let (𝑁hor, 𝜈hor) be the modular pair with symbol 𝜇1/2 +𝑃 +for total Hodge–de +Rham commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ +𝑃, 2Π − id) induced by (Δver, Δhor, ★, 𝜏). Then +(𝑁hor, 𝜈hor) defines a horizontal twist for (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ +𝑃, 2Π − id). +Proof. We use the notation of the proof of Proposition 4.44. Hence, it suffices to show that +𝑁hor satisfies 𝑁hor · e(Ω1 +𝐵) · 𝑁hor ⊆ LU(1) (Ω𝑃). +Let 𝛽 ∈ Ω1 +𝐵 be given. Recall that the pre-Hilbert space Ω𝑃 admits the orthogonal decom- +postion Ω𝑃 = �∞ +𝑗=−∞ +�1 +𝑟=0 +�𝑁 +𝑠=0(Ω𝑟,𝑠 +𝑃 )𝑗. Hence, fix (𝑟, 𝑠, 𝑗) ∈ {0, 1} × {0, . . . , 𝑁} ×Z, so that +e(𝛽) maps (Ω𝑟,𝑠 +𝑃 )𝑗 to (Ω𝑟,𝑠+1 +𝑃 +)𝑗, and let 𝑇𝑟,𝑠 +𝑗 +� 𝑁hore(𝛽)𝑁hor↾(Ω𝑟,𝑠 +𝑃 )𝑗= e( ˆΦ𝑗 +𝑃(𝜇𝑃(𝑗)−1)𝛽)↾(Ω𝑟,𝑠 +𝑃 )𝑗. +It therefore suffices to bound the operator norm of 𝑇𝑟,𝑠 +𝑗 +uniformly in 𝑗 ∈ Z. + +NONCOMMUTATIVE U(1)-GAUGE THEORY +79 +Let 𝐸 � (Ω𝑟,𝑠 +𝑃 )𝑗, 𝐹 � (Ω𝑟,𝑠+1 +𝑃 +)𝑗, 𝑉 � (Ω𝑟,𝑠 +𝑃 )U(1), and 𝑊 � (Ω𝑟,𝑠+1 +𝑃 +)U(1), which we view as +orthogonal direct summands of the pre-Hilbert space Ω𝑃; note that each of these pre-Hilbert +spaces also defines a 𝐵-self-correspondence of finite type by the proof of Proposition 4.29, +where each pre-Hilbert space norm is bounded from above by the corresponding norm as a +𝐵-self-correspondence of finite type. Write ˆ𝐿 ◦ Hor𝜅(𝑃; Ω𝑃, d𝑃; Π) � (𝑃𝑗, 𝜎𝑃;𝑗, ∇𝑃;𝑗), where +we conflate the Hermitian line 𝐵-bimodule L(𝑃)(𝑗) with 𝑃𝑗; recall that Ω1 +𝐵 defines a 𝐵-self- +correspondence of finite type by Proposition 4.5, so that Ω1 +𝐵 ⊗𝐵 𝑃𝑗 and 𝑃𝑗 ⊗𝐵 Ω1 +𝐵 both define +𝐵-self-correspondences of finite type. Hence, we can view each of Ω1 +𝐵 ⊗𝐵 𝐸, (Ω1 +𝐵 ⊗𝐵 𝑃𝑗) ⊗𝐵 𝑉, +(𝑃𝑗 ⊗𝐵 Ω1 +𝐵) ⊗𝐵 𝑉, 𝑃𝑗 ⊗𝐵 (Ω1 +𝐵 ⊗ 𝑉) and 𝑃𝑗 ⊗𝐵 𝑊 as pre-Hilbert spaces with respect to the inner +product defined, mutatis mutandis, by (2.12). Finally, define 𝜏𝑗 : Ω1 +𝐵 ⊗𝐵 𝑃𝑗 → 𝑃𝑗 ⊗𝐵 Ω1 +𝐵 by +∀𝜂 ∈ Ω1 +𝐵 ⊗𝐵 𝑃𝑗, +𝜏𝑗(𝜂) � 𝜎𝑃;𝑗(𝜇𝑃(−𝑗))𝜂) = 𝜎𝑃;𝑗(𝜂)𝜇𝑃(𝑗)−1. +It now follows that 𝑇𝑟,𝑠 +𝑗 +: 𝐸 → 𝐹 factorizes as the composition +𝐸 +𝛽⊗− +−−−→ Ω1 +𝐵 ⊗𝐵 𝐸 +�−→ (Ω1 +𝐵 ⊗𝐵 𝑃𝑗) ⊗𝐵 𝑉 +𝜏𝑗 ⊗id +−−−−→ (𝑃𝑗 ⊗𝐵 Ω1 +𝐵) ⊗𝐵 𝑉 +�−→ 𝑃𝑗 ⊗𝐵 (Ω1 +𝐵 ⊗ 𝑉) +id ⊗𝑚𝑟,𝑠 +−−−−−→ 𝑃𝑗 ⊗𝐵 𝑊 +�−→ 𝐹, +where the first two arrows denoted by � are the usual (inverse) associators, which are unitary +[18, §8.2.12], where 𝑃𝑗 ⊗𝐵 𝑊 +�−→ 𝐹 is given by multiplication in Ω𝑃 and hence unitary by the +proof of Proposition 4.29, and where 𝑚𝑟,𝑠 : Ω1 +𝐵 ⊗ 𝑉 → 𝑊 is given by multiplication in Ω𝑃. +Let us now look at the non-trivial arrows in this composition. First, an explicit calculation +shows that 𝛽 ⊗ − � (𝜉 ↦→ 𝛽 ⊗ 𝜉) is bounded with operator norm ∥𝛽 ⊗ −∥ ≤ ∥𝛽∥, where +∥𝛽∥ = ∥𝑔𝐵(𝛽, 𝛽)∥1/2 is the norm of 𝛽 as an element of the 𝐵-self-correspondence of finite type +Ω𝐵. Next, since 𝜏𝑗 is right 𝐵-linear map between pre-Hilbert 𝐵-modules of finite type, it is +necessarily bounded and adjointable, so that 𝜏𝑗 ⊗ id is bounded as a map between pre-Hilbert +spaces with operator norm ∥𝜏𝑗 ⊗id∥ ≤ ∥𝜏𝑗∥∥id∥ = ∥𝜏𝑗∥ by standard results [18, §8.2.12]. Finally, +since 𝑚𝑟,𝑠 : Ω1 +𝐵 ⊗𝐵 𝑉 → 𝑊is a right 𝐵-linear map of pre-Hilbert 𝐵-modules of finite type, it +is bounded and adjointable, and hence bounded as a map of pre-Hilbert spaces with operator +norm ∥𝑚𝑟,𝑠∥, so that id ⊗𝑚𝑟,𝑠 is also bounded as a map of pre-Hilbert spaces with operator +norm ∥id ⊗𝑚𝑟,𝑠∥ ≤ ∥id∥∥𝑚𝑟,𝑠∥ = ∥𝑚𝑟,𝑠∥. Thus, the operator norm of 𝑇𝑟,𝑠 +𝑗 +is bounded from +above by ∥𝛽∥∥𝜏𝑗∥∥𝑚𝑟,𝑠∥, so that, at last, it suffices to show that ∥𝜏𝑗∥ = 1. +Finally, let 𝜂 ∈ Ω1 +𝐵 ⊗𝐵 𝑃𝑗 be given, so that 𝜂 = �𝑛 +𝑖=1 𝛼𝑖 ⊗ 𝑝𝑖 for 𝛼1, . . . , 𝛼𝑛 ∈ Ω1 +𝐵 and +𝑝1, . . . , 𝑝𝑛 ∈ 𝑃𝑗; hence, let ˜𝜂 � �𝑛 +𝑖=1 𝛼𝑖 · 𝑝𝑖 ∈ (Ω0,1 +𝑃 )𝑗, so that +★(˜𝜂) = +∑︁ +𝑖 +★(𝛼𝑖𝑝𝑖) = − +∑︁ +𝑖 +𝜗 ★𝐵 (𝛼𝑖)𝑝𝑖𝜇𝑃(𝑗)2−𝑁𝜅𝑗 = (−1)𝑁 ∑︁ +𝑖 +★𝐵(𝛼𝑖)𝑝𝑖𝜇𝑃(𝑗)−𝑁𝜗𝜇𝑃(𝑗)2 +by (4.20). On the one hand, +★𝑃((𝜂, 𝜂)) = +∑︁ +𝑖,𝑗 +𝑝∗ +𝑖 𝑔𝐵(𝛼𝑖, 𝛼𝑗)𝑝𝑗𝜗★𝐵(1) = (−1)𝑁∑︁ +𝑖,𝑗 +𝑝∗ +𝑖 𝛼𝑖★𝐵(1)𝑝𝑗𝜇𝑃(𝑗)−𝑁𝜗 = ˜𝜂∗★𝑃(˜𝜂)𝜇𝑃(𝑗)−2, +while on the other, +★𝑃 +�(𝜎𝑃;𝑗(𝜂), 𝜎𝑃;𝑗(𝜂))� = +∑︁ +𝑖,𝑗 +𝑔𝐵(𝛼𝑖, 𝑝∗ +𝑖 𝑝𝑗𝛼𝑗)𝜗★𝐵 (1) = (−1)𝑁 ∑︁ +𝑖,𝑗 +𝛼∗ +𝑖 𝑝∗ +𝑖 𝑝𝑗 ★𝐵 (𝛼𝑗)𝜗 = ˜𝜂∗ ★𝑃 (˜𝜂), +so that +(𝜏𝑗(𝜂), 𝜏𝑗(𝜂)) = (𝜇𝑃(𝑗)−1)∗(𝜎𝑃;𝑗(𝜂), 𝜎𝑃;𝑗(𝜂))𝜇𝑃(𝑗)−1 = (𝜎𝑃;𝑗(𝜂), 𝜎𝑃;𝑗(𝜂))𝜇𝑃(𝑗)−2 = (𝜂, 𝜂). □ + +80 +BRANIMIR ĆAĆIĆ +We conclude with a first step towards relating our constructions to Rieffel’s compact +quantum metric spaces [87]. We show that a faithful projectable commutator representation +of (𝑃; Ω𝑃, d𝑃; Π) equipped with vertical and horizontal twists yields a Lipschitz seminorm +[58, Def. 2.1] on the 𝐶∗-algebra completion of 𝑃 that satisfies a twisted Leibniz inequality [58, +Lemma 4.8]. This, in turn, will recover, up to equivalence of seminorm, Kaad–Kyed’s compact +quantum metric space on quantum SU(2) for a canonical choice of parameters [58, §4]. +Proposition 4.75 (cf. Kaad–Kyed [58, Lemma 48]). Let (𝐻, 𝜋, 𝐷,Γ) be a faithful projectable +commutator representation of (𝑃; Ω𝑃, d𝑃; Π) with vertical twist (𝑁ver, 𝜈ver) and horizontal twist +(𝑁hor, 𝜈hor). Define a U(1)-invariant norms ∥ · ∥𝜏 and ∥ · ∥𝜏,tot on 𝑃 and Ω1 +𝑃, respectively, by +∀𝑝 ∈ 𝑃, +∥𝑝∥𝜏 � max +� +∥𝜈ver(𝑝)∥ + ∥𝜈hor(𝑝)∥, ∥𝜈−1 +ver(𝑝)∥ + ∥𝜈−1 +hor(𝑝)∥ +� +, +∀𝜔 ∈ Ω1 +𝑃, +∥𝜔∥𝜏;tot � ∥𝑁ver(𝜋𝐷 ◦ (id −Π))(𝜔)𝑁ver + 𝑁hor(𝜋𝐷 ◦ Π)(𝜔)𝑁hor∥. +Then ∥ · ∥𝜏 makes 𝑃 into a normed ∗-algebra, while ∥ · ∥𝜏;tot is invariant under the ∗-operation +and satisfies +∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝜔 ∈ Ω1 +𝑃, +∥𝑝1𝜔𝑝2∥𝜏;tot ≤ ∥𝑝1∥𝜏∥𝜔∥𝜏;tot∥𝑝2∥𝜏. +(4.55) +Hence, the U(1)-invariant seminorm 𝐿𝜏 � ∥d𝑃(·)∥𝜏;tot on 𝑃 annihilates C ⊆ 𝑃, is invariant +under the ∗-operation, and satisfies +∀𝑝1, 𝑝2 ∈ 𝑃, +𝐿𝜏(𝑝1𝑝2) ≤ 𝐿𝜏(𝑝1)∥𝑝2∥𝜏 + ∥𝑝1∥𝜏𝐿𝜏(𝑝2). +(4.56) +Lemma 4.76 (cf. Kaad–Kyed [58, Rem. 4.6]). Under the hypotheses of Proposition 4.75, the +U(1)-invariant seminorms ∥ · ∥𝜏,ver and ∥ · ∥𝜏,hor on Ω1 +𝑃 defined by +∥ · ∥𝜏,ver � ∥𝑁ver(𝜋𝐷 ◦ (id −Π))(·)𝑁ver∥, +∥ · ∥𝜏,hor � ∥𝑁hor(𝜋𝐷 ◦ Π)(·)𝑁hor∥ +satisfy the inequality max{∥𝜔∥𝜏,ver, ∥𝜔∥𝜏,hor} ≤ ∥𝜔∥𝜏,tot for all 𝜔 ∈ Ω1 +𝑃. +Proof. Let 𝜔 ∈ Ω1 +𝑃 be given; let 𝜔hor � Π(𝜔), and write (id −Π)(𝜔) = 𝑝𝜗 for unique 𝑝 ∈ 𝑃. +Let 𝑐 � 𝜋𝐷(𝜗)Λ−1 +𝜅 , which is a U(1)-invariant self-adjoint unitary by definition of a projectable +commutator representation. On the one hand, 𝑐 manifestly commutes with the operator +𝜋𝐷(𝑝𝜗) = 𝜋(𝑝)𝑐Λ𝜅. On the other hand, since Ω1 +𝑃,hor = 𝑃 · d𝐵(𝐵) and +[𝐷, 𝜋(𝑏)] = [𝜋𝐷(𝜗)𝜕𝜅 + 𝐷hor, 𝜋(𝑏)] = [𝐷hor, 𝜋(𝑏)] +for all 𝑏 ∈ 𝐵, it follows that 𝑐 anticommutes with 𝜋𝐷(𝜔hor) as well. Setting 𝐸± � 1 +2 (id ±𝑐), +we may now decompose 𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver + 𝑁hor𝜋𝐷(𝜔hor)𝑁hor with respect to the orthogonal +direct sum decomposition 𝐻 = 𝐸+(𝐻) ⊕ 𝐸−(𝐻) as +𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver+𝑁hor𝜋𝐷(𝜔hor)𝑁hor = +� 𝐸+𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸+ +𝐸+𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸− +𝐸−𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸+ +𝐸−𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸− +� +, +so that +∥𝜔∥𝜏;ver = +���� +�𝐸+𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸+ +0 +0 +𝐸−𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸− +����� ≤ ∥𝜔∥𝜏;tot, +∥𝜔∥𝜏,hor = +���� +� +0 +𝐸+𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸− +𝐸−𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸+ +0 +����� ≤ ∥𝜔∥𝜏;tot. +□ +Proof of Proposition 4.75. In what follows, we use the notation of Lemma 4.76. The only non- +trivial points are positive-definiteness of ∥ · ∥𝜏,tot, (4.55), and (4.56); note that ∥ · ∥𝜏,𝜏 is positive- +definite by the proof of Lemma 4.76, while (4.55) implies (4.56) by the usual Leibniz rule for + +NONCOMMUTATIVE U(1)-GAUGE THEORY +81 +d𝑃. Let 𝑝1, 𝑝2 ∈ 𝑃 and 𝜔 ∈ Ω1 +𝑃 be given; set 𝜔ver � (id −Π)(𝜔) and 𝜔hor � Π(𝜔). Then, by +Lemma 4.76, +∥𝑝1𝜔𝑝2∥𝜏;ver = ∥𝑁ver𝜋𝐷(𝑝1𝜔ver𝑝2)𝑁ver∥ ≤ ∥𝜈−1 +ver(𝑝)∥∥𝑁ver𝜋𝐷(𝜔ver)∥∥𝜈ver(𝑝)∥ +≤ ∥𝜈−1 +ver(𝑝)∥∥𝜔∥𝜏,tot∥𝜈ver(𝑝)∥, +and similarly ∥𝑝1𝜔𝑝2∥𝜏;hor ≤ ∥𝜈−1 +hor(𝑝)∥∥𝜔∥𝜏,tot∥𝜈hor(𝑝)∥, so that, in turn, +∥𝑝1𝜔𝑝2∥𝜏;tot ≤ ∥𝑝1𝜔𝑝2∥𝜏;ver + ∥𝑝1𝜔𝑝2∥𝜏;hor +≤ ∥𝜈−1 +ver(𝑝)∥∥𝜔∥𝜏,tot∥𝜈ver(𝑝)∥ + ∥𝜈−1 +hor(𝑝)∥∥𝜔∥𝜏,tot∥𝜈hor(𝑝)∥ +≤ ∥𝑝∥𝜏∥𝜔∥𝜏;tot∥𝑝∥𝜏. +□ +Example 4.77. Continuing from Examples 4.55 and 4.73, we may apply Proposition 4.75 to +(/𝑆𝑞(SU(2)), ˜𝜋, ˜/𝐷𝑞,Γ𝑞) equipped with its unique vertical twist (Λ𝑞−1, Λ𝑞) and unique horizontal +twist (Λ𝑞−1/2, Λ𝑞1/2). We claim that the resulting seminorm 𝐿𝜏 is equivalent to 𝐿𝑞2,𝑞, where +(𝐿𝑡,𝑞)𝑡∈(0,∞) is the family of Lipschitz seminorms on O𝑞(SU(2)) with which Kaad–Kyed make +𝐶𝑞(SU(2)) into a compact quantum metric space [58, Def. 4.6 & Cor. 5.24]. First, note that +∀𝑝 ∈ O𝑞(SU(2)), +∥𝑝∥𝜏 = max +� +∥Λ𝑞(𝑝)∥ + ∥Λ𝑞1/2 (𝑝)∥, ∥Λ−1 +𝑞 (𝑝)∥ + ∥Λ−1 +𝑞1/2 (𝑝)∥ +� += ∥𝑝∥𝑞2,𝑞, +where (∥ · ∥𝑡,𝑞)𝑡∈(0,∞) is the family of norms on O𝑞(SU(2)) of [58, §3.5], so that (4.56) for 𝐿𝜏 is +identical to the inequality of [58, Lemma 4.8] for 𝐿𝑞2,𝑞. Next, using the explicit construction of +Example 4.55 together with the proof of Proposition 4.50, we may now write 𝐿𝜏 = ∥𝜕tot(·)∥, +where 𝜕tot : O𝑞(SU(2)) → LU(1) (/𝑆𝑞(SU(2)) is given by +𝜕tot � Λ𝑞−1i[ ˜/𝐷𝑞,ver, ˜𝜋(·)]Λ𝑞−1 + Λ𝑞−1i[ ˜/𝐷𝑞,hor, ˜𝜋(·)]Λ𝑞−1 += 𝜎2 ⊗ +�Λ𝑞 ◦ 𝜕𝑞2 +0 +0 +Λ𝑞 ◦ 𝜕𝑞2 +� ++ 𝜎3 ⊗ +� +0 +Λ𝑞−1/2 ◦ 𝜕+ +Λ𝑞−1/2 ◦ 𝜕− +0 +� +; +here, by abuse of notation, we identify 𝑀2(O𝑞(SU(2))) � 𝑀2(C) ⊗ O𝑞(SU(2)) with its image +in L(C2 ⊗ O𝑞(SU(2))) via left multiplication of O𝑞(SU(2)) on itself, while, for 𝑡 ∈ (0, ∞), we +define 𝜕𝑡 : O𝑞(SU(2)) → O𝑞(SU(2)) by 𝜕𝑡 � � +𝑗∈Z 2𝜋i[𝑗]𝑡 idO𝑞(SU(2))𝑗. At last, we relate 𝐿𝜏 +to 𝐿𝑞2,𝑞 as follows. On the one hand, note that if 𝐻 is a Z/2Z-graded pre-Hilbert space with an +odd self-adjoint unitary 𝑐 and 𝑆 : 𝐻 → 𝐻 is an odd bounded operator supercommuting with 𝑐, +then ∥𝑆∥ = ∥𝑆0∥ for 𝑆0 � −i𝑐 ◦ 𝑆↾𝐻even= i𝑆 ◦ 𝑐↾𝐻even. On the other, we may construct unitary +𝑈 : O𝑞(SU(2))2 → /𝑆𝑞(SU(2))even by 𝑈 � �� 𝑝1 +𝑝2 +� ↦→ � 1 +0 +� ⊗ � 1 +0 +� ⊗ 𝑝1 + � 0 +1 +� ⊗ � 0 +1 +� ⊗ 𝑝2 +�, +Applying these considerations to 𝑐 = 𝜎1 ⊗ id ⊗ id and 𝑆 = 𝜕tot(𝑝) for 𝑝 ∈ O𝑞(SU(2)) shows +that 𝐿𝜏 = ∥𝜕′ +tot(·)∥, where 𝜕′ +tot : O𝑞(SU(2)) → 𝑀2(O𝑞(SU(2))) is given by +𝜕′ +tot � +� Λ𝑞 ◦ 𝜕𝑞2 +−Λ𝑞−1/2 ◦ 𝜕+ +Λ𝑞−1/2 ◦ 𝜕− +−Λ𝑞 ◦ 𝜕𝑞2 +� +. +But now, for each 𝑡 ∈ (0, ∞), a careful comparison with Kaad–Kyed’s notations [58, §§3.1, 3.5, +4.1] shows that 𝐿𝑡,𝑞 = ∥𝜕𝑡,𝑞(·)∥, where 𝜕𝑡,𝑞 : O𝑞(SU(2)) → 𝑀2(O𝑞(SU(2))) is given by +𝜕𝑡,𝑞 = +�−i𝐾𝑡Λ𝑡1/2 ◦ 𝜕𝑡 +−Λ𝑞−1/2 ◦ 𝜕+ +−Λ𝑞−1/2 ◦ 𝜕− +i𝐾𝑡Λ𝑡1/2 ◦ 𝜕𝑡 +� +, +𝐾𝑡 � +1 +2𝜋(1 + 𝑡−1) . +Hence, an elementary comparison of 𝜕′ +tot with 𝜕𝑞2,𝑞 implies that +∀𝑝 ∈ O𝑞(SU(2)), +1 +1 + 𝐾−1 +𝑞2 +𝐿𝜏(𝑝) ≤ 𝐿𝑞2,𝑞(𝑝) ≤ (1 + 𝐾𝑞2)𝐿𝜏(𝑝). + +82 +BRANIMIR ĆAĆIĆ +References +[1] +B. Abadie, S. Eilers, and R. Exel. Morita equivalence for crossed products by Hilbert 𝐶∗-bimodules. Trans. +Amer. Math. 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Vieweg + Teubner, Wiesbaden, 2011. + +REFERENCES +85 +Department of Mathematics & Statistics, University of New Brunswick, PO Box 4400, Fredericton, NB +E3B 4A3, Canada +Email address: mailto:bcacic@unb.ca + diff --git a/JdAzT4oBgHgl3EQfyP4s/content/tmp_files/load_file.txt b/JdAzT4oBgHgl3EQfyP4s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8e0b3483ad103217458be7ae9da783ada1e0997f --- /dev/null +++ b/JdAzT4oBgHgl3EQfyP4s/content/tmp_files/load_file.txt @@ -0,0 +1,4484 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf,len=4483 +page_content='GEOMETRIC FOUNDATIONS FOR CLASSICAL U(1)-GAUGE THEORY ON NONCOMMUTATIVE MANIFOLDS BRANIMIR ĆAĆIĆ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We systematically extend the elementary differential and Riemannian geometry of classical U(1)-gauge theory to the setting of noncommutative differential geometry in the sense of Connes by combining recent advances in noncommutative Riemannian geometry with the theory of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We show that Hermitian line bimodules with Hermitian bimodule connection over a unital pre-𝐶∗-algebra with ∗-exterior algebra form a coherent 2-group, and we prove that weak monoidal functors between coherent 2-groups canonically define bar or involutive monoidal functors in the sense of Beggs–Majid and Egger, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we prove that a suitable Hermitian line bimodule with Hermitian bimodule connection yields an essentially unique differentiable quantum principal U(1)-bundle with principal connection and vice versa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' here, U(1) is 𝑞-deformed for 𝑞 a numerical invariant of the bimodule connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' From here, we compute moduli spaces of solutions to Maxwell’s equations and we formulate and solve the interrelated lifting problems for noncommutative Riemannian structure in terms of abstract Hodge star operators and formal spectral triples, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' all the while, we account precisely for emergent modular phenomena of geometric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, it follows that no spectral triple on quantum CP1 lifts to a twisted spectral triple for quantum SU(2) with the 3-dimensional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, we may use the canonical lift of the spin Dirac spectral triple on quantum CP1 with respect to the 𝑞-monopole connection to recover Kaad–Kyed’s compact quantum metric space on quantum SU(2) for a canonical choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A coherent 2-group of noncommutative Hermitian line bundles with connection 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Preliminaries on coherent 2-groups 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Picard 2-group of a nc topological space 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The differential Picard 2-group of a noncommutative manifold 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Canonical actions of the differential Picard group 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Reconstruction of noncommutative principal U(1)-bundles with connection 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Monoidal inversion and homomorphisms of coherent 2-groups 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Generalised crossed products via homomorphisms of coherent 2-groups 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Horizontal calculi as generalised crossed products 35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Reconstruction of total calculi 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lifting problems for noncommutative Riemannian structures 46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Basic noncommutative Hodge theory and moduli spaces of U(1)-instantons 46 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The lifting problem for Riemannian structures via Hodge operators 53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Unbounded lifts of commutator representations 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Twisted boundedness of lifted commutator representations 73 References 82 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='01749v1 [math-ph] 4 Jan 2023 2 BRANIMIR ĆAĆIĆ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Introduction The primordial application of noncommutative (nc) geometry to theoretical physics is the conceptually economical construction of physical models as classical physics on nc manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, in Bellissard–Van Elst–Schulz-Baldes’ model of the integer quantum hall effect [15], the nc Brouillin zone accounts for both the magnetic field and disorder in the crystal, while in particle physics [43] and cosmological models [69] using Chamseddine–Connes’s spectral action principle [30], 0-dimensional nc fibres encode the particle content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The prototypical such construction is Connes–Rieffel’s topologically non-trivial Yang–Mills gauge theory on irrational nc 2-tori [35], the first of many nc field theories built from a range of seemingly disparate variations on Connes’s nc differential geometry [31, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, one can approach various aspects or special cases of nc U(1)-gauge theory in terms of quantum principal bundles [22, 39], principal U(1)-spectral triples [42, 19, 26], or even the spectral action principle [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This fragmentary understanding of classical U(1)-gauge theory on nc manifolds is unsat- isfactory for reasons beyond the obvious ones internal to nc geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, consider mathematical analysis of the integer quantum Hall effect in terms of quantum adiabatic trans- port, where one probes the qualitative behaviour of relevant observables by considering the integer quantum Hall effect on general compact Riemann surfaces [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A satisfactory generali- sation to nc compact Riemann surfaces would require a precise extension of the elementary differential and Riemannian geometry of classical U(1)-gauge theory as a coherent whole to the nc setting that is compatible with both nc Kähler geometry [79] and the framework of spectral triples [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our goal here is to effect just such an extension, which would also be applicable to the study of electromagnetism on nc spacetimes [68] and to the differential-geometric refinement of nc 𝑇-duality as applied to the bulk-edge correspondence [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We construct this extension from the ground up in accordance with the philosophy of quantum Riemannian geometry [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, we view a nc manifold as consisting of a unital pre-𝐶∗-algebra equipped with a ∗-exterior calculus, so that a nc Riemannian manifold is a nc manifold in this sense equipped with a compatible nc Riemannian structure, whether it be an abstract Hodge star operator or a spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This, in stark contrast with other areas of nc geometry and operator algebras, requires working exclusively ‘on the nose’—at worst, up to explicit isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Fortunately, in our setting, we may obviate any resulting algebraic difficulties through the systematic use of coherent 2-groups [7], generalised groups whose group law, unit object, and inversion satisfy the group axioms up to coherent isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, borrowing insights from relevant applications of unbounded 𝐾𝐾-theory [19, 48, 26], we minimise the use of functional analysis and obviate further algebraic difficulties through the systematic use of finite tight Parseval frames on (pre-)Hilbert modules [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our results are independent of Schwieger–Wagner’s cohomological classification of princi- pal T𝑁-𝐶∗-algebras [93] and Saldaña’s Tannaka–Krein theorem [72] for differentiable quantum principal bundles d’après Ðurđević [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, the former presages the rôle of coherent 2-groups and their group-cohomological classification in the case of Abelian structure groups, while the latter will be prototypical for any generalisation of our results to non-Abelian or quantum structure groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Overview of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin in §2 by developing the elementary theory of nc Hermitian line bundle with unitary connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior algebra (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Building on a proposal of Beggs–Brzeziński [9], we define Hermitian line 𝐵-bimodules with connection (up to formal refinement) to be strong Morita auto-equivalences of 𝐵 equipped with extendable bimodule connections [13] with respect to (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, building on results of Beggs–Majid [13], we prove that Hermitian line 𝐵-bimodules with connection form a NONCOMMUTATIVE U(1)-GAUGE THEORY 3 coherent 2-group DPic(𝐵), the differential Picard 2-group of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The isomorphism classes of DPic(𝐵) still form a group DPic(𝐵), the differential Picard group, whose canonical (and typically non-trivial) action on the graded centre Z(Ω𝐵) of Ω𝐵 will appear throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By results of Beggs–Majid [13], this DPic(𝐵)-action admits a 1-cocycle of supreme importance: the curvature 2-forms of Hermitian line 𝐵-bimodules with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, we can characterise the fibres of the forgetful map from DPic(𝐵) to the 𝐾0-monoid V(𝐵) of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, in §3, we develop the corresponding elementary theory of nc principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝜅 > 0, we synthesize a definition of 𝜅-differentiable quantum principal U(1)-bundle with connection from relevant work of Brzeziński–Majid [22], Hajac [52], Ðurđević [39], and Beggs–Majid [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' here, the differential calculus on U(1) is deformed to satisfy d𝑧 · 𝑧 = 𝜅𝑧 · d𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we may define a functor that maps a 𝜅-differentiable quantum principal U(1)-bundle with connection to its nc associated Hermitian line bundle with unitary connection of winding number −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in fact, we show that [𝐸, ∇𝐸] ∈ DPic(𝐵) is contained in the essential range of this functor if and only if its curvature 2-form F[𝐸,∇𝐸] satisfies F[𝐸,∇𝐸] ⊳ [𝐸, ∇𝐸] = 𝜅−1F[𝐸,∇𝐸] with respect to the DPic(𝐵)-action on Z(Ω𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We prove that this functor is indeed an equivalence of categories onto its essential range, thereby generalising the familiar dictionary between Hermitian line bundles with unitary connection and principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our proof ultimately depends on applying two apparently novel technical results on coherent 2-groups to weak monoidal functors Z → DPic(𝐵), which typically output the nc Hermitian line bundles with unitary connection associated to a nc principal U(1)-bundle with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The first, that Z is the free coherent 2-group on one generator, is a straightforward corollary of Joyal–Street’s group-cohomological classification of weak monoidal functors between coherent 2-groups [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The second, that every weak monoidal functor between coherent 2-groups is a bar functor or involutive monoidal functor in the sense of Beggs–Majid [13] and Egger [44], respectively, is a non-trivial application of the coherence theorem for coherent 2-groups of Ulbrich [95] and Laplaza [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We may view this pair of results as an abstract distillation of Pimsner’s construction [81]—by applying them to weak monoidal functors from Z to the coherent 2-group Pic(𝐵) of Hermitian line 𝐵-bimodules, one may recover Arici–Kaad–Landi’s characterisation [4] of nc topological principal U(1)-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, in §4, we turn to the nc Riemannian geometry of nc principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that the best-known nc 3-manifolds are total spaces of nc principal U(1)-bundles with principal connection, whose nc Riemannian geometry therefore has implications for the construction of nc Lorentzian 4-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, 3-dimensional quantum SU(2) poses fundamental challenges for all existing frameworks—for instance, it cannot be faithfully represented by a spectral triple [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We therefore draw on a range of advances in nc Riemannian geometry—unbounded 𝐾𝐾-theory [71, 59, 19], nc Kähler geometry [79], and quantum Riemannian geometry [14]—to lift nc Riemannian geometry from well- behaved nc base spaces to nc total spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our guide is the commutative case: a principal U(1)-bundle 𝜋 : 𝑋 → 𝑌 with principal connection Π admits a bijection between metrics on 𝑌 and U(1)-invariant metrics on 𝑋 that make Π orthogonal and the fibres have unit length, which is defined by the constraint that 𝜋 become a Riemannian submersion [2, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, in §§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2, we consider the conceptually prior notion of nc Riemannian ge- ometry via abstract Hodge operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For us, a Riemannian geometry on an nc manifold (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) is a pair (★, 𝜏), where ★ generalises the Hodge star operator and 𝜏 is a faithful state generalising integration against the Riemannian volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This suffices for the for- mulation of (Euclidean) Maxwell’s equations, whose moduli spaces of solutions we construct by combining the results of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 with the relevant nc Hodge decomposition theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, 4 BRANIMIR ĆAĆIĆ we propose a similar definition of total Riemannian geometry for a 𝜅-differentiable quantum principal U(1)-bundle with connection (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), where failure of the Hodge operator to be right 𝑃-linear and ∗-preserving is governed by a commuting pair of modular automorphisms of Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We show that (★, 𝜏) lifts to at most one total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), whose existence we characterize in terms of conformality of the corresponding Hermitian line 𝐵-bimodule with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, the unique lift of canonical Riemanian geometry on quantum CP1 as an nc Kähler manifold to the 𝑞-monopole of Brzeziński–Majid [22] recovers a construction of Zampini [99] for a canonical choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3, we consider Connes’s familiar nc Riemannian geometry via spectral triples [32], which, following Schmüdgen [90], we generalise to bounded commutator representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We propose a definition of projectable commutator representation, where represented 1-forms are only locally bounded in a certain precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We then use a formal version of the unbounded Kasparov product [71, 59] to construct an equivalence of categories between faithful bounded commutator representations of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) and faithful projectable commutator representations of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' isomorphism of the latter is given by U(1)-equivariant unitary equivalence up to perturbation by a suitable relative remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, if (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) is equipped with a liftable Riemannian geometry and (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) is equipped with its unique lift, then the resulting Hodge–de Rham commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) lifts to the resulting total Hodge–de Rham commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4, we draw on Connes–Moscovici’s formalism of twisted spectral triples [34] to control unboundedness of represented 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We consider modular pairs (𝑁, 𝜈), where 𝜈 is a modular automorphism of Ω𝑃 and 𝑁 is a suitable unbounded operator satisfying 𝜈 = 𝑁−1(·)𝑁;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let us say that (𝑁, 𝜈) dampens an unbounded operator 𝑆 whenever 𝑁𝑆𝑁 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we define a vertical or horizontal twist for a faithful projectable commutator representation to be a modular pair that dampens represented vertical or horizontal 1-forms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There is a universal vertical twist, but the existence of horizontal twists is non-trivial and characterizable using a conformal generalisation of metric equicontinuity à la Bellissard–Marcolli–Reihani [16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, a total Hodge–de Rham representation always admits a canonical horizontal twist of conformal origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the case of 3-dimensional quantum SU(2), vertical and horizontal twists are unique but distinct, thereby excluding the existence of non-pathological U(1)-equivariant twisted spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Still, they permit a geometric derivation of Kaad–Kyed’s compact quantum metric space on quantum SU(2) for 𝑡 = 𝑞2 [58] from the spin Dirac spectral triple on quantum CP1 of Dąbrowski–Sitarz [41] via the 𝑞-monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Running examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We consider three main running examples, which we index here for the reader’s convenience: (1) the commutative case—Exx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) the real multiplication instanton—Exx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='62, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='67;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) the 𝑞-monopole—Exx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='61, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='68, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='73, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The author wishes to thank Edwin Beggs, Cole Dunphy, Viqar Hu- sain, Andrey Krutov, Matilde Marcolli, Bram Mesland, Réamonn Ó Buachalla, Adam Rennie, Karen Strung, Nicholas Touikan, and Alessandro Zampini for helpful conversations and cor- respondence, and he especially thanks Timmavajjula Venkata Karthik for numerous technical conversations over the last several years that have indelibly shaped this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The author was supported by nserc Discovery Grant rgpin-2017-04249 and a Harrison McCain Foundation Young Scholar Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A coherent 2-group of noncommutative Hermitian line bundles with connection In this section, we build on work of Beggs–Brzeziński [9] and Beggs–Majid [13] to construct a coherent 2-group of nc Hermitian line bundles with unitary connection over a nc differen- tiable manifold, the differential Picard 2-group, that makes curvature into a canonical group 1-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, we algebraically characterise the fibers of the forgetful functors passing to nc Hermitian line bundles and nc Hermitian vector bundles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us recall some category-theoretic terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A category is essentially small whenever its hom-sets and its class of isomorphism classes are all sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A concrete category is a category C equipped with a faithful functor 𝑈 : C → Set to the category Set of sets and functions, which we view as the forgetful functor mapping objects of C to their underlying sets and arrows of C to their underlying functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Likewise, we define a functor category to be a category C equipped with a faithful functor 𝑈 : C → [A, B], where A and B are categories and [A, B] is the usual functor category whose objects are functors 𝐹 : A → B and whose arrows are natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, a subcategory A of a category B is strictly full whenever it is full—every arrow in B between objects of A is an arrow of A—and closed under isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Preliminaries on coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin by reviewing the elementary theory of coherent 2-groups, which generalise ordinary groups by permitting the group law, unit, and inversion to satisfy the group axioms up to coherent isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, we show that Z is the free coherent 2-group on one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We follow the account of Baez–Lauda [7] but with technical simplications drawn from Laplaza [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that a (weak) monoidal category is a category C with a bifunctor ⊗ : C × C → C, the monoidal product, a distinguished object 1, the unit, and natural isomorphisms 𝜆 � (𝜆𝑎 : 1 ⊗ 𝑎 → 𝑎)𝑎∈Obj(C), 𝜌 � (𝜌𝑎 : 𝑎 ⊗ 1 → 𝑎)𝑎∈Obj(C), 𝛼 � �𝛼𝑎,𝑏,𝑐 : (𝑎 ⊗ 𝑏) ⊗ 𝑐 → 𝑎 ⊗ (𝑏 ⊗ 𝑐)� (𝑎,𝑏,𝑐) ∈Obj(C)3 , respectively, the left unitor, right unitor, and associator, that satisfy certain coherence diagrams [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 428–9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, it is strict whenever its left unitor, right unitor, and associator are given by identity arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, a monoidal subcategory of a monoidal category C is a subcategory D of C that is closed under the monoidal product, contains the unit, and contains all left unitor, right unitor, and associator arrows between its objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a unital associative algebra over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The concrete category Bimod(𝐵) of 𝐵-bimodules and 𝐵-bimodule homomorphisms defines a monoidal category with respect to the usual balanced tensor product of 𝐵-bimodules and of 𝐵-bimodule homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, the associator 𝛼𝐸,𝐹,𝐺 of 𝐵-bimodules 𝐸, 𝐹, and 𝐺 is given by ∀𝑒 ∈ 𝐸, ∀𝑓 ∈ 𝐹, ∀𝑔 ∈ 𝐺, 𝛼𝐸,𝐹,𝐺((𝑒 ⊗ 𝑓) ⊗ 𝑔) � 𝑒 ⊗ (𝑓 ⊗ 𝑔), the unit object is the trivial 𝐵-bimodule 𝐵, and the left and right unitors of a 𝐵-bimodule are induced by its left and right 𝐵-module structures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Just as a group is a monoid with a notion of inversion, so too is a coherent 2-group a monoidal category together with a notion of inversion up to coherent isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 (Sính [94];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Laplaza [66, §4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Baez–Lauda [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A coherent 2-group is an essentially small monoidal category G in which every arrow is invertible together with: (1) a function (𝑔 ↦→ 𝑔) : Obj(G) → Obj(G) called monoidal inversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) a family of arrows ev = (ev𝑔 : 𝑔 ⊗ 𝑔 → 1)𝑔∈Obj(G) in G called evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, a sub-2-group of a coherent 2-group G is a monoidal subcategory H of G that is closed under monoidal inversion and contains {ev𝑔 | 𝑔 ∈ Obj(H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6 BRANIMIR ĆAĆIĆ A group Γ defines a coherent 2-group: take the discrete category on its underlying set with the strict monoidal structure given by the group law and monoidal inversion given by inversion in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This example admits the following wide-ranging generalisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' for a review of the relevant group cohomology, see [56, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3 (see [56, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Γ be a group, let 𝑀 be a Γ-module, and let 𝜔 ∈ 𝑍3(Γ, 𝑀) be a normalised cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a coherent 2-group 2Grp(Γ, 𝑀, 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) The set of objects of 2Grp(Γ, 𝑀, 𝜔) is Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) The set of arrows of 2Grp(Γ, 𝑀, 𝜔) is 𝑀 × Γ, where (𝑚, 𝛾) ∈ 𝑀 × Γ is an automorphism of the object 𝛾;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, composition of arrows is induced by the group law of 𝑀, so that the identity automorphism of an object 𝛾 is (1𝑀, 𝛾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) The monoidal product on objects is given by the group law of Γ, the monoidal product on arrows is given by the group law of 𝑀 ⋊ Γ, the monoidal unit is 1Γ, left unitors and right unitors are identity arrows, and the associator is given by ∀𝛾1, 𝛾2, 𝛾3 ∈ Γ, 𝛼𝛾1,𝛾2,𝛾3 � (𝜔(𝛾1, 𝛾2, 𝛾3), 𝛾1𝛾2𝛾3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4) Monoidal inversion is given by inversion in the group Γ, so that evaluation is induced by the group law of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now take a closer look at monoidal inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 be an object of a monoidal category G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall [45, Deff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1] that an inverse for 𝑔 is a triple (ℎ, e, i) consisting of an object ℎ of G and isomorphisms e : ℎ ⊗ 𝑔 → 1 and i : 1 → 𝑔 ⊗ ℎ in G that make the following diagrams commute: (𝑔 ⊗ ℎ) ⊗ 𝑔 𝑔 ⊗ (ℎ ⊗ 𝑔) 1 ⊗ 𝑔 𝑔 𝑔 ⊗ 1 𝛼𝑔,ℎ,𝑔 𝜆𝑔 𝜌𝑔 i⊗id𝑔 id𝑔 ⊗e (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) (ℎ ⊗ 𝑔) ⊗ ℎ ℎ ⊗ (𝑔 ⊗ ℎ) 1 ⊗ ℎ ℎ ℎ ⊗ 1 𝛼ℎ,𝑔,ℎ−1 𝜆ℎ 𝜌ℎ e⊗idℎ idℎ ⊗i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) Recall, moreover, that an isomorphism of inverses (ℎ1, e1, i1) and (ℎ2, e2, i2) for the object 𝑔 is an isomorphism 𝑢 : ℎ1 → ℎ2 in G that makes the following diagrams commute: ℎ1 ⊗ 𝑔 ℎ2 ⊗ 𝑔 𝑔 𝑢⊗id𝑔 e1 e2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) 𝑔 ⊗ ℎ1 𝑔 ⊗ ℎ2 𝑔 id𝑔 ⊗𝑢 i1 i2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) It is well known that if an object 𝑔 of a monoidal category G has an inverse, then that inverse is unique up to unique isomorphism in the above sense [45, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 (Laplaza [66, §4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Monoidal inversion in G uniquely extends to a functor G → G that makes evaluation in G into a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) There exists a unique natural isomorphism coev = (coev𝑔 : 1G → 𝑔 ⊗ 𝑔)𝑔∈Obj(G), such that, for every 𝑔 ∈ Obj(G), the triple (𝑔, ev𝑔, coev𝑔) defines an inverse for 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) There exists a unique natural isomorphism bb = (bb𝑔 : 𝑔 → 𝑔)𝑔∈Obj(G), such that, for every 𝑔 ∈ Obj(G), the arrow bb𝑔 : 𝑔 → 𝑔 gives an isomorphism of the inverses (𝑔, coev−1 𝑔 , ev−1 𝑔 ) and (𝑔, ev𝑔, coev𝑔) of 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This robust functorial picture of monoidal inversion and evaluation permits a direct statement for general coherent 2-groups of the following elementary structural result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 (Sính [94], see [7, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜋0(G) be the group of isomorphisms classes in G with group law induced by the monoidal product, and let 𝜋1(G) be NONCOMMUTATIVE U(1)-GAUGE THEORY 7 the group of automorphisms of the monoidal unit 1 of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜋1(G) is Abelian and defines a 𝜋0(G)-module with respect to the left action ⊲G given by ∀𝑔 ∈ Obj(G), ∀𝛼 ∈ 𝜋1(G), [𝑔] ⊲G 𝛼 � coev−1 𝑔 ◦(𝜌𝑔 ⊗ id𝑔) ◦ �(id𝑔 ⊗ 𝛼) ⊗ id𝑔 � ◦ (𝜌−1 𝑔 ⊗ id𝑔) ◦ coev𝑔 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, a group Γ viewed as a coherent 2-group Γ satisfies 𝜋0(Γ) = Γ and 𝜋1(Γ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' More generally, given a group Γ, a Γ-module 𝑀, and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), it follows that 𝜋0(2Grp(Γ, 𝑀, 𝜔)) = Γ and 𝜋1(2Grp(Γ, 𝑀, 𝜔)) = 𝑀 × {1Γ} � 𝑀, where the 𝜋0(2Grp(Γ, 𝑀, 𝜔))-module structure on 𝜋1(2Grp(Γ, 𝑀, 𝜔)) reduces to the given Γ-module structure on 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now generalise group homomorphisms to the setting of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that G and G′ are monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A (weak) monoidal functor 𝐹 : G → G′ consists of a func- tor 𝐹 : G → G′ together with an an isomorphism 𝐹 (0) : 𝐹(1) → 1 and a natural isomorphism 𝐹 (2) = � 𝐹 (2) 𝑔,ℎ : 𝐹(𝑔 ⊗ ℎ) → 𝐹(𝑔) ⊗ 𝐹(ℎ) � (𝑔,ℎ) ∈Obj(G)2 satisfying certain coherence diagrams [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 429–430].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given monoidal functors 𝑃 : G → G′ and 𝑄 : G → G′, a natural transfor- mation 𝜙 : 𝑃 ⇒ 𝑄 is monoidal whenever 𝑃 (0) = 𝑄(0) ◦ 𝜙1 and 𝜙𝑔⊗ℎ ◦ 𝑃 (2) 𝑔,ℎ = 𝑄(2) 𝑔,ℎ ◦ (𝜙𝑔 ⊗ 𝜙ℎ) for all objects 𝑔 and ℎ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6 (see [7, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G and G′ be coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Hom(G, G′) is the essentially small functor category whose objects are monoidal functor 𝐹 : G → G′ and whose arrows are monoidal natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A homomorphism from G to G′ is an object of Hom(G, G′), while a 2-isomorphism between homomorphisms 𝑅, 𝑆 : G → G′ is an arrow 𝜂 : 𝑅 ⇒ 𝑆 in Hom(G, G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, given a homomorphism 𝐹 : G → G′, let 𝜋0(𝐹) : 𝜋0(G) → 𝜋0(G′) and 𝜋1(𝐹) : 𝜋1(G) → 𝜋1(G′) denote the respective group homomorphisms induced by 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, let Γ1 and Γ2 be groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A homomorphism of coherent 2-groups 𝑓 : Γ1 → Γ2 is simply a group homomorphism with 𝑓 (0) and 𝑓 (2) given by identity arrows, so that 𝜋0(𝑓) = 𝑓 and 𝜋1(𝑓) = id1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, all 2-homomorphisms in Hom(Γ1,Γ2) are simply identity natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It turns out that a composition of homomorphisms of coherent 2-groups is again a homo- morphism of coherent 2-groups, making the assignments 𝜋0 and 𝜋1 functorial in the sense of mapping compositions to compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, more generally, let G1, G2, and G3 be monoidal categories, and let 𝑃 : G1 → G2 and 𝑄 : G2 → G3 be monoidal functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝑄 ◦ 𝑃 : G1 → G3 defines a monoidal functor with respect to the natural transformations (𝑄 ◦ 𝑃)(0) � 𝑄(0) ◦ 𝑄(𝑃 (0)) and (𝑄 ◦ 𝑃)(2) � � 𝑄(2) 𝑃(𝑔),𝑃(ℎ) ◦ 𝑄(𝑃 (2) 𝑔,ℎ ) � (𝑔,ℎ) ∈Obj(G1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We conclude by using the cohomological classification of coherent 2-groups and their homomorphisms to show that Z is the free coherent 2-group on one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that a monoidal equivalence of monoidal categories G1 and G2 is a monoidal functor 𝑃 : G1 → G2 for which there exist a monoidal functor 𝑄 : G2 → G1 and monoidal natural isomorphisms 𝑃 ◦ 𝑄 ⇒ idG2 and 𝑄 ◦ 𝑃 ⇒ idG1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in fact, it suffices that the underlying functor 𝑃 : G1 → G2 be an equivalence of categories [45, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Coherent 2-groups admit the following classification up to monoidal equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 (Sính [94], see [7, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a normalised co- cycle 𝜔 ∈ 𝑍3(𝜋0(G), ⊲G, 𝜋1(G)), unique up to cohomology, such that G is monoidally equivalent to 2Grp(𝜋0(G), 𝜋1(G), 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, the coherent 2-group G is uniquely determined up to monoidal equivalence by the resulting quadruple (𝜋0(G), 𝜋1(G), ⊲G, [𝜔]), where [𝜔] ∈ 𝐻3(𝜋0(G), 𝜋1(G)) is the cohomology class of 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 8 BRANIMIR ĆAĆIĆ Hence, the Sính invariant of a coherent 2-group G is the complete monoidal equivalence invariant (𝜋0(G), 𝜋1(G), ⊲G, [𝜔]) constructed by Sính’s theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that the Sính invariant is referred to as the Postnikov invariant in some references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, the Sính invariant of a group Γ viewed a strict 2-group is (Γ, 1,Γ × 1 → 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given the additional data of a Γ-module 𝑀 and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), the Sính invariant of 2Grp(Γ, 𝑀, 𝜔) reduces to (Γ, 𝑀, ⊲, [𝜔]), where ⊲ is the given Γ-action on 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Homomorphisms of coherent 2-groups now also admit a cohomological classification—for simplicity, we give the relevant special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8 (Joyal–Street [57, §6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' see [7, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3] and [55, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐺 and Γ be groups, let 𝑀 be a Γ-module, and let 𝜔 ∈ Z3(Γ, 𝑀) be a normalised cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a category H(𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) An object is a pair (𝛼, 𝜅), where 𝛼 : 𝐺 → Γ is a group homomorphism and 𝜅 ∈ 𝐵2(𝐺, 𝑀) is a normalised 2-cochain with respect to 𝛼 that satisfies d𝜅 = (𝛼∗𝜔)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Let (𝛼1, 𝜅1) and (𝛼2, 𝜅2) be objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If 𝛼1 = 𝛼2, then an arrow 𝜛 : (𝛼1, 𝜅1) → (𝛼2, 𝜅2) consists of a normalised 1-cochain 𝜛 ∈ 𝐵1(𝐺, 𝑀), such that d𝜇 = 𝜅1 · 𝜅−1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' else, there are no arrows from (𝛼1, 𝜅1) to (𝛼2, 𝜅2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) Let (𝜇1 : (𝛼1, 𝜅1) → (𝛼2, 𝜅2) and 𝜇2 : (𝛼2, 𝜅2) → (𝛼3, 𝜅3) be arrows with 𝛼1 = 𝛼2 = 𝛼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜇2 ◦ 𝜇1 : (𝛼1, 𝜅2) → (𝛼3, 𝜅3) is given by 𝜇2 · 𝜇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4) The identity arrow of an object (𝛼, 𝜅) is given by the trivial 1-cochain 1 : Γ → 𝜋1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines an equivalence Θ : H(𝐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) → Hom(𝐺, 2Grp(Γ, 𝑀, 𝜔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given an object (𝛼, 𝜅), define Θ(𝛼, 𝜅) : 𝐺 → 2Grp(Γ, 𝑀, 𝜔) by ∀𝑔 ∈ 𝐺, Θ(𝛼, 𝜅)(𝑔) � 𝛼(𝑔);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Θ(𝛼, 𝜅)(0) � (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝑔, ℎ ∈ 𝐺, Θ(𝛼, 𝜅)(2) 𝑔,ℎ � (𝜅(𝑔, ℎ), 𝑔ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given an arrow 𝜇 : (𝛼, 𝜅1) → (𝛼, 𝜅2), define Θ(𝜇) : Θ(𝛼, 𝜅1) ⇒ Θ(𝛼, 𝜅2) by ∀𝑔 ∈ 𝐺, Θ(𝜇)𝑔 � (𝜇(𝑔), 𝛼(𝑔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, given coherent 2-groups G and G′, each object 𝑔 of G yields a corresponding evaluation functor 𝜖𝑔 : Hom(G, G′) → G′ defined by ∀𝑃 ∈ Obj(Hom(G, G′)), 𝜖𝑔(𝑃) � 𝑃(𝑔);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝜂 ∈ Hom(Hom(G, G′)), 𝜖𝑔(𝜂) � 𝜂𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that Z is indeed the free coherent 2-group on one generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The evaluation functor 𝜖1 : Hom(Z, G) → G is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, for every object 𝑔 of G, there exists an essentially unique homomorphism 𝐹 : Z → G that satisfies 𝐹(1) � 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7, we may assume without loss of generality that exist a group Γ, a Γ-module 𝑀, and a normalised cocycle 𝜔 ∈ 𝑍3(Γ, 𝑀), such that G = 2Grp(Γ, 𝑀, 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let Θ : H(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) → Hom(Z, G) be the equivalence of categories of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It therefore suffices to show that 𝜖1 ◦ Θ : H(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) → G is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, we show that the functor 𝜖1 ◦ Θ is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝛾 ∈ Γ = Obj(G) be given, and set 𝛼𝛾 � (𝑘 ↦→ 𝛾𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the group Z has cohomological dimension 1 [20, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (b)], the 3-cocycle 𝛼∗ 𝛾𝜔 on Z is necessarily trivial in cohomology, so that there exists a normalised 2-cochain 𝜅𝛾 ∈ 𝐵2(Z, 𝑀) that satisfies d𝜅𝛾 · 𝛼∗ 𝛾𝜔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It now follows that (𝛼𝛾, 𝜅𝛾) is a well-defined object of H(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) satisfying 𝜖1 ◦ Θ(𝛼𝛾, 𝜅𝛾) = 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that 𝜖1 ◦ Θ is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑚, 𝛾) ∈ 𝑀 × Γ = Hom(G) be given, so that (𝑚, 𝛾) is an automorphism of the object 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By the above argument, let (𝛼𝛾, 𝜅𝛾) be any preimage of NONCOMMUTATIVE U(1)-GAUGE THEORY 9 the object 𝛾 under 𝜖1 ◦ Θ, and let 𝛽(𝑚,𝛾) ∈ 𝑍1(Z, 𝑀) be the unique normalised 1-cocycle with respect to 𝛼𝛾 that satisfies 𝛽(𝑚,𝛾) (1) = 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝛽(𝑚,𝛾) : (𝛼𝛾, 𝜅𝛾) → (𝛼𝛾, 𝜅𝛾) is a well-defined arrow of H(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Γ, 𝑀, 𝜔) satisfying 𝜖1 ◦ Θ(𝛽(𝑚,𝛾)) = (𝑚, 𝛾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, we show that the functor 𝜖1 ◦ Θ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Fix a homomorphism 𝛼 : Z → Γ and normalised 2-cochains 𝜅, 𝜅′ ∈ 𝐵2(Z, 𝑀), such that d𝜅 = d𝜅′ = (𝛼∗𝜔)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' suppose that 𝜇1, 𝜇2 : (𝛼, 𝜅) → (𝛼, 𝜅′) satisfy 𝜖1 ◦ Θ(𝜇1) = 𝜖1 ◦ Θ(𝜇2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This means that 𝜇1, 𝜇2 ∈ 𝐵1(Z, 𝑀) are normalised chains, such that d𝜇1 = 𝜅 · (𝜅′)−1 = d𝜇2 and 𝜇1(1) = 𝜇2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It follows that 𝛽 � 𝜇1 · 𝜇−1 2 is a normalised 1-cocycle on Z that satisfies 𝛽(1) = 1, but the only such 1-cocycle is the trivial 1-cocycle 𝑚 ↦→ 1, which forces 𝜇1 = 𝜇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Picard 2-group of a nc topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a given unital pre-𝐶∗-algebra, which we view as a nc topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now review the theory of nc Hermitian line bundles over 𝐵, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', strong Morita auto-equivalences [88] passed through the algebraic lens of Beggs–Brzeziński [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This is standard material with adaptations to the setting of pre-𝐶∗- algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' following Kajiwara–Watatani [60], we derive substantial technical simplifications from the systematic use of finite pre-Hilbert module frames or bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a right 𝐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that a 𝐵-valued inner product on 𝐸 is a R-bilinear map (·, ·) : 𝐸 × 𝐸 → 𝐵 that is right 𝐵-linear in the second argument and satisfies ∀𝑥, 𝑦 ∈ 𝐸, (𝑦, 𝑥) = (𝑥, 𝑦)∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, we define a cobasis for (·, ·) to be finite family (𝜖𝑖)𝑛 𝑖=1 in 𝐸, such that �𝑛 𝑖=1(𝜖𝑖, 𝜖𝑖) = 1, and we say that (·, ·) is strictly full whenever (·, ·) admits a cobasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that a right 𝐵-bimodule is faithful whenever it admits a strictly full 𝐵-valued inner product [60, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10 (Rieffel [88, §6], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Bass [8, §ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A Hermitian line 𝐵-bimodule is a 𝐵-bimodule 𝐸 together with strictly full inner products on both 𝐸 and 𝐸, respectively, such that ∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, ∥(𝑏𝑥, 𝑏𝑥)∥ ≤ ∥𝑏∥2∥(𝑥, 𝑥)∥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) ∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, ∥(𝑥𝑏, 𝑥𝑏)∥ ≤ ∥𝑏∥2∥(𝑥, 𝑥)∥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) ∀𝑏 ∈ 𝐵, ∀𝑥, 𝑦 ∈ 𝐸, (𝑥, 𝑏𝑦) = (𝑏∗𝑥, 𝑦), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7) ∀𝑏 ∈ 𝐵, ∀𝑥, 𝑦 ∈ 𝐸, (𝑥, 𝑦𝑏) = (𝑥𝑏∗, 𝑦), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8) ∀𝑥, 𝑦, 𝑧 ∈ 𝐸, (𝑥, 𝑦)𝑧 = 𝑥(𝑦, 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9) For example, the trivial Hermitian line 𝐵-bimodule is the trivial 𝐵-bimodule 𝐵 together with the 𝐵-valued inner products on 𝐵 and 𝐵 defined, respectively, by ∀𝑏, 𝑐 ∈ 𝐵, (𝑏, 𝑐) � 𝑏∗𝑐, (𝑏, 𝑐) � 𝑏𝑐∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10) This example admits the following non-trivial generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜙 be an isometric ∗-automorphism of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵𝜙 � {𝑏𝜙 | 𝑏 ∈ 𝐵} be 𝐵 as a free left 𝐵-module together with the right 𝐵-module structure defined by ∀𝑏, 𝑐 ∈ 𝐵, 𝑏𝜙 · 𝑐 � (𝑏𝜙(𝑐))𝜙, and the 𝐵-valued inner products on 𝐵𝜙 and 𝐵𝜙 respectively defined by ∀𝑏, 𝑐 ∈ 𝐵, (𝑏𝜙, 𝑐𝜙) � 𝜙−1(𝑏∗𝑐), (𝑏𝜙, 𝑐𝜙) � 𝑏𝑐∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐵𝜙 is a Hermitian line 𝐵-bimodule with cobases 1𝜙 for 𝐵𝜙 and 1𝜙 for 𝐵𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑋 be a closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that the commutative unital ∗-algebra 𝐶∞(𝑋) of smooth complex-valued functions on 𝑋 defines a unital pre-𝐶∗-algebra with respect 10 BRANIMIR ĆAĆIĆ to the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given a Hermitian line bundle E → 𝑋, the balanced 𝐶∞(𝑋)- bimodule Γ(E) of global smooth sections of Edefines a Hermitian line 𝐶∞(𝑋)-bimodule with respect to the 𝐶∞(𝑋)-valued inner product on Γ(E) induced by the Hermitian metric on Eand the 𝐶∞(𝑋)-valued inner product on Γ(E) � Γ(E) defined by ∀𝜎1, 𝜎2 ∈ Γ(E), (𝜎1, 𝜎2) � (𝜎2, 𝜎1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, cobases for both of these 𝐶∞(𝑋)-valued inner products can be constructed using an atlas of local trivialisations for E → 𝑋 together with a smooth partition of unity subordinate to the corresponding open cover of 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our primary goal for this subsection is the following refinement of standard lore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13 (Rieffel [88, §6], Brown–Green–Rieffel [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Bass [8, §ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Picard 2-group of 𝐵 is the coherent 2-group Pic(𝐵) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) As a category, Pic(𝐵) is the concrete category whose objects are Hermitian line 𝐵-bimod- ules and whose arrows are 𝐵-bimodule isomorphisms 𝑢 : 𝐸 → 𝐹, such that ∀𝑥, 𝑦 ∈ 𝐸, (𝑢(𝑥), 𝑢(𝑦)) = (𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11) (2) The monoidal product of objects 𝐸 and 𝐹 is the balanced tensor product 𝐸 ⊗𝐵 𝐹 together with the 𝐵-valued inner products on 𝐸 ⊗𝐵 𝐹 and 𝐸 ⊗𝐵 𝐹 defined by ∀𝑥1, 𝑦1, 𝑥2, 𝑦2 ∈ 𝐸, (𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) � (𝑦1, (𝑥1, 𝑥2) 𝑦2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12) ∀𝑥1, 𝑦1, 𝑥2, 𝑦2 ∈ 𝐸, (𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) � (𝑥1, (𝑦1, 𝑦2)𝑥2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, the monoidal product of arrows is given by their monoidal product in Bimod(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) The unit object is the trivial Hermitian line 𝐵-bimodule 𝐵, and left unitors, right unitors, and associators are given by the corresponding left unitors, right unitors, and associators in Bimod(𝐵), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4) The monoidal inverse of a Hermitian line 𝐵-bimodule 𝐸 is 𝐸 together with the given 𝐵-valued inner product on 𝐸 and the 𝐵-valued inner product on 𝐸 defined by ∀𝑥, 𝑦 ∈ 𝐸, (𝑥, 𝑦) � (𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14) (5) The evaluation morphism for an object 𝐸 is the arrow ev𝐸 : 𝐸 ⊗𝐵 𝐸 → 𝐵 defined by ∀𝑒1, 𝑒2 ∈ 𝐸, ev𝐸(𝑒1 ⊗ 𝑒2) � (𝑒1, 𝑒2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15) Hence, the Picard group of 𝐵 is the group Pic(𝐵) � 𝜋0(Pic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14 (Bass [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a homomorphism of coherent 2-groups 𝜏 : Aut(𝐵) → Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given 𝜙 ∈ Aut(𝐵), let 𝜏(𝜙) � 𝐵𝜙 be the Hermitian line 𝐵-bimodule of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Set 𝜏 (0) � id𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' given 𝜙, 𝜓 ∈ Aut(𝐵), define 𝜏 (2) 𝜙,𝜓 : 𝜏(𝜙) ⊗𝐵 𝜏(𝜓) → 𝜏(𝜙𝜓) by ∀𝑎, 𝑏 ∈ 𝐵, 𝜏 (2) 𝜙,𝜓 (𝑎𝜙 ⊗ 𝑏𝜓) � (𝑎𝜙(𝑏))𝜙𝜓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall [60, §1] that a basis for a right 𝐵-module 𝐸 with respect to a right 𝐵-valued inner product (·, ·) is a finite family (𝑒𝑖)𝑛 𝑖=1 in 𝐸, such that 𝑥 = �𝑛 𝑖=1 𝑒𝑖(𝑒𝑖, 𝑥) for all 𝑥 ∈ 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, we define a right pre-Hilbert 𝐵-module of finite type to be a right 𝐵-module 𝐸 equipped with a 𝐵-valued inner product ⟨·, ·⟩ that admits a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In turn, we denote by Hilb(𝐵) the concrete category whose objects are right pre-Hilbert 𝐵-modules of finite type and whose arrows are isomorphisms of right 𝐵-modules satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 11 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a unital pre-𝐶∗-algebra, let 𝑛 ∈ N, and let P ∈ 𝑀𝑛(𝐵) be an orthogonal projection, which means that P2 = P = 𝑃∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then P· 𝐵𝑛 defines a right pre-Hilbert 𝐵-module of finite type with respect to the 𝐵-linear inner product defined by ∀(𝑥𝑖)𝑛 𝑖=1, (𝑦𝑖)𝑛 𝑖=1 ∈ P · 𝐵𝑛, �(𝑥𝑖)𝑛 𝑖=1, (𝑦𝑖)𝑛 𝑖=1 � � 𝑛 ∑︁ 𝑖=1 𝑥∗ 𝑖 𝑦𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that if 𝐸 is a right pre-Hilbert 𝐵-module of finite type with 𝐵-valued inner product (·, ·), then 𝐸 is necessarily finitely generated and projective as a right 𝐵-module and (·, ·) is necessarily positive definite in the sense that ∀𝑥 ∈ 𝐸, (𝑥, 𝑥) ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16) {𝑥 ∈ 𝐸 | (𝑥, 𝑥) = 0} = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17) Thus, every right pre-Hilbert 𝐵-module is isomorphic in Hilb(𝐵) to a right pre-Hilbert 𝐵- module of finite type of the kind constructed in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15, so that the category Hilb(𝐵) is essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let 𝐸 be a a right pre-Hilbert 𝐵-module of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By positive-definiteness of the 𝐵-valued inner product (·, ·) on 𝐸, the norm ∥ · ∥ on 𝐸 defined by ∀𝑥 ∈ 𝐸, ∥𝑥∥ � ∥(𝑥, 𝑥)∥1/2 satisfies the following crucial inequalities: ∀𝑥 ∈ 𝐸, ∀𝑏 ∈ 𝐵, ∥𝑥𝑏∥ ≤ ∥𝑥∥ · ∥𝑏∥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) ∀𝑥, 𝑦 ∈ 𝐸, (𝑥, 𝑦)∗(𝑥, 𝑦) ≤ ∥ 𝑦∥2(𝑥, 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19) Hence, one can show that the algebra L(𝐸) of all right 𝐵-linear maps 𝐸 → 𝐸 defines a unital pre-𝐶∗-algebra with respect to the ∗-operation implicitly defined by ∀𝑇 ∈ L(𝐸), ∀𝑥, 𝑦 ∈ 𝐸, (𝑥,𝑇∗ 𝑦) � (𝑇𝑥, 𝑦) and the operator norm induced by the aforementioned norm ∥ · ∥ on 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, given a unital pre-𝐶∗-algebra 𝐴, we define an (𝐴, 𝐵)-correspondence of finite type to be a right pre-Hilbert 𝐵-module of finite type 𝐸 equipped with a isometric unital ∗-homomorphism 𝐴 → L(𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, when 𝐴 = 𝐵, we call 𝐸 a 𝐵-self-correspondence of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a Hermitian line 𝐵-bimodule equipped with 𝐵-valued inner products (·, ·)𝐸 on 𝐸 and (·, ·)𝐸 on 𝐸, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐸 and 𝐸 define 𝐵-self-correspondences of finite type with respect to (·, ·)𝐸 and (·, ·)𝐸, respectively, such that ∀𝑥 ∈ 𝐸, ∥(𝑥, 𝑥)𝐸∥ = ∥(𝑥, 𝑥)𝐸∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 (Rieffel [88, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22], Kajiwara–Watatani [60, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a unital pre-𝐶∗-algebra, and let 𝐸 be a right pre-Hilbert 𝐵-module of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a unique isomorphism of L(𝐸)-bimodules coev𝐸 : L(𝐸) → 𝐸 ⊗𝐵 𝐸, such that ∀𝑥, 𝑦, 𝑧 ∈ 𝐸, coev−1 𝐸 (𝑥 ⊗ 𝑦)𝑧 = 𝑥(𝑦, 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Fix cobases (𝜖𝑖)𝑚 𝑖=1 and (𝑒𝑗)𝑛 𝑗=1 for (·, ·)𝐸 and (·, ·)𝐸, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9), one shows that (𝑒𝑗)𝑛 𝑗=1 is a basis for 𝐸 with respect to (·, ·)𝐸 and that (𝜖𝑖)𝑚 𝑖=1 is a basis for 𝐸 with respect to (·, ·)𝐸, so that 𝐸 is a right pre-Hilbert 𝐵-module of finite type with respect to (·, ·)𝐸, and 𝐸 is a right pre-Hilbert 𝐴-module of finite type with respect to (·, ·)𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7), the map 𝜋𝐸 : 𝐵 → L(𝐸) defines a bounded ∗-homomorphism, which is surjective by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 together with strict fullness of (·, ·)𝐸 and injective by strict 12 BRANIMIR ĆAĆIĆ fullness of (·, ·)𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By symmetry, this also shows that 𝜋𝐸 : 𝐵 → L(𝐸) defines a bounded bijective ∗-homomorphism 𝜋𝐸 : 𝐵 → L(𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, by the fact that (·, ·)𝐸 is positive definite together with the assumption that 𝐵 is a pre-𝐶∗-algebra, the data �𝐸, (·, ·)𝐸, (·, ·)𝐸 � yield a pre-imprimitivity (𝐵, 𝐵)-bimodule in the usual sense [88, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10] with left 𝐵-valued inner product induced by (·, ·)𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20 follows from the corresponding result for pre-imprimitivity bimodules [86, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now prove boundedness of 𝜋−1 𝐸 and 𝜋−1 𝐸 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑡 ∈ L(𝐸) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9), one shows that 𝜋−1 𝐸 (𝑡) = �𝑛 𝑗=1(𝑡𝑒𝑗, 𝑒𝑗), so that ∥𝜋−1 𝐸 (𝑡)∥ ≤ ∑︁𝑛 𝑗=1∥(𝑡𝑒𝑗, 𝑒𝑗)∥ ≤ ∑︁𝑛 𝑗=1∥𝑡𝑒𝑗∥∥𝑒𝑗∥ = ∑︁𝑛 𝑗=1∥𝑡𝑒𝑗∥∥𝑒𝑗∥ ≤ �∑︁𝑛 𝑗=1∥𝑒𝑗∥2� ∥𝑡∥, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19) together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The same argument also shows that 𝜋−1 𝐸 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the maps 𝜋𝐸 and 𝜋𝐸 are bounded bijective ∗-homomorphisms between unital pre-𝐶∗-algebras with bounded inverses, and hence are both isometric ∗-isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ It is easy to check that a 𝐵-self-correspondence of finite type admits at most one 𝐵-valued inner product on 𝐸 making 𝐸 into a Hermitian line 𝐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, suppose that (·, ·)1 and (·, ·)2 are two such 𝐵-valued inner products on 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, for all 𝑧 ∈ 𝐸, ((𝑥, 𝑦)1 − (𝑥, 𝑦)2)𝑧 = 𝑥(𝑦, 𝑧) − 𝑥(𝑦, 𝑧) = 0𝑧, so that (𝑥, 𝑦)1 = (𝑥, 𝑦)2 by strict fullness of either of (·, ·)1 or (·, ·)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, such a 𝐵-valued inner product on 𝐸 exists only if the left 𝐵-module structure 𝐵 → L(𝐸) on 𝐸 is an isometric ∗-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This turns out to be not only necessary but sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be an 𝐵-self-correspondence of finite type with strictly full 𝐵-valued inner product, and let 𝜋𝐸 : 𝐵 → L(𝐸) be the left 𝐵-module structure on 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a 𝐵-valued inner product on 𝐸 making 𝐸 into a Hermitian line 𝐵-bimodule if and only if 𝜋𝐸 is an isometric ∗-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that the left 𝐵-module structure 𝜋𝐸 is an isometric ∗-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7) are already satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 together with bijectivity of 𝜋𝐸, we may define an 𝐵-valued inner product (·, ·) on 𝐸 satsifying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8) by (𝑥, 𝑦) � 𝜋−1 𝐸 (𝑥 ⊗ 𝑦) for 𝑥, 𝑦 ∈ 𝐸;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' indeed, this 𝐵-valued inner product is strictly full since any basis (𝑒𝑖)𝑛 𝑖=1 for 𝐸 yields a cobasis (𝑒𝑖)𝑛 𝑖=1 for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6 follows since, for all 𝑥, 𝑦 ∈ 𝐸 and 𝑏 ∈ 𝐵, by positive definitness of ⟨·, ·⟩ on 𝐸, isometry of 𝜋𝐸, and equations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19, ∥(𝑥𝑏, 𝑥𝑏) 𝑦∥ = ∥𝑥𝑏(𝑥𝑏, 𝑦)∥ = ∥𝑥𝑏𝑏∗(𝑥, 𝑦)∥ ≤ ∥𝑥∥∥𝑏∥2∥(𝑥, 𝑦)∥ ≤ ∥𝑏∥2∥𝑥∥2∥ 𝑦∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ At last, we can prove Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13 exactly as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof of Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, by swapping out Hermitian line 𝐵-bimodules for 𝐵- self-correspondences in the proposed definition of Pic(𝐵), we obtain a more familiar essen- tially small monoidal concrete category Corr(𝐵) whose objects are 𝐵-self-correspondences of finite type [23, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that essential smallness of Corr(𝐵) follows from essential smallness of the category Hilb(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18 now implies that the category Pic(𝐵) is well-defined as a strictly full subcategory of Corr(𝐵), which clearly contains the monoidal unit 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18 together show that monoidal inversion is well-defined as a function Obj(Pic(𝐵)) → Obj(Pic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let 𝐸 and 𝐹 be Hermitian 𝐵-line modules, so that their tensor product 𝐸 ⊗𝐵 𝐹 in Corr(𝐵) is a well-defined 𝐵-self-correspondence of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, the 𝐵- valued inner product on 𝐸 ⊗𝐵 𝐹 is strictly full since cobases (𝜖𝑖)𝑛 𝑖=1 and (𝜙𝑗)𝑞 𝑗=1 for 𝐸 and 𝐹, NONCOMMUTATIVE U(1)-GAUGE THEORY 13 respectively, yield a cobasis (𝜖𝑖 ⊗ 𝜙𝑗)1≤𝑖≤𝑛,1≤𝑗≤𝑞 for 𝐸 ⊗𝐵 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, the 𝐵-valued inner product on the tensor product 𝐹 ⊗𝐵 𝐸 in Corr(𝐵) pulls back under the canonical isomorphism of 𝐵-bimodules (𝑥 ⊗ 𝑦 ↦→ 𝑦 ⊗ 𝑥) : 𝐸 ⊗𝐵 𝐹 → 𝐹 ⊗𝐵 𝐸 to the 𝐵-valued inner product on 𝐸 ⊗𝐵 𝐹 of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' this 𝐵-valued inner product is strictly full since strict cobases (𝑒𝑖)𝑚 𝑖=1 and (𝑓𝑗)𝑝 𝑗=1 for 𝐸 and 𝐹, respectively, yield a cobasis (𝑒𝑖 ⊗ 𝑓𝑗)1≤𝑖≤𝑚,1≤𝑗≤𝑝 for 𝐸 ⊗𝐵 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9) for 𝐸 ⊗𝐵 𝐹 now follows from repeated applications of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝐸 be a Hermitian 𝐵-line module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, the map ev𝐸 is an isomorphism of 𝐵-bimodules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' that ev𝐸 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11) now follows from observing that for all 𝑥1, 𝑥2 ∈ 𝐸 and 𝑦1, 𝑦2 ∈ 𝐸, (𝑥1 ⊗ 𝑦1, 𝑥2 ⊗ 𝑦2) = (𝑦1, (𝑥1, 𝑥2) 𝑦2) = (𝑦1, 𝑥1(𝑥2, 𝑦2)) = (𝑥1, 𝑦1)∗(𝑥2, 𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ This characterization of the monoidal category Pic(𝐵) as a monoidal subcategory of the monoidal category Corr(𝐵) of 𝐵-self-correspondences of finite type yields, with superficial changes, a right action of the Picard group Pic(𝐵) on the 𝐾0-monoid V(𝐵) of isomorphism classes of right pre-Hilbert 𝐵-modules of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, given a right pre-Hilbert 𝐵- module of finite type 𝐸 and a Hermitian line 𝐵-bimodule 𝐹, set [𝐸] ⊳ [𝐹] � [𝐸 ⊗𝐵 𝐹], where the balanced tensor product 𝐸 ⊗𝐵 𝐹 is equipped with the right 𝐵-valued inner product given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We may use this Pic(𝐵)-action to characterise the fibres of the obvious forgetful map Pic(𝐵) → V(𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in turn, this helps us understand the information lost when passing from Pic(𝐵) to the 𝐾-theory of 𝐵 or its 𝐶∗-algebraic completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19 (Bass [8, Propp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ΠV(𝐵) : Pic(𝐵) → V(𝐵) denote the set function induced by the forgetful functor Pic(𝐵) → Hilb(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Pic(𝐵)[𝐵] denote the stabiliser subgroup of Pic(𝐵) with respect to [𝐵] ∈ ran ΠV(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the homomorphism of coherent 2-groups 𝜏 : Aut(𝐵) → Pic(𝐵) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14 yields into a exact sequence of groups 1 → U(Z(𝐵)) → U(𝐵) 𝑢↦→Ad𝑢 −−−−−→ Aut(𝐵) 𝜋0(𝜏) −−−−→ Pic(𝐵)[𝐵] → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that this canonically identifies the outer automorphism group of 𝐵 with a subgroup of Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' What is more surprising is that the entire Picard group Pic(𝐵) acts as isometric ∗-automorphisms on the centre of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20 (Fröhlich [50, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2], Beggs–Brzeziński [9, §10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Fröhlich homomorphism of 𝐵 is the unique group homomorphism Φ : Pic(𝐵) → Aut(Z(𝐵)), such that, for every Hermitian line 𝐵-bimodule 𝐸, the Fröhlich automorphism Φ[𝐸] of [𝐸] satisfies ∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, Φ[𝐸](𝑏)𝑥 = 𝑥𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21) Hence, the canonical left action of 𝜋0(Pic(𝐵)) � Pic(𝐵) on 𝜋1(Pic(𝐵)) = U(Z(𝐵)) is the left action induced by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Relative to the references, it remains to show each Fröhlich automorphism is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a Hermitian line 𝐵-bimodule, and let (𝑒𝑖)𝑛 𝑖=1 be a cobasis for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, ∀𝑏 ∈ 𝐵, ∥Φ−1 [𝐸](𝑏)∥ = ��� ∑︁𝑛 𝑖=1(𝑒𝑖, 𝑏𝑒𝑖) ��� ≤ ∑︁𝑛 𝑖=1∥𝑒𝑖∥∥𝑏𝑒∥ ≤ �∑︁𝑛 𝑖=1∥𝑒𝑖∥2� ∥𝑏∥, so that Φ−1 [𝐸] is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Using 𝐸 instead now shows that 𝜙[𝐸] = 𝜙−1 [𝐸] is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Pic(𝑋) be the Picard group of isomor- phism classes of complex line bundles over 𝑋, which admits a right action of Diff(𝑋) by pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, any two Hermitian metrics on a line bundle are unitarily equiva- lent, so that ([E] ↦→ [Γ(E)]) : Pic(𝑋) → Pic(𝐶∞(𝑋)) is a well-defined homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, the map �𝑓 ↦→ (𝑓 −1)∗� : Diff(𝑋) → Aut(𝐶∞(𝑋)) is a group isomorphism [74], 14 BRANIMIR ĆAĆIĆ so let Ψ : Pic(𝐶∞(𝑋)) → Diff(𝑋) be the resulting homomorphism induced by the Fröhlich homomorphism of 𝐶∞(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, Serre–Swan duality yields a split exact sequence 1 → Pic(𝑋) [E]↦→[Γ(E)] −−−−−−−−−−→ Pic(𝐶∞(𝑋)) Ψ−→ Diff(𝑋) → 1 with right splitting 𝜙 ↦→ 𝜋0(𝜏)((𝜙−1)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, given the resulting isomorphism �(𝜙, [E]) ↦→ [Γ((𝜙−1)∗E)] · 𝜋0(𝜏)((𝜙−1)∗)� : Diff(𝑋) ⋉ Pic(𝑋) → Pic(𝐶∞(𝑋)), we may identify the Fröhlich homomorphism of 𝐶∞(𝑋) with the quotient map ((𝜙, [E]) ↦→ 𝜙) : Diff(𝑋) ⋉ Pic(𝑋) → Diff(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We conclude by noting certain simplications that arise when 𝐵 behaves sufficiently like a 𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This will permit us to introduce our first main running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑛 ∈ N, and let 𝑀𝑛(𝐵) denote the unital ∗-algebra of 𝑛 × 𝑛 matrices with entries in 𝐵, which is defined by analogy with 𝑀𝑛(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' one calls 𝑀𝑛(𝐵) a matrix algebra over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that 𝐵𝑛 defines a right pre-Hilbert 𝐵-module of finite type by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence observe that matrix multiplication on left defines an injective ∗-homomorphism 𝑀𝑛(C) → L(𝐵𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the operator norm on L(𝐵𝑛) pulls back to a 𝐶∗-norm on 𝑀𝑛(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that 𝐵 admits polar decompositions if, for every 𝑛 ∈ N and positive 𝑏 ∈ 𝑀𝑛(𝐵), there exists unique positive element √ 𝑏 ∈ 𝑀𝑛(𝐵) that satisfies ( √ 𝑏)2 = 𝑏 and is invertible in 𝑀𝑛(𝐵) whenever 𝑏 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, given 𝑛 ∈ N, the polar decomposition of invertible 𝑏 ∈ 𝑀𝑛(𝐵) is 𝑏 = sgn(𝑏)|𝑏|, where |𝑏| � √ 𝑏∗𝑏 ∈ 𝑀𝑛(𝐵) is positive and invertible and sgn(𝑏) � 𝑏|𝑏|−1 ∈ 𝑀𝑛(𝐵) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, a unital 𝐶∗-algebra admits polar decompositions by the holomorphic func- tional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' More generally, a unital pre-𝐶∗-algebra 𝐵 admits polar decompositions whenever it and all its matrix algebras are closed under the holomorphic functional calculus in their respective 𝐶∗-closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, recall that a 𝐵-valued inner product on a right 𝐵-module 𝐸 is algebraically full whenever it satisfies SpanC{(𝑥, 𝑦) | 𝑥, 𝑦 ∈ 𝐸} = 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝐵 is unital pre-𝐶∗-algebra that admits polar decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a 𝐵-bimodule, let (·, ·)𝐸 be a 𝐵-valued inner product on 𝐸, and let (·, ·)𝐸 be a 𝐵-valued inner product on 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐸 defines a Hermitian line 𝐵-bimodule with respect to (·, ·)𝐸 and (·, ·)𝐸 if and only if the following conditions are all satisfied: (1) the 𝐵-valued inner product (·, ·)𝐸 is algebraically full and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) the 𝐵-valued inner product (·, ·)𝐸 is algebraically full and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) the 𝐵-valued inner products (·, ·)𝐸 and (·, ·)𝐸 respectively satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The forward implication is trivial, so we prove the backward implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that all three conditions are satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it remains to show that both (·, ·)𝐸 and (·, ·)𝐸 are strictly full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (·, ·)𝐸 is algebraically full, we may choose finite families (𝑥𝑖)𝑛 𝑖=1 and (𝑦𝑖)𝑛 𝑖=1 in 𝐸 that satisfy �𝑛 𝑖=1(𝑥𝑖, 𝑦𝑖)𝐸 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define 𝑋 ∈ 𝑀𝑛(𝐵) by 𝑋 � �(𝑥𝑖, 𝑥𝑗)𝐸 �𝑛 𝑖,𝑗=1, so that, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9), 1 = 𝑛 ∑︁ 𝑖,𝑗=1 (𝑦𝑖, 𝑥𝑖)𝐸(𝑥𝑗, 𝑦𝑗)𝐸 = 𝑛 ∑︁ 𝑖,𝑗=1 (𝑦𝑖, 𝑥𝑖(𝑥𝑗, 𝑦𝑗)𝐸)𝐸 = 𝑛 ∑︁ 𝑖,𝑗=1 (𝑦𝑖, (𝑥𝑖, 𝑥𝑗)𝐸 𝑦𝑗)𝐸 = 𝑛 ∑︁ 𝑖,𝑗=1 (𝑦𝑖, 𝑋𝑖𝑗 𝑦𝑗)𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Applying to [88, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7] to 𝑋 as a bounded operator on 𝐵𝑛 with the 𝐵-valued inner product of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15 shows that 𝑋 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By our hypothesis on 𝐵, there exists 𝑎 = (𝑎𝑖𝑗)𝑛 𝑖,𝑗=1 ∈ 𝑀𝑛(𝐵), such that 𝑎∗𝑎 = 𝑋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it now follows that ��𝑛 𝑘=1 𝑎𝑖𝑘 𝑦𝑘 �𝑛 𝑖=1 is a cobasis for (·, ·)𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' An identical argument shows that (·, ·)𝐸 is strictly full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ NONCOMMUTATIVE U(1)-GAUGE THEORY 15 We now introduce our first main running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜃 ∈ R, so that the corresponding (continuous) nc 2-torus is the universal 𝐶∗-algebra 𝐶𝜃(T2) generated by unitaries 𝑢 and 𝑣 satisfying 𝑣𝑢 = e2𝜋i𝜃𝑢𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The corresponding smooth nc 2-torus 𝐶∞ 𝜃 (T2) is the dense unital ∗- subalgebra of 𝐶𝜃(T2) consisting of Laurent series in 𝑢 and 𝑣 with rapidly decaying coefficients, which admits polar decompositions since it and all its matrix algebras are closed under the holomorphic functional calculus [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜃 ∈ R be a quadratic irrationality, so that the subgroup Γ𝜃 � � 𝑔 ∈ SL(2, Z) ��� 𝑔11𝜃+𝑔12 𝑔21𝜃+𝑔22 = 𝜃, 𝑔21𝜃 + 𝑔22 > 0 � of SL(2, Z) is non-trivial and hence infinite cyclic [53, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Connes’s Heisenberg modules [31] over the unital pre-𝐶∗-algebra 𝐶∞ 𝜃 (T2) yield, in particular, a homomorphism 𝐸 : Γ𝜃 → Pic(𝐶∞ 𝜃 (T2)) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given 𝑔 ∈ Γ𝜃, let 𝐸(𝑔) be the basic Heisenberg module of rank 𝑔21𝜃 + 𝑔22 and degree 𝑔21 [84, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1], which defines a Hermitian line 𝐶∞ 𝜃 (T2)-bimodule by a result of Rieffel [89, Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15] together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Since 𝐸(1) = 𝐶∞ 𝜃 (T2), set 𝐸(0) � id𝐶∞ 𝜃 (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) Given 𝑔, ℎ ∈ Γ𝜃, let 𝐸(2) 𝑔,ℎ : 𝐸(𝑔) ⊗𝐴∞ 𝜃 𝐸(ℎ) → 𝐸(𝑔ℎ) be the isomorphism of 𝐶∞ 𝜃 (T2)- bimodules constructed by Schwarz [91, §3] and Dieng–Schwarz [37], which is an isomor- phism of Hermitian line 𝐶∞ 𝜃 (T2)-bimodules by a result of Vlasenko [97, Thm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, that the functor 𝐸 is monodal with respect to 𝐸(0) and 𝐸(2) reduces to a result of Polishchuk–Schwarz [84, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The differential Picard 2-group of a noncommutative manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we build on results of Beggs–Majid [13] to construct a coherent 2-group of Hermitian line bundles with connection over a manifold, which we term the differential Picard 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with preliminary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that a graded algebra is a unital complex algebra Ω together with a vector space decomposition Ω = �∞ 𝑘=0 Ω𝑘, such that 1 ∈ Ω0 and Ω𝑗 · Ω𝑗+𝑘 ⊆ Ω𝑗+𝑘 for all 𝑗, 𝑘 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, a graded ∗-algebra is a graded algebra Ω = �∞ 𝑘=0 Ω𝑘 with a unit- and grading-preserving complex-antilinear involution ∗ : Ω → Ω, such that ∀𝑗, 𝑘 ∈ N0, ∀𝛼 ∈ Ω𝑗, ∀𝛽 ∈ Ω𝑘, (𝛼𝛽)∗ = (−1)𝑗𝑘𝛽𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, given a unital pre-𝐶∗-algebra 𝐵, a graded ∗-algebra over 𝐵 is a graded ∗-algebra Ω together with a unital ∗-isomorphism 𝐵 → Ω0, which we suppress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in this case, we denote by Aut(Ω) the group of all grading- and ∗-preserving automorphisms 𝜙 of Ω as a unital complex algebra that restrict to an isometric ∗-automorphism of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, suppose that 𝐵 is a unital pre-𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define a ∗-quasi-differential graded algebra or ∗-quasi-dga over 𝐵 to be a pair (Ω, d), where Ω is a graded ∗-algebra over 𝐵 and d : Ω → Ω is a ∗-preserving complex linear map that satisfies d(Ω𝑘) ⊂ Ω𝑘+1 for all 𝑘 ∈ N0 together with the graded Leibniz rule ∀𝑘 ∈ N0, ∀𝛼 ∈ Ω𝑘, ∀𝛽 ∈ Ω, d𝐵(𝛼𝛽) = d𝐵(𝛼)𝛽 + (−1)𝑘𝛼d𝐵(𝛽);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, its graded centre, then, is the graded ∗-subalgebra Z(Ω) of Ω defined by ∀𝑚 ∈ N0, Z(Ω)𝑚 � {𝜔 ∈ Ω𝑚 | ∀𝑛 ∈ N0, ∀𝜉 ∈ Ω𝑛 𝐵, 𝜔𝜉 = (−1)𝑚𝑛𝜉𝜔}, which is closed under d, and its dimension, if it exists, is the largest 𝑁 ∈ N such that Ω𝑁 ≠ 0 and Ω𝑘 = 0 for all 𝑘 > 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we call (Ω, d) a ∗-exterior algebra over 𝐵 whenever the algebra Ω is generated by 𝐵 and d(𝐵) and the map d satisfies d2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 16 BRANIMIR ĆAĆIĆ Finally, we may define a concrete category QDGA whose objects (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d) consist of a unital pre-𝐶∗-algebra 𝐵 together with a choice of ∗-quasi-dga (Ω, d) over 𝐵 and whose arrows 𝑓 : (𝐵1, Ω1, d1) → (𝐵2, Ω2, d2) consist of a grading- and ∗-preserving homomorphism of unital complex algebras 𝑓 : Ω1 → Ω2 that restrict to a bounded (and hence necessarily contractive) ∗-homomorphism 𝑓↾𝐵1: 𝐵1 → 𝐵2 and satisfy 𝑓 ◦d1 = d2 ◦𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, given a ∗-quasi-dga (Ω, d) over a unital pre-𝐶∗-algebra 𝐵, we denote by Aut(Ω, d) the automorphism group of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d) in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' From now on, let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior calculus (Ω𝐵, d𝐵), which we view as a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given a 𝐵-bimodule 𝐸, we shall apply the following Sweedler-type notation to elements of 𝐸 ⊗𝐵 Ω1 𝐵 and Ω1 𝐵 ⊗𝐵 𝐸, respectively: ∀𝜂 ∈ 𝐸 ⊗𝐵 Ω1 𝐵, 𝜂⟨0⟩ ⊗ 𝜂⟨1⟩ � 𝜂;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝜉 ∈ Ω1 𝐵 ⊗𝐵 𝐸, 𝜉⟨−1⟩ ⊗ 𝜉⟨0⟩ � 𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now recall the relevant generalization of unitary connection appropriate to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a right pre-Hilbert 𝐵-module of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Extend the 𝐵-valued inner product (·, ·) on 𝐸 to a real-bilinear map (·, ·) : 𝐸 ⊗𝐵 Ω𝐵 × 𝐸 ⊗𝐵 Ω𝐵 → Ω𝐵 by setting ∀𝑥, 𝑦 ∈ 𝐸, ∀𝛼, 𝛽 ∈ Ω𝐵, (𝑥 ⊗ 𝛼, 𝑦 ⊗ 𝛽) � 𝛼∗(𝑥, 𝑦)𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This extension satisfies ∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, ∀𝛽 ∈ Ω𝐵, (𝜉, 𝜐𝛽) = (𝜉, 𝜐)𝛽, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22) ∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, (𝜐, 𝜉) = (−1) |𝜉 ||𝜐|(𝜉, 𝜐)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23) Following Connes [33, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18], one now defines right Hermitian connection on 𝐸 to be a complex-linear map ∇ : 𝐸 → 𝐸 ⊗𝐵 Ω1 𝐵, such that ∀𝑥 ∈ 𝐸, ∀𝑏 ∈ 𝐵, ∇(𝑥𝑏) = ∇𝑥𝑏 + 𝑥 ⊗ d𝐵𝑏, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24) ∀𝑥, 𝑦 ∈ 𝐸, d𝐵(𝑥, 𝑦) = (∇𝑥, 𝑦 ⊗ 1) + (𝑥 ⊗ 1, ∇𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) One can now show that there exists unique complex-linear ∇ : 𝐸⊗𝐵 Ω𝐵 → 𝐸⊗𝐵 Ω𝐵 extending the right Hermitian connection ∇, such that ∀𝜂 ∈ 𝐸⊗𝐵, ∀𝛽 ∈ Ω𝐵, ∇(𝜂𝛽) = ∇(𝜂)𝛽 + (−1) |𝜂|𝜂d𝛽, ∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, d𝐵(𝜉, 𝜐) = (∇𝜉, 𝜂) + (−1) |𝜉 |(𝜉, ∇𝜂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25 (Beggs–Majid [13, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 & §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a 𝐵-self-correspondence of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A generalised braiding for 𝐸 is an isomorphism 𝜎 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 of graded 𝐵-bimodules that extends 𝜌−1 𝐸 ◦ 𝜆𝐸 : 𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 𝐵 and satisfies ∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝑥 ∈ 𝐸, 𝜎 (𝛼 ⊗ 𝜎 (𝛽 ⊗ 𝑥) ⟨0⟩)𝜎 (𝛽 ⊗ 𝑥) ⟨1⟩ = 𝜎 (𝛼𝛽 ⊗ 𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) Hence, a Hermitian bimodule connection on 𝐸 is a pair (𝜎, ∇), where 𝜎 is a Hermitian generalised braiding on 𝐸 and ∇ is a Hermitian right connection on 𝐸, such that ∀𝛽 ∈ Ω𝐵, ∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, ∇(𝛽𝜉) = d𝐵(𝛽)𝜉 + (−1) |𝛽|𝛽∇𝜉, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) where 𝐸 ⊗𝐵 Ω𝐵 carries the graded Ω𝐵-bimodule structure given by ∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, 𝛼𝜉𝛽 � 𝜎 (𝛼 ⊗ 𝜉⟨0⟩)𝜉⟨1⟩𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The pair (𝜎𝐵, ∇𝐵) � (𝜆−1 Ω𝐵 ◦ 𝜌Ω𝐵, 𝜆−1 Ω𝐵 ◦ d) defines a Hermitian bimodule connection on the trivial Hermitian line 𝐵-bimodule 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' where convenient, we shall abuse notation and identify (𝜎𝐵, ∇𝐵) with (idΩ𝐵, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 17 It is easy enough to check that if 𝐸 is a 𝐵-self-correspondence of finite type with right Hermitian connection ∇𝐸, then there exists at most one Hermitian generalised braiding 𝜎𝐸 on 𝐸 that makes (𝜎𝐸, ∇𝐸) into a Hermitian bimodule connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, in this case, ∀𝜉, 𝜐 ∈ Ω𝐵, d𝐵(𝜉, 𝜐) = (∇𝐸𝜉, 𝜐) + (−1) |𝜉 |(𝜉, ∇𝐸𝜐), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) ∀𝛽 ∈ Ω𝐵, ∀𝜉, 𝜐 ∈ 𝐸 ⊗𝐵 Ω𝐵, (𝛼𝜉, 𝜐) = (𝜉, 𝛼∗𝜐);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26), it suffices to check (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) in the special case where 𝛽 ∈ d(𝐵) and 𝜉, 𝜐 ∈ 𝐸 ⊗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We shall use the following characterisation of Hermitian bimodule connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27 (Beggs–Majid [13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a 𝐵-self-correspondence of finite type, let 𝜎 be a generalised braiding on 𝐸, and let ∇ be a Hermitian right connection on 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝜎, ∇) defines a Hermitian bimodule connection on 𝐸 if and only if the following both hold: ∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, ∇(𝑏𝜉) = 𝜎 (d𝐵𝑏 ⊗ 𝑥) + 𝑏∇𝑥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) ∀𝑏 ∈ 𝐵, ∀𝑥 ∈ 𝐸, ∇2(𝑏𝜉) = 𝑏∇2𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32) We now introduce our first nontrivial family of examples of Hermitian bimodule con- nections on Hermitian line bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜃 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that the smooth 2-torus 𝐶∞ 𝜃 (T2) admits a canonical ∗-exterior calculus (Ω𝜃(T2), d) due to Connes [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝛿1 and 𝛿2 be the unique ∗-derivations on 𝐶∞ 𝜃 (T2), such that, respectively ∀(𝑚, 𝑛) ∈ Z2, 𝛿1(𝑈𝑚𝑉 𝑛) = 2𝜋i𝑚𝑈𝑚𝑉 𝑛, 𝛿2(𝑈𝑚𝑉 𝑛) = 2𝜋i𝑛𝑈𝑚𝑉 𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then let Ω𝜃(T2) be the graded ∗-algebra over 𝐶∞ 𝜃 (T2) generated by central self-adjoint el- ements 𝑒1, 𝑒2 ∈ Ω1 𝜃 (T2) satisfying (𝑒1)2 = (𝑒2)2 = 𝑒1𝑒2 + 𝑒2𝑒1 = 0, and let d be the unique ∗-derivation of degree 1 on Ω𝜃(T2), such that ∀𝑎 ∈ 𝐶∞ 𝜃 (T2), d𝑎 � 𝛿1(𝑎)𝑒1 + 𝛿2(𝑎)𝑒2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' d𝑒1 � 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' d𝑒2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the case where 𝜃 is a quadratic irrationality, the basic Heisenberg modules of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 admit canonical Hermitian bimodule connections due to Connes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28 (Connes [31, Thm 7], Polishchuk–Schwarz [84, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 ∈ Γ𝜃 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Connes constructs maps ∇𝑔,1, ∇𝑔,2 : 𝐸(𝑔) → 𝐸(𝑔) that yield a right Hermitian connection ∇𝑔 : 𝐸(𝑔) → 𝐸(𝑔) ⊗𝐵 Ω1 𝐵 by setting ∀𝑝 ∈ 𝐸(𝑔), ∇𝑔(𝑝) � ∇𝑔,1(𝑝) ⊗ 𝑒1 + ∇𝑔,2(𝑝) ⊗ 𝑒2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, [31, Thm 7] implies that ∀𝑝 ∈ 𝐸(𝑔), ∇2 𝑔(𝑝) = 𝑝 · 2𝜋i 𝑔21 𝑔21𝜃 + 𝑔22 𝑒1𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by [84, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1] together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27, the map ∇𝑔 defines a Hermitian bimodule connection on 𝐸(𝑔) with respect to the generalised braiding 𝜎𝑔 given by ∀𝑖 ∈ {1, 2}, ∀𝑝 ∈ 𝐸(𝑔), 𝜎𝑔(𝑒𝑖 ⊗ 𝑝) � 1 𝑔21𝜃 + 𝑔22 𝑝 ⊗ 𝑒𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The primary technical advantage of bimodule connections is that they are compatible with taking balanced tensor products of bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In fact, they give rise to a monoidal category of 𝐵-self-correspondence of finite type with Hermitian bimodule connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29 (Beggs–Majid [13, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Beggs–Brzeziński [10, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines an essentially small monoidal concrete category DCorr(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) An object of DCorr(𝐵) is a triple (𝐸, 𝜎𝐸, ∇𝐸), where 𝐸 is a 𝐵-self-correspondence of finite type and (𝜎𝐸, ∇𝐸) is a Hermitian bimodule connection on 𝐸;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 18 BRANIMIR ĆAĆIĆ (2) An arrow 𝑢 : (𝐸, 𝜎𝐸, ∇𝐸) → (𝐹, 𝜎𝐹, ∇𝐹) is a 𝐵-bimodule isomorphism 𝑓 : 𝐸 → 𝐹 satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11) and ∇𝐹 ◦ 𝑢 = (𝑢 ⊗ idΩ1 𝐵) ◦ ∇𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) (3) The tensor product of objects (𝐸, 𝜎𝐸, ∇𝐸) and (𝐹, 𝜎𝐹, ∇𝐹) is (𝐸 ⊗𝐵 𝐹, 𝜎𝐸⊗𝐵𝐹, ∇𝐸⊗𝐵𝐹), where 𝐸 ⊗𝐵 𝐹 is the balanced tensor product of 𝐵-bimodules equipped with the 𝐵-valued inner product of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12), and where 𝜎𝐸⊗𝐵𝐹 � 𝛼−1 𝐸,𝐹,Ω𝐵 ◦ (id𝐸 ⊗ 𝜎𝐹) ◦ 𝛼𝐸,Ω𝐵,𝐹 ◦ 𝜎𝐸 ⊗ id𝐹, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) ∇𝐸⊗𝐵𝐹 � 𝛼−1 𝐸,𝐹,Ω𝐵 ◦ �(id𝐸 ⊗ 𝜎𝐹) ◦ 𝛼𝐸,Ω𝐵,𝐹 ◦ (∇𝐸 ⊗ id) + id𝐸 ⊗ ∇𝐹 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35) moreover, the monoidal product of arrows is given by the balanced tensor product of 𝐵-bimodule homomorphisms, and the associators are given by the corresponding associators in Bimod(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4) The unit object of DCorr(𝐵) is (𝐵, 𝜎𝐵, ∇𝐵), where (𝜎𝐵, ∇𝐵) is the Hermitian bimodule con- nection of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, left unitors, and right unitors are given by the corresponding left unitors and right unitors of Bimod(𝐵), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In addition, if 𝑢 : (𝐸, 𝜎𝐸, ∇𝐸) → (𝐹, 𝜎𝐹, ∇𝐹) is an arrow in DPic(𝐵), then ∇𝐹 ◦ (𝑢 ⊗ idΩ𝐵) = (𝑢 ⊗ idΩ𝐵) ◦ ∇𝐸, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36) 𝜎𝐹 ◦ (idΩ𝐵 ⊗ 𝑢) = (𝑢 ⊗ idΩ𝐵) ◦ 𝜎𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Relative to [13, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7] and the discussion before the proof of Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13 (with minor changes), it suffices to check that the tensor product is indeed well-defined on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let 𝐸 be a 𝐵-self-correspondence of finite type with Hermitian bimodule connection (𝜎𝐸, ∇𝐸), and let 𝐹 be a 𝐵-self-correspondence of finite type with Hermitian bimodule connection (𝜎𝐹, ∇𝐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A straightforward calculation shows that ∀𝑥, 𝑣 ∈ 𝐸, ∀𝜐, 𝜏 ∈ 𝐹 ⊗𝐵 Ω𝐵, �(𝑥 ⊗ 𝜐⟨0⟩) ⊗ 𝜐⟨1⟩, (𝑣 ⊗ 𝜏⟨0⟩) ⊗ 𝜏⟨1⟩ � = (𝜐, (𝑥, 𝑣)𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) Relative to [13, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7], it remains to check that 𝜎𝐸⊗𝐵𝐹 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) and that ∇𝐸⊗𝐵𝐹 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25), but this now follows by repeated application of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23), as appropriate, to the Hermitian bimodule connections (𝜎𝐸, ∇𝐸) and (𝜎𝐹, ∇𝐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ We now construct our coherent 2-group of Hermitian line bundles with unitary connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Beggs–Majid [11, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The differential Picard 2-group of the unital pre-𝐶∗-algebra 𝐵 is the coherent 2-group DPic(𝐵) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) As a monoidal category, DPic(𝐵) is the full monoidal subcategory of DCorr(𝐵), whose objects are of the form (𝐸, 𝜎𝐸, ∇𝐸), where 𝐸 is a Hermitian line 𝐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) The monoidal inverse of an object (𝐸, 𝜎𝐸, ∇𝐸) is given by (𝐸, 𝜎𝐸, ∇𝐸), where ∀𝛽 ∈ Ω𝐵, ∀𝑥 ∈ 𝐸, 𝜎𝐸(𝛽 ⊗ 𝑥) � ΥΩ𝐵,𝐸 � 𝜎−1 𝐸 (𝑥 ⊗ 𝛽∗) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39) ∀𝑥 ∈ 𝐸, ∇𝐸𝑥 � ΥΩ𝐵,𝐸 � 𝜎−1 𝐸 (∇𝐸𝑥) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40) here, by abuse of notation, we let ΥΩ𝐵,𝐸 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 denote the isomorphism of 𝐵-bimodules defined by ∀𝑥 ∈ 𝐸, ∀𝛽 ∈ Ω𝐵, ΥΩ𝐵,𝐸(𝛽 ⊗ 𝑥) � 𝑥 ⊗ 𝛽∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41) (3) The evaluation arrows are given by the corresponding evaluation arrows in Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, a Hermitian line 𝐵-bimodule with connection is an object of DPic(𝐵), and an isomorphism of Hermitian line 𝐵-bimodules with connection is an arrow of DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, the differential Picard group of 𝐵 is the group DPic(𝐵) � 𝜋0(DPic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29 and Theorem-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13, it remains to show that monoidal inversion and evaluation in DPic(𝐵) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a Hermitian line 𝐵-bimodule with Hermitian bimodule connection (𝜎𝐸, ∇𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that 𝐸 admits the Hermitian bimodule connection (𝜎𝐸, ∇𝐸) defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By a theorem of Beggs–Majid [11, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3], suitably adapted, we know that 𝜎𝐸 is a 𝐵-bimodule isomorphism, that ∇𝐸 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24), and that the pair (𝜎𝐸, ∇𝐸) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27, it remains to show that 𝜎𝐸 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) and that ∇𝐸 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In turn, by construction of 𝜎𝐸 and ∇𝐸, it therefore suffices to show that, respectively, for all 𝛼, 𝛽 ∈ Ω𝐵 and 𝑥, 𝑦, 𝑧 ∈ 𝐸, 𝜎−1 𝐸 (𝑥 ⊗ 𝛽𝛼) = 𝜎−1 𝐸 (𝑥 ⊗ 𝛽) ⟨−1⟩𝜎−1 𝐸 � 𝜎𝐸(𝑥 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='42) 𝜎𝐸(d𝐵(𝑥, 𝑦) ⊗ 𝑧) = 𝜎𝐸 �(∇𝐸𝑥, 𝑦 ⊗ 1) ⊗ 𝑧� + 𝜎𝐸 �(𝑥 ⊗ 1, ∇𝐸 𝑦) ⊗ 𝑧�, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43) 𝜎−1 𝐸 �∇2 𝐸𝑥� = 𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩𝜎−1 𝐸 � ∇𝐸(𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩) � + d𝐵𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44) First, we check (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝛼, 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) applied to 𝜎𝐸, 𝑒 ⊗ 𝛽𝛼 = 𝜎𝐸 � 𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ � 𝛼 = 𝜎𝐸 � 𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩ ⊗ (𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼) ⟨0⟩ � (𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼) ⟨1⟩ = 𝜎𝐸 � 𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨−1⟩𝜎−1 𝐸 � 𝜎−1 𝐸 (𝑒 ⊗ 𝛽) ⟨0⟩ ⊗ 𝛼 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we check (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑥, 𝑦, 𝑧 ∈ 𝐸 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (𝜎𝐸, ∇𝐸) is a Hermitian bimodule connection, it follows that 𝜎𝐸(d𝐵(𝑥, 𝑦) ⊗ 𝑧) = ∇𝐸((𝑥, 𝑦)𝑧) − (𝑥, 𝑦)∇𝐸𝑧 = ∇𝐸(𝑥)(𝑦, 𝑧) + 𝑥 ⊗ (∇𝐸 𝑦, 𝑧), On the one hand, by definition of ∇𝐸, we see that 𝜎𝐸 �(∇𝐸𝑥, 𝑦) ⊗ 𝑧� = 𝜎𝐸 � 𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩(𝑦, 𝑧) � = ∇𝐸(𝑥)(𝑦, 𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by definition of ∇𝐸 together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) applied to 𝜎𝐸, 𝑥 ⊗ (∇𝐸 𝑦, 𝑧) = 𝑥 ⊗ � 𝜎−1 𝐸 (∇𝐸 𝑦) ⟨−1⟩(𝜎−1 𝐸 (∇𝐸 𝑦) ⟨0⟩ ⊗ 1), 𝑧 ⊗ 1 � = 𝑥 ⊗ � 𝜎−1 𝐸 (∇𝐸 𝑦) ⟨0⟩ ⊗ 1, 𝜎−1 𝐸 (∇𝐸 𝑦) ∗ ⟨−1⟩ ⊗ 𝑧) � = � 𝑥, 𝜎−1 𝐸 (∇𝐸 𝑦) ⟨0⟩ � 𝜎𝐸(𝜎−1 𝐸 (∇𝐸 𝑦) ∗ ⟨−1⟩ ⊗ 𝑧) = 𝜎𝐸 �(𝑥 ⊗ 1, ∇𝐸(𝑦))) ⊗ 𝑧�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, we check (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑥 ∈ 𝐸 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) applied to 𝜎𝐸 and(𝜎𝐸, ∇𝐸), respectively, ∇2 𝐸𝑥 = ∇𝐸 ◦ 𝜎𝐸 � 𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩ � = 𝜎𝐸 � 𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ ∇𝐸(𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩) + d𝐵𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩ � = 𝜎𝐸 � 𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩𝜎−1 𝐸 (∇𝐸(𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩)) � + 𝜎𝐸 � d𝐵𝜎−1 𝐸 (∇𝐸𝑥) ⟨−1⟩ ⊗ 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 20 BRANIMIR ĆAĆIĆ We now show that ev𝐸 : 𝐸 ⊗𝐵 𝐸 → 𝐸 in Pic(𝐵) defines a corresponding arrow in DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since ∇𝐵 = 𝜆−1 Ω𝐵 ◦ d, it suffices to show that for all 𝑥, 𝑦 ∈ 𝐸, d𝐵 ev𝐸(𝑥 ⊗ 𝑦) = 𝜆Ω𝐵 ◦ (ev𝐸 ⊗ idΩ𝐵) ◦ 𝛼−1 𝐸,𝐸,Ω𝐵 � (id𝐸 ⊗ 𝜎𝐸) ◦ 𝛼𝐸,Ω𝐵,𝐸(∇𝐸𝑥 ⊗ 𝑦) + 𝑥 ⊗ ∇𝐸 𝑦 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝑥, 𝑦 ∈ 𝐸 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) applied to 𝜎𝐸 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) applied to ∇𝐸, d𝐵 ev𝐸(𝑥 ⊗ 𝑦) = (∇𝐸𝑥, 𝑦 ⊗ 1) + (𝑥 ⊗ 1, ∇𝐸 𝑦) = � 𝜎−1 𝐸 (∇𝐸𝑥) ⟨0⟩ ⊗ 1, 𝜎−1 𝐸 (∇𝐸𝑥) ∗ ⟨−1⟩(𝑦 ⊗ 1) � + (𝑥 ⊗ 1, ∇𝐸 𝑦) = ev𝐸 � ∇𝐸(𝑥) ⟨0⟩ ⊗ 𝜎𝐸(∇𝐸(𝑥) ⟨1⟩ ⊗ 𝑦) ⟨0⟩ � 𝜎𝐸(∇𝐸(𝑥) ⟨1⟩ ⊗ 𝑦) ⟨−1⟩ + ev𝐸 � 𝑥 ⊗ ∇𝐸(𝑦) ⟨0⟩ � ∇𝐸(𝑦) ⟨1⟩ = 𝜆Ω𝐵 ◦ (ev𝐸 ⊗ idΩ𝐵) ◦ 𝛼−1 𝐸,𝐸,Ω𝐵 � (id𝐸 ⊗ 𝜎𝐸) ◦ 𝛼𝐸,Ω𝐵,𝐸(∇𝐸𝑥 ⊗ 𝑦) + 𝑥 ⊗ ∇𝐸 𝑦 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31 (Connes [31, Thm 7], Polishchuk–Schwarz [84, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a lift of the homomorphism 𝐸 : Γ𝜃 → Pic(𝐶∞ 𝜃 (T2)) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 with respect to a homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ 𝜃 (T2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given 𝑔 ∈ Γ𝜃, let ˆ𝐸(𝑔) � (𝐸(𝑔), 𝜎𝑔, ∇𝑔), where (𝜎𝑔, ∇𝑔) is the Hermitian bimodule connection on 𝐸(𝑔) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Let ˆ𝐸(0) be given by id𝐶∞ 𝜃 (T2) � 𝐸(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) Given 𝑔, ℎ ∈ Γ𝜃, let ˆ𝐸(2) 𝑔,ℎ be given by 𝐸(2) 𝑔,ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, that ˆ𝐸(0) and ˆ𝐸(2) satisfy the required commutative diagrams follows (with superficial changes) from the result of Polishchuk–Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Canonical actions of the differential Picard group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Again, let 𝐵 be a unital pre-𝐶∗- algebra with ∗-exterior algebra (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We show that the differential Picard group DPic(𝐵) defines a generalised diffeomorphism group for (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), whose action on the 𝐾0-monoid V(𝐵) characterizes the fibres of the forgetful map DPic(𝐵) → V(𝐵) and whose action on the graded centre Z(Ω𝐵) admits curvature as a canonical group 1-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first consider naïve diffeomorphisms of the manifold (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Gel’fand duality initially suggests that a diffeomorphism of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) should be an automorphism of the ∗-exterior algebra (Ω𝐵, d𝐵) over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, as we shall see in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35, inner automor- phisms of 𝐵, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', automorphisms of the form 𝑏 ↦→ 𝑢𝑏𝑢∗ for fixed 𝑢 ∈ U(𝐵), will generally only satisfy the following conservative generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define the extended diffeomorphism group of 𝐵 with respect to (Ω𝐵, d) to be the subgroup � Diff(𝐵) of (Ω1 𝐵)sa ⋊ Aut(Ω𝐵) consisting of elements (𝜔, 𝜙) satisfying ∀𝛽 ∈ Ω𝐵, d𝛽 − 𝜙 ◦ d ◦ 𝜙−1(𝛽) = i[𝜔, 𝛽], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45) where [·, ·] denotes the supercommutator in Ω𝐵 with respect to parity of degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' An extended diffeomorphism of 𝐵 with respect to (Ω𝐵, d) is an element of � Diff(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, we say that (𝜔, 𝜙) ∈ � Diff(𝐵) is topologically trivial whenever 𝜙↾𝐵= id𝐵, and we denote by � Diff0(𝐵) the normal subgroup of � Diff(𝐵) consisting of topologically trivial elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21, equip 𝐶∞(𝑋) with the de Rham ∗-exterior calculus (Ω(𝑋, C), d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The isomorphism Diff(𝑋) → Aut(𝐶∞(𝑋)) yields an isomorphism �(𝑓, 𝜔) ↦→ ((𝑓 −1)∗𝜔, (𝑓 −1)∗)� : Diff(𝑋) ⋉ Ω1(𝑋, R) → � Diff(𝐶∞(𝑋)) NONCOMMUTATIVE U(1)-GAUGE THEORY 21 where Diff(𝑋) acts on Ω1(𝑋, R) from the right by pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, it follows that the preimage of � Diff0(𝑋) is {id𝑋} × Ω1(𝑋, R) � Ω1(𝑋, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall the homomorphism 𝜏 : Aut(𝐵) → Pic(𝐵) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a lift of 𝜏 with respect to the forgetful homomorphism DPic(𝐵) → Pic(𝐵) to a homomorphism ˆ𝜏 : � Diff(𝐵) → DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given (𝜔, 𝜙) ∈ � Diff(𝐵), let ˆ𝜏(𝜙,𝜔) � (𝐵𝜙, 𝜎𝜙, ∇(𝜔,𝜙)), where 𝐵𝜙 � 𝜏𝜙 and ∀𝛽 ∈ Ω𝐵, ∀𝑏 ∈ 𝐵, 𝜎𝜙(𝛽 ⊗ 𝑏𝜙) � 1𝜙 ⊗ 𝜙−1(𝛽𝑏), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46) ∀𝑏 ∈ 𝐵, ∇(𝜙,𝜔) (𝑏𝜙) � 1𝜙 ⊗ 𝜙−1(d𝑏 + 𝑏 · i𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='47) (2) Let ˆ𝜏 (0) be given by id𝐵 � 𝜏 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) Given (𝜔1, 𝜙1), (𝜔2, 𝜙2) ∈ � Diff(𝐵), let ˆ𝜏 (2) (𝜔1,𝜙1),(𝜔2,𝜙2) be given by 𝜏 (2) 𝜙1,𝜙2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that the homomorphism ˆ𝜏 : � Diff(𝐵) → DPic(𝐵) is faithful on objects, so that it can be viewed as embedding the group � Diff(𝐵) in the coherent 2-group DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let ΠPic(𝐵) : DPic(𝐵) → Pic(𝐵) be the group homomorphism induced by the forgetful homomorphism DPic(𝐵) → Pic(𝐵), and recall that ΠV(𝐵) : Pic(𝐵) → V(𝐵) is the set map induced by the forgetful functor Pic(𝐵) → Hilb(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, note that the right Pic(𝐵)-action on V(𝐵) of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19 pulls back via ΠPic(𝐵) to a right DPic(𝐵)-action on V(𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in turn, this right DPic(𝐵)-action correctly pulls back via 𝜋0(ˆ𝜏) to the usual pullback action of isometric ∗-automorphisms on V(𝐵) Since this DPic(𝐵)-action is transitive on the range of ΠV(𝐵) ◦ ΠPic(𝐵) : DPic(𝐵) → V(𝐵), we may use the resulting stabilizer group DPic(𝐵)[𝐵] of [𝐵] ∈ ran(ΠV(𝐵) ◦ ΠPic(𝐵)) to characterize the fibres of the forgetful map from the differential Picard group DPic(𝐵) to the 𝐾0-monoid V(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, since ΠPic(𝐵) is a group homomorphism, its kernel yields the fibres of the forgetful map from DPic(𝐵) to the (topological) Picard group Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' As it turns out, the subgroups DPic(𝐵)[𝐵] and ker ΠPic(𝐵) are completely determined by the group homomorphism 𝜋0(ˆ𝜏) : � Diff(𝐵) → DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let DPic(𝐵)[𝐵] denote the stabilizer subgroup of DPic(𝐵) with respect to the isomorphism class [𝐵] ∈ ran(ΠK(𝐵) ◦ ΠPic(𝐵)), and let � Ad : U(𝐵) → � Diff(𝐵) be given by ∀𝑢 ∈ U(𝐵), � Ad𝑢 � (−i d𝐵(𝑢)𝑢∗, Ad𝑢) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48) Then the homomorphisms 𝜋0(ˆ𝜏) and � Ad fit into the exact sequences of groups 1 → U(Z(𝐵) ∩ ker d𝐵) → U(𝐵) � Ad −−→ � Diff(𝐵) 𝜋0(ˆ𝜏) −−−−→ DPic(𝐵)[𝐵] → 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49) 1 → U(Z(𝐵) ∩ ker d𝐵) → U(Z(Ω𝐵)0) � Ad −−→ � Diff0(𝐵) 𝜋0(ˆ𝜏) −−−−→ DPic(𝐵) ΠPic(𝐵) −−−−−→ Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Before continuing, we must show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49) is a well-defined diagram of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A straightforward calculation show that � Ad : U(𝐵) → � Diff(𝐵) is well-defined, so it remains to check that ran 𝜋0(ˆ𝜏) ≤ DPic(𝐵)[𝐵].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, given (𝜔, 𝜙) ∈ � Diff(𝐵), the required isomor- phism 𝑈 : 𝐵 ⊗𝐵 𝐵𝜙 → 𝐵 in Hilb(𝐵) is given by 𝑈 � �𝑏 ⊗ 𝑐𝜙 ↦→ 𝜙−1(𝑏𝑐)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49) is an exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Exactness at U(𝐵) is immediate, so we proceed to checking exactness at � Diff(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let (𝜔, 𝜙) ∈ ker 𝜋0(ˆ𝜏) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, there exists an arrow 𝑈 : (𝐵𝜙, 𝜎𝜙, ∇𝜔,𝜙) → (𝐵, 𝜎𝐵, ∇𝐵) in DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Set 𝑢 � 𝑈(1𝜙);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' we claim that (𝜔, 𝜙) = � Ad𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, observe that 𝑢 ∈ U(𝐵) since the singleton {1𝜙} defines both a basis and strict cobasis for 𝐵𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, observe that 𝜙 = Ad𝑢 since for all 𝛽 ∈ Ω𝐵, 𝛽𝑢 = 𝜆Ω𝐵 ◦ 𝜎0 ◦ (idΩ𝐵 ⊗ 𝑈)(𝛽 ⊗ 1𝜙) = 𝜆Ω𝐵 ◦ 𝜎0 ◦ (idΩ𝐵 ⊗ 𝑈) ◦ 𝜎−1 𝜙 (1𝜙 ⊗ 𝜙−1(𝛽)) = 𝑢𝜙−1(𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 22 BRANIMIR ĆAĆIĆ Finally, given that 𝜙 = Ad𝑢, observe that 𝜔 = −i d(𝑢)𝑢∗ since 0 = 𝜆Ω𝐵 �(𝑈 ⊗ idΩ𝐵)(∇𝜔,𝜙1𝜙) − d𝐵𝑢𝑈(1𝜙)� = 𝜆Ω𝐵 ◦ (𝑈 ⊗ idΩ𝐵) �1𝜙 ⊗ i𝜙−1(𝜔)� − d𝐵𝑢 = i(𝜔 + id𝐵(𝑢)𝑢∗)𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, given 𝑢 ∈ U(𝐵), similar calculations show that (𝑏Ad𝑢 ↦→ 𝑏𝑢) : 𝐵Ad𝑢 → 𝐵 defines an arrow ˆ𝜏� Ad𝑢 → (𝐵, 𝜎0, ∇0) in DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, we check exactness at DPic(𝐵)[𝐵].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connection, and suppose that [(𝐸, 𝜎𝐸, ∇𝐸)] ∈ DPic(𝐵)[𝐵].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Using 𝜆𝐵 : 𝐵 ⊗𝐵 𝐵 → 𝐵, we conclude that there exists an arrow 𝑈 : 𝐵 → 𝐸 in Hilb(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Set 𝑒0 � 𝑈(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since the singleton {1} defines both a basis and strict cobasis for 𝐵, it follows that {𝑒0} defines both a basis and strict cobasis for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We shall use 𝑒0 together with (𝜎𝐸, ∇𝐸) to construct (𝜔, 𝜙) ∈ � Diff(𝐵) and an arrow 𝑉 : ˆ𝜏(𝜙,𝜔) → (𝐸, 𝜎𝐸, ∇𝐸) in DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, define a C-linear map Φ : Ω𝐵 → Ω𝐵 by Φ � (𝛽 ↦→ (𝑒0 ⊗ 1, 𝜎𝐸(𝛽 ⊗ 𝑒0)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' once we know that the degree-preserving map Φ is an element of Aut(Ω𝐵), we shall set 𝜙 � Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, note that Φ is unit-preserving since Φ(1) = (𝑒0, 𝑒0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, note that Φ is multiplicative by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) applied to 𝜎𝐸 and ∗-preserving by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) applied to 𝜎𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, note that Φ is bijective since, for all 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, 𝛽 ⊗ 𝑥 = 𝛽 ⊗ 𝑒0(𝑒0, 𝑥) = 𝜎−1 𝐸 (𝑒0 ⊗ (𝑒0 ⊗ 1, 𝜎−1 𝐸 (𝛽 ⊗ 𝑒0)))(𝑒0, 𝑥) = 𝜎−1 𝐸 (𝑒0 ⊗ Φ(𝛽))(𝑒0, 𝑒), so that, in terms of the arrow ΥΩ𝐵,𝐸 : Ω𝐵 ⊗𝐵 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 in Bimod(𝐵), ∀𝛽 ∈ Ω𝐵, Φ−1(𝛽) = � ΥΩ𝐵,𝐸(𝜎−1 𝐸 (𝑒0 ⊗ 𝛽)), 𝑒0 ⊗ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, note that Φ is isometric on 𝐵 since, for all 𝑏 ∈ 𝐵, ∥Φ(𝑏)∥ = ∥(𝑒0, 𝑏𝑒0)∥ ≤ ∥𝑏∥ · ∥𝑒0∥2 = ∥𝑏∥ · ∥(𝑒0, 𝑒0)∥ = ∥𝑏∥, ∥Φ−1(𝑏)∥ = ∥(𝑒0𝑏, 𝑒0)∥ ≤ ∥𝑏∥ · ∥𝑒0∥2 = ∥𝑏∥ · ∥(𝑒0, 𝑒0)∥ = ∥𝑏∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Having constructed Φ, we now claim that (𝜔, 𝜙, ) � (−iΦ−1((𝑒0 ⊗ 1, ∇𝐸𝑒0)), Φ−1) defines an element of � Diff(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that 𝜔 ∈ Ω1 𝐵 is self-adjoint since 𝜙 ∈ Aut(Ω𝐵) and since 0 = d𝐵(𝑒0, 𝑒0) = (∇𝐸𝑒0, 𝑒0 ⊗ 1) + (𝑒0 ⊗ 1, ∇𝐸𝑒0) = (𝑒0, ∇𝐸𝑒0)∗ + (𝑒0, ∇𝐸𝑒0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, it remains to show that (𝜙, 𝜔) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝛽 ∈ Ω𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜎𝐸(d𝐵𝜙(𝛽) ⊗ 𝑒0) = ∇𝐸(𝜎𝐸(𝜙(𝛽) ⊗ 𝑒0)) − (−1) |𝛽|𝜎𝐸 � 𝜙(𝛽) ⊗ ∇𝐸(𝑒0) ⟨0⟩ � ∇𝐸(𝑒0) ⟨1⟩ = ∇𝐸(𝑒0 ⊗ 𝛽) − (−1) |𝛽|𝜎𝐸(𝜙(𝛽) ⊗ 𝑒0)(𝑒0, ∇𝐸𝑒0) = ∇𝐸(𝑒0)𝛽 + 𝑒0 ⊗ d𝛽 − (−1) |𝛽|𝑒0 ⊗ 𝜔(𝑒0, ∇𝐸𝑒0) = 𝑒0 ⊗ (d𝐵𝛽 + [(𝑒0, ∇𝐸𝑒0), 𝛽]) = 𝜎𝐸((𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)]) ⊗ 𝑒0), so that, indeed, for every 𝑥 ∈ 𝐸, d𝐵𝜙(𝛽)⊗𝑥 = d𝐵𝜙(𝛽)⊗𝑒0(𝑒0, 𝑥) = (𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)])⊗𝑒0(𝑒0, 𝑥) = (𝜙(d𝐵𝛽) + i[𝜔, 𝜙(𝛽)])⊗𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 23 Finally, define an arrow 𝑉 : 𝐵𝜙 → 𝐸 in Pic(𝐵) by 𝑉 (1𝜙) � 𝑒0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' we claim that it yields an arrow 𝑉 : ˆ𝜏𝜔,𝜙 → (𝐸, 𝜎𝐸, ∇𝐸) in DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, for all 𝑏 ∈ 𝐵, ∇𝐸 �𝑉 (𝑏𝜙)� = 𝜎𝐸(d𝑏 ⊗ 𝑒0) + 𝑏∇𝐸𝑒0 = 𝑒0 ⊗ ((𝑒0 ⊗ 1, 𝜎𝐸(d𝑏 ⊗ 𝑒0))) + 𝑏𝑒0 ⊗ (𝑒0 ⊗ 1, ∇𝐸𝑒0) = 𝑒0 ⊗ 𝜙−1(d𝑏) + 𝑏𝑒0 ⊗ i𝜙−1(𝜔) = (𝑉 ⊗ id)�∇(𝜔,𝜙)𝑏𝜙 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ We have just seen that the generalised diffeomorphism group � Diff(𝐵) embeds via ˆ𝜏 in the differential Picard 2-group DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following refinement of Proposition-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20 yields a surprising ‘moral converse’: the entire differential Picard group DPic(𝐵) acts canonically as automorphisms on the graded centre of Ω𝐵 in a manner that will turn out to be explicitly compatible with the embedding of � Diff(𝐵) in DPic(𝐵) by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 (Beggs–Majid [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Fröhlich homomorphism of 𝐵 with respect to (Ω𝐵, d) is the unique group homomorphism ˆΦ : DPic(𝐵) → Aut(Z(Ω𝐵), d), such that, for every Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸), ∀𝛽 ∈ Z𝐵(Ω𝐵), ∀𝑥 ∈ 𝐸, ˆΦ[𝐸,∇𝐸](𝛽) ⊗ 𝑥 = 𝜎−1 𝐸 (𝑥 ⊗ 𝛽);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51) in this case, we call ˆΦ[𝐸,∇𝐸] the Fröhlich automorphism of (𝐸, 𝜎𝐸, ∇𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Relative to [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9], it remains to check that for each (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)), the automorphism ˆΦ[𝐸,∇𝐸] of the graded algebra Z𝐵(Ω𝐵) is ∗-preserving and commutes with d𝐵 on Z(Ω𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' observe that the restriction ˆΦ[𝐸,∇𝐸]↾Z(𝐵)= Φ[𝐸] is isometric by Proposition- Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)) be given, and fix a basis (𝑒𝑖)𝑛 𝑖=1 and a strict cobasis (𝜖𝑗)𝑚 𝑗=1 for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35, mutatis mutandis, ∀𝛽 ∈ Z(Ω𝐵), ˆΦ[𝐸,∇𝐸](𝛽) = ∑︁𝑛 𝑖=1 � ΥΩ𝐵,𝐸(𝜎−1 𝐸 (𝑒𝑖 ⊗ 𝛽)), 𝑒𝑖 ⊗ 1 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52) ∀𝛽 ∈ Z(Ω𝐵), ˆΦ−1 [𝐸,∇𝐸](𝛽) = ∑︁𝑚 𝑗=1 �𝜖𝑗 ⊗ 1, 𝜎𝐸(𝛽 ⊗ 𝜖𝑗)�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='53) hence, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='53) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) applied to 𝜎𝐸, the map ˆΦ[𝐸,∇𝐸] is ∗-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, let 𝛽 ∈ Z(Ω𝐵) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, for all 𝑥 ∈ 𝐸, 𝑥 ⊗ d𝐵 ˆΦ−1 [𝐸,∇𝐸](𝛽) = ∇𝐸 �𝑥 ⊗ Φ[𝐸,∇𝐸](𝛽)� − ∇𝐸(𝑥)Φ−1 [𝐸,∇𝐸](𝛽) = ∇𝐸(𝛽(𝑥 ⊗ 1)) − 𝛽∇𝐸𝑥 = 𝑥 ⊗ ˆΦ−1 [𝐸,∇𝐸](d𝐵𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The canonical left action of 𝜋0(DPic(𝐵)) � DPic(𝐵) on the Abelian group 𝜋1(DPic(𝐵)) = U(Z(𝐵) ∩ ker d𝐵) is the left action induced by ˆΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We can now introduce curvature as a 1-cocycle for this action of DPic(𝐵) on Z(Ω𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For convenience, let us define a pre-symplectic form on 𝐵 to be self-adjoint 𝛽 ∈ Z(Ω𝐵)2 satisfying d𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we denote by S(𝐵) the real subspace of all pre-symplectic forms on 𝐵, which we endow with the right action of DPic(𝐵) defined by ∀[𝐸, ∇𝐸] ∈ DPic(𝐵), ∀𝜔 ∈ S(𝐵), 𝜔 ⊳ [𝐸, ∇𝐸] � ˆΦ−1 [𝐸,∇𝐸](𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='54) Moreover, recall that if Γ is a group and 𝑀 is a right Γ-module (written additively), then a map 𝑐 : Γ → 𝑀 is a right 1-cocycle whenever 𝑐(𝛾𝜂) = 𝑐(𝛾) ⊳ 𝜂 + 𝑐(𝜂) for all 𝛾, 𝜂 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 24 BRANIMIR ĆAĆIĆ Proposition-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38 (Beggs–Majid [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The curvature 1-cocycle of 𝐵 with respect to (Ω𝐵, d) is the unique right 1-cocycle F : DPic(𝐵) → S(𝐵), such that, for every Hermitian 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸), ∀𝜉 ∈ 𝐸 ⊗𝐵 Ω𝐵, ∇2 𝐸𝜉 = 𝜉 · iF[𝐸,∇𝐸];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55) in this case, we call F[𝐸,∇𝐸] ∈ S(𝐵) the curvature 2-form of [𝐸, ∇𝐸].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let (𝐸, 𝜎𝐸, ∇𝐸) ∈ Obj(DPic(𝐵)) be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' fix a basis (𝑒𝑖)𝑚 𝑖=1 and a cobasis (𝜖𝑗)𝑛 𝑗=1 for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The map ∇2 𝐸 : 𝐸 → 𝐸 ⊗𝐵 Ω𝐵 is right 𝐵-linear by repeated applications of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24) and left 𝐵-linear by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, ∇2 𝐸𝑥 = ∇2 𝐸 ��𝑛 𝑗=1(𝑥, 𝜖𝑗)𝜖𝑗 � = �𝑚 𝑗=1(𝑥, 𝜖𝑗)∇2 𝐸𝜖𝑗 = 𝑥⊗�𝑚 𝑗=1(𝜖𝑗 ⊗1, ∇2 𝐸𝜖𝑗) for every 𝑥 ∈ 𝐸, so that the 2-form F[𝐸,∇𝐸] � −i �𝑚 𝑗=1(𝜖𝑗 ⊗ 1, ∇2 𝐸𝜖𝑗) ∈ Ω2 𝐵 satisfies ∀𝑥 ∈ 𝐸, ∇2 𝐸𝑥 = 𝑥 ⊗ iF[𝐸,∇𝐸].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56) On the one hand, F[𝐸,∇𝐸] is the unique 2-form satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56) since, for any such 2-form 𝜛, 𝜛 = ∑︁𝑛 𝑗=1(𝜖𝑗, 𝜖𝑗)𝜛 = ∑︁𝑛 𝑗=1(𝜖𝑗 ⊗ 1, 𝜖𝑗 ⊗ 𝜛) = −i ∑︁𝑛 𝑗=1(𝜖𝑗 ⊗ 1, ∇2 𝐸𝜖𝑗) = F[𝐸,∇𝐸];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in fact, this uniqueness implies that F[𝐸,∇𝐸] depends only on [𝐸, ∇𝐸] ∈ DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other, F[𝐸,∇𝐸] is self-adjoint by construction from (𝜖𝑗)𝑛 𝑗=1 and repeated applications of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that F[𝐸,∇𝐸] is central, is closed, and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, by repeated applica- tions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27), it follows that ∇2 𝐸𝜎𝐸(𝛽 ⊗ 𝑥) = 𝜎𝐸(𝛽 ⊗ 𝑥) · iF[𝐸,∇𝐸] for all 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, so that by invertibility of the map 𝜎𝐸, the 2-form F[𝐸,∇𝐸] satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55) together with repeated applications of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24), it follows that for every 𝛽 ∈ Ω𝐵 and 𝑥 ∈ 𝐸, 𝑥 ⊗ F[𝐸,∇𝐸]𝛽 = −i∇2 𝐸(𝑥 ⊗ 𝛽) = 𝑥 ⊗ 𝛽F[𝐸,∇𝐸], so that F[𝐸,∇𝐸] is indeed central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, F[𝐸,∇𝐸] is closed since for every 𝑥 ∈ 𝐸, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55), 𝑥 ⊗ i d𝐵F[𝐸,∇𝐸] = ∇𝐸(𝑥 ⊗ iF[𝐸,∇𝐸]) − ∇𝐸(𝑥) · iF[𝐸,∇𝐸] = ∇𝐸(∇2 𝐸𝑥) − ∇2 𝐸(∇𝐸𝑥) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, by [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9], mutatis mutandis, the map [𝐸, ∇𝐸] ↦→ F[𝐸,∇𝐸] satisfies ∀(𝐸, 𝜎𝐸, ∇𝐸), (𝐹, 𝜎𝐹, ∇𝐹) ∈ Obj(DPic(𝐵)), F[𝐸⊗𝐵𝐹,∇𝐸⊗𝐵𝐹 ] = ˆΦ−1 [𝐹,∇𝐹 ](F[𝐸,∇𝐸]) + F[𝐹,∇𝐹 ], which is precisely the required right 1-cocycle identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Ω2(𝑋, R)cl be the real vector space of closed real 2-forms on 𝑋, which admits a right action of Diff(𝑋) by pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let Ψ : DPic(𝐶∞(𝑋)) → Diff(𝑋) be the homomorphism induced by the Fröhlich homomorphism of 𝐶∞(𝑋) with respect to the de Rham calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other, recall that the ordinary differential cohomology group ˇ𝐻2(𝑋) is the group of isomorphism classes of Hermitian line bundles on 𝑋 with unitary connection [54, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, by Serre–Swan, 1 → ˇ𝐻2(𝑋) [E,∇E]↦→[Γ(E),∇E] −−−−−−−−−−−−−−−→ DPic(𝐶∞(𝑋)) Ψ−→ Diff(𝑋) → 1 defines a split exact sequence with canonical right splitting 𝜙 ↦→ [ˆ𝜏(0, (𝜙−1)∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given the resulting isomorphism Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → DPic(𝐶∞(𝑋)) defined by (𝜙, [E, ∇E]) ↦→ [Γ((𝜙−1)∗E), (𝜙−1)∗∇E][ˆ𝜏(0, (𝜙−1)∗)] we may identify the Fröhlich homomorphism Φ with the quotient map ((𝜙, [E, ∇E]) ↦→ 𝜙) : Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → Diff(𝑋) and the curvature 1-cocycle F : DPic(𝐶∞(𝑋)) → Ω2(𝑋, R)cl with the familiar-looking map �(𝜙, [E, ∇E]) ↦→ 𝜙∗ tr(∇2 E)� : Diff(𝑋) ⋉ ˇ𝐻2(𝑋) → Ω2(𝑋, R)cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 25 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The homomorphism ˆ𝜏 : � Diff(𝐵) → DPic(𝐵) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34 satisfies ∀(𝜔, 𝜙) ∈ � Diff(𝐵), ˆΦ ◦ 𝜋0(ˆ𝜏)(𝜔, 𝜙) = 𝜙↾Z(Ω𝐵), F ◦ 𝜋0(ˆ𝜏)(𝜔, 𝜙) = 𝜙−1�d𝜔 − i𝜔2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41 (Connes [31, Thm 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31, observe that Z(Ω𝜃(T2)) is the complex Graßmann algebra in the self-adjoint generators 𝑒1 and 𝑒2 of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ 𝜃 (T2)) satisfies ∀𝑔 ∈ Γ𝜃, ˆΦ ◦ 𝜋0( ˆ𝐸)(𝑔) = 2 � 𝑘=0 (𝑔21𝜃 + 𝑔22)𝑘 idZ(Ω𝜃)𝑘, F ◦ 𝜋0( ˆ𝐸)(𝑔) = 2𝜋𝑔21 𝑔21𝜃 + 𝑔22 𝑒1𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Reconstruction of noncommutative principal U(1)-bundles with connection In this section, we generalise the familiar correspondence between Hermitian line bundles with unitary connection and principal U(1)-bundles with principal connection to the nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This generalisation takes the form of an explicit equivalence of categories that can be viewed as an adaptation of Pimsner’s construction [81] from the 𝐶∗-algebraic literature to the setting of nc differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, we say that a representation 𝑈 : U(1) → GL(𝑉) is of finite type whenever 𝑉 = �alg 𝑘∈Z 𝑉𝑘, where 𝑉𝑘 � {𝑣 ∈ 𝑉 | ∀𝑧 ∈ U(1), 𝑈𝑧𝑣 = 𝑧𝑘𝑣} for all 𝑘 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Monoidal inversion and homomorphisms of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First we leverage the coherence theorem for coherent 2-groups of Ulbrich [95] and Laplaza [66] to show that every homomorphism of coherent 2-groups canonically defines a bar functor or involutive monoidal functor in the sense of Beggs–Majid and Egger [44], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This will obviate any difficulties related to reconstructing ∗-structures on nc principal U(1)-bundle with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We first recall the additional categorical structure that will fully capture the behaviour of monoidal inversion in a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 (Beggs–Majid [12], Egger [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A strong bar category is a monoidal category G together with a functor · : G → G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' an isomorphism ★ : 1 → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' and natural isomorphisms bb = � bb𝑔 : 𝑔 → 𝑔 � 𝑔∈Obj(G) and Υ = � Υ𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 � (𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ) ∈Obj(G)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' such that bb𝑔 = bb𝑔 for every 𝑔 ∈ Obj(G) and the following coherence diagrams commute for all 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝑘 ∈ Obj(G): 1 1 1 ★ bb1 ★ 𝑔 ⊗ 1 1 ⊗ 𝑔 𝑔 1 ⊗ 𝑔 Υ𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 𝜌−1 𝑔 𝜌𝑔 ★⊗id𝑔 1 ⊗ 𝑔 𝑔 ⊗ 1 𝑔 𝑔 ⊗ 1 Υ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 𝜆−1 𝑔 𝜆𝑔 id𝑔 ⊗★ (𝑔 ⊗ ℎ) ⊗ 𝑘 𝑘 ⊗ 𝑔 ⊗ ℎ 𝑘 ⊗ (ℎ ⊗ 𝑔) 𝑔 ⊗ (ℎ ⊗ 𝑘) ℎ ⊗ 𝑘 ⊗ 𝑔 (𝑘 ⊗ ℎ) ⊗ 𝑔 Υ𝑔⊗ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 id ⊗Υ𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ Υ𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ⊗𝑘 Υℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 ⊗id𝑔 𝛼𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 𝛼𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 𝑔 ⊗ ℎ 𝑔 ⊗ ℎ 𝑔 ⊗ ℎ ℎ ⊗ 𝑔 bb𝑔 ⊗ bbℎ Υ𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ bb𝑔⊗ℎ Υℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' given a unital pre-𝐶∗-algebra 𝐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' the monoidal category Bimod(𝐵) defines a strong bar category with ★ : 𝐵 → 𝐵 and natural isomorphisms bb and Υ defined as follows: ∀𝑏 ∈ 𝐵,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★(𝑏) � 𝑏∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝐸 ∈ Obj(Bimod(𝐵)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝑥 ∈ 𝐸,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' bb𝐸(𝑥) � 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝐸,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝐹 ∈ Obj(Bimod(𝐵)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝑥 ∈ 𝐸,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀𝑦 ∈ 𝐹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Υ𝐸,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑥 ⊗ 𝑦) � 𝑦 ⊗ 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 26 BRANIMIR ĆAĆIĆ Next, let us recall the coherence theorem for coherent 2-groups on which everything will hinge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a structural arrow of a coherent 2-group G to be an element of the smallest subclass Str(G) of Hom(G) that: (1) contains the identity arrows, associators, left unitors, and right unitors of G as a monoidal category and the evaluation arrows of G as a coherent 2-group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) is closed under composition and inversion in G as a category, the monoidal product in G as a monoidal category, and monoidal inversion in G as a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, given endofunctors 𝑃, 𝑄 : G → G of a coherent 2-group G, we say that a natural transformation 𝜂 : 𝑃 ⇒ 𝑄 is structural whenever, for every 𝑔 ∈ Obj(G), the arrow 𝜂𝑔 is structural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, given a coherent 2-group G, the natural isomorphisms coev and bb of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 are both structural [66, Lemm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 (Ulbrich [95], Laplaza [66, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For every pair of objects (𝑔, ℎ) in G, there exists at most one structural arrow 𝑔 → ℎ in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our first application of the coherence theorem is that a coherent 2-group canonically defines a strong bar category with respect to monoidal inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Laplaza [66, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 310]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exist a unique structural isomorphism ★ : 1 → 1 and a unique structural natural isomorphism Υ = � Υ𝑔,ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 � 𝑔,ℎ∈Obj(G) making G into a strong bar category with respect to monoidal inversion and the natural isomorphism bb of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, construct a structural arrow ★ : 1 → 1 by setting ★ � 𝜆1 ◦ coev1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, given objects 𝑔 and ℎ of G, construct a structural arrow Υ𝑔,ℎ : 𝑔 ⊗ ℎ → ℎ ⊗ 𝑔 as follows: first, construct a structural arrow � coev𝑔⊗ℎ : 1 → (𝑔 ⊗ ℎ) ⊗ (ℎ ⊗ 𝑔) by setting � coev𝑔⊗ℎ � 𝛼𝑔⊗ℎ,ℎ,𝑔 ◦ � 𝛼−1 𝑔,ℎ,ℎ ⊗ id𝑔 � �(id𝑔 ⊗ coevℎ) ⊗ id𝑔 � ◦ � 𝜌−1 𝑔 ⊗ id𝑔 � coev𝑔, and then set Υ𝑔,ℎ � 𝜆ℎ⊗𝑔 ◦ (ev𝑔⊗ℎ ⊗ idℎ⊗𝑔) ◦ 𝛼−1 𝑔⊗ℎ,𝑔⊗ℎ,ℎ⊗𝑔 ◦ (id𝑔⊗ℎ ⊗ � coev𝑔⊗ℎ) ◦ 𝜌−1 𝑔⊗ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The claim now follows by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The structural isomorphism ★ : 1 → 1 is the unique isomorphism of the inverses (1, 𝜆1, 𝜆−1 1 ) and (1, ev1, coev1) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Likewise, given 𝑔, ℎ ∈ Obj(G), the structural isomorphism Υ𝑔,ℎ is the unique isomorphism of the inverses (ℎ⊗𝑔, �ev𝑔⊗ℎ, � coev𝑔⊗ℎ) and (𝑔 ⊗ ℎ, ev𝑔⊗ℎ, coev𝑔⊗ℎ) of 𝑔⊗ℎ, where �ev𝑔⊗ℎ : (ℎ⊗𝑔)⊗(𝑔⊗ℎ) → 1 and � coev𝑔⊗ℎ : 1 → (𝑔 ⊗ ℎ) ⊗ (𝑔 ⊗ ℎ) are the unique such structural arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For example, let 𝐵 be a unital pre-𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the canonical strong bar category structure on Pic(𝐵) of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3 is that induced by the aforementioned strong bar category structure on Bimod(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now come to the main definition of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 (Beggs–Majid [12], Egger [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G and G′ be strong bar categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) A bar functor 𝐹 : G → G′ consists of a monoidal functor 𝐹 : G → G′ together with a natural isomorphism 𝐹 (−1) = � 𝐹 (−1) 𝑔 : 𝐹(𝑔) → 𝐹(𝑔) � 𝑔∈Obj(G) making the following diagrams commute for all 𝑔, ℎ ∈ Obj(G): NONCOMMUTATIVE U(1)-GAUGE THEORY 27 𝐹(1) 𝐹(1) 𝐹(1) 1 1 𝐹 (−1) 1 𝐹(★−1) ★−1 𝐹 (0) 𝐹 (0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) 𝐹(𝑔) 𝐹(𝑔) 𝐹(𝑔) 𝐹(𝑔) 𝐹(bb𝑔) 𝐹 (−1) 𝑔 bb𝐹(𝑔) 𝐹 (−1) 𝑔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) 𝐹(𝑔) ⊗ 𝐹(ℎ) 𝐹(ℎ) ⊗ 𝐹(𝑔) 𝐹(ℎ) ⊗ 𝐹(𝑔) 𝐹(𝑔 ⊗ ℎ) 𝐹(𝑔 ⊗ ℎ) 𝐹(ℎ ⊗ 𝑔) Υ𝐹(𝑔),𝐹(ℎ) 𝐹 (−1) 𝑔 ⊗𝐹 (−1) ℎ 𝐹 (−1) 𝑔⊗ℎ 𝐹(Υ𝑔,ℎ) 𝐹 (2) 𝑔,ℎ 𝐹 (2) ℎ,𝑔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) (2) Given bar functors 𝑃, 𝑄 : G → G′, a monoidal natural transformation 𝜙 : 𝑃 ⇒ 𝑄 is a bar natural transformation whenever 𝑄(−1) 𝑔 𝜙𝑔 = 𝜙𝑔 ◦ 𝑃 (−1) 𝑔 for all 𝑔 ∈ Obj(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given a homomorphism of coherent 2-groups 𝐹, the following will supply the missing natural isomorphism 𝐹 (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6 (Baez–Lauda [7, Thm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐹 : G → G′ be a homomorphism of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a unique natural transformation 𝐹 (−1) = � 𝐹 (−1) 𝑔 : 𝐹(𝑔) → 𝐹(𝑔) � 𝑔∈Obj(G), such that the following diagrams commute for every object 𝑔 of G: 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 1 𝐹(1) 𝐹(𝑔 ⊗ 𝑔) 𝐹 (−1) 𝑔 ⊗id𝑔 𝐹 (0) 𝐹(ev𝑔) ev𝑔 𝐹 (2) 𝑔,𝑔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 1 𝐹(1) 𝐹(𝑔 ⊗ 𝑔) id𝑔 ⊗𝐹 (−1) 𝑔 𝐹 (0) 𝐹(coev𝑔) coev𝑔 𝐹 (2) 𝑔,𝑔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) We now prove that the natural isomorphism of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6 makes every homomorphism of coherent 2-groups into a bar functor and every 2-isomorphism of coherent 2-groups into a bar natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G and G′ be coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Let 𝐹 : G → G′ be a monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐹 defines a bar functor with respect to the canonical natural isomorphism 𝐹 (−1) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Let 𝑃, 𝑄 : G → G′ be monoidal functors, so that 𝑃 and 𝑄 uniquely define bar functors satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then every monoidal natural transformation 𝜂 : 𝑃 ⇒ 𝑄 is a bar natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8 (Baez–Lauda [7, Proof of Thm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G be a coherent 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For every object 𝑔 of 𝐺 and every inverse (ℎ, e, i) of 𝑔, there exists a unique isomorphism of the inverses (𝑔, ev𝑔, coev𝑔) and (ℎ, e, i) of 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, for every inverse (ℎ, e, i) of 𝑔 and every arrow 𝑢 : 𝑔 → ℎ, the arrow 𝑢 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) with respect to the inverses (𝑔, ev𝑔, coev𝑔) and (ℎ, e, i) of 𝑔 if and only if 𝑢 is the unique isomorphism of the inverses (𝑔, ev𝑔, coev𝑔) and (ℎ, e, i) of 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let G and G′ be coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐹 : G → G′ be a monoidal functor, and let 𝐹 (−1) be the natural isomorphism of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 and ℎ be objects of G, and let e𝑔,ℎ : (ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ) → 1 and e𝐹(𝑔),𝐹(ℎ) : (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) → 1 be the unique such structural arrows in G and G′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following diagram commutes: (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) 𝐹(ℎ ⊗ 𝑔) ⊗ 𝐹(𝑔 ⊗ ℎ) 1 𝐹(1) 𝐹((ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ)) e𝐹(𝑔),𝐹(ℎ) (𝐹 (−1) ℎ ⊗𝐹 (−1) 𝑔 ) ⊗id 𝐹 (2) ℎ,𝑔 ⊗𝐹 (2) 𝑔,ℎ 𝐹 (2) ℎ⊗𝑔,𝑔⊗ℎ 𝐹(e𝑔,ℎ) 𝐹 (0) 28 BRANIMIR ĆAĆIĆ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔, ℎ ∈ Obj(G) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Observe that we can construct e𝑔,ℎ and e𝐹(𝑔),𝐹(ℎ) as e𝑔⊗ℎ = evℎ ◦ 𝜌ℎ ◦ (id ⊗ ev𝑔) ⊗ id ◦ 𝛼−1 ℎ,𝑔,𝑔, e𝐹(𝑔) ⊗𝐹(ℎ) = ev𝐹(ℎ) ◦ 𝜌𝐹(ℎ) ◦ (id ⊗ ev𝐹(𝑔)) ⊗ id ◦ 𝛼−1 𝐹(ℎ),𝐹(𝑔),𝐹(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, our claim follows from commutativity of the following diagram, where, for visual clarity, we replace the symbol ⊗ with ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 𝐹(𝑔)) · (𝐹(𝑔) · 𝐹(ℎ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 𝐹(𝑔)) · (𝐹(𝑔) · 𝐹(ℎ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ · 𝑔) · (𝐹(𝑔) · 𝐹(ℎ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ · 𝑔) · 𝐹(𝑔 · ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹((ℎ · 𝑔) · (𝑔 · ℎ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='((𝐹(ℎ) · 𝐹(𝑔)) · 𝐹(𝑔)) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='((𝐹(ℎ) · 𝐹(𝑔)) · 𝐹(𝑔)) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ · 𝑔) · 𝐹(𝑔)) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · (𝐹(𝑔) · 𝐹(𝑔))) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · (𝐹(𝑔) · 𝐹(𝑔))) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹((ℎ · 𝑔) · 𝑔) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(((ℎ · 𝑔) · 𝑔) · ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 𝐹(𝑔 · 𝑔)) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ · (𝑔 · 𝑔)) · 𝐹(𝑔) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹((ℎ · (𝑔 · 𝑔)) · ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 𝐹(1)) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ · 1) · 𝐹(𝑔) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹((ℎ · 1) · ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 1) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹(ℎ) · 1) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) · 𝐹(ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ · ℎ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='(𝐹 (−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹 (−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=') ·id ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹 (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ·id id ·𝐹 (2) 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ 𝐹 (2) ℎ·𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔·ℎ (𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ·id)·id (𝐹 (−1) ℎ (𝐹 (−1) 𝑔 id)) ·id 𝐹 (2) (ℎ·𝑔)·𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ 𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔·𝑔 ·id 𝐹 (2) ℎ·(𝑔·𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ·id 𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 ·id 𝐹 (2) ℎ·1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ (𝐹 (−1) ℎ id) ·id 𝐹 (−1) ℎ id 𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ 𝐹 (0) 𝛼−1 𝐹(ℎ)·𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) 𝛼−1 𝐹(ℎ)·𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) 𝛼−1 𝐹(ℎ·𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) 𝐹(𝛼−1 ℎ·𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ) 𝛼𝐹(ℎ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔) ·id 𝛼𝐹(ℎ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(𝑔) ·id 𝐹 (2) ℎ·𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ·id (id ·ev𝐹(𝑔))·id (id ·𝐹 (2) 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 )·id 𝐹(𝛼ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔)·id 𝐹(𝛼ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ·id) (id ·𝐹(ev𝑔)) ·id 𝐹(id · ev𝑔) ·id 𝐹((id · ev𝑔)·id) (id ·𝐹 (0)) ·id 𝐹(𝜌ℎ) ·id 𝐹(𝜌ℎ·id) 𝜌𝐹(ℎ) 𝜌𝐹(ℎ) ·id ev𝐹(ℎ) 𝐹(evℎ) However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' this now follows from applying naturality of 𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' monoidality of 𝐹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' naturality of 𝐹 (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' and commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) as appropriate to each sub-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝐹 : G → G′ be a monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Before continuing, let us recall the construction of the natural isomorphism 𝐹 (−1) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑔 ∈ Obj(G), and ∗�ev𝐹(𝑔) � (𝐹 (0))−1 ◦ 𝐹(ev𝑔) ◦ 𝐹 (2) 𝑔,𝑔 and � coev𝐹(𝑔) � (𝐹 (2) 𝑔,𝑔 )−1 ◦ 𝐹(coev𝑔) ◦ 𝐹 (0), so that (𝐹(𝑔), �ev𝐹(𝑔), � coev𝐹(𝑔)) is an inverse for 𝐹(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8, we may define 𝐹 (−1) as follows: for each 𝑔 ∈ Obj(G), let 𝐹 (−1) 𝑔 : 𝐹(𝑔) → 𝐹(𝑔) be the unique isomorphism of the inverses (𝐹(𝑔), ev𝐹(𝑔), coev𝐹(𝑔)) and (𝐹(𝑔), �ev𝐹(𝑔), � coev𝐹(𝑔)) of 𝐹(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8, it suffices to show that the composite ar- row 𝐹(★)◦(𝐹 (0))−1◦★−1◦𝐹 (0) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) with respect to the inverses (𝐹(1), ev𝐹(1), coev𝐹(1)) and (𝐹(1), �ev𝐹(1), � coev𝐹(1)) of 𝐹(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in turn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' is proved by the commutativity of the fol- lowing diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' which follows by applying bifunctoriality of ⊗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' coherence in G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' coherence in G′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' monoidality of 𝐹,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' or naturality of 𝐹 (2) as appropriate to each sub-diagram: 𝐹(1) ⊗ 𝐹(1) 1 ⊗ 𝐹(1) 1 ⊗ 𝐹(1) 𝐹(1) ⊗ 𝐹(1) 𝐹(1) ⊗ 𝐹(1) 1 ⊗ 1 1 ⊗ 1 𝐹(1 ⊗ 1) 1 𝐹(1) 𝐹(1 ⊗ 1) 𝐹 (0) ⊗id ★−1 ⊗id 𝐹 (0) ⊗id 𝐹(★) ⊗id ★−1 ⊗id 𝐹 (0) 𝐹(ev1) ev𝐹(1) 𝜆𝐹(1) 𝐹 (2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 𝐹 (2) 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 𝐹 (0) ⊗𝐹 (0) id ⊗𝐹 (0) ev1 𝜆1 𝐹(𝜆1) 𝐹(★⊗id) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let us show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 ∈ Obj(G) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8, it suf- fices to show that the arrow 𝐹(bb𝑔) ◦ bb−1 𝐹(𝑔) ◦(𝐹 (−1) 𝑔 )−1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) with respect to the inverses (𝐹(𝑔), ev𝐹(𝑔), coev𝐹(𝑔)) and (𝐹(𝑔), �ev𝐹(𝑔), � coev𝐹(𝑔)) of 𝐹(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This is now proved by the commutativity of the following diagram, which follows by applying bifunctoriality of ⊗, NONCOMMUTATIVE U(1)-GAUGE THEORY 29 coherence in G, coherence in G′, commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5), naturality of ev, and naturality of 𝐹 (2) as appropriate to each sub-diagram: 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔) ⊗ 𝐹(𝑔) 𝐹(𝑔 ⊗ 𝑔) 1 𝐹(1) 𝐹(𝑔 ⊗ 𝑔) (𝐹 (−1) 𝑔 )−1 ⊗id bb−1 𝐹(𝑔) ⊗ id 𝐹(bb𝑔) ⊗id bb𝐹(𝑔) ⊗ id 𝐹 (0) 𝐹(ev𝑔) ev𝐹(𝑔) 𝐹 (−1) 𝑔 ⊗𝐹 (−1) 𝑔 id ⊗𝐹 (−1) 𝑔 𝐹 (2) 𝑔,𝑔 𝐹 (2) 𝑔,𝑔 𝐹(coev𝑔) 𝐹(bb𝑔 ⊗ id) ev𝐹(𝑔) coev𝐹(𝑔) Finally, let us show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔, ℎ ∈ Obj(G) be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' for convenience, let e𝑔,ℎ and e𝐹(𝑔),𝐹(ℎ) be defined as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8, it suffices to show that the arrow 𝐹(Υ−1 𝑔,ℎ) ◦ 𝐹 (2) ℎ,𝑔 ◦ 𝐹 (−1) ℎ ⊗ 𝐹 (−1) 𝑔 Υ𝐹(𝑔),𝐹(ℎ) ◦ (𝐹 (2) 𝑔,ℎ )−1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) with respect to the inverses (𝐹(𝑔 ⊗ ℎ), ev𝐹(𝑔⊗ℎ), coev𝐹(𝑔⊗ℎ)) and (𝐹(𝑔 ⊗ ℎ), �ev𝐹(𝑔⊗ℎ), � coev𝐹(𝑔⊗ℎ)) of 𝐹(𝑔 ⊗ ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This, in turn, is proved by commutativity of the following diagram, which follows by applying coherence in G, coherence in G′, bifunctoriality of ⊗, naturality of ev, naturality of 𝐹 (2) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9 as appropriate to each sub-diagram: 𝐹(𝑔 ⊗ ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) 1 𝐹(𝑔) ⊗ 𝐹(ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) 𝐹(𝑔) ⊗ 𝐹(ℎ) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ 𝐹(𝑔 ⊗ ℎ) (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ 𝐹(𝑔 ⊗ ℎ) (𝐹(ℎ) ⊗ 𝐹(𝑔)) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) 𝐹(ℎ ⊗ 𝑔) ⊗ 𝐹(𝑔 ⊗ ℎ) 𝐹(ℎ ⊗ 𝑔) ⊗ (𝐹(𝑔) ⊗ 𝐹(ℎ)) 𝐹((ℎ ⊗ 𝑔) ⊗ (𝑔 ⊗ ℎ)) 𝐹(1) 𝐹(𝑔 ⊗ ℎ) ⊗ 𝐹(𝑔 ⊗ ℎ) 𝐹(𝑔 ⊗ ℎ ⊗ (𝑔 ⊗ ℎ)) ev𝐹(𝑔⊗ℎ) id ⊗𝐹 (2) 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ ev𝐹(𝑔)⊗𝐹(ℎ) ⊗ id e𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) id ⊗𝐹 (2) 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ 𝐹(e𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ) 𝐹 (2) 𝑔⊗ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔⊗ℎ (𝐹 (2) 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ )−1 ⊗id 𝐹 (0) Υ𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) ⊗id Υ𝐹(𝑔),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝐹(ℎ) ⊗id (𝐹 (−1) ℎ ⊗𝐹 (−1) 𝑔 ) ⊗id (𝐹 (−1) ℎ ⊗𝐹 (−1) 𝑔 ) ⊗id 𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ⊗id 𝐹 (2) ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔 ⊗id 𝐹(Υ−1 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ ) ⊗id 𝐹(ev𝑔⊗ℎ) 𝐹(Υ−1 𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ℎ ⊗id) 𝐹 (2) ℎ⊗𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑔⊗ℎ Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝑃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝑄 : G → G′ be monoidal functors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' and let 𝜂 : 𝑃 ⇒ 𝑄 be a monoidal natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 (−1) and 𝑄(−1) be constructed as above, so that 𝑃 and 𝑄 define bar func- tors satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 ∈ Obj(G) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' To show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) commutes, it suffices to show that 𝜙−1 𝑔 ◦𝑄(−1) 𝑔 𝜙𝑔 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) with respect to the inverses (𝑃(𝑔), ev𝑃(𝑔), coev𝑃(𝑔)) and (𝑃(𝑔), �ev𝑃(𝑔), � coev𝑃(𝑔)) of 𝑃(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In turn, it suffices to show that the following diagram commutes: 𝑃(𝑔) ⊗ 𝑃(𝑔) 𝑄(𝑔) ⊗ 𝑄(𝑔) 𝑄(𝑔) ⊗ 𝑄(𝑔) 𝑃(𝑔) ⊗ 𝑃(𝑔) 𝑄(1) 𝑄(𝑔 ⊗ 𝑔) 1 𝑃(1) 𝑃(𝑔 ⊗ 𝑔) 𝜙𝑔 ⊗𝜙𝑔 𝑄(−1) 𝑔 ⊗id 𝜙𝑔 ⊗𝜙𝑔 𝑄(ev𝑔) 𝑃(ev𝑔) 𝑃 (0) ev𝑃(𝑔) ev𝑄(𝑔) 𝑄(2) 𝑔,𝑔 𝑃 (2) 𝑔,𝑔 𝑄(0) 𝜙1 𝜙𝑔⊗𝑔 However, this diagram commutes by applying naturality of ev, commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) for 𝑄, naturality of 𝜙, and monoidality of 𝜙 as appropriate to each sub-diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10 (Buss–Meyer–Zhu [23, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐵 be a unital pre-𝐶∗-algebra, let Γ be a group, and let 𝐹 : Γ → Pic(𝐵) be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The disjoint union F � � 𝛾∈Γ 𝐹(𝛾) 30 BRANIMIR ĆAĆIĆ defines a pre-Fell bundle over Γ in the sense of Exel [47, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2] with respect to the fibrewise multiplication F× F→ Fand the fibrewise ∗-operation F→ Fdefined, respectively, by ∀𝛾, 𝜂 ∈ Γ, ∀𝑝 ∈ 𝐹(𝛾), ∀𝑞 ∈ 𝐹(𝜂), 𝑝𝑞 � 𝐹 (2) 𝛾,𝜂 (𝛾 ⊗ 𝜂), ∀𝛾 ∈ Γ, ∀𝑝 ∈ 𝐹(𝛾), 𝑝∗ � 𝐹 (−1) 𝛾 (𝑝), where 𝐹 (−1) is the natural transformation of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 as applied to Hom(Γ, Pic(𝐵)) recovers Buss–Meyer–Zhu’s construction [23, Proof of Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3] of the fibrewise ∗-operation on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Generalised crossed products via homomorphisms of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this section, we revisit the well-known theory of nc topological principal U(1)-bundles [1, 9, 4] from the perspective of coherent 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This will permit us to generalise Abadie–Eilers– Exel’s framework [1] of generalized crossed products via Pimsner’s construction [81] to the setting of nc differential geometry by replacing the Picard 2-group with the differentiable Picard 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, let 𝐵 be a unital pre-𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us define a U(1)-pre-𝐶∗-algebra of finite type is a unital pre-𝐶∗-algebra 𝑃 equipped with a strongly continuous U(1)-action of finite type by isometric ∗-automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, the spectral subspace 𝑃U(1) = 𝑃0 is a unital ∗-subalgebra of 𝑃, and the decomposition of complex vector spaces 𝑃 = ⊕𝑘∈Z𝑃𝑘 defines a Z-grading of the unital ∗-algebra 𝑃 in the sense that 𝑃𝑚 · 𝑃𝑛 ⊆ 𝑃𝑚+𝑛 for all 𝑚, 𝑛 ∈ Z and ∗(𝑃𝑚) ⊆ 𝑃−𝑚 for all 𝑚 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This permits the following minimalistic definition of topological quantum principal U(1)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Arici–Kaad–Landi [4, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A topological quantum principal U(1)-bundle is a pre-𝐶∗-algebra (𝑃, 𝛼) of finite type, such that there exist finite families (𝑒𝑖)𝑚 𝑖=1 and (𝜖𝑗)𝑛 𝑗=1 in 𝑃1 satisfying �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 and �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This definition is slightly unconventional but may be related to more familiar definitions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝑃, 𝛼) is a topological quantum principal U(1)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by an observation of Năstăsescu–Van Ostaeyen [78, Lemma i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2], the U(1)-action 𝛼 is principal in the sense that SpanC{𝑧𝑘 ⊗ 𝑝𝑞 | 𝑘 ∈ Z, 𝑝 ∈ 𝑃𝑘, 𝑞 ∈ 𝑃} = O(U(1)) ⊗C 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by an observation of Ulbrich [96, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1], it follows that the Z-grading 𝑃 = � 𝑘∈Z 𝑃𝑘 of 𝑃 is strong in the sense that 𝑃𝑚 · 𝑃𝑛 = 𝑃𝑚+𝑛 for all 𝑚, 𝑛 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The familiar fact that 𝛼 is principal if and only if the Z-grading of 𝑃 is strong [78, Lemma i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2] yields the familiar algebraic definition of (topological) quantum principal U(1)-bundle in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜋 : 𝑋 → 𝑌 be a compact differentiable principal U(1)-bundle with principal right U(1)-action 𝜎 : U(1) → Diff(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝐶∞ alg(𝑋) � alg � 𝑘∈Z {𝜔 ∈ 𝐶∞(𝑋) | ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧𝑘𝜔}, which is norm-dense in 𝐶(𝑋) since it is Fréchet-dense in 𝐶∞(𝑋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐶∞ alg(𝑋) defines a topological quantum principal U(1)-bundle with respect to 𝛼 � (𝑧 ↦→ (𝜎𝑧−1)∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, the pullback homomorphism 𝜋∗ : 𝐶∞(𝑌) → 𝐶∞(𝑋)U(1) is an isometric ∗-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, one can use an atlas of local principal U(1)-bundle trivialisations for 𝜋 : 𝑋 → 𝑌 together with a subordinate smooth partition of unity on 𝑌 to construct a finite family (𝑒𝑖)𝑚 𝑖=1 in 𝐶∞ alg(𝑋)1 satisfying �𝑛 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = �𝑛 𝑖=1 𝑒∗ 𝑖 𝑒𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following introduces our second main running example, the first genuinely nc example of a topological quantum principal U(1)-bundle in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 31 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13 (Brzeziński–Majid [22, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑞 ∈ (0, ∞) \\ {1}, so that the corresponding quantum special unitary group á la Woronowicz [98] is the universal 𝐶∗-algebra 𝐶𝑞(SU(2)) generated by elements 𝑎 and 𝑐 satisfying 𝑎𝑐 = 𝑞𝑐𝑎, 𝑎𝑐∗ = 𝑞𝑐∗𝑎, 𝑐∗𝑐 = 𝑐𝑐∗, 𝑎∗𝑎 + 𝑐∗𝑐 = 1, 𝑎𝑎∗ + 𝑞2𝑐𝑐∗ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' the corresponding unital pre-𝐶∗-algebra O𝑞(SU2) is the dense unital ∗-subalgebra of 𝐶𝑞(SU2) consisting of complex polynomials in 𝑎, 𝑎∗, 𝑐, and 𝑐∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then O𝑞(SU(2)) defines a topological quantum principal U(1)-bundle with respect to the unique U(1)-action of finite type 𝛼 satisfying 𝛼𝑧(𝑎) = 𝑧𝑎 and 𝛼𝑧(𝑐) = 𝑧𝑐 for all 𝑧 ∈ U(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, the families (𝑎, 𝑞𝑐) and (𝑎, 𝑐) in O𝑞(SU(2))1 respectively satisfy 𝑎𝑎∗ + (𝑞𝑐)(𝑞𝑐)∗ = 1 and 𝑎∗𝑎 + 𝑐∗𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, the U(1)-action 𝛼 satisfies O𝑞(SU(2))U(1) = O𝑞(CP1), where O𝑞(CP1), the algebraic standard Podleś sphere [82], is the unital ∗-subalgebra of O𝑞(SU2) consisting of complex polynomials in the elements 𝑐∗𝑐, 𝑎𝑐∗, and 𝑐𝑎∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We note that our rather strict definition of topological quantum principal U(1)-bundle reduces to a simpler definition whenever the ∗-subalgebra of U(1)-invariant elements is sufficiently like a 𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a unital pre-𝐶∗-algebra with U(1)-action of finite type 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that the fixed-point subalgebra 𝑃U(1) admits polar decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝑃, 𝛼) is a topological quantum principal U(1)-bundle if and only if 𝑃1 · 𝑃−1 = 𝑃U(1) and 𝑃−1 · 𝑃1 = 𝑃U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For each 𝑘 ∈ Z, the spectral subspace 𝑃𝑘 defines a 𝑃U(1)-bimodule with positive definite 𝑃U(1)-valued inner product (·, ·)𝑘 � ((𝑝, 𝑞) ↦→ 𝑝∗𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, for each 𝑘 ∈ Z, the 𝑃U(1)- valued inner products (·, ·)𝑘 and (·, ·)−𝑘 satisfy 𝑝 · (𝑞, 𝑟)𝑘 = (𝑝∗, 𝑞∗)−𝑘 · 𝑟 for all elements 𝑝, 𝑞, 𝑟 ∈ 𝑃𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we may apply the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23, mutatis mutandis, to 𝑃1 and 𝑃−1, where 𝑃−1 admits the isomorphism of 𝐵-bimodules (𝑝 ↦→ 𝑝∗) : 𝑃−1 → 𝑃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Recall that 𝐵 is a given unital pre-𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us define the concrete category Circ(𝐵) of topological quantum principal U(1)-bundles over 𝐵 and their isomorphisms as follows: (1) an object of Circ(𝐵) is a topological quantum principal U(1)-bundle together with an isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) an arrow 𝑓 : 𝑃 → 𝑄 in Circ(𝐵) is a U(1)-equivariant isometric ∗-isomorphism, such that 𝑓 ◦ 𝜄𝑃 = 𝜄𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' One can now make precise sense of associated line bundles in the nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15 (Exel [46, §2], Schwieger–Wagner [93, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a functor L : Circ(𝐵) → Hom(Z, Pic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a homomorphism L(𝑃) : Z → Pic(𝐵) as follows: (a) given 𝑘 ∈ Z, let L(𝑃)(𝑘) � 𝑃𝑘 as a complex vector space with the 𝐵-bimodule structure ∀𝑎, 𝑏 ∈ 𝐵, ∀𝑝 ∈ 𝑃𝑘, 𝑎𝑝𝑏 � 𝜄𝑃(𝑎)𝑝𝜄𝑃(𝑏) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) and the 𝐵-valued inner products on 𝑃𝑘 and 𝑃𝑘 defined, respectively, by ∀𝑝, 𝑞 ∈ 𝑃𝑘, (𝑝, 𝑞) � 𝜄−1 𝑃 (𝑝∗𝑞), (𝑝, 𝑞) � 𝜄−1 𝑃 (𝑝𝑞∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7) (b) set L(𝑃)(0) � 𝜄−1 𝑃 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (c) given 𝑚, 𝑛 ∈ Z, let L(𝑃)(2) 𝑚,𝑛 : L(𝑃)(𝑚) ⊗ L(𝑃)(𝑛) → L(𝑃)(𝑚 + 𝑛) be induced by multiplication in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 32 BRANIMIR ĆAĆIĆ (2) Let 𝑓 : 𝑃 → 𝑄 be an isomorphism of topological quantum principal U(1)-bundles over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define the corresponding 2-isomorphism L(𝑓) : L(𝑃) ⇒ L(𝑄) by ∀𝑘 ∈ Z, L(𝑓)𝑘 � 𝑓↾𝑃𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This is mostly a straightforward exercise in checking definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 ∈ Obj(Pic(𝐵)) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' When checking that the functor L(𝑃) : Z → Pic(𝐵) is well defined, the only non- trivial point is strict fullness of all 𝐵-valued inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, let (𝑒𝑖)𝑚 𝑖=1 and (𝜖𝑗)𝑛 𝑗=1 be families in 𝑃1 respectively satisfying �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 and �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1, and define 𝑒𝐼 � 𝑒𝑖1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝑒𝑖𝑘 for all 𝑘 ∈ N and 𝐼 = (𝑖1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑖𝑘) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑚}𝑘 and 𝜖𝐽 � 𝜖𝑗1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝜖𝑗𝑘 for all 𝑘 ∈ N and 𝐽 = (𝑗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑗𝑘) ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑛}𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, for each 𝑘 ∈ N, it follows that (𝑒∗ 𝐼)𝐼 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=',𝑚}𝑘 is a cobasis for L(𝑃)(−𝑘), that (𝜖∗ 𝐽)𝐽∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=',𝑛}𝑘 is a cobasis for L(𝑃)(−𝑘), that (𝜖𝐽)𝐽∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=',𝑛}𝑘 is a cobasis for L(𝑃)(𝑘), and that (𝑒𝐼)𝐼 ∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=',𝑚}𝑘 is a cobasis for L(𝑃)(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' From here, monoidality of L(𝑃) follows from elementary algebraic properties of 𝑃: coherence with respect to unitors follows from multiplicativity of the isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1), while coherence with respect to associators follows from associativity of multiplication in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Similarly, if 𝑓 : 𝑃 → 𝑄 is an arrow in Pic(𝐵), then L(𝑓) intertwines L(𝑃)(0) and L(𝑄)(0) since 𝑓 intertwines the given isometric ∗-isomorphisms 𝜄𝑃 : 𝐵 → 𝑃U(1) and 𝜄𝑄 : 𝐵 → 𝑄U(1), while coherence of L(𝑓) with respect to L(𝑃)(2) and L(𝑄)(2) follows from multiplicativity of 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ For example, let 𝜋 : 𝑋 → 𝑌 be a compact differentiable principal U(1)-bundle with principal U(1)-action 𝜎 : U(1) → Diff(𝑋), so that 𝐶∞ alg(𝑋) defines an object of Circ(𝐶∞(𝑌)) with respect to 𝜋∗ : 𝐶∞(𝑌) → 𝐶∞ alg(𝑋)U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Serre–Swan duality for smooth Hermitian vector bundles on 𝑌, for each 𝑘 ∈ Z, the Hermitian line 𝐵-bimodule L𝑘(𝐶∞ alg(𝑋)) recovers the associated Hermitian line bundle to 𝑌 of winding number −𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Likewise, in the setting of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13, the homomorphism E : Z → Pic(O𝑞(CP1)) given by E � L(O𝑞(SU(2))) recovers (up to a sign convention) the canonical line bundles on O𝑞(CP1) as studied by Landi–Reina–Zampini [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In fact, it follows from a result of Carotenuto–Ó Buachalla [29, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4] that the homomorphism Eexhausts the left O𝑞(SU(2))- covariant Hermitian line O𝑞(CP1)-bimodules up to O𝑞(SU(2))-covariant isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now recover the known result that the functor Lextracting associated line bundles is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' As a preliminary, recall that a conditional expectation of a unital pre-𝐶∗-algebra 𝐴2 onto a unital pre-𝐶∗-algebra 𝐴1 with respect to an isometric ∗- homomorphism 𝜄 : 𝐴1 → 𝐴2 is a contractive unit-preserving and ∗-preserving 𝐴1-bimodule map E : 𝐴2 → 𝐴1, such that E((𝐴2)+) ⊆ (𝐴1)+ and E ◦ 𝜄 = id𝐴1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, we say that E is faithful whenever it satisfies {𝑎 ∈ (𝐴2)+ | E(𝑎) = 0} = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a complex-linear map E𝑃 : 𝑃 → 𝐵 by setting E𝑃↾𝑃𝑗� �𝑝 ↦→ 𝜄−1 𝑃 �𝛿 𝑗,0𝑝�� for all 𝑗 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then E is a U(1)-invariant faithful conditional expectation of 𝑃 onto 𝐵 with respect to 𝜄𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜎 denote the U(1)-action on 𝑃, and let 𝑚 denote the normalised Haar measure on U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that E𝑃 is manifestly U(1)-invariant, unit-preserving, ∗-preserving, and 𝐵-bilinear and that it satisfies E𝑃 ◦ 𝜄𝑃 = id𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜎 is of finite type, we may use Bochner integration on U(1) to write E𝑃 = � 𝑝 ↦→ 𝜄𝑃 �∫ U(1) 𝜎𝑧(𝑝) d𝑚(𝑧) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜎 acts isometrically on 𝑃, it follows that E is contractive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since 𝜎 acts by unital ∗-automorphisms and by our convention for positive cones, it follows that the E𝑃 maps positive elements to positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 33 Let us now show that E is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 Let 𝑝 ∈ 𝑃+ \\ {0}, so that there exists a bounded state 𝜙 : 𝑃 → C, such that 𝜙(𝑝) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (𝑧 ↦→ 𝜙(𝜎𝑧(𝑝))) : U(1) → [0, ∞) is continuous, there exists an open neighbourhood 𝐼 of 1, such that 𝜙(𝜎𝑧(𝑝)) > 1 2𝜙(𝑝) for all 𝑧 ∈ 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by norm-continuity of E𝑃, (𝜙 ◦ 𝜄𝑃)(E𝑃(𝑝)) = ∫ U(1) (𝜙 ◦ 𝜎𝑧)(𝑝) d𝑚(𝑧) ≥ 1 2𝜙(𝑝)𝑚(𝐼) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 (Buss–Meyer–Zhu [19, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3], Schwieger–Wagner [93, Thmm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a a weak inverse Σ : Hom(Z, Pic(𝐵)) → Circ(𝐵) of the functor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given a homomorphism 𝐹 : Z → Pic(𝐵), construct a topological quantum principal U(1)- bundle Σ(𝐹) over 𝐵 as follows: (a) define the unital ∗-algebra Σ(𝐹) by equipping the complex vector space � 𝑘∈Z 𝐹(𝑘) with the multiplication and ∗-operation defined, respectively, by ∀𝑚, 𝑛 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), ∀𝑞 ∈ 𝐹(𝑛), 𝑝𝑞 � 𝐹 (2) 𝑚,𝑛(𝑝 ⊗ 𝑞), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9) ∀𝑚 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), 𝑝∗ � 𝐹 (−1) 𝑚 (𝑝);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10) (b) equip Σ(𝐹) with the unique 𝐶∗-norm ∥ · ∥Σ(𝐹), such that ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), ∥𝑝∥2 Σ(𝐹) = ∥(𝑝, 𝑝)∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11) (c) define a U(1)-action of finite type 𝛼 on Σ(𝐹) by ∀𝑧 ∈ U(1), ∀𝑚 ∈ Z, ∀𝑝 ∈ 𝐹(𝑚), 𝛼𝑧(𝑝) � 𝑧𝑚𝑝;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12) (d) set 𝜄Σ(𝐹) � (𝐹 (0))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given a 2-isomorphism 𝜂 : 𝑅 → 𝑆, construct an isomorphism Σ(𝜂) : Σ(𝑅) → Σ(𝑆) of topological quantum principal U(1)-bundles over 𝐵 by ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑅(𝑘), Σ(𝜂)𝑘(𝑝) � 𝜂𝑘(𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13) Hence, in particular, the category Circ(𝐵) is essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We supply a proof that we can (and shall) adapt to other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that Σ is well-defined on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐹 : Z → Pic(𝐵) be a given homomorphism, which is a bar functor by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This now implies that Σ(𝐹) is a unital ∗-algebra and that 𝜄Σ(𝐹) is a ∗-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, coherence of 𝐹 with respect to associators implies associativity of Σ(𝐹), while coherence of 𝐹 with respect to unitors implies that Σ(𝐹) is unital and that 𝜄Σ(𝐹) is a unital homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' From there, commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) imply that the ∗-operation is antimultiplicative, involutive, and unital, respectively, while commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) also implies that 𝜄Σ(𝐹) is a ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, recall from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10 that 𝐹 canonically defines a a pre-Fell bundle Fover Z in the sense of Exel [47, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it follows immediately that Σ(𝐹) is precisely the ∗-algebra of compactly supported cross-sections of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, by [47, Propp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (iv) & 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8], the 𝐶∗-norm on the reduced cross-sectional 𝐶∗-algebra [47, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6] of the Fell bundle completion [47, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7] of Fyields the unique 𝐶∗-norm on Σ(𝐹) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since 𝐹 (0) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11), this implies that 𝜄Σ(𝐹) is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, by [85, Thm 3], it follows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12) correctly defines a U(1)- action of finite type on the unital pre-𝐶∗-algebra Σ(𝐹);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' that (Σ(𝐹);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝛼) defines a topological quantum principal U(1)-bundle over 𝐵 now follows from the existence of strict cobases for 𝐹(1) and 𝐹(1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let us show that Σ is well-defined on arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜂 : 𝑅 → 𝑆 be a 2-isomorphism between homomorphisms 𝑅, 𝑆 : Z → Pic(𝐵), so that 𝜂 is automatically a bar natural transfor- mation by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This now implies that the U(1)-equivariant vector space isomorphism 1This elementary argument, which is surely folkloric, was found in an anonymous answer to a MathOverflow question (https://mathoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='net/q/72624).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 34 BRANIMIR ĆAĆIĆ Σ(𝜂) : Σ(𝑅) → Σ(𝑆) is a unital ∗-isomorphism intertwining 𝜄Σ(𝑅) and 𝜄Σ(𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, coherence of 𝜂 with respect to 𝑅(2) and 𝑆(2) implies that Σ(𝜂) is multiplicative, that 𝜂1 intertwines 𝑅(0) and 𝑆(0) implies that Σ(𝜂) is unital and intertwines 𝜄Σ(𝑅) and 𝜄Σ(𝑆), and the fact that 𝜂 is a bar natural transformation implies that Σ(𝜂) is ∗-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since, for each 𝑘 ∈ Z, the arrows 𝜂𝑘 and 𝜂−1 𝑘 both satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11), it follows that the bar natural transformation 𝜂 induces a isomor- phism of the Fell bundle completions of the pre-Fell bundles induced by 𝑅 and 𝑆 respectively, so that Σ(𝜂) is isometric by [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, functoriality of Σ is easily checked, so it remains to construct natural isomorphisms 𝜇 : idCirc(𝐵) ⇒ Σ ◦ L and 𝜈 : idHom(Z,𝑃𝑖𝑐(𝐵)) ⇒ L ◦ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the Z-grading 𝑃 = � 𝑘∈Z 𝑃𝑘 is strong, the spectral subspaces of 𝑃 define a pre-Fell bundle over Z fibrewise-isometrically isomorphic (mutatis mutandis) over 𝜄−1 𝑃 to the pre-Fell bundle over Z induced by L(𝑃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note, moreover, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, that this Z-grading is topological in the sense of Exel [46, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2] and that averaging over the U(1)-action yields a faithful conditional expectation of the 𝐶∗-completion of 𝑃 onto the 𝐶∗-completion of 𝐵, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' [3, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by [47, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3], there exists unique U(1)-equivariant isometric ∗-isomorphism 𝜇𝑃 : 𝑃 → Σ ◦ L(𝑃) that satisfies 𝜇𝑃 ◦ 𝜄𝑃 = 𝜄Σ◦L(𝑃), namely, set 𝜇𝑃 ↾𝑃𝑘� id for 𝑘 ∈ Z \\ {0} and 𝜇𝑃 ↾𝑃0� 𝜄−1 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Naturality of 𝜇 � (𝜇𝑃 : 𝑃 → Σ ◦ L(𝑃))𝑃∈Obj(Circ(𝐵)) now follows by uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, given monoidal 𝐹 : Z → Pic(𝐵), define 𝜈𝐹 : 𝐹 ⇒ L ◦ Σ(𝐹) as follows: for each 𝑘 ∈ Z, let (𝜈𝐹)𝑘 : 𝐹(𝑘) → (L◦ Σ(𝐹))(𝑘) be the inclusion of 𝐹(𝑘) in Σ(𝐹) as a direct sum of 𝐵- bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Naturality of 𝜈 � (𝜈𝐹 : 𝐹 ⇒ L◦ Σ(𝐹))𝐹∈Obj(Hom(Z,Pic(𝐵))) follows from the fact that direct sums in Bimod(𝐵) are coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑇 be a compact Abelian group with Pontrjagin dual ˆ𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' suppose that 𝐵 admits polar decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The results above generalise to yield an equivalence of categories between Hom( ˆ𝑇, Pic(𝐵)) and an analogous category of topological quantum principal 𝑇- bundles over 𝐵, thereby recovering the relevant classification results of Schwieger–Wagner [93] in a manner that is adaptable to nc differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The construction of the natural isomorphism 𝜇 : idCirc(𝐵) ⇒ Σ◦Lin the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 implies the following useful characterisation of relevant 𝐶∗-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19 (Arici–Kaad–Landi [4, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a topological quantum principal U(1)-bundle on 𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let ∥ · ∥ denote its 𝐶∗-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ∥ · ∥′ be a U(1)-invariant 𝐶∗-norm on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ∥ · ∥′ = ∥ · ∥ if and only if ∥ · ∥′↾𝑃U(1) = ∥ · ∥↾𝑃U(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 with Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9 recovers Arici–Kaad–Landi’s characterisation of topological quantum principal U(1)-bundles in terms of Pimsner’s construction [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20 (Arici–Kaad–Landi [4, §3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Abadie–Eilers–Exel [1, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1], Beggs–Brzez- iński [9, Thm 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor 𝜖1 ◦ L : Circ(𝐵) → Pic(𝐵) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21 (Abadie–Eilers–Exel [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐸 be a Hermitian line 𝐵-bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The crossed product of 𝐵 by 𝐸 is the essentially unique topological quantum principal U(1)-bundle 𝐵 ⋊𝐸 Z over 𝐵, such that L(𝐵 ⋊𝐸 Z)(1) � 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' One may justify this terminology as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜙 ∈ Aut(𝐵), so that its algebraic crossed product 𝐵 ⋊alg 𝜙 Z is the unital ∗-algebra obtained from 𝐵 by adjoining a unitary 𝑈 satisfying 𝑈𝑏𝑈∗ = 𝜙(𝑏) for all 𝑏 ∈ 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝐵 ⋊alg 𝜙 Z defines topological quantum U(1)-bundle over 𝐵 when equipped with the reduced crossed product 𝐶∗-norm and the unique U(1)-action of finite type 𝛼, such that 𝛼𝑧↾𝐵= id and 𝛼𝑧(𝑈) = 𝑧𝑈 for all 𝑧 ∈ U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since �𝑏𝜙 ↦→ 𝑈𝜙−1(𝑏)� : 𝐵𝜙 → L(𝐵 ⋊𝜙 Z)(1) NONCOMMUTATIVE U(1)-GAUGE THEORY 35 is an isomorphism of Hermitian line 𝐵-bimodules, we may therefore take 𝐵⋊𝜏(𝜙) Z � 𝐵⋊alg 𝜙 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Horizontal calculi as generalised crossed products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' As promised, we now adapt the considerations of the last subsection to the setting of nc differential geometry by replacing the Picard 2-group with the differential Picard 2-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, in the absence of additional constraints, we can only reconstruct the horizontal calculus of a quantum principal U(1)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, let 𝐵 be a given unital pre-𝐶∗-algebra with ∗-exterior algebra (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a U(1)-pre-𝐶∗-algebra of finite type with U(1)-action 𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define a U(1)-∗- quasi-dga of finite type over 𝑃 to be a ∗-quasi-dga (Ω, d) over 𝑃 together with a pointwise extension of 𝛼 to a group homomorphism ˆ𝛼 : U(1) → Aut(Ω, d), such that, for each 𝑘 ∈ N0, the restriction of ˆ𝛼 to a 𝑈-action on the complex vector space Ω𝑘 is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, we call (Ω, d) a U(1)-∗-exterior algebra of finite type over 𝑃 whenever the underlying ∗-quasi-dga is a ∗-exterior algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we denote by CDGAU(1) the concrete category whose objects (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d) consist of a U(1)-pre-𝐶∗-algebra of finite type 𝑃 together with a U(1)- ∗-quasi-dga of finite type (Ω, d) over 𝑃 and whose arrows 𝑓 : (𝑃1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω1, d1) → (𝑃2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω2, d2) are U(1)-equivariant morphisms of ∗-quasi-dga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following definition characterises the differentiable structure that a Hermitian line 𝐵-bimodule with connection can generally induce on the corresponding topological quantum principial U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22 (Ðurđević [38, §2], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ćaćić [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵 with isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A horizontal calculus for 𝑃 is a U(1)-∗-quasi-dga (Ω𝑃,hor, d𝑃,hor) of finite type over 𝑃 together with an isomorphism of quasi-∗-dga ˆ𝜄𝑃 : (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) → (𝑃U(1), (Ω𝑃,hor)U(1), d𝑃,hor↾(Ω𝑃,hor)U(1) ) extending the isometric ∗-isomorphism 𝜄𝑃 : 𝐵 → 𝑃U(1), such that Ω𝑃,hor = 𝑃 · (Ω𝑃,hor)U(1) · 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23 (Majid [67, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Ω𝑞,hor(SU(2)) be the graded ∗-algebra over O𝑞(SU(2)) generated by 𝑒+ ∈ Ω1 𝑞,hor(SU(2)) and 𝑒− � −(𝑒+)∗ satisfying 𝑒±𝑎 = 𝑞−1𝑎𝑒±, 𝑒±𝑎∗ = 𝑞𝑎∗𝑒±, 𝑒±𝑐 = 𝑞−1𝑐𝑒±, 𝑒±𝑐∗ = 𝑞𝑐∗𝑒±, (𝑒±)2 = 0, 𝑒−𝑒+ + 𝑞−2𝑒+𝑒− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define complex-linear maps 𝜕± : O𝑞(SU(2)) → O𝑞(SU(2) by 𝜕+(𝑎) � −𝑞𝑐∗, 𝜕+(𝑎∗) � 0, 𝜕+(𝑐) � 𝑎∗, 𝜕+(𝑐∗) � 0, 𝜕−(𝑎) � 0, 𝜕−(𝑎∗) � 𝑐, 𝜕−(𝑐) � 0, 𝜕−(𝑐∗) � −𝑞−1𝑎, together with the twisted Leibniz rule ∀𝑥 ∈ O𝑞(SU(2)), ∀𝑗 ∈ Z, ∀𝑦 ∈ O𝑞(SU(2))𝑗, 𝜕±(𝑥𝑦) = 𝜕±𝑥𝑦𝑞−𝑗 + 𝑥𝜕±(𝑦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, define d𝑞,hor : O𝑞(SU(2)) → Ω1 𝑞,hor(SU(2)) by setting ∀𝑝 ∈ O𝑞(SU(2)), d𝑞,hor(𝑝) � 𝜕+(𝑝)𝑒+ + 𝜕−(𝑝)𝑒−, and extend d𝑞,hor to Ω𝑞,hor(SU(2)) by setting d𝑞,hor(𝑒±) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, extend the U(1)- action from O𝑞(SU(2)) to Ω𝑞,hor(SU(2)) by setting 𝛼𝑧(𝑒±) = 𝑧±2𝑒± for all 𝑧 ∈ U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (Ω𝑞,hor(SU(2)), d𝑃,hor) defines a horizontal calculus for the topological quantum principal U(1)-bundle O𝑞(SU(2)) over O𝑞(CP1) with respect to the ∗-exterior algebra (Ω𝑞(CP1), d) � � Ω𝑞,hor(SU(2))U(1), d𝑞,hor↾Ω𝑞,hor(SU(2))U(1) � on O𝑞(CP1), which, by Majid’s result, recovers the 2-dimensional calculus on O𝑞(CP1) first constructed by Podleś [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 36 BRANIMIR ĆAĆIĆ Let us now define the concrete category DCirchor(𝐵) of horizontally differentiable quantum principal U(1)-bundles over 𝐵 and their isomorphisms as follows: (1) an object (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) consists of a topological quantum principal U(1)-bundle 𝑃 over 𝐵 together with a horizontal calculus (Ω𝑃,hor, d𝑃,hor) on 𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) an arrow 𝑓 : (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) → (𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑄,hor, d𝑄,hor) is an isomorphism of U(1)-∗-quasi- dga, such that ˆ𝜄𝑄 ◦ 𝑓 = 𝑓 ◦ ˆ𝜄𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It is useful to observe that the forgetful functor DCirchor(𝐵) → Circ(𝐵) is faithful: an arrow 𝑓 : (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) → (𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑄,hor, d𝑄,hor) in DCirc(𝐵) is uniquely determined by the corresponding arrow 𝑓 ↾𝑃: 𝑃 → 𝑄 in Circ(𝐵) precisely because Ω𝑃,hor is generated as an algebra by 𝑃 and ˆ𝜄𝑃(d(𝐵)) ⊂ d𝑃,hor(𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We can now make precise sense of associated line bundles with connection in the nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ćaćić–Mesland [26, Appx B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Observe that Ω𝑃,hor defines a 𝐵-bimodule with respect to 𝜄𝑃 : 𝐵 → 𝑃U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a unique U(1)-equivariant isomorphism of 𝐵-bimodules ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵, such that ∀𝑝 ∈ 𝑃, ∀𝛽 ∈ Ω𝐵, ˆℓ−1 𝑃 (𝑝 ⊗ 𝛽) = 𝑝ˆ𝜄𝑃(𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14) (2) Let 𝑘 ∈ Z be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define functions 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 : Ω𝐵 ⊗𝐵 L(𝑃)(𝑘) → L(𝑃)(𝑘) ⊗𝐵 Ω𝐵 and ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 : L(𝑃)(𝑘) → L(𝑃) ⊗𝐵 Ω1 𝐵 by ∀𝛽 ∈ Ω𝐵, ∀𝑝 ∈ 𝑃𝑘, 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝛽 ⊗ 𝑝) � ˆℓ𝑃(ˆ𝜄𝑃(𝛽)𝑝), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15) ∀𝑝 ∈ 𝑃𝑘, ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝑝) � ˆℓ𝑃 �d𝑃,hor(𝑝)�, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘, ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘) defines a Hermitian bimodule connection on the Hermitian line 𝐵-bimodule L(𝑃)(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that ˆℓ𝑃 is well-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' uniqueness and U(1)-equivariance will then follow from the explicit form of ˆℓ−1 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑘 ∈ Z, let (𝑒𝑖)𝑚 𝑖=1 be a basis for L(𝑃)(𝑘), and define ˆℓ𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 : (Ω𝑃,hor)𝑘 → L(𝑃)(𝑘) ⊗𝐵 Ω𝐵 by ˆℓ𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 � �𝜔 ↦→ �𝑚 𝑖=1 𝑒𝑖 ⊗ ˆ𝜄−1 𝑃 (𝑒∗ 𝑖 𝜔)�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' that ˆℓ𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 is an isomorphism of 𝐵-bimodules with inverse given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14) now follows from observing that (𝑒𝑖)𝑚 𝑖=1 satisfies 1 = 𝜄𝑃 ��𝑚 𝑖=1(𝑒𝑖, 𝑒𝑖)� = �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We may now set ˆℓ𝑃 � � 𝑘∈Z ˆℓ𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now fix 𝑘 ∈ Z and show that (𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘, ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘) defines a Hermitian bimodule connection on the Hermitian line 𝐵-bimodule L(𝑃)(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑒𝑖)𝑚 𝑖=1 be a basis and let (𝜖𝑗)𝑛 𝑗=1 be a strict cobasis for L(𝑃)(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 and observe that �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 𝜄𝑃 ��𝑛 𝑗=1(𝜖𝑗, 𝜖𝑗) � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, the fact that �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1 implies that 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 is indeed an isomorphism of graded 𝐵-bimodules with inverse 𝜎−1 𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 = � 𝑝 ⊗ 𝛽 ↦→ �𝑛 𝑗=1 ˆ𝜄−1 𝑃 � 𝑝𝛽𝜖∗ 𝑗 � ⊗ 𝜖𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, the fact that � 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 implies, that for all 𝛼, 𝛽 ∈ Ω𝐵 and 𝑝 ∈ 𝑃𝑘, ˆℓ−1 𝑃 ◦ 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝛼𝛽 ⊗ 𝑝) = ˆ𝜄𝑃(𝛼𝛽)𝑝 = ˆℓ−1 𝑃 � 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝛼 ⊗ 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝛽 ⊗ 𝑝) ⟨0⟩)𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝛽 ⊗ 𝑝) ⟨1⟩ � , which yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 is a well-defined Hermitian generalised braiding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it remains to show that ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘 is a right Hermitian connection satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) with respect to 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, we may again use the maps ˆℓ𝑃 and ˆ𝜄𝑃 together with the equality �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 to derive (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) from the Leibniz rule for d𝑃,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Beggs–Majid [13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56], Saldaña [72, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor L of Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15 lifts with respect to the obvious forgetful functors DCirchor(𝐵) → Circ(𝐵) and Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)) to the functor ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 37 (1) Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a homomorphism ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) : Z → DPic(𝐵) as follows: (a) given 𝑘 ∈ Z, let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(𝑘) � (L(𝑃)(𝑘), 𝜎𝑃,𝑘, ∇𝑃,𝑘), where (𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘, ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘) is the Hermitian bimodule connection of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (b) let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(0) be the unique lift of id𝑃0 � L(𝑃)(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (c) given 𝑚, 𝑛 ∈ Z, let ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(2) 𝑚,𝑛 be the unique lift of L(𝑃)(2) 𝑚,𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given an isomorphism 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) of horizontally differen- tiable quantum principal U(1)-bundles over 𝐵, let ˆL(𝑓) : ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) ⇒ ˆL(𝑄, Ω𝑄,hor, d𝑄,hor) be the unique lift of the 2-isomorphism L(𝑓) : L(𝑃) ⇒ L(𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)- bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For notational simplicity, we set 𝐹 � L(𝑃) and denote our would-be homomorphism ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) by ˆ𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor ˆ𝐹 : Z → DPic(𝐵) is well defined by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' that the arrow 𝐹 (0) : 𝐹(0) → 𝐵 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) follows from the fact that ˆ𝜄𝑃 ◦ d = d𝑃,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑚, 𝑛 ∈ Z, the arrow 𝐹 (2) 𝑚,𝑛 : 𝐹(𝑚) ⊗𝐵 𝐹(𝑛) → 𝐹(𝑚 + 𝑛) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) by applying the isomorphism ˆℓ−1 𝑃 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 to both sides of the desired equality and then applying the Leibniz rule for d𝑃,hor in Ω𝑃,hor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' thus, the natural isomorphism ˆ𝐹 (2) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Commutativity of the relevant commutative diagrams now follows from observing that the forgetful functor DPic(𝐵) → Pic(𝐵) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) be an isomorphism of horizontally differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Again, for notational simplicity, set 𝑅 � L(𝑃), ˆ𝑅 � ˆL(𝑃, Ω𝑃,hor, d𝑃,hor), 𝑆 � L(𝑄), and ˆ𝑆 � ˆL(𝑄, Ω𝑄,hor, d𝑄,hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Observe that 𝑓 ⊗ idΩ𝐵 necessarily intertwines the isomorphisms ˆℓ𝑃 and ˆℓ𝑄 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, so that for each 𝑘 ∈ Z, the arrow L(𝑓)𝑘 : 𝑅(𝑘) → 𝑆(𝑘) in Pic(𝐵) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) precisely since d𝑄,hor ◦ 𝑓 = 𝑓 ◦ d𝑃,hor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it follows that ˆL(𝑓) : ˆ𝑅 → ˆ𝑆 is well defined as a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Once more, commutativity of the relevant commutative diagrams now follows from observing that the forgetful functor DPic(𝐵) → Pic(𝐵) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 (Landi–Reina–Zampini [65], Khalkhali–Landi–Van Suijlekom [61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, we now equip O𝑞(CP1) with Podleś’s 2-dimensional calculus (Ω𝑞(CP1), d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The homomorphism E � L(O𝑞(SU(2))) lifts to the corresponding homo- morphism ˆE : Z → DPic(O𝑞(CP2)) by setting ˆE � ˆL(O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, given 𝑘 ∈ Z, it follows that ˆE(𝑘) = (E(𝑘), 𝜎𝑘, ∇𝑘), where ∇𝑘 and 𝜎𝑘 respectively recover the canonical connection [65, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1] and ‘twisted flip’ [61, §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5-6] on E(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we show that the functor ˆLis, indeed, an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ˆ𝐹 : Z → DPic(𝐵) be a homomorphism with image 𝐹 : Z → Pic(𝐵) under the forgetful functor Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)), and let 𝑃 � Σ(𝐹) be the topological quantum principal U(1)-bundle over 𝐵 induced by 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a horizontal calculus (Ω𝑃,hor, d𝑃,hor) on 𝑃: (1) define the graded ∗-algebra Ω𝑃,hor by equipping the complex vector space 𝑃 ⊗𝐵 Ω𝐵 with the multiplication and ∗-operation defined, respectively, by ∀𝛼, 𝛽 ∈ Ω𝐵, ∀𝑝 ∈ Z, ∀𝑘 ∈ Z, ∀𝑞 ∈ 𝐹(𝑛), (𝑝 ⊗ 𝛼)(𝑞 ⊗ 𝛽) � 𝑝𝜎𝐹(𝑘) (𝛼 ⊗ 𝑞)𝛽, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17) ∀𝛼 ∈ Ω𝐵, ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), (𝑝 ⊗ 𝛼)∗ � 𝜎𝐹(−𝑘) (𝛼∗ ⊗ 𝑝∗), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) and with the grading induced by the grading on Ω𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 38 BRANIMIR ĆAĆIĆ (2) define d𝑃,hor : Ω𝑃,hor → Ω𝑃,hor by ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝐹(𝑘), ∀𝛽 ∈ Ω𝐵, d𝑃,hor(𝑝 ⊗ 𝛽) � ∇𝐹(𝑘) (𝑝) ⊗ 𝛽 + 𝑝 ⊗ d𝛽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19) (3) extend the canonical U(1)-action 𝛼 on 𝑃 pointwise to ˆ𝛼 : U(1) → Aut(Ω𝑃,hor) by ∀𝑧 ∈ U(1), ∀𝑝 ∈ 𝑃, ∀𝛽 ∈ Ω𝐵, ˆ𝛼𝑧(𝑝 ⊗ 𝛽) � 𝛼𝑧(𝑝) ⊗ 𝛽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20) (4) let ˆ𝜄𝑃 : (Ω𝐵, d) → (ΩU(1) 𝑃,hor, d𝑃,hor↾ΩU(1) 𝑃,hor) be induced by (𝐹 (0) ⊗ id) ◦ 𝜆Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The construction of Ω𝑃,hor, ˆ𝛼, and ˆ𝜄𝑃 from ˆ𝐹 follows, mutatis mutandis, from the explicit construction of 𝑃 � Σ(𝐹), 𝛼, and 𝜄𝑃 from 𝐹 in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, recall that ˆ𝐹 canonically defines a bar functor by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, each definitional commutative diagram satisfied by the bar functor ˆ𝐹 yields a corresponding commutative diagram satis- fied by the family of Hermitian generalised braidings (𝜎𝐹(𝑘))𝑘∈Z, which, in turn, yields the corresponding properties of Ω𝑃,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us now turn to d𝑃,hor, which is U(1)-equivariant and complex-linear by construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it also clearly satisfies d𝑃,hor ◦ ˆ𝜄𝑃 = ˆ𝜄𝑃 ◦d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑚, 𝑛 ∈ Z, the fact that 𝐹 (2) 𝑚,𝑛 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) implies that d𝑃,hor satisfies the Leibniz rule on (Ω𝑃,hor)𝑚 · (Ω𝑃,hor)𝑛 = (Ω𝑃,hor)𝑚+𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, given 𝑘 ∈ Z, the fact that 𝐹 (−1) 𝑘 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) implies that d𝑃,hor is ∗-preserving on the subspace ∗�(Ω𝑃,hor)𝑘 � = (Ω𝑃,hor)−𝑘 of Ω𝑝,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor Σ : Hom(Z, Pic(𝐵)) → Circ(𝐵) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 lifts with respect to the forgetful functors DCirchor(𝐵) → Circ(𝐵) and Hom(Z, DPic(𝐵)) → Hom(Z, Pic(𝐵)) to the weak inverse ˆΣ : Hom(Z, DPic(𝐵)) → DCirchor(𝐵) of ˆL defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given a homomorphism ˆ𝐹 : Z → DPic(𝐵) descending to 𝐹 : Z → Pic(𝐵), let ˆΣ( ˆ𝐹) � (Σ(𝐹), ΩΣ(𝐹),hor, dΣ(𝐹),hor), where (ΩΣ(𝐹),hor, dΣ(𝐹),hor) is the horizontal calculus of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given a 2-isomorphism ˆ𝜂 : ˆ𝑅 → ˆ𝑆 between homomorphisms ˆ𝑅, ˆ𝑆 : Z → DPic(𝐵) that descends to a 2-isomorphism 𝜂 : 𝑅 → 𝑆 between homomorphisms 𝑅, 𝑆 : Z → Pic(𝐵), let ˆΣ( ˆ𝜂) : ˆΣ( ˆ𝑅) → ˆΣ( ˆ𝑆) be the unique lift of Σ(𝜂) : Σ(𝑅) → Σ(𝑆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, in particular, the category DCirchor(𝐵) is essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We have seen that Σ is well defined on objects, so let us check that it is well defined on arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ˆ𝜂 : ˆ𝑅 → ˆ𝑆 be an arrow in Hom(Z, DPic(𝐵)) descending to 𝜂 : 𝑅 → 𝑆 in Hom(Z, Pic(𝐵)), so that ˆ𝜂 and 𝜂 define bar natural transformations by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We can extend Σ(𝜂) : Σ(𝑅) → Σ(𝑆) to a U(1)-equivariant isomorphism of graded 𝐵-bimodules ˆΣ( ˆ𝜂) : ΩΣ(𝑅),hor → ΩΣ(𝑆),hor by setting ˆΣ � Σ(𝜂) ⊗ idΩ𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Coherence of 𝜂 with respect to 𝑅(2) and 𝑆(2) implies that ˆΣ( ˆ𝜂) is multiplicative, that 𝜂1 intertwines 𝑅(0) and 𝑆(0) implies that ˆΣ( ˆ𝜂) is unital, and the fact that ˆ𝜂 is a bar functor implies that ˆΣ(𝜂) is ∗-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, given 𝑘 ∈ Z, the fact that 𝜂(𝑘) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) implies that ˆΣ(𝜂) satisfies ˆΣ(𝜂) ◦ dΣ(𝑅),hor = dΣ(𝑆),hor ◦ ˆΣ(𝜂) on (ΩΣ(𝑅),hor)𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The rest now follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17, mutatis mutandis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Building on a proposal of Ðurđević [40, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4], Saldaña proves analogues of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27 [72, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11] and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28 [72, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12] for quantum principal bundles with structure quantum group given by a Hopf ∗-algebra in terms of certain heavily structured functors that resemble bar functors à la Beggs–Majid [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By contrast, in the special case of quantum principal U(1)-bundles, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 allows us to use monoidal functors simpliciter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, after suitable generalisation, the same will still be true in the somewhat more general case where the structure quantum group is a group ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 39 By combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28 with Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9, we obtain the promised generalisation of Abadie–Eilers–Exel’s generalised crossed product construction to the setting of nc differential geometry in the absence of any further constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor 𝜖1 ◦ ˆL : DCirchor(𝐵) → DPic(𝐵) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The horizontal crossed product of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) by a Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) is the essentially unique horizontally differentiable quantum principal U(1)-bundle (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z over (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d), such that ˆL � (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z � (1) � (𝐸, 𝜎𝐸, ∇𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' One may justify this terminology as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝜔, 𝜙) be an extended diffeomorphism of 𝐵, so that 𝐵 ⋊alg 𝜙 Z admits the horizontal calculus (Ω𝐵 ⋊alg 𝜙 Z, d(𝜔,𝜙)), where the graded ∗-algebra Ω𝐵 ⋊𝜙 Z is obtained from Ω𝐵 by adjoining a unitary 𝑈 ∈ (Ω𝐵 ⋊𝜙 Z)0 that satisfies 𝑈𝜙𝛽𝑈−1 𝜙 = 𝜙(𝛽) for all 𝛽 ∈ Ω𝐵, the ∗-derivation d(𝜔,𝜙) is uniquely determined by requiring that d(𝜔,𝜙)↾Ω𝐵� d𝐵 and d(𝜔,𝜙) (𝑈𝜙) � i𝜔𝑈𝜙, and the U(1)-action ˆ𝛼 on Ω𝐵 ⋊alg 𝜙 Z is uniquely determined by setting ˆ𝛼𝑧↾Ω𝐵= idΩ𝐵 and 𝛼𝑧(𝑈𝜙) � 𝑧𝑈𝜙 for all 𝑧 ∈ U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (𝑏𝜙 ↦→ 𝑈𝜙−1(𝑏)) : ˆ𝜏(𝜔, 𝜙) → ˆL(𝐵 ⋊alg 𝜙 Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵 ⋊alg 𝜙 Z, d(𝜔,𝜙))(1) is an isomorphism of Hermitian line 𝐵-bimodules with connection, we may therefore take (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor ˆ𝜏(𝜔,𝜙) Z � (𝐵 ⋊alg 𝜙 Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵 ⋊alg 𝜙 Z, d(𝜔,𝜙)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We conclude this subsection by discussing curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In general, the curvature of a ∗-quasi- dga (Ω, d) is the map d2, which vanishes for a ∗-exterior algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the curvature (in this sense) of a horizontally differentiable quantum principal U(1)-bundle (𝑃, Ω𝑃,hor, d𝑃,hor) over 𝐵 is the map d2 𝑃,hor, which is U(1)-equivariant ∗-derivation that vanishes on Ω𝐵 and hence, in particular, is left and right Ω𝐵-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Passing this notion of curvature through the lens of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28 yields the following more refined definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition-Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ðurđević [38, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a hori- zontally differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Its Fröhlich automorphism is the unique U(1)-equivariant automorphism ˆΦ𝑃 of the U(1)- ∗-quasi-dga of finite type (Z(Ω𝐵), d↾Z(Ω𝐵)), such that ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, ∀𝛽 ∈ Z(Ω𝐵), ˆ𝜄𝑃 � ˆΦ𝑘 𝑃(𝛽) � 𝑝 = 𝑝ˆ𝜄𝑃(𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21) (2) Its curvature 1-cocycle is the unique group 1-cocycle F𝑃 : Z → S(𝐵) for the right Z-action generated by ˆΦ−1 𝑃 , such that ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, d2 𝑃,hor(𝑝) = 𝑝 · ˆ𝜄𝑃(iF𝑃(𝑘)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22) Hence, its curvature data is the pair (Φ𝑃, F𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25 together with Proposition-Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38, we can and must take ˆΦ𝑃 � Φ � ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) � (1), F𝑃 � F � ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Suppose that (𝑃, Ω𝑃,hor, d𝑃,hor) is a horizontally differentiable quantum principal U(1)- bundle over 𝐵 with curvature data (Φ𝑃, F𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28, every homo- morphism ˆ𝐹 : Z → DPic(𝐵) that is 2-isomorphic to ˆL(𝑃, Ω𝑃,hor, d𝑃,hor) satisfies ˆΦ ◦ 𝜋0( ˆ𝐹)(1) = ˆΦ𝑃, F ◦ 𝜋0( ˆ𝐹(1)) = F𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 40 BRANIMIR ĆAĆIĆ On the other hand, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30, every Hermitian line 𝐵-bimodule (𝐸, 𝜎𝐸, ∇𝐸) that is isomorphic to ˆL(𝑃, Ω𝑃,hor, d𝑃,hor)(1) satisfies ˆΦ[𝐸,∇𝐸] = ˆΦ𝑃, F[𝐸,∇𝐸] = F𝑃(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in other words, for every Hermitian line 𝐵-bimodule (𝐸, 𝜎𝐸, ∇𝐸), the resulting horizontal crossed product (𝐵, Ω𝐵, d) ⋊(𝐸,𝜎𝐸,∇𝐸) Z has curvature data (Φ[𝐸,∇𝐸], F[𝐸,∇𝐸]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33 (Landi–Reina–Zampini [65, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall (O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let (ΦO𝑞(SU(2)), FO𝑞(SU(2))) be its curva- ture data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Using the pbw basis for O𝑞(SU(2)), one shows that Z(Ω𝑞(CP1)) = C[i𝑒+𝑒−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the generators 𝑎, 𝑐 ∈ O𝑞(SU(2))1 satisfy 𝑎𝑎∗ + (𝑞𝑐)(𝑞𝑐)∗ = 1 and 𝑎∗𝑎 + 𝑐∗𝑐 = 1, we may use them to compute ˆΦO𝑞(SU(2)) (i𝑒+𝑒−) = 𝑞2i𝑒+𝑒−, FO𝑞(SU(2)) (1) = 𝑞−2i𝑒+𝑒−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Reconstruction of total calculi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we leverage structural results of Ðurđević [39] and Beggs–Majid [14] to obtain the promised nc generalisation of the classical correspondence between Hermitian line bundles with unitary connection and principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Once more, let 𝐵 be a unital pre-𝐶∗-algebra with ∗-exterior algebra (Ω𝐵, d𝐵), which we view as a fixed nc base manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, given 𝑞 ∈ (0, ∞), we define the corresponding 𝑞-integers by setting [𝑘]𝑞 � 1−𝑞𝑘 1−𝑞 for 𝑘 ∈ Z when 𝑞 ≠ 1 and [𝑘]𝑞 � 𝑘 for 𝑘 ∈ Z when 𝑞 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin by noting that U(1) does not always appear in nc differential geometry with its usual smooth structure as a Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Instead, we must allow for all possible 1-dimensional bi-invariant ∗-exterior algebras on the unital pre-𝐶∗-algebra O(U(1)) of trigonometric polyno- mials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' These, in turn, are exhausted up to isomorphism by the family ((Ω𝜅(U(1)), d𝜅))𝜅∈(0,∞) of ∗-exterior algebras on O(U(1)) whose construction is conveniently generalised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define 𝜅-deformed Chevalley–Eilenberg extension to be the faithful functor CE𝜅 : QDGAU(1) → QDGAU(1) constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given a U(1)-∗-quasi-dga of finite type (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d), let CE𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d) � (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' CE𝜅(Ω), CE𝜅(d)) , where CE𝜅(Ω) is the graded ∗-algebra obtained from Ω by adjoining a self-adjoint element 𝑒𝜅 of degree 1 satisfying the relations 𝑒2 𝜅 = 0 and ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 𝑘, 𝑒𝜅𝜔 = (−1)𝑛𝜅−𝑘𝜔𝑒𝜅, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24) where CE𝜅(d) is defined by setting CE𝜅(d)(𝑒𝜅) � 0 and ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 𝑘, CE𝜅(d)(𝜔) � (−1)𝑛𝜅−𝑘2𝜋i[𝑘]𝜅𝜔𝑒𝜅 + d𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) and where the U(1)-action on CE𝜅(Ω) is the unique extension of the U(1)-action on Ω leaving 𝑒𝜅 invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given a morphism 𝑓 : (𝑃, Ω𝑃, d𝑃) → (𝑄, Ω𝑄, d𝑄) of U(1)-∗-quasi-dga, let CE𝜅(𝑓) : CE𝜅(𝑃, Ω𝑃, d𝑃) → CE𝜅(𝑄, Ω𝑄, d𝑄) be the unique extension of 𝑓 : Ω𝑃 → Ω𝑄 satisfying CE𝜅(𝑓)(𝑒𝜅) = 𝑒𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, given 𝜅 > 0, the ∗-exterior algebra (Ω𝜅(U(1)), d𝜅) � (CE𝜅(O(U(1))), CE𝜅(0)) on O(U(1)) is the essentially unique ∗-exterior algebra on O(U(1)) of dimension 1 that satisfies the relation d𝜅(𝑧) · 𝑧 = 𝜅𝑧 · d𝜅(𝑧), where d𝜅(𝑧) = 2𝜋i𝑒𝜅 · 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that 𝜅 = 1 recovers the usual de Rham calculus on U(1) as a Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In general, differentiability of a U(1)-action with respect to the ∗-exterior algebra (Ω𝜅(U(1)), d𝜅) can now be characterised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 41 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ðurđević [39, §3], Beggs–Brzeziński [10, §7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑃 be a U(1)-pre-𝐶∗- algebra of finite type and let (Ω, d) be a U(1)-∗-exterior algebra over 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that (Ω, d) is 𝜅-vertical whenever there exists a (necessarily unique) lift of id𝑃 to a morphism of U(1)- ∗-quasi-dga ver : (𝑃, Ω, d) → CE𝜅(𝑃, Ω, d), the vertical coevaluation on (Ω, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, we define horizontal form in Ω to be an element of the U(1)-invariant graded ∗-subalgebra Ωhor � {𝜔 ∈ Ω | ver(𝜔) = 𝜔} of Ω, and a basic form to be an element of the U(1)-invariant and d-invariant graded ∗-subalgebra Ωbas � (Ωhor)U(1) of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, given 𝜅 > 0, we can make precise sense of nc differentiable principal U(1)-bundles, where U(1) carries the bi-invariant ∗-exterior algebra (Ω𝜅(U(1)), d𝜅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 (Brzeziński–Majid [22, §4], Hajac [52], Ðurđević [39, §3], Beggs–Brzeziński [10, §7], Beggs–Majid [14, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ćaćić [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A 𝜅-differentiable quantum principal U(1)-bundle over 𝐵 is a triple (𝑃, Ω𝑃, d𝑃), where 𝑃 is a topological quantum principal U(1)- bundle over 𝐵 and (Ω𝑃, d𝑃) is a 𝜅-vertical U(1)-∗-exterior algebra over 𝑃 together with an isomorphism ˆ𝜄𝑃 : (Ω𝐵, d𝐵) → (Ω𝑃,bas, d𝑃↾Ω𝑃,bas) of ∗-quasi-dga extending 𝜄𝑃, such that Ω𝑃,hor = 𝑃 · Ω𝑃,bas · 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12, let 𝜕 𝜕𝑡 be the fundamental vector field of the U(1)-action on 𝑋, and let Ωalg(𝑋) � �alg 𝑘∈Z{𝜔 ∈ Ω(𝑋) | ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧−𝑘𝜔}, which we equip with the U(1)-action 𝑧 ↦→ (𝜎𝑧−1)∗ and the de Rham exterior derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝐶∞ alg(𝑋), Ωalg(𝑋), d) defines a 1-differentiable quantum principal U(1)-bundle over (𝐶∞(𝑌), Ω(𝑌), d) with respect to 𝜋∗ : Ω(𝑌) → Ωalg(𝑋)U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note, in particular, that the vertical coevaluation reduces to the map Ωalg(𝑋) → Ω(U(1))U(1) �⊗C Ωalg(𝑋) that dualises contraction of differential forms with the fundamental vector field 𝜕 𝜕𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following necessary and sufficient conditions are of both theoretical and practical importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that they involve the strong connection condition first identified by Hajac [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38 (Beggs–Majid [14, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='53 & Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), let 𝑃 be a topological quantum principal U(1)-bundle over 𝐵, let (Ω𝑃, d𝑃) be a 𝜅-vertical U(1)-∗-exterior algebra over 𝑃, and let ˆ𝜄𝑃 : (Ω𝐵, d𝐵) → (Ω𝑃,bas, d𝑃↾Ω𝑃,bas) be an injective morphism of ∗-quasi- dga extending 𝜄𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝑃, Ω𝑃, d𝑃) defines a 𝜅-differentiable quantum principal U(1)-bundle over 𝐵 with respect to ˆ𝜄𝑃 if and only if Ω𝑃,hor = 𝑃 · ˆ𝜄𝑃(Ω𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) Moreover, if, for each 𝑛 ∈ N0, the left 𝐵-module Ω𝑛 𝐵 is flat, then (𝑃, Ω𝑃, d𝑃) defines a 𝜅- differentiable quantum principal U(1)-bundle over 𝐵 with respect to ˆ𝜄𝑃 if and only if Ω1 𝑃,hor = 𝑃 · ˆ𝜄𝑃(Ω1 𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) We now recall the notions of principal Ehresmann connection and connection 1-form appropriate to our nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39 (Brzeziński–Majid [22, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 & Appx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' a], Hajac [52, §4], Ðurđević [39, §4], Beggs–Majid [14, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum principal U(1)-bundle over 𝐵 with respect to (Ω𝐵, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) A connection on (𝑃, Ω𝑃, d𝑃) is a surjective U(1)-equivariant grading- and ∗-preserving algebra homomorphism Π : Ω𝑃 → Ω𝑃,hor, such that Π2 = Π and ∀𝜔 ∈ Ω1 𝑃, (id −Π)(𝜔)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28) 42 BRANIMIR ĆAĆIĆ (2) A connection 1-form on (𝑃, Ω𝑃, d𝑃) is U(1)-invariant self-adjoint 𝜗 ∈ Ω1 𝑃, such that ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑛 𝑃)𝑘, 𝜗𝜔 = (−1)𝑛𝜅−𝑘𝜔𝜗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) ver(𝜗) = 𝑒𝜅 + 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the terminology of Brzeziński–Majid [22, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 & Appx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' a], Hajac [52, §4], and Beggs–Majid [14, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5], the restriction of a connection Π to 1-forms is a ∗-preserving strong bimodule connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the terminology of Ðurđević [39, §4], the datum of a connection 1-form is equivalent to the datum of a multiplicative regular connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The bijection between principal connections and connection 1-forms persists in the nc context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Brzeziński–Majid [22, Propp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 & 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10], Ðurđević [39, Proof of Thm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For every connection Π on (𝑃, Ω𝑃, d𝑃), there exists a unique connection 1-form 𝜗, such that ∀𝑘 ∈ Z, ∀𝑝 ∈ 𝑃𝑘, (id −Π) ◦ d𝑃(𝑝) = 2𝜋i[𝑘]𝜅𝜅−𝑘𝑝𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) Conversely, for every connection 1-form 𝜗 on (𝑃, Ω𝑃, d𝑃), there exists a unique connection Π that satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with preliminary observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By a lemma of Beggs–Majid [14, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='59], the vertical coevaluation of (𝑃, Ω𝑃, d𝑃) satisfies ∀𝑛 ∈ N, (ver − id)(Ω𝑛 𝑃) ⊆ Ω𝑛−1 𝑃,hor · 𝑒𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32) Together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26), this yields a short exact sequence 0 → Ω𝑃,hor → Ω𝑃 ver−id −−−−→ Ω𝑃,hor · 𝑒𝜅 → 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) of ∗-closed U(1)-invariant Ω𝑃,hor-sub-bimodules of Ω𝑃 and U(1)-equivariant left and right Ω𝑃,hor-linear maps preserving both the ambient ∗-operation and N0-grading on CE𝜅(Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, suppose that Π is a connection on 𝑃, Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Π is a U(1)-equivariant left and right Ω𝑃,hor-linear left splitting of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) preserving both the ambient ∗-operation and N0-grading in CE𝜅(Ω𝑃), so that (ver − id) ↾ran(id −Π): ran(id −Π) → Ω𝑃,hor · 𝑒𝜅 is a U(1)- equivariant isomorphism of Ω𝑃,hor-bimodules preserving both the ambient ∗-operation and the ambient N0-grading in CE𝜅(Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝜗 � �(ver − id)↾ran(id −Π) �−1 (𝑒𝜅), which is thus a U(1)-invariant self-adjoint element of Ω1 𝑃 satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) by construction and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) applied to d𝑃(𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It remains to show that 𝜗 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜗2 = 0 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28), it suffices to show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) holds for horizontal 𝜔, but this now follows from the fact that (ver − id) ↾ran(id −Π) is an isomorphism of Ω𝑃,hor-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let us show that 𝜗 is uniquely determined by Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝜖𝑗)𝑛 𝑗=1 be a finite family in 𝑃1 satisfying �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜗 = ∑︁𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗𝜗 = 𝜅 2𝜋i ∑︁𝑛 𝑗=1 𝜖∗ 𝑗 (2𝜋i[1]𝜅𝜅−1𝜖𝑗𝜗) = (id −Π) � 𝜅 ∑︁𝑛 𝑗=1 𝜖∗ 𝑗 d𝑃(𝜖𝑗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, suppose that 𝜗 is a connection 1-form on (𝑃, Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by construc- tion of CE𝜅(Ω𝑃), the element 𝑒𝜅 freely generates the left Ω𝑃,hor-submodule Ω𝑃 ·𝑒𝜅 ⊆ CE𝜅(Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30), the element 𝜗 satisfies the same relations in Ω𝑃 that 𝑒𝜅 satisfies in CE𝜅(Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the identity map idΩ𝑃 extends to a surjective U(1)-equivariant algebra homomorphism 𝜓𝜗 : CE𝜅(Ω𝑃) → Ω𝑃 intertwining ∗-operations and N0-gradings by setting 𝜓𝜗(𝑒𝜅) � 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that Π � idΩ𝑃 −𝜓𝜗 ◦ (ver − idΩ𝑃) defines a connection satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) with respect to 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 43 First, by construction, the map Π is U(1)-equivariant and unital, is left and right Ω𝑃,hor- linear, and intertwines ∗-operations and N0-gradings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, by definition of Ω𝑃,hor, it follows that Π↾Ω𝑃,hor= idΩ𝑃,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32) together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30), it follows that (ver − id) ◦ Π = (ver − id) − (ver − id) ◦ 𝜓𝜗 ◦ (ver − id) = 0, so that ran Π ⊂ Ω𝑃,hor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' from this, it follows that Π2 = Π, and hence, in particular, that ran(id −Π) = Ω𝑃,ℎ𝑜𝑟 · 𝜗, so that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28) follows since 𝜗2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Multiplicativity now follows from left and right Ω𝑃,hor-linearity of Π together with the decomposition Ω𝑃 = Ω𝑃,hor ⊕ Ω𝑃,hor · 𝜗 of Ω𝑃,hor-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, that Π is uniquely determined by 𝜗 follows from multiplicativity of Π together with the fact that 𝑃 and d𝑃(𝑃) generate Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Hence, just as in the classical case, one may now use the connection 1-form to define the curvature 2-form of a principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑃, Ω𝑃, d𝑃) be a 𝜅-differentiable quantum principal U(1)-bundle over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Π be a connection on (𝑃, Ω𝑃, 𝜄𝑃) with connection 1-form 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The curvature of Π is the closed self-adjoint 2-form FΠ � −ˆ𝜄−1 𝑃 (d𝑃(𝜗)) ∈ Z(Ω𝐵)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻∗𝑋 → 𝑋 be the horizontal cotangent bundle of the total space 𝑋, whose fibre at 𝑥 ∈ 𝑋 is the annihilator of 𝜕 𝜕𝑡 at 𝑥, so that Ωalg(𝑋)hor = � 𝑘∈Z � 𝜔 ∈ Γ �� 𝐻∗𝑋 ⊗ C � ��� ∀𝑧 ∈ U(1), (𝜎𝑧)∗𝜔 = 𝑧−𝑘𝜔 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let Π be a principal connection on 𝜋 : 𝑋 → 𝑌, which we view as a U(1)-equivariant real vector bundle endomorphism Π : 𝑇∗𝑋 → 𝑇∗𝑋 satisfying Π2 = Π and ran Π = 𝐻∗𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Π induces a connection on (𝐶∞ alg(𝑋), Ωalg(𝑋), d), whose connection 1-form and curvature 2-form respectively recover the usual connection 1-form and curvature 2-form of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now leverage structural results of Ðurđević [39] to obtain the promised correspondence between nc Hermitian line bundles with connection and nc principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We may define the concrete category Gauge𝜅(𝐵) of 𝜅-differentiable quantum principal U(1)-bundle with connection over 𝐵 and their isomorphisms as follows: (1) an object is a triple (𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) consisting of a 𝜅-differentiable quantum principal U(1)- bundle (𝑃, Ω𝑃, d𝑃) over 𝐵 and a connection Π𝑃 on (𝑃, Ω𝑃, d𝑃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) an arrow 𝑓 : (𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑃) → (𝑄, Ω𝑄, d𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑄) is an isomorphism of curved U(1)-∗-dga 𝑓 : (𝑃, Ω𝑃, d𝑃) → (𝑄, Ω𝑄, d𝑄) that satisfies 𝑓 ◦ ˆ𝜄𝑃 = ˆ𝜄𝑄 and 𝑓 ◦ Π𝑃 = Π𝑄 ◦ 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we may define a functor Hor𝜅 : Gauge𝜅(𝐵) → DCirchor(𝐵) as follows: (1) given an object (𝑃, Ω, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), let Hor𝜅(𝑃, Ω, d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) � �𝑃, Ωhor, Π ◦ d↾Ωhor �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) given an arrow 𝑓 : (𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑃) → (𝑄, Ω𝑄, d𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑄), let Hor𝜅(𝑓) : Hor𝜅(𝑃, Ω𝑃, d𝑃, Π𝑃) → Hor𝜅(𝑄, Ω𝑄, d𝑄, Π𝑄) be given by the map 𝑓↾Ω𝑃,hor: Ω𝑃,hor → Ω𝑄,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the functor Hor𝜅 takes a 𝜅-differentiable quantum principal U(1)-bundle with connec- tion and extracts the horizontal calculus induced by the choice of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A straight- forward calculation shows that the essential range of this functor satisfies a simple algebraic constraint: the curvature 2-form solves the eigenvector equation for the Fröhlich automor- phism with respect to 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 44 BRANIMIR ĆAĆIĆ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ðurđević [39, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let (𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) be a 𝜅- differentiable quantum principal U(1)-bundle with connection over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let FΠ be the curvature 2-form of Π, and let ( ˆΦ𝑃,Π, F𝑃,Π) be the curvature data of Hor𝜅(𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), so that ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝛽 ∈ (Ω𝑛 𝑃,hor)𝑘, (Π ◦ d𝑃)2(𝛽) = 𝛽 · ˆ𝜄𝑃 �i F𝑃,Π(𝑘)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then F𝑃,Π : Z → S(𝐵) is given by F𝑃,Π = �𝑘 ↦→ 2𝜋[𝑘]𝜅𝜅−𝑘FΠ �, so that, in particular, ˆΦ𝑃,Π �F𝑃,Π(1)� = 𝜅F𝑃,Π(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑃, Ω𝑃,hor, d𝑃,hor) be a horizontally differentiable quantum principal U(1)-bundle over 𝐵 with curvature data ( ˆΦ𝑃, F𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that (𝑃, Ω𝑃,hor, d𝑃,hor) is flat when- ever F𝑃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' When F𝑃(1) is an eigenvector of ˆΦ𝑃, the vertical deformation parameter 𝜅𝑃 ∈ R× of (𝑃, Ω𝑃,hor, d𝑃,hor) is defined to be the corresponding eigenvalue of ˆΦ𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Remarkably, this single algebraic constraint suffices to characterize the essential range of the functor Hor𝜅, which therefore yields an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46 (Ðurđević [39, Thm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12 & §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let DCirchor,𝜅(𝐵) denote the strictly full subcategory of DCirchor(𝐵) whose objects are either flat or have vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Hor𝜅 defines an equivalence Gauge𝜅(𝐵) → DCirchor,𝜅(𝐵) with weak inverse Tot𝜅 : DCirchor,𝜅(𝐵) → Gauge𝜅(𝐵) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given a horizontally differentiable quantum principal U(1)-bundle (𝑃, Ω𝑃,hor, d𝑃,hor) over 𝐵 with curvature 1-cocycle FΠ, let Tot𝜅(𝑃, Ω𝑃,hor, d𝑃,hor) � (𝑃, CE𝜅(Ω𝑃,hor), CE𝜅(d𝑃,hor) + iΠ, Π𝜅), where iΠ : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑃,hor) is the complex-linear map defined by ∀𝜔1, 𝜔2 ∈ Ω𝑃,hor, iΠ(𝜔1 + 𝜔2𝑒𝜅) � − 𝜅 2𝜋 𝜔2FΠ(1), and where Π𝜅 : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑃,hor) is the unique algebra homomorphism satisfying Π𝜅↾Ω𝑃,hor= idΩ𝑃,hor and Π𝜅(𝑒𝜅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given an isomorphism 𝑓 : (𝑃, Ω𝑃,hor, d𝑃,hor) → (𝑄, Ω𝑄,hor, d𝑄,hor) of horizontally differen- tiable quantum principal U(1)-bundles over 𝐵, let Tot𝜅(𝑓) : Tot𝜅(𝑃, Ω𝑃,hor, d𝑃,hor) → Tot𝜅(𝑄, Ω𝑄,hor, d𝑄,hor) be given by the map CE𝜅(𝑓) : CE𝜅(Ω𝑃,hor) → CE𝜅(Ω𝑄,hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, a canonical natural isomorphism Ð : idGauge𝜅 (𝐵) ⇒ Tot𝜅 ◦ Hor𝜅 is defined as follows: given a 𝜅-differentiable quantum principal U(1)-bundle with connection (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃, Π) over 𝐵, define Ð(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Ω𝑃,d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Π) : (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) → Tot𝜅 ◦ Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) by ∀𝜔 ∈ Ω𝑃, Ð(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Ω𝑃,d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Π) (𝜔) � (ver − id) ◦ (id −Π)(𝜔) + Π(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) By combining this theorem with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28, Proposition-Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32, and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9, we generalise of Abadie–Eilers–Exel’s generalised crossed product construction to a precise nc generalisation of the classical correspondence between Hermitian line bundles with unitary connection and principal U(1)-bundles with principal connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that (𝐸, 𝜎𝐸, ∇𝐸) is flat whenever F[𝐸,∇𝐸] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' When F[𝐸,∇𝐸] is an eigenvector of the automorphism ˆΦ[𝐸,∇𝐸], the vertical deformation parameter 𝜅[𝐸,∇𝐸] ∈ R× of (𝐸, 𝜎𝐸, ∇𝐸) is defined to be the corresponding eigenvalue of ˆΦ[𝐸,∇𝐸].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 45 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let DPic𝜅(𝐵) be the strictly full subcategory of DPic(𝐵) whose objects are either full or have vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then DPic𝜅(𝐵) is the essential image of DCirchor,𝜅(𝐵) under the equivalence 𝜖1 ◦ ˆL : DCirchor(𝐵) → DPic(𝐵), so that 𝜖1 ◦ ˆL◦ Hor𝜅 : DCirc𝜅,tot(𝐵) → DPic𝜅(𝐵) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let (𝐸, 𝜎, ∇) be a Hermitian line 𝐵-bimodule with connection that is flat or has vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define the 𝜅-total crossed product of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) by (𝐸, 𝜎𝐸, ∇𝐸) to be the essentially unique 𝜅-differentiable quantum principal U(1)-bundle with connection (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊𝜅,tot (𝐸,𝜎𝐸,∇𝐸) Z on 𝐵, such that ( ˆL◦ Hor𝜅) � (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊𝜅,tot (𝐸,𝜎𝐸,∇𝐸) Z � � (𝐸, 𝜎𝐸, ∇𝐸);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in this case, we define a ∗-exterior algebra (Ω𝐵, d𝐵) ⋊𝜅,tot (𝐸,𝜎𝐸) Z and connection Π(𝐸,𝜎𝐸,∇𝐸) by � 𝐵 ⋊𝐸 Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (Ω𝐵, d𝐵) ⋊𝜅,tot (𝐸,𝜎𝐸) Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π(𝐸,𝜎𝐸,∇𝐸) � � (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊𝜅,tot (𝐸,𝜎𝐸,∇𝐸) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, given 𝜅 ∈ (0, ∞) and (𝐸, 𝜎𝐸, ∇𝐸) that is either flat or has vertical deformation parameter 𝜅, we may always take (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊𝜅,tot (𝐸,𝜎𝐸,∇𝐸) Z � Tot𝜅((𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z), where (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z is any horizontal crossed product of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) by (𝐸, 𝜎𝐸, ∇𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (𝐸, 𝜎𝐸, ∇𝐸) is flat if and only if (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z is flat and that (𝐸, 𝜎𝐸, ∇𝐸) has vertical deformation parameter 𝜅 if and only if (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d) ⋊hor (𝐸,𝜎𝐸,∇𝐸) Z has vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There are certain examples that are naturally described in terms of homomorphisms from Z to DPic(𝐵) or that give rise to homomorphisms of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In such cases, it convenient to have a straightforward algebraic characterization of the essential range of the composite functor ˆL◦ Hor𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜅 ∈ (0, ∞), and let Hom𝜅(Z, DPic(𝐵)) be the essential image of the subcategory DCirchor,𝜅(𝐵) under the equivalence ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then a homomorphism ˆ𝐹 : Z → DPic(𝐵) defines an object of Hor𝜅(Z, DPic(𝐵)) if and only if ˆ𝐹(1) is flat or has vertical deformation parameter 𝜅, so that, in the latter case, ∀𝑚 ∈ Z, F ◦ 𝜋0( ˆ𝐹)(𝑚) = 𝜅−𝑚+1[𝑚]𝜅F[ ˆ𝐹(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given the discussion after Proposition-Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32, it remains to check (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Sup- pose that ˆ𝐹 : Z → DPic(𝐵) is a homomorphism, such that ˆ𝐹(1) has vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The right 1-cocycle identity for F : DPic(𝐵) → S(𝐵) specialises to ∀𝑚, 𝑛 ∈ Z, F ◦ 𝜋0( ˆ𝐹)(𝑚 + 𝑛) = ˆΦ−𝑛 [ ˆ𝐹(1)] � F ◦ 𝜋0( ˆ𝐹)(𝑚) � + F ◦ 𝜋0( ˆ𝐹)(𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By induction together with the equation ˆΦ[ ˆ𝐹(1)](F[ ˆ𝐹(1)]) = 𝜅F[ ˆ𝐹(1)], it follows that F ◦ 𝜋0( ˆ𝐹) satisfies F ◦ 𝜋0( ˆ𝐹) = � 𝑚 ↦→ [𝑚]𝜅−1F[ ˆ𝐹(1)] � , which, in turn, yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 (Ðurđević [39, §4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23), it follows that (O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) has deformation parameter 𝑞2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35), the homomorphism ˆE : Z → DPic(O𝑞(CP1)) of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 satisfies ∀𝑚 ∈ Z, F ◦ 𝜋0( ˆE)(𝑚) = [𝑚]𝑞2𝑞−2𝑚i𝑒+𝑒−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, the results of Ðurđević show that (O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑞) � Tot𝑞2 (O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor)) 46 BRANIMIR ĆAĆIĆ recovers the 3-dimensional calculus (Ω𝑞(SU(2)), d𝑞) on O𝑞(SU(2)) of Woronowicz [98] and the non-universal 𝑞-monopole connection Π𝑞 of Brzeziński–Majid [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In other words, we may obtain Ω𝑞(SU(2)) from Ω𝑞,hor(SU(2)) by adjoining the skew-adjoint U(1)-invariant 1-form 𝑒0 = 2𝜋i𝑞−2𝑒𝑞2 subject to the relations (𝑒0)2 = 0 and ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 𝑞,hor(SU(2))𝑘, 𝑒0𝜔 = (−1)𝑛𝑞−2𝑘𝜔𝑒0, and we may obtain d𝑞 from d𝑞,hor by setting d𝑞(𝑒0) � 𝑞−2𝑒+𝑒− and ∀(𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑛 𝑞,hor(SU(2))𝑘, d𝑞(𝜔) � (−1)𝑛[𝑘]𝑞−2𝜔𝑒0 + d𝑞,hor(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' From now on, we shall refer to (O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑞) as the 𝑞-monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the map (𝑔 ↦→ 𝑔21𝜃 + 𝑔22) : Γ𝜃 → R× is an injective group homomorphism [53, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10], there exists a unique generator 𝛾 of the infinite cyclic group Γ𝜃 satisfying 𝛾21𝜃 + 𝑔22 > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, let 𝜖𝜃 � 𝛾21𝜃 + 𝛾22, which recovers the norm-positive fundamental unit of the real quadratic field generated by 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, since ˆΦ[ ˆ𝐸(𝛾)](𝑒1𝑒2) = (𝛾21𝜃 + 𝛾22)2𝑒1𝑒2 = 𝜖2 𝜃 𝑒1𝑒2, it follows that ˆ𝐸(𝛾) has vertical deformation parameter 𝜖2 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the composite homomor- phism ˆ𝐸◦(𝑘 ↦→ 𝛾𝑘) is an object of Hom𝜖2 𝜃 (Z, DPic(𝐶∞ 𝜃 (T2))), so that ˆΣ � ˆ𝐸 ◦ (𝑘 ↦→ 𝛾𝑘) � defines an object of DCirchor,𝜖2 𝜃 (𝐶∞ 𝜃 (T2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, at last, we define the real multiplication instanton to be the 𝜖2 𝜃-differentiable quantum principal U(1)-bundle over 𝐶∞ 𝜃 (T2) given by (𝑃𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃𝜃, d𝑃𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑃𝜃) � Tot𝜖2 𝜃 ◦ˆΣ � ˆ𝐸 ◦ (𝑘 ↦→ 𝛾𝑘) � which recovers a construction of Ćaćić [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that 𝐶∗-algebraic completion of 𝑃𝜃 is part of a family of Cuntz–Pimsner algebras first considered by Nawata [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lifting problems for noncommutative Riemannian structures In the commutative case, a Riemannian metric on the base space of a principal U(1)-bundle with principal connection lifts canonically to the total space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this section, we use study the analogous lifting problems for two closely interrelated notions of Riemannian structure on nc manifolds, which are based, respectively, on generalised Hodge star operators and formal spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, we show that these lifted Riemannian structures inexorably involve modular phenomena in both vertical and horizontal directions that are generally non- trivial and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Along the way, we construct moduli spaces of U(1)-instantons, show that quantum SU(2) qua total space of the 𝑞-monopole does not admit a non-pathological U(1)- equivariant twisted spectral triple, and obtain a geometric formal derivation of Kaad–Kyed’s compact quantum metric space [58] on quantum SU(2) for a canonical choice of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For the entirety of this section, let 𝐵 be a unital separable pre-𝐶∗-algebra with ∗-differential calculus (Ω𝐵, d𝐵), which we assume has dimension 𝑁 ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝛾𝐵 : Ω𝐵 → Ω𝐵 denote the Z/2Z-grading on Ω𝐵 by parity of degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, given a horizontal quantum principal U(1)-bundle (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' d𝑃,hor) over 𝐵, we suppress the isomorphism ˆ𝜄𝑃 : Ω𝐵 → ΩU(1) 𝑃,hor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Basic noncommutative Hodge theory and moduli spaces of U(1)-instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin by considering the bare minimum of Riemannian geometry required for classical U(1)- gauge theory on nc manifolds: the Hodge star operator and integration against the Riemannian volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Such an approach was first pursued by Kustermans–Murphy–Tuset for quantum groups [63] and then by Majid [67] and Zampini [99] for quantum CP1 and quantum SU(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it has attained its fullest expression in the context of nc Kähler geometry in the sense of Ó NONCOMMUTATIVE U(1)-GAUGE THEORY 47 Buachalla [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We combine the relevant nc Hodge decomposition theorem with the results of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 to construct moduli spaces of solutions to Maxwell’s equations, and we obtain a robust nc generalisation of the notion of conformal orientation-preserving diffeomorphism to the entire differential Picard group that makes conformal factors into a multiplicative group 1-cocycle on the resulting conformal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with a straightforward generalisation of the Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 (Kustermans–Murphy–Tuset [63], Majid [67], Zampini [99], Ó Buachalla [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A Hodge operator on (Ω𝐵, d𝐵) is a ∗-preserving 𝐵-bimodule morphism ★ : Ω𝐵 → Ω𝐵, such that, for every 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, the restriction of ★ to Ω𝑘 𝐵 satisfies ★(Ω𝑘 𝐵) ⊆ Ω𝑁−𝑘 𝐵 , ★2↾Ω𝑘 𝐵= (−1)𝑘(𝑁−𝑘) idΩ𝑘 𝐵, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) ∀𝜔, 𝜂 ∈ Ω𝑘 𝐵, 𝜔 · ★(𝜂) = ★−1(𝜔) · 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) Hence, the inverse metric induced by ★ is the right 𝐵-valued inner product 𝑔 on Ω𝐵 given by ∀𝜔, 𝜂 ∈ Ω𝐵, 𝑔(𝜔, 𝜂) � ★(𝜔∗ · ★(𝜂)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3) By combining a generalised Hodge star operator with a suitable generalisation of integra- tion against the corresponding Riemannian volume form, we obtain our first notion of nc Riemannian structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, following Connes [33] and Kustermans–Murphy–Tuset [62], we impose Stokes’s theorem for divergence as a requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Kustermans–Murphy–Tuset [63], Ó Buachalla [79], Saldaña [73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A Rie- mannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) is a pair (★, 𝜏), where ★ is a Hodge operator on (Ω𝐵, d𝐵) whose inverse metric 𝑔 admits a basis as a right 𝐵-valued inner product on Ω𝐵 and satisfies ∀𝑏 ∈ 𝐵, ∀𝜔 ∈ Ω𝐵, 𝑔(𝑏𝜔, 𝑏𝜔) ≤ ∥𝑏∥2𝑔(𝜔, 𝜔), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4) and where 𝜏 is a bounded state on 𝐵 that satisfies ∀𝜔 ∈ Ω𝑁−1 𝐵 , (𝜏 ◦ ★ ◦ d𝐵)(𝜔) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5) ∀𝑏 ∈ 𝐵, sup{𝜏(𝑎∗𝑏∗𝑏𝑎) | 𝑎 ∈ 𝐴, 𝜏(𝑎∗𝑎) ≤ 1} = ∥𝑏∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝑋 is orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Equip 𝑋 with an orientation and a Riemannian metric 𝑔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let ★𝑔 and vol𝑔 respectively denote the resulting Hodge star operator and Riemannian volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (★𝑔, ∫ 𝑋 (·) vol𝑔) defines a Riemannian geometry on (𝐶∞(𝑋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω(𝑋, C), d), whose inverse metric is the usual inverse Riemannian metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, the inner product of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7) is the usual Riemannian 𝐿2 inner product on differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that a basis for Ω(𝑋, C) with respect to the inverse metric can be constructed from local orthonormal frames using a smooth partition of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ℎ𝑞 denote Woronowicz’s Haar state on O𝑞(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (O𝑞(CP1), Ω𝑞(CP1), d𝑞) is a nc Kähler manifold à la Ó Buachalla [79, §§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4], it admits a canonical Riemannian geometry (★𝑞, ℎ𝑞↾O𝑞(CP1)), where ★𝑞(1) � i𝑒+𝑒− and ★𝑞 restricts to ±i id on O𝑞(SU(2))∓2 · 𝑒±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that ★𝑞 recovers Zampini’s modification [99, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14] of Majid’s Hodge operator [67, §4] for the choice of parameter 𝛼′′ = −𝑞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) is satisfied by a theorem of Nagy [75], which shows that the Haar state ℎ𝑞 remains faithful on 𝐶𝑞(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Just as in the classical case, we may now equip Ω𝐵 with an 𝐿2-inner product and compute the (formal) adjoint of the exterior derivative d𝐵 in terms of the Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 48 BRANIMIR ĆAĆIĆ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 (Ó Buachalla [79, §§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2–3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★, 𝜏) be a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝑔 be the resulting inverse metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Ω𝐵 defines a 𝐵-self-correspondence of finite type with respect to 𝑔 that decomposes as an orthogonal direct sum Ω𝐵 = �𝑁 𝑘=0 Ω𝑘 𝐵 of sub-𝐵-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the C-vector space Ω𝐵 defines a separable pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 defined by ∀𝜔, 𝜂 ∈ Ω𝐵, ⟨𝜔, 𝜂⟩𝜏 � 𝜏(𝑔(𝜔, 𝜂)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7) with respect to which the left 𝐵-module structure on Ω𝐵 defines an isometric ∗-representation of 𝐵, the direct sum decomposition Ω𝐵 = �𝑁 𝑘=0 Ω𝑘 𝐵 is orthogonal, the Hodge operator ★ is unitary, and the operator d𝐵 is adjointable with adjoint d∗ 𝐵 = ★−1 ◦ d𝐵 ◦ ★ ◦ 𝛾𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Relative to the references (compare the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29 below), it remains to show that Ω𝐵 is separable as a pre-Hilbert space and that the left 𝐵-module structure is isometric as a ∗-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔅 be the 𝐶∗-algebraic completion of 𝐵, so that 𝜏 extends to a state on 𝔅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, and let (𝑒𝑖)𝑛 𝑖=1 be a basis for Ω𝑚 𝐵 with respect to 𝑔, so that the matrix 𝑋 � �𝑔(𝑒𝑖, 𝑒𝑗)�𝑛 𝑖,𝑗=1 ∈ 𝑀𝑛(𝐵) is positive with unique positive square root √ 𝑋 ∈ 𝑀𝑛(𝔅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑎 � (𝑎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑎𝑛) ∈ 𝐵𝑛 ⊂ 𝔅𝑛 and set 𝜔 � �𝑛 𝑖=1 𝑒𝑖𝑎𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ⟨𝜔, 𝜔⟩𝜏 = 𝜏 �∑︁𝑛 𝑖,𝑗=1 𝑎∗ 𝑖 𝑔(𝑒𝑖, 𝑒𝑗)𝑎𝑗 � = 𝜏((𝑎, 𝑋𝑎)𝐵𝑛) ≤ 𝜏 ���� √ 𝑋 ��� 2 (𝑎, 𝑎)𝔅𝑛 � ≤ ∥𝑋∥ ∑︁𝑛 𝑖=1∥𝑎𝑖∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝐵 is separable as a normed vector space and since (𝑒𝑖)𝑚 𝑖=1 generates Ω𝑚 𝐵 as a right 𝐵- module, it follows that Ω𝑚 𝐵 is separable as a pre-Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the finite orthogonal direct sum Ω𝐵 = �𝑁 𝑚=0 Ω𝑚 𝐵 is also separable as a pre-Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us now show that the left 𝐵-module structure 𝜋 : 𝐵 → L(Ω𝐵) is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑏 ∈ 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the 𝜋 is bounded, it necessarily contractive, so that, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6), it follows that ∥𝑏∥2 ≥ ∥𝜋(𝑏)2∥ ≥ sup{⟨𝜋(𝑏)𝑎, 𝜋(𝑏)𝑎⟩𝜏 | 𝑎 ∈ 𝐵, ⟨𝑎, 𝑎⟩𝜏 ≤ 1} = ∥𝑏∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Given a Riemannian geometry (★𝐵, 𝜏) on our nc manifold (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), we may now consider the Yang–Mills equation d∗ 𝐵F[𝐸,∇𝐸] = 0 for unknown [𝐸, ∇𝐸] ∈ DPic(𝐵), where d𝐵F[𝐸,∇𝐸] = 0 is automatically satisfied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in fact, for any closed self-adjoint j ∈ Z(Ω𝐵)3, we may consider the (Euclidean) Maxwell’s equation d∗ 𝐵F[𝐸,∇𝐸] = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following nc Hodge decomposition theorem will allow us to apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35 to the construction of moduli spaces of solutions in a fixed topological sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that a symmetric operator 𝑆 on a pre-Hilbert space His Fréchet- diagonalisable with spectral gap whenever the following all hold: (1) there is a countable maximal orthonormal subset of Hconsisting of eigenvectors of 𝑆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) the vector space Hdefines a Fréchet space with respect to the countable family of pre- Hilbert space norms (∥ · ∥𝑘)𝑘∈N0 defined by ∀𝑘 ∈ N0, ∀𝜉 ∈ H, ∥𝜉∥𝑘 � �∑︁𝑘 𝑚=0⟨𝑆𝑚𝜉, 𝑆𝑚𝜉⟩ �1/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8) (3) the non-zero eigenvalues of 𝑆 are bounded away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 (Ó Buachalla [79, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2], Ó Buachalla–Šťoviček–Van Roosmalen [80, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Kustermans–Murphy–Tuset [63, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' suppose that d𝐵 + d∗ 𝐵 is diagonalisable or Fréchet-diagonalisable with spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For each 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, the pre-Hilbert space Ω𝑚 𝐵 decomposes orthogonally as Ω𝑚 𝐵 = d𝐵(Ω𝑚−1 𝐵 ) ⊕ �Ω𝑚 𝐵 ∩ ker(d𝐵 + d∗ 𝐵)2� ⊕ d∗ 𝐵(Ω𝑚+1 𝐵 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 49 The case where d𝐵 + d∗ 𝐵 is diagonalisable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', admits an Hamel basis for Ω𝐵 consisting of eigenvectors) is given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The case where d𝐵 + d∗ 𝐵 is Fréchet-diagonalisable with spectral gap is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑉 be a pre-Hilbert space, let 𝑠 : 𝑉 → 𝑉 be an adjointable complex-linear map satisfying 𝑠2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝑆 � 𝑠 + 𝑠∗ is Fréchet-diagonalisable with spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝑉 decomposes orthogonally as 𝑉 = ran 𝑠 ⊕ ker 𝑆 ⊕ ran 𝑠∗, where ker 𝑠 = ran 𝑠 ⊕ ker 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔙 denote the Hilbert space completion of 𝑉, let (𝜉𝑖)𝑖∈N be an orthonormal basis for 𝔙 consisting of eigenvectors of 𝑆 in 𝑉, and let Vdenote the algebraic span of (𝜉𝑖)𝑖∈N, so that 𝑆 is essentially self-adjoint on V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since 𝑆 is assumed to be Fréchet-diagonalisable with spectral gap, it follows that 𝑉 is the Fréchet space of smooth vectors for the unique self-adjoint extension 𝑆 of 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, if we set 𝔙 � 𝔙0 and, for 𝑘 ∈ N, set 𝔙𝑘 to be the Hilbert space completion of 𝑉 with respect to the inner product ⟨·, ·⟩𝑘 � � (𝑣, 𝑤) ↦→ �𝑘 𝑚=0⟨𝑆𝑚𝑣, 𝑆𝑚𝑤⟩ � , then the Fréchet space 𝑉 is the projective limit of the decreasing countable family of Hilbert spaces (𝔙𝑘)𝑘∈N0 with contractive inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, a subspace 𝑋 of 𝑉 is Fréchet-closed if and only if 𝑋 = �∞ 𝑘=0 𝑋 𝔙𝑘, where, for each 𝑘 ∈ N0, we denote by 𝑋 𝔙𝑘 the closure of 𝑋 in 𝔙𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝑃 : 𝔙 → 𝔙be the orthogonal projection onto ker 𝑆, so that id −𝑃 is the orthogonal projection onto the closure in 𝔙 of ran 𝑆 For each 𝑖 ∈ N, let 𝜆𝑖 denote the eigenvalue of 𝑆 corresponding to 𝜉𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, let I � {𝑖 ∈ N | 𝜆𝑖 ≠ 0}, so that 𝐶 � sup𝑖∈I 𝜆−1 𝑖 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we may define a Fréchet-continuous map 𝑅 : 𝑉 → 𝑉 by 𝑅 � �𝑣 ↦→ � 𝑖∈I 𝜆−1 𝑖 ⟨𝜉𝑖, 𝑣⟩𝜉𝑖 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since id −𝑅𝑆 = id −𝑆𝑅 = 𝑃↾𝑉, it follows, in particular, that ran 𝑆 is Fréchet-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let us look at the kernel of 𝑠 and the range of 𝑠∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' recall that 𝑠2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, one can show that ⟨𝑆𝑘𝑠𝑣, 𝑆𝑘𝑠𝑣⟩ + ⟨𝑆𝑘𝑠∗𝑣, 𝑆𝑘𝑠∗𝑣⟩ = ⟨𝑆𝑘+1𝑣, 𝑆𝑘+1𝑣⟩ for all 𝑘 ∈ N and 𝑣 ∈ 𝑉, so that 𝑠 is Fréchet continuous, and hence ker 𝑠 is Fréchet-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, one can also show that ⟨𝑆𝑘𝑠𝑣, 𝑆𝑘𝑠∗𝑤⟩ = 0 for all 𝑘 ∈ N and 𝑣, 𝑤 ∈ 𝑉, so that ran 𝑆 = ran 𝑠 ⊕ ran 𝑠∗, where, for each 𝑘 ∈ N the direct sum is orthogonal with respect to the inner product ⟨·, ·⟩𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it now follows that ran 𝑠 = �∞ 𝑘=0 ran 𝑠𝔙𝑘 is Fréchet-closed since ran 𝑠 ⊕ ran 𝑠∗ = ran 𝑆 = ∞ � 𝑘=0 ran 𝑆 𝔙𝑘 = ∞ � 𝑘=0 ran 𝑠𝔙𝑘 ⊕ ran 𝑠∗𝔙𝑘 = ∞ � 𝑘=0 ran 𝑠𝔙𝑘 ⊕ ∞ � 𝑘=0 ran 𝑠∗𝔙𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, by the proof of [80, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3], we know that V ⊆ ker 𝑠 ⊕ran 𝑠∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since Vis Fréchet– dense in 𝑉 and ker 𝑠 and ran 𝑠∗ are both Fréchet–closed, it follows that 𝑉 = ker 𝑠 ⊕ ran 𝑠∗, which suffices by the proof of [80, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The canonical Riemannian geometry (★, 𝜏) on 𝐶∞ 𝜃 (T2) with respect to the canonical ∗-exterior algebra (Ω𝜃(T2), d) is given by ★(1) � 𝑒1𝑒2, ★(𝑒1) � 𝑒2, ★(𝑒2) � −𝑒1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∀(𝑚, 𝑛) ∈ Z2, 𝜏(𝑈𝑚𝑉 𝑛) � 𝛿𝑚,0𝛿 𝑛,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' so that 𝜏 is the canonical U(1)-invariant trace on 𝐶∞ 𝜃 (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since Ω𝜃(T2) = 𝐶∞ 𝜃 (T2) ·C[𝑒1, 𝑒2], where 𝐶∞ 𝜃 (T2) is the Fréchet completion of � 𝑚,𝑛∈Z C · 𝑢𝑚𝑣𝑛 with respect to the family of seminorms induced by −(𝛿2 1 + 𝛿2 2) = (d + d∗)2↾𝐶∞ 𝜃 (T2), an elementary calculation shows that d + d∗ is Fréchet-diagonalisable with spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we construct moduli spaces of U(1)-instantons—more generally, solutions to Maxwell’s equations—with fixed topological sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the classical case, the Picard group of line bundles up to isomorphism is canonically isomorphic to integral singular cohomology in degree 2, so that a choice of topological sector can be equivalently specified with respect to either group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, in the absence of well-defined singular cohomology for nc topological spaces, we define a choice of topological sector to be a choice of 𝑐 ∈ Pic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 50 BRANIMIR ĆAĆIĆ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), and suppose that d𝐵 + d∗ 𝐵 is diagonalisable or Fréchet-diagonalisable with spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let � Diff00(𝐵) � � (𝜔, 𝜙) ∈ � Diff0(𝐵) ��� d𝐵𝜔 − i𝜔2 = 0 � ≤ � Diff0(𝐵), and suppose that � Diff0(𝐵) itself satisfies ∀(𝜔, 𝜙) ∈ � Diff0(𝐵), d𝐵𝜔 − i𝜔2 ∈ d𝐵 �Z(Ω𝐵)1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9) Finally, let 𝑐 ∈ Pic(𝐵) and j ∈ Ω1 𝐵 be given, let A(𝑐, j) � � [𝐸, ∇𝐸] ∈ DPic(𝐵) �� [𝐸] = 𝑐, d∗ 𝐵F[𝐸,∇𝐸] = j � , and suppose that A(𝑐, j) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the curvature 1-cocycle F is constant on A(𝑐, j) and the group � Diff00(𝐵)/� Ad�U(Z(𝐵)0)� acts freely and transitively on A(𝑐, j) by ∀(𝜔, 𝜙) ∈ � Diff00(𝐵), ∀[𝐸, ∇𝐸] ∈ A(𝑐, j), [𝜔, 𝜙] ⊲ [𝐸, ∇𝐸] � [ˆ𝜏(𝜔, 𝜙)][𝐸, ∇𝐸].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, we show that the curvature 1-cocycle F : DPic(𝐵) → Ω2 𝐵 ∩ ker d𝐵 descends to a map c : ran ΠPic(𝐵) → 𝐻2 dR(𝐵), where ΠPic(𝐵) : DPic(𝐵) → Pic(𝐵) is the forgetful homomorphism and 𝐻2 dR(𝐵) � (Ω2 𝐵 ∩ ker d𝐵)/d𝐵(Ω1 𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let [𝐸, ∇𝐸], [𝐹, ∇𝐹] ∈ DPic(𝐵), and suppose that [𝐸] = [𝐹].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35, there exists (𝜔, 𝜙) ∈ � Diff0(𝐵), such that [𝐸, ∇𝐸][𝐹, ∇𝐹]−1 = [ˆ𝜏(𝜔, 𝜙)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' But now, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9), d𝐵𝜔 − i𝜔2 = d𝐵𝛽 for some 𝛽 ∈ Z(Ω𝐵)1, so that F[𝐸,∇𝐸] − F[𝐹,∇𝐹 ] = ˆΦ−1 [𝐹,∇𝐹 ] �F[ˆ𝜏(𝜔,𝜙)] � = ˆΦ−1 [𝐹,∇𝐹 ] �𝜙−1(d𝐵𝛽)� = d𝐵 � ˆΦ−1 [𝐹,∇𝐹 ] ◦ 𝜙−1(𝛽) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 together with our hypothesis on d𝐵 + d∗ 𝐵, we obtain a vector space isomorphism �𝜔 ↦→ ([𝜔], d∗ 𝐵𝜔)� : ker � d𝐵↾Ω2 𝐵 � → 𝐻2 dR(𝐵) ⊕ d∗ 𝐵(Ω1 𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the map F is constant on A(𝑐, j) with value F𝑐,j uniquely determined by ([F𝑐,j], d∗ 𝐵F𝑐,j) = (c([𝐸]), j), so that, in particular, A(𝑐, j) = � [𝐸, ∇𝐸] ∈ DPic(𝐵) �� [𝐸] = 𝑐, F[𝐸,∇𝐸] = F𝑐,j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let 𝑐 ∈ Pic(𝐵), j ∈ Ω1 𝐵, and [𝐸, ∇𝐸] ∈ DPic(𝐵) with F[𝐸,∇𝐸] = 0 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' we show that A([𝐸] · 𝑐, j) = [𝐸] · A(𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that [𝐹, ∇𝐹] ∈ A(𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then [𝐸, ∇𝐸][𝐹, ∇𝐹] satisfies both ΠPic(𝐵) ([𝐸, ∇𝐸][𝐹, ∇𝐹]) = [𝐸]𝑐 and d∗ 𝐵 �F[𝐸,∇𝐸] [𝐹,∇𝐹 ] � = d∗ 𝐵 � ˆΦ[𝐹,∇𝐹 ](0) + F[𝐹,∇𝐹 ] � = j, so that [𝐸, ∇𝐸][𝐹, ∇𝐹] ∈ A([𝐸]𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since F[𝐸,∇𝐸]−1 = − ˆΦ[𝐸,∇𝐸] �F[𝐸,∇𝐸] � = 0, we similarly find that [𝐸, ∇𝐸]−1[𝐺, ∇𝐺] ∈ A(𝑐, j) for every [𝐺, ∇𝐺] ∈ A([𝐸] · 𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝑐 ∈ Pic(𝐵) and j ∈ Ω1 𝐵 be given, and suppose that A(𝑐, j) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let (𝜔, 𝜙) ∈ � Diff00(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since [ˆ𝜏(𝜔, 𝜙)] = [𝐵] = 1 and F[ˆ𝜏(𝜔,𝜙)] = 0, it now follows that [ˆ𝜏(𝜔, 𝜙)] · A(𝑐, j) = A(𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, let [𝐸, ∇𝐸], [𝐹, ∇𝐹] ∈ A(𝑐, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ΠPic(𝐵) ([𝐸, ∇𝐸][𝐹, ∇𝐹]−1) = 𝑐𝑐−1 = 1 and F[𝐸,∇𝐸] [𝐹,∇𝐹 ]−1 = ˆΦ[𝐹,∇𝐹 ] �F[𝐸,∇𝐸] − F[𝐹,∇𝐹 ] � = ˆΦ[𝐹,∇𝐹 ] �F𝑐,j − F𝑐,j � = 0, hence [𝐸, ∇𝐸][𝐹, ∇𝐹]−1 ∈ 𝜋0(ˆ𝜏) � � Diff00(𝐵) � by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻1(𝑋, R)Z ≤ 𝐻1(𝑋, R) be the lattice of integral classes, so that 𝐻1(𝑋, R)Z = {[−i d𝑓 · 𝑓 −1] | 𝑓 ∈ 𝐶∞(𝑋, U(1))} by the isomorphism ([𝑓] ↦→ 𝑓∗) : [𝑋, U(1)] → Hom(𝜋1(𝑋), 𝜋1(U(1))) � 𝐻1(𝑋, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that d + d∗ is the usual Hodge–de Rham operator, which is Fréchet–diagonalisable with spectral gap by the theory of elliptic regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Fix c ∈ 𝐻2(𝑋, Z) and j ∈ Ω1(𝑋, R) and let 𝔄(c, j) � � [E, ∇E] ∈ ˇ𝐻2(𝑋) �� 𝑐1(E) = c, d∗�tr ∇2 E � = j � , where 𝑐1 : Pic(𝑋) → 𝐻2(𝑋, Z) denotes the integral first Chern class on line bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' thus, the set 𝔄(c, j) is the moduli space of gauge equivalence classes of solutions in the instanton NONCOMMUTATIVE U(1)-GAUGE THEORY 51 sector c to Maxwell’s equations with current 1-form j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑐 ∈ Pic(𝐶∞(𝑋)) be the preimage of (id, c) under the isomorphism Pic(𝑋) → Diff(𝑋) ⋉ 𝐻2(𝑋, Z) induced by the integral first Chern class and the isomorphism Diff(𝑋) ⋉ Pic(𝑋) → Pic(𝐶∞(𝑋)) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝔄(c, j) is non-empty if and only if A(𝑐, j) is non-empty, in which case the bijection ([E, ∇E] ↦→ [Γ(E), ∇E]) : 𝔄(c, j) → A(𝑐, j) and the group isomorphism �[𝛽] + 𝐻1(𝑋, R)Z ↦→ [(𝛽, id)]� : 𝐻1(𝑋, R)/𝐻1(𝑋, R)Z → � Diff00(𝐶∞(𝑋))/� Ad(U(𝐶∞(𝑋))) combine to recover 𝔄(c, j) as a torsor for the torus 𝐻1(𝑋, R)/𝐻1(𝑋, R)Z � T dim 𝐻1(𝑋,R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall from Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31 the homomorphism 𝐸 : Γ𝜃 → Pic(𝐶∞ 𝜃 (T2)) and its lift ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ 𝜃 (T2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, note that � Diff0(𝐶∞(T2)) = SpanR{𝑒1, 𝑒2} × {id} � R2 by a computation of Ćaćić– Karthik [25], so that � Diff0(𝐶∞ 𝜃 (T2)) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, note that Z(𝐶∞ 𝜃 (T2)) = C since 𝜃 is irrational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10, for each 𝑔 ∈ Γ𝜃, the set A([𝐸(𝑔)], 0) is a 2- dimensional real affine space with basepoint [ ˆ𝐸(𝑔)], on which the curvature 1-cocycle F takes the constant value 2𝜋𝑔21 𝑔21𝜃+𝑔22 𝑒1𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note the contrast with Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11 as applied to T2, where a moduli space A(𝑐, j) � 𝔄(c, j), if non-empty, is a T2-torsor instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now generalise the notion of conformal orientation-preserving diffeomorphism to our nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For convenience, let Z>0(𝐵) denote the multiplicative group of all positive invertible elements of Z(Ω𝐵)0, so that Z>0(𝐵) admits a canonical right action of the differential Picard group DPic(𝐵) defined by ∀𝜇 ∈ Z>0(𝐵), ∀[𝐸, ∇𝐸] ∈ DPic(𝐵), 𝜇 ⊳ [𝐸, ∇𝐸] � ˆΦ−1 [𝐸,∇𝐸](𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10) Note from Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39 and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35 that the dynamical content of Hermitian line 𝐵-bimodule with connection is encoded by its generalised braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we promote the behaviour of the usual Hodge star operator under orientation-preserving conformal diffeomorphisms [17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='h] into the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal when there exists (necessarily unique) 𝜇 ∈ Z>0(𝐵), the conformal factor of (𝐸, 𝜎𝐸, ∇𝐸), such that ∀𝑥 ∈ 𝐸, ∀𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ∀𝛼 ∈ Ω𝑘 𝐵, 𝜎𝐸(★𝐵(𝛼) ⊗ 𝑥) = 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ � 𝜇𝑁−2𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='11) We denote by DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) the strictly full subcategory of DPic(𝐵) whose objects are ★𝐵- conformal, we denote by DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) the corresponding subset of DPic(𝐵), and we define 𝜇 : DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) → Z>0(𝐵) to be the function that maps [𝐸, ∇𝐸] ∈ DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) to the conformal factor 𝜇[𝐸,∇𝐸] of any (and hence every) representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the classical case, orientation-preserving conformal diffeomorphisms form a group and their conformal factors define a multiplicative 1-cocycle on this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The same is true in the noncommutaitve setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that ★𝐵 is a Hodge operator on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) defines a sub-2-group of DPic(𝐵), the subset DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) defines a subgroup of DPic(𝐵), and the function 𝜇 : DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) → Z>0(𝐵) defines a group 1-cocycle with respect to the restriction to DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) of the right DPic(𝐵)-action on Z>0(𝐵) defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 52 BRANIMIR ĆAĆIĆ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, note that the monoidal unit (𝐵, 𝜎𝐵, ∇𝐵) is trivially ★𝐵-conformal with confor- mal factor 𝜇[𝐵,∇𝐵] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, suppose that (𝐸, 𝜎𝐸, ∇𝐸) and (𝐹, 𝜎𝐹, ∇𝐹) are ★𝐵- conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, given 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, 𝛼 ∈ Ω𝑘 𝐵, 𝑥 ∈ 𝐸, and 𝑦 ∈ 𝐹, 𝜎𝐸⊗𝐵𝐹 (★𝐵(𝛼) ⊗ (𝑥 ⊗ 𝑦)) = � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜎𝐹 � ★𝐵(𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩)𝜇𝑁−2𝑘 [𝐸,∇𝐸] ⊗ 𝑦 � ⟨0⟩ � ⊗ 𝜎𝐹 � ★𝐵(𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩)𝜇𝑁−2𝑘 [𝐸,∇𝐸] ⊗ 𝑦 � ⟨1⟩ = � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜎𝐹 � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ ⊗ 𝑦 � ⟨0⟩ � ⊗ ★𝐵 � 𝜎𝐹 (𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ ⊗ 𝑦) ⟨1⟩ � ˆΦ−1 [𝐹,∇𝐹 ] (𝜇𝑁−2𝑘 [𝐸,∇𝐸])𝜇𝑁−2𝑘 [𝐹,∇𝐹 ] = 𝜎𝐸⊗𝐵𝐹 (𝛼 ⊗ (𝑥 ⊗ 𝑦)) ⟨0⟩ ⊗ ★𝐵 � 𝜎𝐸⊗𝐵𝐹 (𝛼 ⊗ (𝑥 ⊗ 𝑦)) ⟨1⟩ � � ˆΦ−1 [𝐹,∇𝐹 ] (𝜇[𝐸,∇𝐸])𝜇[𝐹,∇𝐹 ] �𝑁−2𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the subcategory DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) is closed under the monoidal product and the map 𝜇 satisfies the required 1-cocycle identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, suppose that (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, given 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, 𝛼 ∈ Ω𝑘 𝐵, and 𝑥 ∈ 𝐸, 𝜎𝐸(★𝐵(𝛼) ⊗ 𝑥) = 𝜎−1 𝐸 (𝑥 ⊗ ★𝐵(𝛼)∗) ⟨0⟩ ⊗ 𝜎−1 𝐸 (𝑥 ⊗ ★𝐵(𝛼)∗) ∗ ⟨−1⟩ = 𝜎−1 𝐸 (𝑥 ⊗ 𝛼∗) ⟨0⟩𝜇−𝑁+2𝑘 [𝐸,∇𝐸] ⊗ ★𝐵 � 𝜎−1 𝐸 (𝑥 ⊗ 𝛼∗) ∗ ⟨−1⟩ � = 𝜇−𝑁+2𝑘 [𝐸,∇𝐸]𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ � = 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨0⟩ ⊗ ★𝐵 � 𝜎𝐸(𝛼 ⊗ 𝑥) ⟨1⟩ � ˆΦ[𝐸,∇𝐸](𝜇−1 [𝐸,∇𝐸])𝑁−2𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the the subcategory DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) is also closed under monoidal inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Given a Hodge operator ★𝐵 on (Ω𝐵, d𝐵), we may therefore call DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) the conformal sub-2-group of DPic(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, for every Hermitian line bundle with unitary connection (E, ∇E) on 𝑋, (Γ(E), flip, ∇E) is ★𝑔-conformal with confor- mal factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other, for every 𝜙 ∈ Diff(𝑋), ˆ𝜏(0, (𝜙−1)∗) is ★𝑔-conformal if and only if 𝜙 ∈ Conf+(𝑋, 𝑔), in which case 𝜇 ◦ 𝜋0(ˆ𝜏)(0, (𝜙−1)∗) = √︃ 𝜙∗𝑔 𝑔 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the isomorphism of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39 restricts to an isomorphism Conf+(𝑋, 𝑔) ⋉ ˇ𝐻2(𝑋) → DPic(𝐶∞(𝑋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝑔), with respect to which 𝜇 : DPic(𝐶∞(𝑋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝑔) → 𝐶∞(𝑋, (0, ∞)) reduces to the map � (𝜙, [E, ∇E]) ↦→ √︃ 𝜙∗𝑔 𝑔 � : Conf+(𝑋, 𝑔) ⋉ ˇ𝐻2(𝑋) → 𝐶∞(𝑋, (0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In light of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28 and Proposition-Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32, we may equivalently consider conformality of horizontally differentiable quantum principal U(1)-bundles over 𝐵 with respect to a given Hodge operator on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) be a hori- zontally differentiable quantum principal U(1)-bundle over 𝐵 with Fröhlich automorphism ˆΦ𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' define a right Z-action on Z>0(𝐵) by ∀𝜇 ∈ Z>0(𝐵), ∀𝑘 ∈ Z, 𝜇 ⊳ 𝑘 � ˆΦ−𝑘 𝑃 (𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12) We say that (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal if there exists a (necessarily unique) group 1-cocycle 𝜇𝑃 : Z → Z>0(𝐵), the conformal factor of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor), such that ∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝛼 ∈ Ω𝑚 𝐵, ∀𝑝 ∈ 𝑃𝑗, ★𝐵(𝛼)𝑝 = ˆℓ𝑃(𝛼𝑝) ⟨0⟩★𝐵 � ˆℓ𝑃(𝛼𝑝) ⟨1⟩ � 𝜇𝑃(𝑗)𝑁−2𝑘, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13) NONCOMMUTATIVE U(1)-GAUGE THEORY 53 where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is the 𝐵-bimodule isomorphism of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We denote by DCirchor(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) the strictly full subcategory of DCirchor(𝐵) with ★𝐵-conformal objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let ★𝐵 be a Hodge operator on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Hom(Z, DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵)) is the essential image of DCirchor(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) under the functor ˆL : DCirchor(𝐵) → Hom(Z, DPic(𝐵)), so that the composite functor 𝜖1 ◦ ˆL : DCirchor(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) → DPic(𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★𝐵) is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, if (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is a ★𝐵-conformal horizontally differentiable quantum principal U(1)-bundle over 𝐵, then its conformal factor 𝜇𝑃 satisfies 𝜇𝑃 = 𝜇 ◦ 𝜋0 � ˆL(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='14) Thus, a Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal if and only if (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) � (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) ⋊(𝐸,𝜎𝐸,∇𝐸) Z is ★𝐵-conformal, in which case, the conformal factor 𝜇𝑃 of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is uniquely determined by 𝜇𝑃(1) = 𝜇[𝐸,∇𝐸].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The lifting problem for Riemannian structures via Hodge operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now at- tack the problem of lifting Riemannian geometries in terms of Hodge operators to the total spaces of nc principal U(1)-bundles with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The existence of such lifts will be entirely governed by conformality in our nc sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' As a result, the resulting lifted Riemannian geometries necessarily involve modular phenomena in both vertical and horizontal directions that are generally non-trivial and distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, let 𝜅 > 0, let (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) be a 𝜅-differentiable quantum principal U(1)- bundle with connection over 𝐵, let 𝜗 be the connection 1-form of Π, and let ˆΦ𝑃 be the Fröhlich automorphism of Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) = (𝑃, Ω𝑃,hor, d𝑃,hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with a general definition of U(1)-equivariant Hodge operator on a total space that draws on standard requirements imposed in the classical case: that the canonical surjection onto the base be a Riemannian submersion, that the principal Ehresmann connection be fibrewise orthogonal, that the fibres all have unit length, and that the total space have the ’fibre-first’ orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We go beyond the definitions proposed by Kustermans–Murphy–Tuset [63] by carefully controlling failure of the Hodge operator to be right linear and ∗-preserving in terms of (possibly distinct) modular automorphisms in the vertical and horizontal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, we define a modular automorphism of Ω𝑃 is a U(1)-equivariant automor- phism Δ of Ω𝑃 as a unital graded C-algebra satisfying Δ↾ΩU(1) 𝑃 = id and ∀𝑗 ∈ Z, ∀𝑝 ∈ 𝑃𝑗, 𝑝∗Δ(𝑝) ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15) for example, given 𝑡 ∈ (0, ∞), we may define a modular automorphism Λ𝑡 of Ω𝑃 by ∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ Ω𝑚 𝑗 , Λ𝑡(𝜔) � 𝑡−𝑗𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16) On the other hand, we use the connection Π to define a convenient bigrading (Ω𝑗,𝑘 𝑃 )(𝑗,𝑘) ∈N2 0 of Ω𝑃 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For each 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, let Ω0,𝑘 𝑃 � Π(Ω𝑘 𝑃) = Ω𝑘 𝑃,hor, Ω1,𝑘 𝑃 � (id −Π)(Ω𝑘+1 𝑃 ) = 𝜗 · Ω𝑘 𝑃,hor, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17) and for (𝑗, 𝑘) ∉ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, set Ω𝑗,𝑘 𝑃 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the family (Ω𝑗,𝑘 𝑃 )(𝑗,𝑘) ∈N2 0 satisfies: ∀𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 + 1}, 1 � 𝑗=0 𝑚−1 � 𝑘=0 Ω𝑗,𝑘 𝑃 = Ω𝑚 𝑃 , ∀(𝑗1, 𝑘1), (𝑗2, 𝑘2) ∈ N2 0, Ω𝑗1,𝑘1 𝑃 Ω𝑗2,𝑘2 𝑃 = Ω𝑗1+𝑗2,𝑘1+𝑘2 𝑃 , ∀(𝑗, 𝑘) ∈ N2 0, � Ω𝑗,𝑘 𝑃 �∗ = Ω𝑗,𝑘 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 54 BRANIMIR ĆAĆIĆ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (Δver, Δhor) be a commuting pair of modular automorphisms of Ω𝑃 that commute with Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to Π is a U(1)-equivariant left 𝑃-linear map that commutes with both Δver and Δhor, satisfies ∀(𝑗, 𝑘) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ★ � Ω𝑗,𝑘 𝑃 � ⊆ Ω1−𝑗,𝑁−𝑘 𝑃 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) ∀𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 + 1}, ★2↾Ω𝑚 𝑃 = (−1)𝑚(𝑁+1−𝑚) idΩ𝑚 𝑃 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19) and satisfies, for every (𝑗, 𝑘) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ∀𝑝 ∈ 𝑃, ∀𝜔 ∈ Ω𝑗,𝑘 𝑃 , ★(𝜔𝑝) = ★(𝜔) · (Δ2𝑘−𝑁 hor Δ2𝑗−1 ver )(𝑝), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20) ∀𝜔 ∈ Ω𝑗,𝑘 𝑃 , ★(𝜔)∗ = ★ � (Δ2𝑘−𝑁 hor Δ2𝑗−1 ver )(𝜔)∗� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21) ∀𝜔 ∈ Ω𝑗,𝑘 𝑃 , ★ � 𝛿 𝑗,0𝜔 � = (−1)𝑁−𝑘★(𝜔𝜗)𝜗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22) ∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 𝑃 , 𝜔 · ★(𝜂) = ★−1(𝜔) · (Δ2𝑘−𝑁 hor Δ2𝑗−1 ver )(𝜂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23) Hence, in this case, the inverse metric induced by the (Δver, Δhor)-modular Hodge operator ★ is the R-bilinear map 𝑔 : Ω𝑃 × Ω𝑃 → 𝑃 defined by ∀𝜔, 𝜂 ∈ Ω𝑃, 𝑔(𝜔, 𝜂) � ★(𝜔∗ · ★(𝜂)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24) Notwithstanding the appearance of modular automorphisms, the following properties of inverse metrics will suffice for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Δver and Δhor be a commuting pair of modular automorphisms of Ω𝑃 that commute with Π, and let ★ be a (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the inverse metric 𝑔 : Ω𝑃 × Ω𝑃 → 𝑃 is U(1)-equivariant in the sense that ∀(𝑚, 𝑗), (𝑛, 𝑘) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 𝑃 )𝑗, ∀𝜂 ∈ (Ω𝑛 𝑃)𝑘, 𝑔(𝜔, 𝜂) ∈ 𝑃−𝑗+𝑘, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25) makes Π into an orthogonal projection in the sense that ∀𝜔, 𝜂 ∈ Ω𝑃, 𝑔(𝜔, 𝜂) = 𝑔(Π(𝜔), Π(𝜂)) + 𝑔((id −Π)(𝜔), (id −Π)(𝜂)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26) and satisfies, for each (𝑗, 𝑘) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 𝑃 , ∀𝑝 ∈ 𝑃, 𝑔(𝜔, 𝜂 · 𝑝) = 𝑔(𝜔, 𝜂) · (Δ2𝑗 ver ◦ Δ2𝑘 hor)(𝑝), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27) ∀𝜔, 𝜂 ∈ Ω𝑗,𝑘 𝑃 , 𝑔(𝜔, 𝜂)∗ = (Δ2𝑗 ver ◦ Δ2𝑘 hor)(𝑔(𝜂, 𝜔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The non-trivial claims are equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let 𝜔, 𝜂 ∈ Ω𝑃 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since (id −Π)(Ω𝑃)2 = 0, it now follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) that ★(𝑔(𝜔, 𝜂)) = (Π(𝜔∗) + (id −Π)(𝜔∗)) ★ (Π(𝜂) + (id −Π)(𝜂)) = (Π(𝜔∗)) ★ (Π(𝜂)) + (id −Π)(𝜔∗)) ★ ((id −Π)(𝜂)) = ★(𝑔(Π(𝜔), Π(𝜂)) + 𝑔((id −Π)(𝜔), (id −Π)(𝜂))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, let (𝑗, 𝑘) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} and 𝜔, 𝛽 ∈ Ω𝑗,𝑘 𝑃 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23), ★(𝑔(𝜔, 𝜂)∗) = Δver ◦ Δ𝑁 hor(★(𝑔(𝜔, 𝜂))∗) = Δver ◦ Δ𝑁 hor � (−1)(𝑗+𝑘) (𝑁+1−𝑗−𝑘)★(𝜂)∗𝜔 � = Δver ◦ Δ𝑁 hor � (−1)(𝑗+𝑘) (𝑁+1−𝑗−𝑘) (★ ◦ Δ2𝑗−1 ver ◦ Δ2𝑘−𝑁 hor )(𝜂∗) · 𝜔 � = Δ2𝑗 ver ◦ Δ2𝑘 hor(𝜂∗ · ★(𝜔)) = ★ � Δ2𝑗 ver ◦ Δ2𝑘 hor(𝑔(𝜂, 𝜔)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ NONCOMMUTATIVE U(1)-GAUGE THEORY 55 At last, the following definitions give our proposed notion of lifted Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) to be a quadruple (Δver, Δhor, ★, 𝜏), where (Δver, Δhor) is a commuting pair of modular automorphisms of Ω𝑃 that commute with Π, where ★ is a (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to Π whose inverse metric restricts, for each (𝑚, 𝑗) ∈ N0 × Z, to a 𝐵-valued inner product on (Ω𝑚 𝑃 )𝑗 admits a basis and satisfies ∀𝑏 ∈ 𝐵, ∀𝜔 ∈ (Ω𝑚 𝑃 )𝑗, 𝑔(𝑏𝜔, 𝑏𝜔) ≤ ∥𝑏∥2𝑔(𝜔, 𝜔), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) and where 𝜏 is a U(1)-equivariant bounded state on 𝑃 that satisfies ∀𝜔 ∈ Ω𝑁 𝑃 , (𝜏 ◦ ★ ◦ d)(𝜔) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) ∀𝑝 ∈ 𝑃, sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1} = ∥𝑝∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), and suppose that ∀𝛽 ∈ Ω𝐵, ★(𝜗𝛽) = ★𝐵(𝛽), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32) ∀𝑏 ∈ 𝐵, 𝜏(𝑏) = 𝜏𝐵(𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33) We call (★𝐵, 𝜏𝐵) a restriction of (Δver, Δhor, ★, 𝜏) to (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) and we call (Δver, Δhor, ★, 𝜏) a lift of (★𝐵, 𝜏𝐵) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Our definitions are justified by the following existence and uniqueness theorem, which characterizes existence of lifts in terms of conformality and demonstrates the inexorability of non-trivial modular phenomena outside of an narrow range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Λ𝜅 denote the modular automorphism of Ω𝑃 defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a unique restriction (★𝐵, 𝜏𝐵) of (Δver, Δhor, ★, 𝜏) to (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, it follows that Δver = Λ𝜅 and that (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor 𝜇𝑃 satisfying ∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 𝑃 )𝑗, Δhor(𝜔) = 𝜔𝜇𝑃(𝑗) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) (2) Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), and suppose that (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor 𝜇𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, define a modular automorphism Δhor of Ω𝑃 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' There exists a unique (Λ𝜅, Δhor)-modular Hodge operator ★ on (Ω𝑃, d𝑃) with respect to Π and faithful U(1)-equivariant bounded state 𝜏 on 𝑃 making (Λ𝜅, Δhor, ★, 𝜏) into a lift of (★𝐵, 𝜏𝐵) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), namely ∀𝑝 ∈ 𝑃, ∀𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ∀𝛽 ∈ Ω𝑘 𝐵, ★𝑃(𝑝𝛽) � (−1)𝑘𝑝𝜗★𝐵(𝛽), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35) ∀𝑝 ∈ 𝑃, ∀𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, ∀𝛽 ∈ Ω𝑘 𝐵, ★𝑃(𝑝𝜗𝛽) � 𝑝★𝐵(𝛽), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36) ∀𝑗 ∈ Z, ∀𝑝 ∈ 𝑃𝑗, 𝜏𝑃(𝑝) � 𝜏𝐵 � 𝛿 𝑗,0𝑝 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For every modular automorphism Δ of Ω𝑃, there exists a unique group 1-cocycle 𝜇 : Z → Z>0(𝐵) for the right Z-action defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12), such that ∀(𝑚, 𝑗) ∈ N0 × Z, ∀𝜔 ∈ (Ω𝑚 𝑃 )𝑗, Δ(𝜔) = 𝜔𝜇(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) Conversely, for every group 1-cocycle 𝜇 : Z → Z>0(𝐵), Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38 defines a modular automorphism Δ of Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 56 BRANIMIR ĆAĆIĆ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Δ be a modular automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For each 𝑗 ∈ Z, the map Δ restricts to a 𝐵-bimodule morphism L(𝑃)(𝑗) → L(𝑃)(𝑗), so that, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, there exists unique 𝜇(𝑗) ∈ Z(𝐵) that satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) for 𝑚 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' given any cobasis (𝑒𝑖)𝑁 𝑖=1 for L(𝑃)𝑗, it follows that 0 ≤ �𝑁 𝑖=1 𝑒∗ 𝑖 Δ(𝑒𝑖) = �𝑁 𝑖=1 𝑒∗ 𝑖 𝑒𝑖𝜇(𝑗) = 𝜇(𝑗) and 𝜇(𝑗)𝛼 = �𝑁 𝑖=1 𝑒∗ 𝑖 Δ(𝑒𝑖)𝛼 = �𝑁 𝑖=1 𝑒∗ 𝑖 Δ(𝑒𝑖𝛼) = 𝛼𝜇𝑃(𝑗) for all 𝛼 ∈ Ω𝐵, so that 𝜇𝑃(𝑗) ∈ Z(Ω𝐵)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑗, 𝑘 ∈ Z, for all 𝑥 ∈ 𝑃𝑗, and 𝑦 ∈ 𝑃𝑘, we find that 𝑥𝑦𝜇(𝑗 + 𝑘) = Δ(𝑥𝑦) = Δ(𝑥)Δ(𝑦) = 𝑥𝜇𝑃(𝑗) · 𝑦 · 𝜇𝑃(𝑘) = 𝑥𝑦Φ−𝑘 𝑃 (𝜇(𝑗))𝜇(𝑘), so that 𝜇𝑃(𝑗 + 𝑘) = ˆΦ−𝑘 𝑃 (𝜇𝑃(𝑗))𝜇𝑃(𝑘) by the equality 𝑃𝑗+𝑘 = 𝑃𝑗 · 𝑃𝑘 together with uniqueness of 𝜇𝑃(𝑗 + 𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since Δ acts as the identity on 𝑃0, it follows that 𝜇(0) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, for each 𝑗 ∈ Z, it follows that 𝜇(𝑗) ∈ Z>0(𝐵) with inverse Φ−𝑗 𝑃 (𝜇(−𝑗))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In sum, we obtain a unique group 1-cocycle 𝜇 : Z → Z(𝐵)× ≥0 satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) for 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, since Ω𝑃 = 𝑃 · Ω𝐵 ⊕ 𝑃 · 𝜗 · Ω𝐵 and since Δ acts as the identity on Ω𝐵 and on 𝜗, it follows that 𝜇 : Z → Z>0(𝐵) is the unique group 1-cocycle satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Reversing this argument almost suffices to show that a group 1-cocycle 𝜇 : Z → Z>0(𝐵) defines a modular automorphism Δ by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' all that is left is that 𝑝∗Δ(𝑝) = 𝑝∗𝑝 · 𝜄𝑃(𝜇𝑃(𝑗)) = 𝑝∗Φ𝑗 𝑃(𝜇𝑃(𝑗))𝑝 ≥ 0 for all 𝑗 ∈ Z and 𝑝 ∈ 𝑃𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin by showing that Δver = Λ𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23, let 𝜇 : Z → Z>0(𝐵) be the unique group 1-cocycle satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) with respect to Δ = Δver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Δ2 ver = Λ2 𝜅 since ★(𝑝) = (−1)𝑁★(𝑝𝜗)𝜗 = (−1)𝑁★(𝜗) · (Δ−𝑁 hor ◦ Δver ◦ Λ−1 𝜅 )(𝑝) · 𝜗 = ★(1) · (Δ−𝑁 hor ◦ Δver ◦ Λ−2 𝜅 )(𝑝) = ★�Δ2 ver ◦ Λ−2 𝜅 (𝑝)� for every 𝑝 ∈ 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑗 ∈ Z and let (𝑒𝑖)𝑁 𝑖=1 be a cobasis for L(𝑃)𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜇(𝑗) = 𝜅−𝑗 by positivity of 𝜇(𝑗) together with the calculation 𝜅−2𝑗 = �𝑁 𝑖=1 𝑒∗ 𝑖 Λ2 𝜅(𝑒𝑖) = �𝑁 𝑖=1 𝑒∗ 𝑖 Δ2 ver(𝑒𝑖) = 𝜇(𝑗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that there is a unique Hodge operator ★𝐵 on 𝐵 with respect to (Ω𝐵, d𝐵) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19), there exists a unique C-linear map ★𝐵 : Ω𝐵 → Ω𝐵 satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32), which is given by ★𝐵(𝛽) � ★(𝜗𝛽) for all 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} and 𝛽 ∈ Ω𝑘 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The map ★𝐵 is left 𝐵-linear by construction and U(1)-equivariant by U(1)-equivariance of ★ and U(1)-invariance of 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, since both Λ𝜅 and Δhor act as the identity on Ω𝐵 and on 𝜗 and since 𝜗 supercommutes with Ω𝐵, it follows that ★𝐵 is right 𝐵-linear by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20), is ∗-preserving by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21), satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19), and satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that the pair (★𝐵, 𝜏↾𝐵) defines a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, since both Λ𝜅 and Δhor act trivially on Ω𝐵, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19 together with the 𝑗 = 0 case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29) shows that the inverse metric induced by ★𝐵 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' indeed, for each 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑀}, one can obtain a basis for Ω𝑚 𝐵 from a basis for (Ω𝑚 𝑃 )U(1) by applying Π retaining any non-zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, for every 𝛽 ∈ Ω𝑁−1 𝐵 , since d𝑃(𝜗𝛽) = d𝑃(𝜗)𝛽 − 𝜗d𝐵(𝛽) = −FΠ𝛽 − 𝜗d𝐵(𝛽) = −𝜗d𝐵(𝛽) for FΠ the curvature 2-form of the connection Π, it follows that 𝜏 ◦ ★𝐵 ◦ d𝐵(𝛽) = 𝜏(−★(𝜗d𝐵(𝛽))) = 𝜏 ◦ ★ ◦ d(𝜗𝛽) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23, let 𝜇𝑃 : Z → Z>0(𝐵) be the unique group 1-cocycle satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) and the fact that Δhor(𝜗) = 𝜗, that (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor 𝜇𝑃 is now equivalent to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Uniqueness of 𝜏𝐵 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let (★𝐵, 𝜏𝐵) be a Riemannian geometry on 𝐵 with respect to (Ω𝐵, d𝐵), and suppose that (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃,hor, d𝑃,hor) is ★𝐵-conformal with conformal factor ★𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38, the modular automorphism Δhor of Ω𝑃 constructed from 𝜇𝑃 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that there is a unique (Δver, Δhor)-modular Hodge operator on (Ω𝑃, d𝑃) with respect to Π satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36) define the unique NONCOMMUTATIVE U(1)-GAUGE THEORY 57 U(1)-equivariant left 𝑃-linear map ★ : Ω𝑃 → Ω𝑃 satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, the map ★ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='18) by construction, satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='13), satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1) and left 𝑃-linearity, and satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='21) by the fact that ★𝐵 is ∗-preserving together with left 𝑃-linearity of ★ and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, the map ★ satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23) by a case-by-case application of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2) together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20) and left 𝑃-linearity of ★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let us show that the inverse metric 𝑔 induced by ★ satisfies the requirements in the definition of total Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔𝐵 denote the inverse metric induced by ★𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑚, 𝑗) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} × Z be given, and let (·, ·)𝑗 denote the 𝐵-valued inner product on L(𝑃)(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑝1, 𝑝2 ∈ 𝑃𝑗 and 𝛼1, 𝛼2 ∈ Ω𝑚 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, ★(𝑔(𝑝1𝛼1, 𝑝2𝛼2)) = 𝛼∗ 1𝑝∗ 1★(𝑝2𝛼2) = 𝛼∗ 1𝑝∗ 1𝑝2(−1)𝑁★𝐵(𝛼2)𝜗 = ★�𝑔𝐵(𝛼1, (𝑝1, 𝑝2)𝑗𝛼2)�, while on the other, a similar calculation shows that 𝑔(𝑝1𝛼1𝜗, 𝑝2𝛼2𝜗) = 𝑔𝐵(𝛼1, (𝑝1, 𝑝2)𝑗𝛼2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the 𝐵-bimodule isomorphism ˆℓ𝑃 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 induces 𝐵-bimodule isomorphisms �𝜔 ↦→ ˆℓ𝑃(𝜔)� : (Ω𝑚 𝑃,hor)𝑗 → L(𝑃)(𝑗)⊗𝐵Ω𝑚 𝐵, �𝜔𝜗 ↦→ ˆℓ𝑃(𝜔)� : 𝜗(Ω𝑚 𝑃,hor)𝑗 → L(𝑃)(𝑗)⊗𝐵Ω𝑚 𝐵 that respectively realise the restrictions of 𝑔 to (Ω𝑚 𝑃,hor)𝑗 and 𝜗(Ω𝑚 𝑃,hor)𝑗 as pullbacks of the 𝐵-valued inner product on the tensor product L(𝑃)(𝑗) ⊗𝐵 Ω𝑚 𝐵 of 𝐵-self-correspondences of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, both (Ω𝑚 𝑃,hor)𝑗 = Π((Ω𝑚 𝑃 )𝑗) and 𝜗(Ω𝑚 𝑃,hor)𝑗 = (id −Π)((Ω𝑚+1 𝑃 )𝑗) defines 𝐵-self-correspondences of finite type with respect to 𝑔, which suffices for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let us show that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37) defines the unique U(1)-equivariant bounded state 𝜏 on 𝑃 satisfying 𝜏↾𝐵= 𝜏𝐵 and satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall the bounded faithful conditional expectation E𝑃 : 𝑃 → 𝐵 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, the map 𝜏 : 𝑃 → C defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37) can now be rewritten as 𝜏 = 𝜏𝐵 ◦ E𝑃, which therefore defines a faithful bounded U(1)-equivariant state on 𝑃 restricting to 𝜏𝐵 on 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, by continuity and U(1)-invariance, any faithful bounded U(1)-equivariant state 𝜏′ on 𝑃 satisfying 𝜏′↾𝐵= 𝜏𝐵 must satisfy 𝜏′ = 𝜏′ ◦ E𝑃 = 𝜏𝐵 ◦ E𝑃 = 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, we show that 𝜏 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30) with respect to ★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, let 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗, and 𝛽 ∈ Ω𝑁 𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝛿 𝑗,0d𝑃(𝑝𝛽) = 𝛿 𝑗,0 �2𝜋i[𝑗]𝜅𝜗𝑝𝛽 + d𝑃,hor(𝑝)𝛽 + 𝑝d𝐵𝛽� = 0, it follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35) that 𝜏 ◦ ★ ◦ d(𝑝𝛽) = 𝜏𝐵 ◦ ★�𝛿 𝑗,0d(𝑝𝛽)� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, let 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗, and 𝛼 ∈ Ω𝑁−1 𝐵 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝛿 𝑗,0d𝑃(𝑝𝛼𝜗) = d𝑃 � (𝛿 𝑗,0𝑝)𝛼𝜗 � = d𝐵 � (𝛿 𝑗,0𝑝)𝛼 � 𝜗 + (−1)𝑁 (𝛿 𝑗,0𝑝)𝛼FΠ = d𝐵 � (𝛿 𝑗,0𝑝)𝛼 � 𝜗, it follows by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37) that 𝜏 ◦ ★ ◦ d(𝑝𝛼𝜗) = 𝜏𝐵 ◦ ★ � d𝑃(𝛿 𝑗,0𝑝𝛼𝜗) � = (−1)𝑁𝜏𝐵 ◦ ★𝐵 ◦ d𝐵 � (𝛿 𝑗,0𝑝)𝛼 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, either way, the composition 𝜏 ◦ ★ ◦ d↾Ω𝑁 𝑃 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us now show that 𝜏 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define ∥ · ∥′ : 𝑃 → [0, ∞) by ∀𝑝 ∈ 𝑃, (∥𝑝∥′)2 � sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since ∥ · ∥′ is the operator norm on 𝑃 with respect to the gns representation of 𝑃 induced by the faithful bounded state 𝜏, it follows that ∥ · ∥′ is a 𝐶∗-norm bounded from above by ∥ · ∥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since 𝜏 is U(1)-invariant, it follows that ∥ · ∥′ is a U(1)-invariant 𝐶∗-norm on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19, it suffices to show that 𝜏 satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31) on 𝑃U(1) = 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' But now, given 𝑏 ∈ 𝐵, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6) applied to 𝜏𝐵 that (∥𝑏∥′)2 = sup{𝜏(𝑞∗𝑝∗𝑝𝑞) | 𝑞 ∈ 𝑃, 𝜏(𝑞∗𝑞) ≤ 1} ≥ sup{𝜏(𝑐∗𝑝∗𝑝𝑐) | 𝑞 ∈ 𝐵, 𝜏(𝑐∗𝑐) ≤ 1} = ∥𝑏∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ The construction of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23 will be used frequently enough to warrant the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 58 BRANIMIR ĆAĆIĆ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Equip Z>0(𝐵) with the right Z-action constructed from ˆΦ𝑃 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The symbol of a modular automorphism Δ is the unique group 1-cocycle 𝜇 : Z → Z>0(𝐵) that satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38) with respect to Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, we may use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 to rewrite Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connection that is flat or has vertical deformation parameter 𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (★𝐵, 𝜏𝐵) admits a lift (Δver, Δhor, ★, 𝜏) to (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, Σ𝐵)⋊𝜅,tot (𝐸,𝜎𝐸,∇𝐸)Z if and only if (𝐸, 𝜎𝐸, ∇𝐸) is ★𝐵-conformal, in which case the lift is unique, Δver = Λ𝜅, and Δhor has symbol 𝜇◦(𝑚 ↦→ [𝐸, ∇𝐸]𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, observe that (O𝑞(SU(2)), Ω𝑞,hor(SU(2)), d𝑞,hor) = Hor𝜅(O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞, Π𝑞) is ★𝑞-conformal with conformal factor 𝜇O𝑞(SU(2)) = (𝑘 ↦→ 𝑞−𝑘);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' compare [99, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, recall that the usual basis for the free left O𝑞(SU(2))-module Ω𝑞(SU(2)) is {𝑒0, 𝑒+, 𝑒−}, where 𝑒0 � 2𝜋i𝑞−2𝑒𝑞2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22, the unique lift of (★𝑞, ℎ𝑞↾O𝑞(CP1)) to (O𝑞(SU(2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑞(SU(2)), d𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑞) is (Λ𝑞2, Λ𝑞, �★𝑞, ℎ𝑞), where �★𝑞 is uniquely determined by �★𝑞(1) � 𝑞2 2𝜋 𝑒0𝑒+𝑒−, �★𝑞(𝑒0) � − 2𝜋 𝑞2 𝑒+𝑒−, �★𝑞(𝑒+) � − 𝑞6 2𝜋 𝑒0𝑒+, �★𝑞(𝑒−) � 1 2𝜋𝑞2 𝑒0𝑒−, and ℎ𝑞 is Woronowicz’s Haar state on O𝑞(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In fact, the operator �★𝑞 recovers the Hodge operator of Zampini [99, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20] for the choice of parameters (𝛼′, 𝛽, 𝜈, 𝛾) = ( −2𝜋 𝑞4 , 1, 𝑞−2, 4𝜋2 𝑞4 ), which satisfies his canonical constraints [99, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7] with respect to the choice of parameter 𝛼′′ = −𝑞2 from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' it also necessarily recovers the Hodge operator of Kustermans– Murphy–Tuset [63, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1 et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='] up to suitable rescaling in each respective degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that Λ𝑞2 ≠ Λ𝑞 since 𝑞 ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41, the homomorphism ˆ𝐸 of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31 defines a homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ 𝜃 (T2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ★) satisfying ∀𝑔 ∈ Γ𝜃, 𝜇 ◦ 𝜋0( ˆ𝐸)(𝑔) = (𝑔21𝜃 + 𝑔22)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='17 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='22, the unique lift of (★, 𝜏) from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9 to the real multiplication instanton (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃) is (Λ𝜖2 𝜃, Λ𝜖𝜃, �★,�𝜏), where �★ is determined by �★(1) � 𝑒0𝑒1𝑒2, �★(𝑒0) � 𝑒1𝑒2, �★(𝑒1) � −𝑒0𝑒2, �★(𝑒2) � 𝑒0𝑒1, and ˜𝜏 is determined by ˜𝜏↾𝐶∞ 𝜃 (T2)= 𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that Λ𝜖2 𝜃 ≠ Λ𝜖𝜃 since 𝜖𝜃 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In fact, these examples typify the important special case where the base manifold is even- dimensional, the curvature 2-form is a symplectic form, and the Riemannian volume form is a constant multiple of the appropriate power of the curvature 2-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝑁 is even and that there exists 𝑐 ∈ R \\ {0}, such that ★𝐵(1) = 𝑐F𝑁/2 Π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If (★𝐵, 𝜏𝐵) admits a lift (Δver, Δhor, ★, 𝜏) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), then (Δver, Δhor) = (Λ𝜅, Λ𝜅1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜗 be the connection 1-form of the connection Π, let 𝜇𝑃 be the conformal factor of Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) with respect to the Hodge operator ★𝐵, and let (𝜖𝑖)𝑁 𝑖=1 be a finite family in 𝑃1 satisfying �𝑁 𝑖=1 𝜖∗ 𝑖 𝜖𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜇𝑃(1) = 𝜅−1/2 since ★𝑃(1) = 𝑁 ∑︁ 𝑖=1 𝜖∗ 𝑖 ★𝑃 (1)(Δ−𝑁 hor ◦ Δ−1 ver)(𝜖𝑖) = 𝑁 ∑︁ 𝑖=1 𝜖∗ 𝑖 𝜗𝑐F𝑁/2 Π 𝜖𝑖𝜅𝜇𝑃(1)−𝑁 = ★𝑃(1)𝜅−𝑁/2𝜇𝑃(1)−𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ NONCOMMUTATIVE U(1)-GAUGE THEORY 59 We conclude this section by showing that modular phenomena are no obstacle to equipping Ω𝑃 with an 𝐿2-inner product and computing the formal adjoint of the exterior derivative d𝑃 in terms of the Hodge star operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In fact, one can straighforwardly generalise the nc Hodge decomposition of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='7 to total Riemannian geometries by combining the abstract Hodge decomposition of Ó Buachalla–Sťoviček–Van Roosmalen [80, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3] with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8, but we shall not need it in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) with inverse metric 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then Ω𝑃 defines a pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 defined by ∀𝜔, 𝜂 ∈ Ω𝑃, ⟨𝜔, 𝜂⟩ � 𝜏(𝑔(𝜔, 𝜂)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39) Moreover, with respect to this pre-Hilbert space structure, the U(1)-action on Ω𝑃 defines a unitary representation of finite type, the direct sum decomposition Ω𝑃 = �1 𝑗=0 �𝑁 𝑘=0 Ω𝑗,𝑘 𝑃 is orthogonal, the left 𝑃-module structure on Ω𝑃 defines a U(1)-equivariant isometric ∗-representation on 𝑃, the connection Π defines an orthogonal projection, the operator ★𝑃 is unitary, and the operator d𝑃 is adjointable with adjoint d∗ 𝑃 = ★−1 ◦ d𝑃 ◦ ★ ◦ 𝜒𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall the faithful conditional expectation E𝑃 : 𝑃 → 𝐵 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, so that the state 𝜏 satisfies 𝜏 = 𝜏 ◦ E𝑃 = 𝜏𝐵 ◦ E𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let (𝑚, 𝑗) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} × Z be given, so that 𝑔 makes (Ω𝑚 𝑃 )𝑗 and hence its direct summands (Ω0,𝑚 𝑃 )𝑗 = Π((Ω𝑚 𝑃 )𝑗) and (Ω1,𝑚−1 𝑃 )𝑗 = (id −Π)((Ω𝑚 𝑃 )𝑗) into 𝐵-self-correspondences of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the state 𝜏 is faithful and positive, ⟨·, ·⟩𝜏 restricts to U(1)-invariant positive- definite inner products on both (Ω0,𝑚 𝑃 )𝑗 and (Ω1,𝑚−1 𝑃 )𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' moreover, the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 shows that both (Ω0,𝑚 𝑃 )𝑗 and (Ω1,𝑚−1 𝑃 )𝑗 are separable as pre-Hilbert spaces by separability of 𝐵 as a pre-𝐶∗-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' But now, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37), for every (𝑚, 𝑗), (𝑛, 𝑘) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} × Z, 𝜔 ∈ (Ω0,𝑚 𝑃,hor)𝑗, and 𝜂 ∈ (Ω0,𝑛 𝑃,hor)𝑘, we find that ⟨𝜔, 𝜂⟩𝜏 = 𝜏(𝑔(𝜔, 𝜂)) = 𝜏(𝛿𝑚,𝑛𝑔(𝜔, 𝜂)) = 𝜏�𝛿𝑚,𝑛𝛿 𝑗,𝑘𝑔(𝜔, 𝜂)� and similarly that ⟨𝜔𝜗, 𝜂𝜗⟩𝜏 = 𝜏(𝛿𝑚,𝑛𝑔(𝜔𝜗, 𝜂𝜗)), while ⟨𝜔, 𝜂𝜗⟩𝜏 = ⟨𝜔𝜗, 𝜂⟩𝜏 = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This shows that Ω𝑃 = �1 𝑗=0 �𝑁 𝑘=0 �∞ ℓ=−∞(Ω𝑗,𝑘 𝑃 )ℓ is an orthogonal decomposition with respect to ⟨·, ·⟩𝜏, so that ⟨·, ·⟩𝜏 defines a U(1)-invariant positive-definite inner product on Ω𝑃, with respect to which Ω𝑃 is separable and the U(1)-action is unitary and of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that the left 𝑃-module structure on Ω𝑃 yields a U(1)-equivariant isometric ∗-representation of 𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that U(1)-equivariance is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let us show that each 𝑝 ∈ 𝑃 acts as an adjointable operator on Ω𝑃,hor with formal adjoint given by 𝑝∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let 𝑝 ∈ 𝑃 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, ★(𝑔(𝑝𝜔, 𝜂)) = (𝑝𝜔)∗ · ★(𝜂) = 𝜔∗𝑝∗★(𝜂) = 𝜔∗★(𝑝∗𝜂) = ★(𝑔(𝜔, 𝑝∗𝜂)) for all 𝜔, 𝜂 ∈ Ω𝑃, so that ⟨𝑝𝜔, 𝜂⟩𝜏 = ⟨𝜔, 𝑝∗𝜂⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let us show that each 𝑝 ∈ 𝑃 acts as a bounded operator on Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let 𝑝 ∈ 𝑃 be given, and write 𝑝 = � 𝑘∈Z ˆ𝑝(𝑘), where ˆ𝑝(𝑘) ∈ 𝑃𝑘 for each 𝑘 ∈ Z, so that 𝐸(𝑝∗𝑝) = 𝐸�� 𝑘,ℓ ∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(ℓ)� = � 𝑘∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(𝑘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑚, 𝑗) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} × Z and let 𝜔 ∈ (Ω𝑚 𝑃 )𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, by adjointability of 𝑝, Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29, the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5, and contractivity of 𝐸, (E𝑃 ◦ 𝑔)(𝑝𝜔, 𝑝𝜔)) = (E𝑃 ◦ 𝑔) � 𝜔, ∑︁ 𝑘,ℓ ∈Z ˆ𝑝(𝑘)∗ ˆ𝑝(ℓ)𝜔 � = 𝑔(𝜔, E𝑃(𝑝∗𝑝)𝜔) ≤ ∥𝑝∥2𝑔(𝜔, 𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the left 𝑃-module structure defines a bounded ∗-representation of 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' That this ∗- representation is isometric follows, mutatis mutandis, from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that d𝑃 is adjointable with adjoint ★−1 ◦d𝑃 ◦★◦ 𝜒𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 +1}, let 𝜔 ∈ Ω𝑚−1 𝑃 , and let 𝜂 ∈ Ω𝑚 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, since 𝜏𝑃 ◦ ★ ◦ d𝑃(𝜔∗𝜂) = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30), it follows that d𝑃(𝜔)∗★(𝜂) = d𝑃(𝜔∗𝜂) + (−1)𝑚𝜔∗(d𝑃 ◦ ★)(𝜂) = d𝑃(𝜔∗𝜂) + 𝜔∗★�(★−1 ◦ d𝑃 ◦ ★ ◦ 𝛾𝑃)(𝜂)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 60 BRANIMIR ĆAĆIĆ Finally, we show that ★𝑃 is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑗, 𝑘) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝜔, 𝜂 ∈ Ω𝑗,𝑘 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ★(𝜔)∗ · ★(★(𝜂)) = ★ � (Δ1−2𝑗 ver ◦ Δ𝑁−2𝑘 hor )(𝜔∗) � (−1)𝑚(𝑁+1−𝑚)𝜂 = (Δ1−2𝑗 ver ◦ Δ𝑁−2𝑘 hor ) � ★−1(𝜔∗) · (Δ2𝑘−𝑁 hor Δ2𝑗−1 ver )(𝜂) � = (Δ1−2𝑗 ver ◦ Δ𝑁−2𝑘 hor )(𝜔∗ · ★(𝜂)), which suffices to show that ⟨★(𝜔), ★(𝜂)⟩𝜏 = ⟨𝜔, 𝜂⟩𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Unbounded lifts of commutator representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now consider the analogous lifting problem for Connes’s nc Riemannian geometry in terms of spectral triples [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this approach, analogues of Dirac-type operators—for example, the Hodge–de Rham operator d + d∗ on a compact oriented Riemannian manifold—simultaneously encode differential calculus (to first order), index theory, Riemannian geometry, and metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Follow- ing Schmüdgen [90], we restrict our attention to the first aspect and consider commutator representations of ∗-exterior algebras through degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' However, the resulting lifted commu- tator representations also generally involve non-trivial modular phenomena in the form of unboundedness of represented 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Just as before, let 𝜅 > 0, let (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) be a 𝜅-differentiable quantum principal U(1)- bundle with connection over 𝐵, let 𝜗 be the connection 1-form of Π, and let ˆΦ𝑃 be the Fröhlich automorphism of Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) = (𝑃, Ω𝑃,hor, d𝑃,hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, given a pre-Hilbert space 𝐻, let L(𝐻) denote the unital pre-𝐶∗-algebra of bounded adjointable operators on 𝐻, which is Z/2Z-graded as a ∗-algebra whenever the 𝐻 is as a pre-Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with a simplified version of the notion of spectral triple, which we shall apply to the nc base manifold (Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In short, it generalises Clifford actions of 1-forms in terms of bounded commutators with an abstract Dirac-type operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For a detailed introduction, see the survey article of Carey–Phillips–Rennie [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='30 (Baaj–Julg [6], Connes [32], Schmüdgen [90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻 be a separable Z/2Z- graded pre-Hilbert space equipped with a bounded ∗-homomorphism 𝜋 : 𝐵 → L(𝐻)odd and an odd formally self-adjoint C-linear map 𝐷 : 𝐻 → 𝐻, so that L(𝐻) defines a 𝐵-bimodule with respect to 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We call (𝐻, 𝜋, 𝐷) a bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) whenever there exists a (necessarily unique) 𝐵-bimodule homomorphism 𝜋𝐷 : Ω1 𝐵 → L(𝐻), such that ∀𝑏 ∈ 𝐵, 𝜋𝐷 ◦ d𝐵(𝑏) = i[𝐷, 𝜋(𝑏)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40) In this case, we say that (𝐻, 𝜋, 𝐷) is faithful whenever 𝜋 is isometric and 𝜋𝐷 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔅 denote the 𝐶∗-algebra completion of 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A bounded commutator rep- resentation (𝐻, 𝜋, 𝐷) of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) defines an even spectral triple for 𝔅 if and only if 𝐷 is essentially self-adjoint and has compact resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32 (D˛abrowski–Sitarz [41], Majid [67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4 and con- sider the best-known spectral triple on O𝑞(CP1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let /𝑆𝑞,±(CP1) � O𝑞(SU(2))∓1 with the inner product ⟨·, ·⟩ defined by ⟨𝑠1, 𝑠2⟩ � ℎ𝑞(𝑠∗ 1𝑠2), for all 𝑠1, 𝑠2 ∈ /𝑆𝑞,±(CP1), where ℎ𝑞 : O𝑞(SU(2)) → C, as usual, is Woronowicz’s Haar state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 together with Nagy’s result [75] on faithfulness of ℎ𝑞 on 𝐶𝑞(SU(2)), each of /𝑆𝑞,±(CP1) defines a separable pre-Hilbert space admitting isometric 𝜋± : O𝑞(CP1) → L(/𝑆𝑞,±(CP1)), respectively, given by left multiplication in O𝑞(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let /𝑆𝑞(CP1) � /𝑆𝑞,+(CP1) ⊕ /𝑆𝑞,−(CP1) as an orthogonal direct sum with Z/2Z-grading id ⊕(− id) and define 𝜋 : O𝑞(CP1) → L(/𝑆𝑞(CP1)) by setting 𝜋 � (𝑏 ↦→ 𝜋+(𝑏) ⊕ 𝜋−(𝑏)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let /𝐷𝑞 : /𝑆𝑞(CP1) → /𝑆𝑞(CP1) be Majid’s spin Dirac operator [67, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5], which is constructed from the maps 𝜕+ and 𝜕− of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 NONCOMMUTATIVE U(1)-GAUGE THEORY 61 by /𝐷𝑞 � � 0 𝑞−1𝜕+ 𝑞𝜕− 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) is a faithful bounded commutator representation of (Ω𝑞(CP1), d𝑞) that recovers Majid’s 𝑞-deformed Clifford action [67, §5] as the induced map 𝜋 /𝐷𝑞 : Ω1 𝑞(CP1) → L(/𝑆𝑞(CP1)) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, a straightforward calculation now shows that (/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) recovers the spectral triple on O𝑞(CP1) constructed by Dąbrowski–Sitarz [41] as reformulated by Neshveyev–Tuset [77, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following familiar proposition shows that nc Riemannian geometry in terms of abstract Dirac-type operators generalises nc Riemannian geometry in terms of abstract Hodge star operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, it shows that the Hodge–de Rham operator of a compact oriented Riemannian manifold still makes perfect sense in the nc setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Das–Ó Buachalla–Somberg [36, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★, 𝜏) be a Riemannian geom- etry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), so that Ω𝐵 defines a Z/2Z-graded separable pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 induced by (★, 𝜏) and the Z/2Z-grading 𝛾𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜋 : 𝐵 → L(Ω𝐵) denote the bounded ∗-representation of 𝐵 on Ω𝐵 defined by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The triple (Ω𝐵, 𝜋, d𝐵 + d∗ 𝐵) defines a faithful bounded commutator representation of (Ω𝐵, d𝐵) that satisfies ∀𝛼 ∈ Ω1 𝐵, ∀𝛽 ∈ Ω𝐵 𝜋d𝐵+d∗ 𝐵 (𝛼)𝛽 = i𝛼 · 𝛽 + i★−1(𝛼 · (★ ◦ 𝛾𝐵)(𝛽)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Under the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33, given 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} and 𝜔 ∈ Ω𝑘 𝐵, define e(𝜔) : Ω𝐵 → Ω𝐵 to be left multiplication by 𝜔 in Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, for all 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} and 𝜔 ∈ Ω𝑘 𝐵, the map e(𝜔) defines a bounded operator on the pre-Hilbert space Ω𝐵 with adjoint e(𝜔)∗ = (−1)𝑘★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑘 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} and 𝜔 ∈ Ω𝑘 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑔 be the inverse metric induced by (★, 𝜏), so that Ω𝐵 defines a 𝐵-self-correspondence of finite type with respect to 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the right 𝐵-linear map e(𝜔) is adjointable and bounded as an operator on (Ω𝐵, 𝑔) with operator norm ∥e(𝜔)∥ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, given 𝑗 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, 𝛼 ∈ Ω𝑗 𝐵, and 𝛽 ∈ Ω𝑘+𝑗 𝐵 , we see that (e(𝜔)𝛼)∗★(𝛽) = (−1)𝑗𝑘𝛼∗𝜔∗★(𝛽) = 𝛼∗ · ((−1)𝑘★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑘 𝐵)(𝛽), so that e(𝜔)∗ = (−1)𝑘★−1◦e(𝜔∗)◦★◦𝛾𝐵 for e(𝜔) as operators on the 𝐵-self-correspondence of finite type Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' But now, recall that ⟨·, ·⟩𝜏 = 𝜏 ◦ 𝑔, which immediately implies that e(𝜔)∗ remains the adjoint of e(𝜔) as an operator on the pre-Hilbert space Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then e(𝜔) is also bounded with operator norm at most ∥e(𝜔)∥, since, for all 𝛼 ∈ Ω𝐵, ⟨e(𝜔)𝛼, e(𝜔)𝛼⟩𝜏 = 𝜏(𝑔(e(𝜔)𝛼, e(𝜔)𝛼)) ≤ 𝜏�∥e(𝜔)∥2𝑔(𝛼, 𝛼)� = ∥e(𝜔)∥2⟨𝛼, 𝛼⟩𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In light of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34, it suffices to show that ∀𝑏 ∈ 𝐵, ∀𝛽 ∈ Ω𝐵, [d𝐵 + d∗ 𝐵, 𝜋(𝑏)]𝛽 = d𝐵(𝑏) · 𝛽 + ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝑏 ∈ 𝐵, 𝑘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, and 𝛽 ∈ Ω𝑘 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, the Leibniz rule for d𝐵 immediately implies that [d𝐵, 𝜋(𝑏)]𝛽 = d𝐵(𝑏) · 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, together with left 𝐵-linearity of 𝛾𝐵 and ★, it also implies that [d∗ 𝐵, 𝜋(𝑏)]𝛽 = ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)), since d∗ 𝐵(𝑏 · 𝛽) = ★−1 ◦ d𝐵 ◦ ★ ◦ 𝛾𝐵(𝑏 · 𝛽) = ★−1 ◦ d𝐵(𝑏 · (★ ◦ 𝛾𝐵)(𝛽)) = ★−1(d𝐵(𝑏) · (★ ◦ 𝛾𝐵)(𝛽)) + 𝑏 · d∗ 𝐵𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, in the notation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34, we find that i[d𝐵 + d∗ 𝐵, 𝜋(𝑏)] = ie(d𝐵𝑏) + i(★−1 ◦ e(d𝐵𝑏) ◦ ★ ◦ 𝛾𝐵) = ie(d𝐵𝑏) + (ie(d𝐵𝑏∗))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ 62 BRANIMIR ĆAĆIĆ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★, 𝜏) be a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The Hodge–de Rham commutator representation induced by (★, 𝜏) is the faithful bounded commutator representation (Ω𝐵, 𝜋𝐵, d𝐵 + d∗ 𝐵) of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) constructed from (★, 𝜏) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now turn to the construction of commutator representations for (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, we seek a canonical construction for lifting faithful bounded commutator repre- sentations of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following generalisation of Schmüdgen’s no-go theorem for quantum SU(2) with Woronowicz’s 3-dimensional calculus [90] shows that faithful bounded commutator representations of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃) do not exist when 𝜅 ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This forces us to consider commutator representations where 1-forms may be represented by unbounded operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Schmüdgen [90, Lemma 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a bounded com- mutator representation of (𝑃, Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If 𝜅 ≠ 1, then (id −Π)(Ω1 𝑃) ⊆ ker 𝜋𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑒𝑖)𝑚 𝑖=1 and (𝜖𝑗)𝑛 𝑗=1 be finite families in 𝑃1 satsifying �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 and �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1, respectively, and define bounded completely positive maps 𝜙± : LU(1) (𝐻) → LU(1) (𝐻) by ∀𝑇 ∈ LU(1) (𝐻), 𝜙+(𝑇) � ∑︁𝑚 𝑖=1 𝜋(𝑒𝑖)𝑇𝜋(𝑒∗ 𝑖 ), 𝜙−(𝑇) � ∑︁𝑛 𝑗=1 𝜋(𝜖∗ 𝑗 )𝑇𝜋(𝜖𝑗), which are unit preserving and hence contractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜅−1 �𝑚 𝑖=1 𝑒𝑖𝜗𝑒∗ 𝑖 = 𝜗 = 𝜅 �𝑚 𝑗=1 𝜖∗ 𝑗 𝜗𝜖𝑗, it follows that ∥𝜋𝐷(𝜗)∥ = 𝜅∓1∥𝜙± ◦ 𝜋𝐷(𝜗)∥ ≤ 𝜅∓1∥𝜋𝐷(𝜗)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, if 𝜅 ≠ 1, then 𝜋𝐷(𝜗) = 0, so that 𝜋𝐷 vanishes on (id −Π)(Ω1 𝑃) = 𝑃 · 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37 (Schmüdgen [90, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51, let us suppose that (𝐻, 𝜋, 𝐷) is a bounded commutator representation of (O𝑞(SU(2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑞(SU(2)), d𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, since 𝑞2 ≠ 1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 shows that 𝜋𝐷(𝑒0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, since 𝑞 ≠ 1, the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36, mutatis mutandis, shows that 𝜋𝐷(𝑒±) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it follows that 𝜋𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that this recasts Schmüdgen’s original argument [90, Lemma 6 et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='] in way that does not use unitarity of � 𝑎 −𝑞𝑐∗ 𝑐 𝑎∗ � ∈ 𝑀2(O𝑞(SU(2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52, let us suppose that (𝐻, 𝜋, 𝐷) is a bounded commutator representation of (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, since 𝜖2 𝜃 ≠ 1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 shows that 𝜋𝐷(𝑒0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, since 𝜖𝜃 ≠ 1, the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36, mutatis mutandis, shows that 𝜋𝐷(𝑒1) = 0 and 𝜋𝐷(𝑒2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it follows that 𝜋𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This catastrophic failure of bounded commutator representations to accommodate im- portant examples of nc differentiable principal U(1)-bundles forces us to consider a more general notion of commutator representation where elements of Ω1 𝑃 may be represented by unbounded operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The type of unboundedness that arises can be characterized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped with a unitary representation 𝑉 : U(1) → L(𝐻)even of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that an operator 𝑇 : 𝐻 → 𝐻 is locally bounded whenever it satisfies both of the following conditions: (1) for all 𝑗, 𝑘 ∈ Z, the map 𝑃𝑗𝑇𝑃𝑘↾𝐻𝑘: 𝐻𝑘 → 𝐻𝑗 is bounded and adjointable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) the set {𝑐 ∈ Z | ∃𝑘 ∈ Z, 𝑃𝑘+𝑐𝑇𝑃𝑘 ≠ 0} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we denote by LU(1) loc (𝐻) the Z/2Z-graded unital ∗-algebra of locally bounded operators on 𝐻, where the ∗-operation is given by taking operator adjoints, and where the Z/2Z-grading is induced by the Z/2Z-grading on 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, set LU(1) (𝐻) � L(𝐻) ∩ LU(1) loc (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped with a unitary representation 𝑉 : U(1) → U(1)(L(𝐻))even of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then each (𝑇𝑗)𝑗∈Z ∈ � 𝑗∈Z L(𝐻𝑗) NONCOMMUTATIVE U(1)-GAUGE THEORY 63 yields a U(1)-equivariant operator � 𝑗∈Z 𝑇𝑗 ∈ LU(1) loc (𝐻)U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, given 𝜅 > 0, we may define even U(1)-equivariant operators Λ𝜅, 𝜕𝜅 ∈ LU(1) loc (𝐻), respectively, by Λ𝜅 � � 𝑗∈Z 𝜅−𝑗 id𝐻𝑗, 𝜕𝜅 � � 𝑗∈Z 2𝜋i[𝑗]𝜅 id𝐻𝑗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='42) so that Λ𝜅 is formally self-adjoint while 𝜕𝜅 is formally skew-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now weaken the definition of bounded commutator representation accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐴 be a U(1)-pre-𝐶∗-algebra of finite type, and let (Ω, d) be a U(1)-∗- quasi-dga of finite type over 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝐻 be a separable Z/2Z-graded pre-Hilbert space equipped with a unitary representation𝑉 : U(1) → L(𝐻)even of finite type, a U(1)-equivariant bounded ∗-automorphism 𝜋 : 𝐴 → LU(1) (𝐻)even, and a U(1)-invariant odd formally self-adjoint C- linear map 𝐷 : 𝐻 → 𝐻, so that LU(1) loc (𝐻)odd defines a 𝐴-bimodule with respect to 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We call (𝐻, 𝜋, 𝐷) a locally bounded commutator representation of (𝐴;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω, d) whenever there exists a (necessarily unique) 𝐴-bimodule homomorphism 𝜋𝐷 : Ω1 → LU(1) loc (𝐻)odd, such that ∀𝑎 ∈ 𝐴, 𝜋𝐷 ◦ d(𝑎) = i[𝐷, 𝜋(𝑎)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43) In this case, we say that (𝐻, 𝜋, 𝐷) is faithful whenever 𝜋 is isometric and 𝜋𝐷 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we propose a refined notion of locally bounded commutator representation for 𝜅-differentiable quantum principal U(1)-bundles over 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the case where 𝜅 = 1, it reduces to a multigraded variation on a Dąbrowski–Sitarz’s definition of principal U(1)-spectral triples [42] in the spirit of Ćaćić–Mesland [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) is a quadruple of the form (𝐻, 𝜋, 𝐷,Γ), where: (1) the datum (𝐻, 𝜋, 𝐷) is a locally bounded commutator representation of (𝑃, Ω𝑃, d𝑃), such that (𝑝 ⊗ 𝜉 ↦→ 𝜋(𝑝)𝜉) : 𝑃 ⊗𝐵 𝐻U(1) → 𝐻 is bijective and 𝜋𝐷(𝜗)2 = Λ2 𝜅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) the datum Γ ∈ LU(1) (𝐻) is an even U(1)-invariant self-adjoint unitary commuting with ran 𝜋 and anticommuting with 𝜋𝐷(𝜗), such that the horizontal Dirac operator 𝐷hor � 1 2 (𝐷 + Γ𝐷Γ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44) supercommutes with 𝜋𝐷(𝜗) and the remainder 𝑍 � 1 2 (𝐷 − Γ𝐷Γ) + i𝜋𝐷(𝜗)𝜕𝜅 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45) is bounded and supercommutes with ran 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, we say that (𝐻, 𝜋, 𝐷,Γ) is faithful whenever (𝐻, 𝜋, 𝐷) is faithful and the maps �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : 𝐵 → L(𝐻U(1)), �𝛽 ↦→ 𝜋𝐷(𝛽)↾𝐻U(1) � : Ω1 𝐵 → L(𝐻U(1)) are isometric and injective, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔓 denote the 𝐶∗-algebra completions of 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A projectable commutator representation (𝐻, 𝜋, 𝐷,Γ) of 𝑃 can be viewed as defining a formal U(1)-equivariant un- bounded 𝐾𝐾1-cycle (𝑃, 𝐻, 𝐷) for (𝔓, C), where the U(1)-invariant odd self-adjoint unitary −iΓ𝜋𝐷(𝜗)Λ−1 𝜅 generates the requisite 1-multigrading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If 𝜅 = 1, the horizontal Dirac operator 𝐷hor has bounded commutators with 𝜋(𝑃), and the operator 𝐷 is essentially self-adjoint with compact resolvent, then (𝑃, 𝐻, 𝐷) gives rise to a genuine U(1)-equivariant odd spectral triple for 𝔓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Otherwise, the formal unbounded 𝐾𝐾1-cycle (𝑃, 𝐻, 𝐷) generally lies outside the current scope of unbounded 𝐾𝐾-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 64 BRANIMIR ĆAĆIĆ The following proposition shows that a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) induces a canonical projectable commutator representation just as a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) induces a bounded commutator representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This demonstrates the non- triviality of our definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, view Ω𝑃 as a Z/2Z-graded separable pre-Hilbert space with respect to the inner product ⟨·, ·⟩𝜏 induced by (Δver, Δhor, ★, 𝜏) and the Z/2Z-grading 𝛾𝑃, so that the U(1)-action ˆ𝜎 on Ω𝑃 defines a unitary U(1)-representation of finite type by even operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜋 : 𝑃 → L(Ω𝑃) denote the isometric ∗-representation of 𝑃 on Ω𝑃 defined by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (Ω𝑃, 𝜋, d𝑃 +d∗ 𝑃, 2Π−id) defines a faithful projectable commutator representation of (𝑃, Ω𝑃, d𝑃, Π) that satisfies ∀𝜔 ∈ Ω1 𝑃, ∀𝜂 ∈ Ω𝑃, 𝜋d𝑃+d∗ 𝑃 (𝜔)𝜂 = i𝜔 · 𝜂 + ★−1(i𝜔 · (★ ◦ 𝛾𝑃)(𝜂)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46) Moreover, the remainder 𝑍 of (Ω𝑃, 𝜋, d𝑃 + d∗ 𝑃, 2Π − id) is given by ∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝛼1, 𝛼2 ∈ Ω𝐵, 𝑍(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −𝑝2FΠ𝛼2 − 𝑝1𝜗★−1 𝐵 (FΠ★𝐵(𝛼1)), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='47) where FΠ is the curvature 2-form of the connection Π and where (★𝐵, 𝜏𝐵) is the restriction of (Δver, Δhor, ★, 𝜏) to (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Under the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44, let 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 + 1} and 𝜔 ∈ Ω𝑚 𝑃 be given, and let e(𝜔) : Ω𝑃 → Ω𝑃 be left multiplication by 𝜔 in Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then e(𝜔) ∈ LU(1) loc (𝐻) and e(𝜔)∗ = (−1)𝑚★−1 ◦ e(𝜔∗) ◦ ★ ◦ 𝛾𝑚 𝑃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34 applies verbatim except for showing that e(𝜔) ∈ LU(1) loc (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, suppose that 𝑚 = 1 and 𝜔 = 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑚, 𝑗) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 + 1} × Z and let 𝜂 ∈ (Ω𝑚 𝑃 )𝑗, so that 𝜂 = 𝜂1 + 𝜗𝜂2 for 𝜂1 ∈ (Ω𝑚 𝑃,hor)𝑗 and 𝜂2 ∈ (Ω𝑚−1 𝑃,hor)𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then e(𝜔)(𝜂) = 𝜗𝜂1 ∈ (Ω𝑚+1 𝑃 )𝑗, so that ⟨e(𝜔)(𝜂), e(𝜔)(𝜂)⟩𝜏 = ⟨𝜗𝜂1, 𝜗𝜂1⟩𝜏 = 𝜅−2𝑗⟨𝜂1, 𝜂1⟩𝜏 ≤ 𝜅−2𝑗⟨𝜂, 𝜂⟩𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, given 𝑗, 𝑘 ∈ Z, we see that E𝑗e(𝜔)E𝑘 ≠ 0 only if 𝑗 = 𝑘, in which case ∥E𝑗e(𝜔)E𝑗∥ ≤ 𝜅−𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the operator e(𝜔) is locally bounded when 𝜔 = 𝜗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, suppose that 𝜔 ∈ Ω𝑚 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑟, 𝑠, 𝑗) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} × Z be given, and recall from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29 that both (Ω𝑟,𝑠 𝑃 )𝑗 and (Ω𝑟,𝑠+𝑚 𝑃 )𝑗 are 𝐵-self-correspondences of finite type with respect to the inverse metric 𝑔 induced by ★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑆𝑟,𝑠 𝑗 denote the restriction of e(𝜔) to (Ω𝑟,𝑠 𝑃 )𝑗, whose range is therefore contained in (Ω𝑟,𝑠+𝑚 𝑃 )𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝑆𝑟,𝑠 𝑗 : (Ω𝑟,𝑠 𝑃 )𝑗 → (Ω𝑟,𝑠+𝑚 𝑃 )𝑗 is right 𝐵-linear, it is bounded as a map of right pre-Hilbert 𝐵-modules, and hence, since ⟨·, ·⟩𝜏 = 𝜏 ◦ 𝑔, as a map of pre-Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, given 𝑗, 𝑘 ∈ Z, it follows that E𝑗e(𝜔)E𝑘 ≠ 0 only if 𝑗 = 𝑘, in which case ∥E𝑗e(𝜔)E𝑗∥ ≤ sup{∥𝑆𝑟,𝑠 𝑗 ∥ | (𝑟, 𝑠) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the operator e(𝜔) is locally bounded when 𝜔 ∈ Ω𝑚 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us finally consider the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Without loss of generality, there exist 𝑝1, 𝑝2 ∈ 𝑃, 𝛼1 ∈ Ω𝑚 𝐵, and 𝛼2 ∈ Ω𝑚−1 𝐵 , such that 𝜔 = 𝑝1𝛼1 + 𝑝2𝜗𝛼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that 𝜋(𝑝1), 𝜋(𝑝2) ∈ LU(1) (Ω𝑃) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, e(𝜔) = 𝜋(𝑝1)e(𝛼1) + 𝜋(𝑝2)e(𝜗)e(𝛼2) ∈ LU(1) loc (Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, let e : Ω𝑃 → LU(1) loc (Ω𝑃) be the U(1)-equivariant C-linear map defined by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45 together with linearity, let i : Ω𝑃 → LU(1) loc (Ω𝑃) be the U(1)-equivariant C-linear map defined by i(𝜔) � (−1)𝑚★−1 ◦ e(𝜔) ◦ ★ ◦ 𝛾𝑚 𝑃 = e(𝜔∗)∗ for all 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁 + 1} and 𝜔 ∈ Ω1 𝑃, let 𝑐 � i(e− i), and set 𝐷 � d𝑃 + d∗ 𝑃, By analogy, define maps e𝐵, i𝐵, 𝑐𝐵 : Ω𝐵 → L(Ω𝐵) and set 𝐷𝐵 � d𝐵 + d∗ 𝐵, so that 𝜋𝐷𝐵 = 𝑐𝐵↾Ω1 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝜗 denote the connection 1-form of Π, let ∇ � ˆℓ𝑃 ◦ Π ◦ d𝑃↾𝑃, where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is NONCOMMUTATIVE U(1)-GAUGE THEORY 65 the U(1)-equivariant isomorphism of 𝐵-bimodules of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, let 𝐷ver � −i𝜋𝐷(𝜗)𝜕𝜅, and let Γ � 2Π − id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, after substituting Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29 for Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='45 for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='34, the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33 shows that (Ω𝑃, 𝜋, 𝐷) defines a faithful locally bounded commutator representation satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24 combined with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46 yields bijectivity of the multiplication map (𝑝 ⊗ 𝜔 ↦→ 𝑝𝜔) : 𝑃 ⊗𝐵 ΩU(1) 𝑃 → Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us now consider the operator 𝜋𝐷(𝜗), the would-be horizontal Dirac operator 𝐷hor, and the would-be remainder 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Before continuing, note that e(𝜗) maps Ω𝑃,hor to 𝜗 · Ω𝑃,hor and vanishes on 𝜗 · Ω𝑃,hor = Ω⊥ 𝑃,hor, so that its adjoint i(𝜗) maps 𝜗 · Ω𝑃,hor to Ω𝑃,hor and vanishes on Ω𝑃,hor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' since Γ acts as id on Ω𝑃,hor and as − id on 𝜗 · Ω𝑃,hor, this already suffices to show that Γ anticommutes with 𝜋𝐷(𝜗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let 𝑝 ∈ 𝑃, 𝑚 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, and 𝛼 ∈ Ω𝑚 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' we find that e(𝜗)(𝑝𝛼) = Λ𝜅(𝑝)𝜗𝛼 and 𝐷(𝑝𝛼) = d𝑃(𝑝𝛼) + (−1)𝑚★−1 ◦ d𝑃(𝑝𝜗 ★𝐵 (𝛼)) = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 + ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝛼 + 𝑝d𝐵(𝛼) + ★−1� −∇(𝑝) ⟨0⟩𝜗∇(𝑝) ⟨1⟩ ★𝐵 (𝛼) − 𝑝FΠ★𝐵(𝛼) + 𝑝𝜗(d𝐵 ◦ ★𝐵)(𝛼) � = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 + ∇(𝑝) ⟨0⟩e𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝d𝐵(𝛼) − ∇(𝑝) ⟨0⟩i𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝜗i𝐵(FΠ)(𝛼) + 𝑝d∗ 𝐵(𝛼) = � −i∇(𝑝) ⟨0⟩𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝𝐷𝐵(𝛼) � + ((Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 − 𝑝𝜗i𝐵(FΠ)(𝛼)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' so that 𝐷hor(𝑝𝛼) = −i∇(𝑝) ⟨0⟩𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) + 𝑝𝐷𝐵(𝛼),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' and hence 𝑍(𝑝𝛼) = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝜗𝛼 − 𝑝𝜗i𝐵(FΠ)(𝛼) − i𝑐(𝜗)𝜕𝜅(𝑝𝛼) = −𝑝𝜗i𝐵(FΠ)(𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' i(𝜗)(𝑝𝜗𝛼) = (−1)𝑚★−1(𝜗 · 𝑝★𝐵(𝛼)) = (−1)𝑚★−1(Λ𝜅(𝑝)𝜗★𝐵(𝛼)) = Λ𝜅(𝑝)𝛼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝐷(𝑝𝜗𝛼) = d𝑃(𝑝𝜗𝛼) + (−1)𝑚+1★−1 ◦ d𝑃(𝑝★𝐵(𝛼)) = ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝜗𝛼 − 𝑝FΠ𝛼 − 𝑝𝜗d𝐵(𝛼) + (−1)𝑚+1★−1� (Λ𝜅 ◦ 𝜕𝜅)(𝑝)★𝐵(𝛼) + ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩★𝐵(𝛼) + 𝑝(d𝐵 ◦ ★𝐵)(𝛼) � = ∇(𝑝) ⟨0⟩∇(𝑝) ⟨1⟩𝜗𝛼 − 𝑝FΠ𝛼 − 𝑝𝜗d𝐵(𝛼) − (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 + ∇(𝑝) ⟨0⟩𝜗i𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝜗d∗ 𝐵(𝛼) = (−(Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 − 𝑝FΠ𝛼) + � i∇(𝑝) ⟨0⟩𝜗𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝐷𝐵(𝛼) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' so that 𝐷hor(𝑝𝜗𝛼) = i∇(𝑝) ⟨0⟩𝜗𝑐𝐵(∇(𝑝) ⟨1⟩)(𝛼) − 𝑝𝐷𝐵(𝛼),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' and hence 𝑍(𝑝𝜗𝛼) = (Λ𝜅 ◦ 𝜕𝜅)(𝑝)𝛼 − 𝑝FΠ𝛼 − i𝑐(𝜗)𝜕𝜅(𝑝𝜗𝛼) = −𝑝e𝐵(FΠ)(𝛼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, for all 𝑝1, 𝑝2 ∈ 𝑃 and 𝛼1, 𝛼2 ∈ Ω𝐵, 𝜋𝐷(𝜗)(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = Λ𝜅(𝑝2)𝛼2 + Λ𝜅(𝑝1)𝜗𝛼1, 𝐷hor(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −i∇(𝑝1) ⟨0⟩𝑐𝐵(∇(𝑝1) ⟨1⟩)(𝛼1) + 𝑝1𝐷𝐵(𝛼1) + i∇(𝑝2) ⟨0⟩𝜗𝑐𝐵(∇(𝑝2) ⟨1⟩)(𝛼2) − 𝑝2𝜗𝐷𝐵(𝛼2) 𝑍(𝑝1𝛼1 + 𝑝2𝜗𝛼2) = −𝑝2e𝐵(FΠ)(𝛼2) − 𝑝1𝜗i𝐵(FΠ)(𝛼1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' These expressions for 𝜋𝐷(𝜗), 𝐷hor, and 𝑍 now make clear that 𝜋𝐷(𝜗)2 = Λ2 𝜅, that 𝐷hor super- commutes with 𝜋𝐷(𝜗), and that 𝑍 supercommutes with ran 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 66 BRANIMIR ĆAĆIĆ Finally, we show that 𝑍 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall the faithful conditional expectation E𝑃 : 𝑃 → 𝐵 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16, so that 𝑃 ⊗ C2 defines a countably generated right pre-Hilbert 𝐵-module with respect to the 𝐵-valued inner product (·, ·) given by ∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝑣1, 𝑣2 ∈ C2, (𝑝1 ⊗ 𝑣1, 𝑝2 ⊗ 𝑣2) � ⟨𝑣1, 𝑣2⟩E𝑃(𝑝∗ 1𝑝2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, (𝑃 ⊗ C2) ⊗𝐵 Ω𝐵 defines a pre-Hilbert space with respect to the inner product defined, mutatis mutandis, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46, and the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29, we may define unitary 𝑀 : (𝑃 ⊗ C2) ⊗𝐵 Ω𝐵 → Ω𝑃 by ∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝛼 ∈ Ω𝐵, 𝑀 ��𝑝1 𝑝2 � ⊗ 𝛼 � � (𝑝1 + 𝑝2𝜗)𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the left 𝐵-linear maps e(FΠ) and i(FΠ) are both bounded as operators on the pre- Hilbert space Ω𝐵, we may now apply standard Hilbert 𝐶∗-module lore to conclude that 𝑍 = −𝑀 ��0 1 0 0 � ⊗ e(FΠ) + �0 0 1 0 � ⊗ i(FΠ) � 𝑀∗ is bounded as an operator on the pre-Hilbert space Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We conclude by showing that the maps � 𝑏 ↦→ 𝜋(𝑝)↾ΩU(1) 𝑃 � : 𝐵 → L(ΩU(1) 𝑃 ), � 𝛽 ↦→ 𝜋d𝑃+d∗ 𝑃 (𝛽)↾ΩU(1) 𝑃 � : Ω1 𝐵 → L(ΩU(1) 𝑃 ) are isometric and injective, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝑏 ∈ 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, the operator 𝜋(𝑏)↾ΩU(1) 𝑃 block-diagonal with respect to orthogonal decomposition ΩU(1) 𝑃 = Ω𝐵 ⊕ 𝜗Ω𝐵, where 𝜋(𝑏)↾Ω𝐵 is simply left multiplication by 𝑏 on Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33, left multiplication of 𝐵 on Ω𝐵 defines an isometric ∗-homorphism 𝐵 → L(Ω𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it follows that ∥𝑏∥ = ∥𝜋(𝑝)↾Ω𝐵 ∥ ≤ ∥𝜋(𝑏)↾ΩU(1) 𝑃 ∥ ≤ ∥𝜋(𝑏)∥ ≤ ∥𝑏∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let 𝛽 ∈ Ω1 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, both e(𝛽) ↾ΩU(1) 𝑃 and e(𝛽∗) ↾ΩU(1) 𝑃 are both block-diagonal with respect to the orthogonal decomposition ΩU(1) 𝑃 = Ω𝐵 ⊕ 𝜗Ω𝐵, where e(𝛽)↾Ω𝐵= e𝐵(𝛽) and e(𝛽∗)↾Ω𝐵= e𝐵(𝛽∗), so that 𝑐(𝛽)↾ΩU(1) 𝑃 is similarly block-diagonal with 𝑐(𝛽)↾Ω𝐵= 𝑐𝐵(𝛽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='33, the map 𝑐𝐵 : Ω1 𝐵 → L(Ω𝐵) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it follows that 𝑐(𝛽)↾ΩU(1) 𝑃 = 0 only if 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The total Hodge–de Rham commutator representation induced by (Δver, Δhor, ★, 𝜏) is the faithful projectable commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 +d∗ 𝑃, 2Π−id) of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) constructed from (Δver, Δhor, ★, 𝜏) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) lives up to our terminology by canonically projecting to a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This, in turn, will make precise the notion of lifting a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, define the concrete category BCRep(𝐵) of faithful bounded commutator representations of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) and their isomorphisms as follows: (1) an object is a faithful bounded commutator represention (𝐻, 𝜋, 𝐷) of (Ω𝐵, d𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1) → (𝐻2, 𝜋2, 𝐷2) is a unitary 𝑈 : 𝐻1 → 𝐻2 that satisfies 𝑈𝜋1(·)𝑈∗ = 𝜋2, 𝑈𝐷1𝑈∗ = 𝐷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, given 𝜅 > 0 and (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) a 𝜅-differentiable quantum principal U(1)-bundle with connection over 𝐵, define the concrete category PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) of faithful projectable commutator representations of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) and their isomorphisms as follows: NONCOMMUTATIVE U(1)-GAUGE THEORY 67 (1) an object is a faithful projectable commutator representation (𝐻, 𝜋, 𝐷,Γ) of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) an arrow (𝑈, 𝑍) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2) consists of an even U(1)-equivariant unitary 𝑈 : 𝐻1 → 𝐻2 and odd U(1)-invariant self-adjoint 𝑍 ∈ LU(1) (𝐻1) supercommut- ing with ran 𝜋 and Γ, such that 𝑈𝜋1(·)𝑈∗ = 𝜋1, 𝑈(𝐷1 − 𝑍)𝑈∗ = 𝐷2, 𝑈Γ1𝑈∗ = Γ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) given objects (𝐻1, 𝜋1, 𝐷1,Γ1), (𝐻2, 𝜋2, 𝐷2,Γ2), (𝐻3, 𝜋3, 𝐷3,Γ3𝑥), and arrows (𝑈1, 𝑍1) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2), (𝑈2, 𝑍2) : (𝐻2, 𝜋2, 𝐷2,Γ2) → (𝐻3, 𝜋3, 𝐷3,Γ3), the composition (𝑈2, 𝑍2) ◦ (𝑈1, 𝑍1) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻3, 𝜋3, 𝐷3,Γ3) is given by (𝑈2, 𝑍2) ◦ (𝑈1, 𝑍1) � �𝑈2𝑈1,𝑈∗ 1𝑍2𝑈1 + 𝑍1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4) the identity arrow of an object (𝐻, 𝜋, 𝐷,Γ) is given by (id, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that an arrow (𝑈, 𝑍) : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2) in PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) encodes U(1)- equivariant unitary equivalence of (𝐻1, 𝜋1, 𝐷1,Γ1) and (𝐻2, 𝜋2, 𝐷2,Γ2) after perturbation by the relative remainder 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following defines a functor 𝜄∗ 𝑃 : PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) → BCRep(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given an object (𝐻, 𝜋, 𝐷,Γ), let 𝜄∗ 𝑃(𝐻, 𝜋, 𝐷,Γ) � �𝑃𝐻U(1), 𝑃𝜋(·)𝑃, 𝑃𝐷hor𝑃�, where we set 𝑃 � 1 2 (id +Γ)↾𝐻U(1) and 𝐷hor is the horizontal Dirac operator of (𝐻, 𝜋, 𝐷,Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1,Γ1) → (𝐻2, 𝜋2, 𝐷2,Γ2), let 𝜄∗ 𝑃𝑈 be given by 𝑃2𝑈𝑃1, where 𝑃1 � 1 2 (id +Γ1)↾𝐻U(1) 1 and 𝑃2 � 1 2 (id +Γ2)↾𝐻U(1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This is a routine verification except for one subtlety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷,Γ) be a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It remains to show that the bounded commutator representation (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ 𝑃(𝐻, 𝜋, 𝐷,Γ) of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Observe that 𝐻U(1) admits the orthogonal decomposition 𝐻U(1) = 𝐻𝐵 ⊕ Γ𝐻𝐵, where Γ restricts to a unitary 𝑉 : 𝐻𝐵 → Γ𝐻𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it follows that 𝜋(𝑏)↾𝐻U(1) = 𝜋𝐵(𝑏) ⊕ (𝑉𝜋𝐵(𝑏)𝑉∗) for all 𝑏 ∈ 𝐵, so that 𝜋𝐵 is isometric since �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : 𝐵 → L(𝐻U(1)) is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A qualitatively identical argument shows that 𝜋𝐷 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We say that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) is a lift of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) whenever 𝜄∗ 𝑃( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) and (𝐻, 𝜋, 𝐷) are isomorphic in BCRep(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (Δver, Δhor, ★, 𝜏) be a total Riemannian geometry on (𝑃, Ω𝑃, d𝑃, Π), and let (★𝐵, 𝜏𝐵) be its restriction to a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the total Hodge– de Rham commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ 𝑃, 2Π − id) induced by (Δver, Δhor, ★, 𝜏) is a lift of the Hodge–de Rham commutator representation (Ω𝐵, 𝜋𝐵, d𝐵+d∗ 𝐵) induced by (★𝐵, 𝜏𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, the inclusion map ˆ𝜄𝑃 : Ω𝐵 ∼−→ ΩU(1) 𝑃,hor = Π(ΩU(1) 𝑃 ) defines an isomorphism ˆ𝜄𝑃 : (Ω𝐵, 𝜋𝐵, d𝐵 + d∗ 𝐵) → 𝜄∗ 𝑃(Ω𝑃, 𝜋𝑃, d𝑃 + d∗ 𝑃, 2Π − id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we show that every faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) has an essentially unique lift to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', up to U(1)-equivariant unitary equivalence after perturbation by a relative remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that this excludes the use of Schwieger–Wagner’s lifting construction [92], even after modification to permit 𝜅 ≠ 1, since it requires unnatural choices of representation-theoretic data that need not even yield locally bounded commutator representations of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 68 BRANIMIR ĆAĆIĆ We first show that lifts always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' When 𝜅 = 1, the right 𝐵-module Ω1 𝐵 is free with basis consisting of self-adjoint elements of Z(Ω𝐵)1, the Fröhlich automorphism ˆΦ𝑃 is the identity map, and the faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) takes a certain restrictive form, our construction recovers a lifting construction for spectral triples first proposed by Gabriel–Grensing [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, recall the self-adjoint Pauli matrices 𝜎1 � �0 1 1 0 � , 𝜎2 � �0 −i i 0 � , 𝜎3 � �1 0 0 −1 � = −i𝜎1𝜎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a map ∇ : 𝑃 → 𝑃 ⊗𝐵 Ω𝐵 by ∇ � ˆℓ𝑃 ◦ Π ◦ d𝑃↾𝑃, where ˆℓ𝑃 : Ω𝑃,hor → 𝑃 ⊗𝐵 Ω𝐵 is the 𝐵-bimodule isomorphism of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24, and let E𝑃 : 𝑃 → 𝐵 be the faithful conditional expectation of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Equip the left 𝑃-module 𝑃 ⊗ C2 with the right 𝐵-module structure ∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀𝑏 ∈ 𝐵, (𝑝 ⊗ 𝑥) · 𝑏 � 𝑝𝑏 ⊗ 𝑥, equip 𝐻 with the left 𝐵-module structure defined by 𝜋, and equip (𝑃 ⊗ C2) ⊗𝐵 𝐻 with the inner product ⟨·, ·⟩ defined by ∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝑥1, 𝑥2 ∈ C2, ∀ℎ1, ℎ2 ∈ 𝐻, ⟨𝑝1 ⊗ 𝑥1 ⊗ ℎ1, 𝑝2 ⊗ 𝑥2 ⊗ ℎ2⟩ � ⟨𝑥1, 𝑥2⟩⟨ℎ1, 𝜋(E𝑃(𝑝∗ 1𝑝2))ℎ2⟩, the Z/2Z-grading id ⊗𝜎3 ⊗ 𝜒𝐻 and the linear U(1)-representation induced by the U(1)-action on 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, define an operator (id ⊗𝜎3) ⊗∇ 𝐷 on (𝑃 ⊗ C2) ⊗𝐵 𝐻 by ∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀ℎ ∈ 𝐻, (id ⊗𝜎3) ⊗∇ 𝐷(𝑝 ⊗ 𝑥 ⊗ ℎ) � −i∇(𝑝) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋𝐷 � ∇(𝑝) ⟨1⟩ � ℎ + 𝑝 ⊗ 𝜎3𝑥 ⊗ 𝐷ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the data �(𝑃 ⊗ C2) ⊗𝐵 𝐻, id ⊗ id ⊗𝜋(·), i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2 ⊗ id +(id ⊗𝜎3) ⊗∇ 𝐷, id ⊗𝜎3 ⊗ id� define a lift of (𝐻, 𝜋, 𝐷) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) with horizontal Dirac operator (id ⊗𝜎3) ⊗∇ 𝐷 and remainder 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), and let (𝐸, 𝜎, ∇) be a Hermitian line 𝐵-bimodule with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Equip 𝐸 ⊗𝐵 𝐻 with the positive definite inner product defined, mutatis mutandis, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) For every 𝑥 ∈ 𝐸, we obtain contractive 𝜙𝐸[𝑥] : 𝐸 ⊗𝐵 𝐻 → 𝐻 by setting ∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, 𝜙𝐸[𝑥](𝑦 ⊗ ℎ) = 𝜋((𝑥, 𝑦)𝐸)ℎ Hence, in particular, the pre-Hilbert space 𝐸 ⊗𝐵 𝐻 is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) We obtain formally self-adjoint id ⊗∇𝐷 : 𝐸 ⊗𝐵 𝐻 → 𝐸 ⊗𝐵 𝐻 by setting ∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, (id ⊗∇𝐷)(𝑦 ⊗ ℎ) = −i∇(𝑦) ⟨0⟩ ⊗ 𝜋𝐷(∇(𝑦) ⟨1⟩)ℎ + 𝑦 ⊗ 𝐷ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (3) For every 𝛼 ∈ Ω1 𝐵, we obtain bounded 𝜌𝐸[𝛼] : 𝐸 ⊗𝐵 𝐻 → 𝐸 ⊗𝐵 𝐻 by setting ∀𝑦 ∈ 𝐸, ∀ℎ ∈ 𝐻, 𝜌𝛼(𝑦 ⊗ ℎ) = 𝜎 (𝛼 ⊗ 𝑦) ⟨0⟩ ⊗ 𝜋(𝜎 (𝛼 ⊗ 𝑦) ⟨1⟩)ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Before continuing, let us fix a basis (𝑒𝑖)𝑁 𝑖=1 for 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall, moreover, that by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5, mutatis mutandis, every positive 𝑋 ∈ 𝑀𝑛(𝐵) satisfies ∀ℎ = (ℎ𝑖)𝑁 𝑖=1 ∈ 𝐻𝑁, ⟨ℎ, 𝜋𝑛(𝑋)ℎ⟩ ≤ ∥𝑋∥ ∑︁𝑁 𝑖=1∥ℎ𝑖∥2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48) NONCOMMUTATIVE U(1)-GAUGE THEORY 69 where 𝜋𝑛 : 𝑀𝑛(𝐵) → L(𝐻𝑛) is the bounded ∗-homomorphism canonically induced by 𝜋 : 𝐵 → L(𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that this applies, in particular, to the matrix 𝑋 � ((𝑒𝑖, 𝑒𝑗))𝑁 𝑖,𝑗=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝑥 ∈ 𝐸 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define 𝜓𝐸[𝑥] : 𝐻 → 𝐸 ⊗𝐵 𝐻 by 𝜓𝐸[𝑥] � (ℎ ↦→ 𝑥 ⊗ ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A standard calculation show that 𝜙𝐸[𝑥] = 𝜓𝐸[𝑥]∗ and that 𝜓𝐸[𝑥] is bounded with operator norm ∥𝜓𝐸[𝑥]∥ = ∥𝜋(⟨𝑥, 𝑥⟩)∥1/2 ≤ 1, so that 𝜙𝐸[𝑥] is contractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since (𝑒𝑖)𝑁 𝑖=1 is a basis for 𝐸, it now follows that ∀𝜉 ∈ 𝐸 ⊗𝐵 𝐻, 𝜉 = ∑︁𝑁 𝑖=1 𝑒𝑖 ⊗ 𝜙𝐸[𝑒𝑖]𝜉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49) Next, let 𝑉 be a countable dense subset of 𝐻;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' we claim that {�𝑁 𝑖=1 𝑒𝑖 ⊗ 𝑣𝑖 | 𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑣𝑁 ∈ 𝑉} is dense in 𝐸 ⊗𝐵 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜉 ∈ 𝐸 ⊗𝐵 𝐻 and 𝜖 > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑋 � ((𝑒𝑖, 𝑒𝑗))𝑁 𝑖,𝑗=1, and choose 𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑣𝑁 ∈ 𝑉, such that ∥𝜙𝐸[𝑒𝑖]𝜉 − 𝑣𝑖∥2 < 𝜖2 𝐶𝑁+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48), �����𝜉 − 𝑁 ∑︁ 𝑖=1 𝑒𝑖 ⊗ 𝑣𝑖 ����� 2 = 𝑁 ∑︁ 𝑖,𝑗=1 ⟨𝜙𝑒𝑖𝜉 − 𝑣𝑖, 𝜋((𝑒𝑖, 𝑒𝑗))(𝜙𝐸[𝑒𝑗]𝜉 − 𝑣𝑗⟩ ≤ ∥𝑋∥ 𝑁 ∑︁ 𝑖=1 ∥𝜙𝐸[𝑒𝑖]𝜉 − 𝑣𝑖∥2 < 𝜖2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, that id ⊗∇𝐷 is well-defined and formally self-adjoint is well-known in the literature on unbounded 𝐾𝐾-theory—see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', [19, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝛼 ∈ Ω1 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then right 𝐵-linearity of the generalised braiding 𝜎 guarantees that 𝜌𝐸[𝛼] is a well-defined map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='49), for every 𝜉 ∈ 𝐸 ⊗𝐵 𝐻, it follows that ∥𝜌𝐸[𝛼]𝜉∥ ≤ 𝑁 ∑︁ 𝑖,𝑗=1 ��𝑒𝑖 ⊗ 𝜋𝐷 �(𝑒𝑖, 𝜎 (𝛼 ⊗ 𝑒𝑗))�𝜙𝐸(𝑒𝑗)𝜉 �� ≤ �� � 𝑁 ∑︁ 𝑖,𝑗=1 ∥𝑒𝑖∥ · ∥𝜋𝐷 �(𝑒𝑖, 𝜎 (𝛼 ⊗ 𝑒𝑗))�∥�� � ∥𝜉∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For convenience, we shall permit the following abuse of notation: given 𝑗 ∈ Z, we conflate the isotypical subspace 𝑃𝑗 with the Hermitian line 𝐵-bimodule 𝑃𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, we shall also use the notation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us first show that ( ˜𝐻, ˜𝜋, ˜𝐷) � �(𝑃 ⊗ C2) ⊗𝐵 𝐻, id ⊗ id ⊗𝜋(·), i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2 ⊗ id +(id ⊗𝜎3) ⊗∇ 𝐷� defines a faithful locally bounded commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔅 be the 𝐶∗-algebraic completion of 𝐵, and let 𝜏 : ˜𝐻 → C2 ⊗ (𝑃 ⊗𝐵 𝐻) be the canonical unitary defined by 𝜏 � (𝑝 ⊗ 𝑥 ⊗ ℎ ↦→ 𝑥 ⊗ 𝑝 ⊗ ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, we show that ˜𝐻 is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 that the pre-Hilbert space 𝑃 ⊗𝐵 𝐻 = � 𝑗∈Z 𝑃𝑗 ⊗𝐵 𝐻 is separable, so that ˜𝐻 � (𝑃 ⊗𝐵 𝐻)2 is also separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It is now straightforward to check that 𝜒 ˜𝐻 � id⊗𝜎3⊗ 𝜒𝐻 defines a Z2-grading on ˜𝐻 and that 𝜎·⊗id ⊗ id defines a unitary U(1)-representation of finite type on ˜𝐻 with ˜𝐻𝑗 = (𝑃𝑗 ⊗ C2) ⊗𝐵 𝐻 � (𝑃 ⊗𝐵 𝐻)2 for 𝑗 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let us show that ˜𝜋 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It suffices to show that the left 𝑃-module structure on 𝑃 ⊗𝐵 𝐻 defines a bounded ∗-homomorphism 𝜆 : 𝑃 → L(𝑃 ⊗𝐵 𝐻), since this will imply boundedness of ˜𝜋 = 𝜏∗ ◦ (id ⊗𝜆(·)) ◦ 𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' the other properties of ˜𝜋 will follow by routine checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In turn, the only non-trivial points are that 𝜆 is well-defined and bounded as a map of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑝 ∈ 𝑃 be given, so that there exists 𝑁 ∈ N, such that 𝑝 ∈ �𝑁 𝑗=−𝑁 𝑃𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, we may unique write 𝑝 = �𝑁 𝑗=−𝑁 ˆ𝑝(𝑗), where ˆ𝑝(𝑗) ∈ 𝑃𝑗 for each 𝑗 ∈ {−𝑁, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁}, so that E𝑃(𝑝∗𝑝) = �𝑁 𝑗=−𝑁 ˆ𝑝(𝑗)∗ ˆ𝑝(𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑘 ∈ N and 𝜉 ∈ 𝑃𝑗 ⊗𝐵 𝐻 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑒𝑖)𝑀 𝑖=1 be a basis for 𝑃𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, in the notation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51, ∥𝜆(𝑝)𝜉∥2 = 𝑀 ∑︁ 𝑚,𝑛=1 ⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(E𝑃(𝑒∗ 𝑚𝑝∗𝑝𝑒𝑛))𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ = 𝑀 ∑︁ 𝑚,𝑛=1 ⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(𝑒∗ 𝑚E𝑃(𝑝∗𝑝)𝑒𝑛)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 70 BRANIMIR ĆAĆIĆ Now, let 𝑏 � √︁ E𝑃(𝑝∗𝑝) ∈ 𝔅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' After passing to Hilbert 𝐶∗-module and Hilbert space comple- tions, we may apply [64, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 42] to conclude that ∑︁𝑀 𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋(𝑒∗ 𝑚E𝑃(𝑝∗𝑝)𝑒𝑛)⟩ = ∑︁𝑁 𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋((𝑏𝑒𝑚, 𝑏𝑒𝑛))𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ ≤ ∥𝑏∥2 ∑︁𝑁 𝑚,𝑛=1⟨𝜙𝑃𝑗 [𝑒𝑚]𝜉, 𝜋((𝑒𝑚, 𝑒𝑛)𝑗)𝜙𝑃𝑗 [𝑒𝑛]𝜉⟩ ≤ ∥𝑝∥2∥𝜉∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let us show that ˜𝐷 is U(1)-invariant, odd, and formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define operators 𝑆 and 𝑇 satisfying ˜𝐷 = 𝑆 +𝑇 by 𝑆 � i(Λ𝜅 ◦𝜕𝜅) ⊗ 𝜎2 ⊗ id and 𝑇 � (id ⊗𝜎3) ⊗∇ 𝐷, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, the block-diagonal operator 𝑆 = � 𝑗∈Z(−2𝜋[𝑗]𝜅𝜅−𝑗 id) ⊗ 𝜎2 ⊗ id is U(1)- invariant, odd, and formally self-adjoint by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand,the operator 𝑇 is likewise U(1)-invariant and odd by construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' by U(1)-invariance, it follows that 𝑇 = � 𝑗∈Z 𝑇↾ ˜𝐻𝑗, where 𝑇↾ ˜𝐻𝑗= 𝜏∗ ◦ � 𝜎3 ⊗ (id ⊗∇𝑃,𝑗𝐷) � 𝜏↾ ˜𝐻𝑗 is formally self-adjoint for each 𝑗 ∈ Z by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, we show that ˜𝜋 ˜𝐷 : Ω1 𝑃 → LU(1) loc ( ˜𝐻) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, recall that (id −Π)(Ω1 𝑃) is freely generated as a left 𝑃-module by the connection 1-form 𝜗 of Π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, we may define a map ˜𝜋ver : (id −Π)(Ω1 𝑃) → LU(1) loc ( ˜𝐻) by ∀𝑝 ∈ 𝑃, ˜𝜋ver(𝑝𝜗) � ˜𝜋(𝑝) · (Λ𝜅 ⊗ 𝜎2 ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, since multiplication (𝑝 ⊗ 𝛼 ↦→ 𝑝𝛼) : 𝑃 ⊗𝐵 Ω1 𝐵 → Π(Ω1 𝑃) is a 𝐵-bimodule isomorphism, we may apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 to define a map ˜𝜋hor : Π(Ω1 𝑃) → LU(1) loc ( ˜𝐻) by ∀𝑝 ∈ 𝑃, ∀𝛼 ∈ Ω1 𝐵, ∀𝑗 ∈ Z, ∀ ˜𝜋hor(𝑝𝛼)↾ ˜𝐻𝑗� ˜𝜋(𝑝) ◦ 𝜏∗ ◦ (𝜎3 ⊗ 𝜌𝑃𝑗 [𝛼]) ◦ 𝜏↾ ˜𝐻𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since ˜𝐷 = 𝑆 + 𝑇, it now suffices to show that ∀𝑝 ∈ 𝑃, i[𝑆, ˜𝜋(𝑝)] = ˜𝜋ver ◦ (id −Π) ◦ d𝑃(𝑝), i[𝑇, ˜𝜋(𝑝)] = ˜𝜋hor ◦ Π ◦ d𝑃(𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝑗, 𝑘 ∈ Z, 𝑝 ∈ 𝑃𝑗, 𝑞 ∈ 𝑃𝑘, 𝑥 ∈ C2, and ℎ ∈ 𝐻 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, [𝑆, ˜𝜋(𝑝)](𝑞 ⊗ 𝑥 ⊗ ℎ) = −2𝜋[𝑗 + 𝑘]𝜅𝜅−𝑗−𝑘𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ + 2𝜋[𝑘]𝜅𝜅−𝑘𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ = 2𝜋[𝑗]𝜅𝜅−𝑗𝑝𝑞 ⊗ 𝜎2𝑥 ⊗ ℎ − i˜𝜋ver(2𝜋i[𝑗]𝜅𝜅−𝑗𝑝𝜗)(𝑞 ⊗ 𝑥 ⊗ ℎ) = −i( ˜𝜋ver ◦ (id −Π) ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, since Π ◦ d𝑃 is a derivation, it follows that ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗+𝑘(𝑝𝑞) = ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝) ⟨0⟩𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝)) ⟨1⟩ ⊗ 𝑞) ⟨0⟩ ⊗ 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨1⟩ + 𝑝∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝑞) ⟨0⟩ ⊗ ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝑞) ⟨1⟩, so that i𝑇(𝑝𝑞 ⊗ 𝑥 ⊗ ℎ) = ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝) ⟨0⟩𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋(𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝑝) ⟨1⟩ ⊗ 𝑞) ⟨1⟩)ℎ + 𝑝∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝑞) ⟨0⟩ ⊗ 𝜎3𝑥 ⊗ 𝜋(∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑘(𝑞) ⟨1⟩)ℎ + 𝑝𝑞 ⊗ 𝜎3𝑥 ⊗ 𝐷ℎ = ( ˜𝜋hor ◦ Π ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ) + i˜𝜋(𝑝)𝑇(𝑞 ⊗ 𝑥 ⊗ ℎ), and hence i[𝑇, ˜𝜋(𝑝)](𝑞 ⊗ 𝑥 ⊗ ℎ) = ( ˜𝜋hor ◦ Π ◦ d𝑃)(𝑝)(𝑞 ⊗ 𝑥 ⊗ ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, we show that ( ˜𝐻, ˜𝜋, ˜𝐷) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We first show that ˜𝜋 is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since ˜𝜋 is bounded, faithful, and U(1)-equivariant, it suffices by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='19 to show that ˜𝜋 ↾𝐵 is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let 𝑏 ∈ 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜏∗(𝐵 ⊗𝐵 𝐻) � 𝐻 is an orthogonal direct summand of ˜𝐻 NONCOMMUTATIVE U(1)-GAUGE THEORY 71 and 𝜋 is isometric, it follows that ∥𝑏∥ ≥ ∥ ˜𝜋(𝑏)∥ ≥ ∥𝜏 ˜𝜋(𝑏)𝜏∗↾𝐵⊗𝐵𝐻∥ = ∥𝜋(𝑏)∥ = ∥𝑏∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let us show that ˜𝜋 ˜𝐷 is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' to do so, it suffices to show that ˜𝜋ver and ˜𝜋hor are both injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, ˜𝜋ver is injective since ˜𝜋 is injective and ˜𝜋ver(𝜗) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other, to show that ˜𝜋hor is injective, it suffices to show injectivity of 𝑓 : 𝑃 ⊗𝐵 Ω1 𝐵 → EndC(𝐻, 𝑃 ⊗𝐵 𝐻) defined by 𝑓 (𝑝 ⊗ 𝛽)ℎ � 𝜋(𝑝)𝜋𝐷(𝛽)ℎ for 𝑝 ∈ 𝑃, 𝛽 ∈ Ω1 𝐵, and ℎ ∈ 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, fix 𝑗 ∈ Z, and note that 𝑓𝑗↾𝑃𝑗 ⊗𝐵Ω1 𝐵= 𝑟𝑗◦𝑠𝑗, where 𝑠𝑗 : 𝑃𝑗⊗𝐵Ω1 𝐵 → 𝑃𝑗⊗𝐵L(𝐻) and 𝑟𝑗 : 𝑃𝑗⊗𝐵L(𝐻) → EndC(𝐻, 𝑃𝑗⊗𝐵 𝐻) are given by 𝑠𝑗 � id ⊗𝜋𝐷 and 𝑟𝑗 � (𝑝 ⊗ 𝑆 ↦→ 𝜓𝑃𝑗 [𝑝]𝑆), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝑠𝑗 is injective by flatness of the projective right 𝐵-module 𝑃𝑗, while 𝑟𝑗 is injective by existence of the left inverse 𝑇 ↦→ �𝑁 𝑖=1 𝑒𝑖 ⊗ 𝜙𝑃𝑗 [𝑒𝑖]𝑇, where (𝑒𝑖)𝑁 𝑖=1 is any basis for the Hermitian line 𝐵-bimodule 𝑃𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, the map 𝑓𝑗↾𝑃𝑗 ⊗𝐵Ω1 𝐵: 𝑃𝑗 ⊗𝐵 Ω1 𝐵 → EndC(𝐻, 𝑃𝑗 ⊗𝐵 𝐻) is also injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let ˜Γ � id ⊗𝜎3 ⊗ id, which, by construction, is an even U(1)-invariant self-adjoint unitary commuting with ran ˜𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us check that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) defines a lift of (𝐻, 𝜋, 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let 𝑀 : 𝑃 ⊗𝐵 ˜𝐻U(1) → ˜𝐻 be given by 𝑀 � (𝑝 ⊗ 𝜉 ↦→ ˜𝜋𝜉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a left 𝐵-linear unitary Φ : C2 ⊗ 𝐻 → ˜𝐻U(1) by Φ � (𝑥 ⊗ ℎ ↦→ 1 ⊗ 𝑥 ⊗ ℎ), and observe that ∀𝑝 ∈ 𝑃, ∀𝑥 ∈ C2, ∀ℎ ∈ 𝐻, 𝜏 ◦ 𝑀 ◦ (id ⊗Φ)(𝑝 ⊗ 𝑥 ⊗ ℎ) = 𝑥 ⊗ 𝑝 ⊗ ℎ, so that 𝜏◦𝑀◦(id ⊗Φ) : 𝑃 ⊗𝐵 (C2 ⊗ 𝐻) → C2 ⊗ (𝑃 ⊗𝐵 𝐻) is manifestly bijective, which implies that 𝑀 is bijective as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, note that ˜𝑝𝑖 ˜𝐷(𝜗)2 = Λ2 𝜅 since ˜𝜋 ˜𝐷(𝜗) = ˜𝜋ver(𝜗) = Λ𝜅 ⊗ 𝜎2 ⊗ id, which also shows that ˜Γ anticommutes with ˜Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, observe that ˜Γ anticommutes with 𝑆 and commutes with 𝑇, so that ˜𝐷hor = 𝑇 and 𝑍 = 𝑆 + i˜𝜋 ˜𝐷(𝜗)𝜕𝜅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, since 𝜏∗(𝐵 ⊗ 𝐻) � 𝐻 is an orthogonal direct summand of 𝐻U(1), the proof that ˜𝜋 is isometric also shows that the map (𝑏 ↦→ ˜𝜋(𝑏)↾ ˜𝐻U(1) ) : 𝐵 → L( ˜𝐻U(1)) is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Likewise, the proof that ˜𝜋hor is injective, specialised to 𝑗 = 0, shows that the map (𝛽 ↦→ ˜𝜋 ˜𝐷(𝛽)↾ ˜𝐻U(1) ) : Ω1 𝐵 → LU(1) ( ˜𝐻U(1)) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, an arrow 𝑉 : (𝐻, 𝜋, 𝐷) → 𝜄∗ 𝑃( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) is given by 𝑉 � �ℎ ↦→ 1 ⊗ � 1 0 � ⊗ ℎ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Having proved existence of lifts, we now show that they are indeed unique up to U(1)- equivariant unitary equivalence after perturbation by a relative remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The functor 𝜄∗ 𝑃 of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='47 is an equivalence of categories with weak inverse (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' : BCRep(𝐵) → PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) Given an object (𝐻, 𝜋, 𝐷), let (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (𝐻, 𝜋, 𝐷) be the projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) constructed from (𝐻, 𝜋, 𝐷) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) Given an arrow 𝑈 : (𝐻1, 𝜋1, 𝐷1) → (𝐻2, 𝜋2, 𝐷2), let (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (𝑈) � (id ⊗ id ⊗ 𝑈, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, in particular, every bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵) has an essentially unique lift to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It remains to construct 𝑈 : idPCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='Π) ⇒ (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ◦ 𝜄∗ 𝑃 and 𝑉 : idBCRep(𝐵) ⇒ 𝜄∗ 𝑃 ◦ (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='. First, let (𝐻, 𝜋, 𝐷,Γ) be an object of PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝐷hor be its horizontal Dirac operator and 𝑍 its remainder, and let (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ 𝑃(𝐻, 𝜋, 𝐷,Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define an even U(1)-equivariant unitary Υ : (𝑃 ⊗ C2) ⊗𝐵 𝐻𝐵 → 𝐻 by ∀𝑝 ∈ 𝑃, ∀ �𝑣1 𝑣2 � ∈ C2, ∀ℎ ∈ 𝐻𝐵, Υ � 𝑝 ⊗ �𝑣1 𝑣2 � ⊗ ℎ � � 𝜋(𝑝) �𝑣1 id −i𝑣2Γ𝜋𝐷(𝜗)Λ−1 𝜅 � ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A straightforward if tedious calculation generalising the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44 now shows, in the notation of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50, that Υ∗(−i𝜋𝐷(𝜗)𝜕𝜅)Υ = i(Λ𝜅 ◦𝜕𝜅) ⊗𝜎2 ⊗id, Υ∗𝐷horΥ = (id⊗𝜎3) ⊗∇ 𝐷𝐵, Υ∗ΓΥ = id⊗𝜎3 ⊗id, so that we may define 𝑈(𝐻,𝜋,𝐷,Γ) : (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ◦ 𝜄∗ 𝑃(𝐻, 𝜋, 𝐷,Γ) → (𝐻, 𝜋, 𝐷,Γ) by 𝑈(𝐻,𝜋,𝐷,Γ) � (Υ∗, 𝑍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let (𝐻, 𝜋, 𝐷) be an object of BCRep(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50, we may define 𝑉(𝐻,𝜋,𝐷) : (𝐻, 𝜋, 𝐷) → 𝜄∗ 𝑃 ◦ (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (𝐻, 𝜋, 𝐷) by 𝑉(𝐻,𝜋,𝐷) � �ℎ ↦→ 1 ⊗ � 1 0 � ⊗ ℎ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ 72 BRANIMIR ĆAĆIĆ Note that Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='48 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 combine to yield a formalisation of the con- structions of Bellissard–Marcolli–Reihani [16] and Gabriel–Grensing [51] for (generalised) crossed product spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐸, 𝜎𝐸, ∇𝐸) be a Hermitian line 𝐵-bimodule with connec- tion, such that 𝜖1 ◦ ˆL◦ Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) � (𝐸, 𝜎𝐸, ∇𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the 𝜅-total crossed product of (𝐻, 𝜋, 𝐷) by (𝐸, 𝜎𝐸, ∇𝐸) is the canonical lift (𝐻, 𝜋, 𝐷)⋊𝜅,tot (𝐸,𝜎𝐸,Z) Z � (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (𝐻, 𝜋, 𝐷) of (𝐻, 𝜋, 𝐷) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃) of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We continue from Remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The right pre-Hilbert 𝐵-module 𝑃 ⊗ C2 � � 𝑗∈Z L(𝑃)(𝑗)2 gives rise to a formal U(1)-equivariant unbounded 𝐾𝐾1-cycle (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2) for (𝔓, 𝔅), where id ⊗ 𝜎1 generates the 1-multigrading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This defines a genuine U(1)-equivariant unbounded 𝐾𝐾1-cycle for (𝔓, 𝔅) if and only if 𝜅 = 1, in which case, it recovers a well-known construction of Carey–Neshveyev–Nest–Rennie [27, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10] up to 1-multigrading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in all cases, its formal bounded transform recovers, up to 1-multigrading, the canonical representative of the extension class [𝜕] ∈ 𝐾𝐾1(𝔓, 𝔅) of 𝔓 as a Pimsner algebra [4, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, we may now reinterpret Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 in terms of formal unbounded Kasparov products [71, 59]: (1) given an object (𝐻, 𝜋, 𝐷) of BCRep(𝐵), we may write (𝑃, ˜𝐻, ˜𝐷) � (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∇) ⊗𝐵 (𝐵, 𝐻, 𝐷);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) given an object (𝐻, 𝜋, 𝐷,Γ) of PCRep(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), the natural isomorphism 𝑈(𝐻,𝜋,𝐷,Γ) of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 yields (𝑃, 𝐻, 𝐷 − 𝑍) � (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦ 𝜕𝜅) ⊗ 𝜎2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' ∇) ⊗𝐵 (𝐵, 𝐻𝐵, 𝐷𝐵), where 𝑍 is the remainder of (𝐻, 𝜋, 𝐷,Γ) and (𝐻𝐵, 𝜋𝐵, 𝐷𝐵) � 𝜄∗ 𝑃(𝐻, 𝜋, 𝐷,Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In both cases, ∇ is the represented connection on (𝑃, 𝑃 ⊗ C2, i(Λ𝜅 ◦𝜕𝜅) ⊗ 𝜎2) constructed from Π in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In the second case, if (𝑃, 𝐻, 𝐷) defines a genuine U(1)-equivariant unbounded 𝐾𝐾1-cycle for (𝔓, C), then this formal unbounded Kasparov product defines a genuine unbounded Kasparov product by results of Ćaćić–Mesland [26, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Otherwise, the 𝐾𝐾-theoretic significance of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (★𝐵, 𝜏𝐵) be a Riemannian geometry on (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), and suppose that (★𝐵, 𝜏𝐵) lifts to a total Riemannian geometry (Δver, Δhor, ★, 𝜏) on (𝑃, Ω𝑃, d𝑃, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then the total Hodge–de Rham commutator representation (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝜋𝑃, d𝑃 +d∗ 𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2Π−id) induced by (Δver, Δhor, ★, 𝜏) is the essentially unique lift of the Hodge–de Rham commutator representation (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 𝜋𝐵, d𝐵 + d∗ 𝐵) induced by (★𝐵, 𝜏𝐵) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='32, we shall use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50 to construct 𝜄∗ O𝑞(SU(2)) (/𝑆𝑞(CP1), 𝜋, /𝐷1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, let /𝑆𝑞(SU(2)) � C2 ⊗ C2 ⊗ O𝑞(SU(2)) with the inner product ⟨·, ·⟩ given by ∀𝑥1, 𝑥2, 𝑦1, 𝑦2 ∈ C2, ∀𝑝1, 𝑝2 ∈ O𝑞(SU(2)), ⟨𝑥1 ⊗ 𝑦1 ⊗ 𝑝1, 𝑥2 ⊗ 𝑦2 ⊗ 𝑝2⟩ � ⟨𝑥1, 𝑥2⟩⟨𝑦1, 𝑦2⟩ℎ𝑞(𝑝∗ 1𝑝2), the Z2-grading 𝜎3 ⊗ 𝜎3 ⊗ id, and the unitary U(1)-representation of finite type ˜𝑈 defined by setting ˜𝑈 � �𝑧 ↦→ id ⊗ � 𝑧 0 0 𝑧−1 � ⊗ 𝛼𝑧 �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, define Λ𝑞2 and 𝜕𝑞2 on /𝑆𝑞(SU(2)) in terms of ˜𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, let ˜𝜋 : O𝑞(SU(2)) → LU(1) (/𝑆𝑞(SU(2)))even be induced by multiplication from the left in O𝑞(SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, let �/𝐷𝑞 � �/𝐷𝑞,ver + �/𝐷𝑞,hor, where the operators �/𝐷𝑞,ver and �/𝐷𝑞,hor on /𝑆𝑞(SU(2)) are given by �/𝐷𝑞,ver � i(𝜎2 ⊗id ⊗ id)◦Λ𝑞2 ◦𝜕𝑞2, �/𝐷𝑞,hor � 𝜎3 ⊗ � 1 2 (𝜎1 − i𝜎2) ⊗ 𝑞𝜕− + 1 2 (𝜎1 + i𝜎2) ⊗ 𝑞𝜕+ � , NONCOMMUTATIVE U(1)-GAUGE THEORY 73 and let Γ𝑞 � 𝜎3 ⊗ id ⊗ id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the maps (𝑝 ⊗ 𝑥 ↦→ 𝑝 · 𝑥) : O𝑞(SU(2)) ⊗O𝑞(CP1) /𝑆𝑞,±(CP1) are left O𝑞(SU(2))-module isomorphisms by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='15, we may construct an even U(1)-equivariant unitary Φ : (O𝑞(SU(2)) ⊗ C2) ⊗O𝑞(CP1) /𝑆𝑞(CP1) → /𝑆𝑞(SU(2)) by ∀𝑝 ∈ O𝑞(SU(2)), ∀𝑥 ∈ C2, ∀ � 𝑠+𝑠− � ∈ /𝑆𝑞(CP1), Φ�𝑝 ⊗ 𝑥 ⊗ � 𝑠+𝑠− �� � 𝑥 ⊗ �� 1 0 � ⊗ 𝑝 · 𝑠+ + � 0 1 � ⊗ 𝑝 · 𝑠− � , which yields the desired isomorphism of projective commutator representations (Φ, 0) : 𝜄O𝑞(SU(2)) (/𝑆𝑞(CP1), 𝜋, /𝐷𝑞) → � /𝑆𝑞(SU(2)), ˜𝜋, �/𝐷𝑞,Γ𝑞 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note, in particular, that � /𝑆𝑞(SU(2)), ˜𝜋, �/𝐷𝑞,Γ𝑞 � is faithful since (/𝑆𝑞(CP1), 𝜋, /𝐷1) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Twisted boundedness of lifted commutator representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We have solved the lifting problem for faithful bounded commutator representations, but at a cost: the resulting faithful projectable commutator representations generally involve unbounded represented 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Here, we control this unboundedness in the spirit of Connes–Moscovici’s twisted spectral triples [34] by allowing for possibly distinct vertical and horizontal twists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' One upshot is that quantum SU(2) as the total space of the 𝑞-monopole does not admit a non-pathological U(1)-equivariant twisted spectral triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The other is that Kaad–Kyed’s compact quantum metric space [58] on quantum SU(2) for a canonical choice of parameters can be geometrically derived, up to equivalence of Lipschitz seminorms, from the spin Dirac spectral triple on quantum CP1 using the 𝑞-monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Once more, let 𝜅 > 0, let (𝑃, Ω𝑃, d𝑃, Π) be a 𝜅-differentiable quantum principal U(1)- bundle over 𝐵, let 𝜗 be the connection 1-form of Π, let ˆΦ𝑃 be the Fröhlich automorphism of Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) = (𝑃, Ω𝑃,hor, d𝑃,hor), and let Φ𝑃 be the Fröhlich automorphism of the Hermitian line 𝐵-bimodule L(𝑃)(1), so that ˆΦ𝑃 and Φ𝑃 agree on Z(Ω𝐵)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, recall that ˆΦ𝑃 induces the right Z-action on Z>0(𝐵) � (Z(Ω𝐵)0)× + defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12), which therefore extends, mutatis mutandis to a right Z-action on Z(𝐵)× + in terms of Φ𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We begin with the analogue for locally bounded commutator representations of modular automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a locally bounded commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' A modular symmetry of (𝐻, 𝜋, 𝐷) is an even positive U(1)-invariant invertible operator 𝑁 ∈ LU(1) loc (𝐻) the restricts to the identity on 𝐻U(1), commutes with 𝜋(𝐵), and satisfies 𝑁 ran(𝜋)𝑁−1 = ran(𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝔓 denote the 𝐶∗-completion for 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a locally bounded commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃), that 𝜈 is a modular automorphism of Ω𝑃, and that 𝑁 is a modular symmetry of (𝐻, 𝜋, 𝐷), such that 𝑁−1𝜋(·)𝑁 = 𝜋 ◦ 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝐷𝑁 � 𝑁𝐷𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since, for all 𝑝 ∈ 𝑃, 𝑁[𝐷, 𝜋(𝑝)]𝑁 = 𝐷𝑁𝜋(𝜈(𝑝)) − 𝜋 �𝜈−1(𝑝)�𝐷𝑁 = 𝐷𝑁𝜋(𝑝) − 𝜋 �𝜈−2(𝑝)�𝐷𝑁, it follows that (𝑃, 𝐻, 𝐷𝑁) defined a U(1)-equivariant even 𝜈−2-twisted spectral triple for 𝔓 only if 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In light of the above remark, the following theorem will exclude the existence of non- pathological U(1)-equivariant twisted spectral triples that faithfully represent the total spaces of the 𝑞-monopole or the real multiplication instanton of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, it will imply that faithful projectable commutator representations of these two examples cannot naturally be accommodated by the theory of twisted spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 74 BRANIMIR ĆAĆIĆ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that Z(𝐵) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a locally bounded commutator rep- resentation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' suppose that 𝜋 is injective, the subspace 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻, and there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷), such that 𝑁 · 𝜋𝐷(Ω1 𝑃)𝑁 ⊆ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜂 ∈ Ω1 𝑃,hor \\ {0} and 𝑡 ∈ (0, ∞) \\ {𝜅}, and suppose that ∀𝑝 ∈ 𝑃, 𝜂 · 𝑝 = Λ𝑡(𝑝) · 𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50) Then (id −Π)(Ω1 𝑃) ⊆ ker 𝜋𝐷 or 𝜋𝐷(𝜂) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a U(1)-equivariant commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If 𝜋 is injective, the map �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : Z(𝐵) → L(𝐻U(1)) is isometric, and 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻, then for every modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷), there exists a unique right 1-cocycle 𝜇 : Z → Z(𝐵)× +, such that 𝑁 = � 𝑗∈Z 𝜋(𝜇(−𝑗))↾𝐻𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51) Conversely, for every 1-cocycle 𝜇 : Z → Z(𝐵)× +, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51) defines a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We prove the non-trivial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝜋 is injective, the map 𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) restricts to an isometry on Z(𝐵), and 𝜋(𝑃)·𝐻U(1) is dense in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑁 be a modular symmetry of (𝐻, 𝜋, 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜋 is injective, there exists a unique U(1)-equivariant algebra automorphism Δ of 𝑃, such that 𝜋 ◦ Δ = 𝑁−1𝜋(·)𝑁;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' in particular, Δ↾𝐵= id𝐵 since 𝑁 commutes with 𝜋(𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23, mutatis mutandis, there exists a unique 1-cocycle 𝜇 : Z → Z(𝐵)×, such that Δ(𝑝) = 𝑝 · 𝜇(𝑗) for all 𝑗 ∈ Z and 𝑝 ∈ 𝑃𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, for all 𝑗 ∈ Z, 𝑝 ∈ 𝑃𝑗 and ℎ ∈ 𝐻U(1), 𝑁𝜋(𝑝)ℎ = 𝑁𝜋(𝑝)𝑁−1ℎ = 𝜋(Δ−1(𝑝))ℎ = 𝜋(𝑝𝜇(𝑗)−1)ℎ = 𝜋(𝜇(−𝑗))𝜋(𝑝)ℎ since ˆΦ𝑗 𝑃(𝜇(𝑗)−1) = 𝜇(−𝑗), so that (𝑁, 𝜇) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='51) since 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let (𝜖𝑖)𝑁 𝑖=1 be a finite family in 𝑃1 satisfying �𝑁 𝑖=1 𝜖∗ 𝑖 𝜖𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 0 ≤ ∑︁𝑁 𝑖=1 𝜋(𝜖𝑖)∗𝑁𝜋(𝜖𝑖) = ∑︁𝑁 𝑖=1 𝜋(𝜖𝑖)∗𝜋(𝜖𝑖𝜇(1)−1)𝑁 = 𝜋(𝜇(1)−1)𝑁, so that 𝜋(𝜇(1)↾𝐻U(1))≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, since �𝑏 ↦→ 𝜋(𝑏)↾𝐻U(1) � : Z(𝐵) → L(𝐻U(1)) is isometric, it follows that 𝜇(1) ≥ 0, so that 𝜇 takes its values in the subgroup Z(𝐵)× ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that Z(𝐵) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜂 ∈ Ω1 𝑃,hor \\ {0} and 𝑡 ∈ (0, ∞) \\ {𝜅}, and suppose that 𝜂 and 𝑡 satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a locally bounded commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃), such that 𝜋 is injective and 𝜋(𝑃) · 𝐻U(1) is dense in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For every modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷), the operator 𝑁𝜋𝐷(𝜔)𝑁 is bounded only if 𝑁 = Λ𝑡−1/2 or 𝜋𝐷(𝜔) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑁 be a modular symmetry of (𝐻, 𝜋, 𝐷);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' suppose that 𝑁𝜋𝐷(𝜔)𝑁 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since Z(𝐵) = C, it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23 that there exists unique 𝑠 ∈ (0, ∞), such that 𝑁 = Λ𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Now, let (𝑒𝑖)𝑚 𝑖=1 and (𝜖𝑗)𝑛 𝑗=1 be finite families in 𝑃1, such that �𝑚 𝑖=1 𝑒𝑖𝑒∗ 𝑖 = 1 and �𝑛 𝑗=1 𝜖∗ 𝑗 𝜖𝑗 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, let 𝜙± : LU(1) (𝐻) → LU(1) (𝐻) be the unit-preserving contractions from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='36 induced by (𝑒𝑖)𝑚 𝑖=1 and (𝜖𝑗)𝑛 𝑗=1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then 𝜙+(𝑁𝜋𝐷(𝜂)𝑁) = ∑︁𝑚 𝑖=1 𝜋(𝑒𝑖)Λ𝑠𝜋𝐷(𝜂)Λ𝑠𝜋(𝑒∗ 𝑖 ) = 𝜋 �∑︁𝑚 𝑖=1 𝑒𝑖 · (Λ𝑠 ◦ Λ𝑡 ◦ Λ𝑠)(𝑒∗ 𝑖 ) � 𝑁𝜋𝐷(𝜂)𝑁 = 𝑠2𝑡𝑁𝜋𝐷(𝜂)𝑁, while a similar calculation shows that 𝜙−(𝑁𝜋𝐷(𝜂)𝑁) = (𝑠2𝑡)−1𝑁𝜋𝐷(𝜂)𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, ∥𝑁𝜋𝐷(𝜂)𝑁∥ = (𝑠2𝑡)∓1∥𝜙±(𝑁𝜋𝐷(𝜂)𝑁)∥ ≤ (𝑠2𝑡)∓1∥𝑁𝜋𝐷(𝜂)𝑁∥, so that 𝑁𝜋𝐷(𝜂)𝑁 = 0 or 𝑠2𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ NONCOMMUTATIVE U(1)-GAUGE THEORY 75 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑁 be a modular symmetry of the locally bounded commutator representation (𝐻, 𝜋, 𝐷) that satisfies 𝑁 · 𝜋𝐷(Ω1 𝑃) · 𝑁 ⊆ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝜋𝐷(𝜂) ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60 applied to 𝜂, it follows that 𝑁 = Λ𝑡−1/2 ≠ Λ𝜅−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60 applied to 𝜗, it follows that 𝜋𝐷(𝜗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜗 generates (id −Π)(Ω1 𝑃) as a left 𝑃-module, it follows that (id −Π)(Ω1 𝑃) ⊆ ker 𝜋𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='37, let (𝐻, 𝜋, 𝐷) be a locally bounded commuta- tor representation of (O𝑞(SU(2)), Ω𝑞(SU(2)), d𝑞), such that 𝜋 is injective and the subspace 𝜋(O𝑞(SU(2))) · 𝐻U(1) is dense in 𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' If there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷) sat- isfying 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻), then (id −Π𝑞)(Ω1 𝑃) ⊆ ker 𝜋𝐷 or Ω1 𝑞,hor(SU(2)) ⊆ ker 𝜋𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that 𝑁 is such a modular symmetry and (id −Π𝑞)(Ω1 𝑞(SU(2)))\\ker 𝜋𝐷 ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (𝜂, 𝑡) � (𝑒±, 𝑞) satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50), where 𝑞 ≠ 𝑞2, so that 𝜋𝐷(𝑒±) = 0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since {𝑒+, 𝑒−} generates Ω1 𝑞,hor(SU(2)) as a left O𝑞(SU(2))-module, it follows that Ω1 𝑞,hor(SU(2)) ⊆ ker 𝜋𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='38, let (𝐻, 𝜋, 𝐷) be a U(1)-equivariant commu- tator representation of (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃), such that 𝜋 is injective and 𝜋(𝑃𝜃) · 𝐻U(1) is dense in 𝐻, If there exists a modular symmetry 𝑁 of (𝐻, 𝜋, 𝐷) satisfying 𝑁 ran(𝜋𝐷)𝑁 ⊆ LU(1) (𝐻), then (id −Π𝑃𝜃)(Ω1 𝑃𝜃) ⊆ ker 𝜋𝐷 or Ω1 𝑃𝜃,hor ⊆ ker 𝜋𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, note that the left 𝑃-module Ω1 𝑃,hor is freely generated by 𝑒1, 𝑒2 ∈ Z(Ω𝜃(T2))1, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50) is satisfied by (𝜂, 𝑡) = (𝑒𝑖, 𝜖𝜃) for 𝑖 = 1, 2 by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜖𝜃 ≠ 𝜖2 𝜃, we may apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='58 exactly as in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since a single modular symmetry cannot generally be used to control the unboundedness of represented 1-forms, we are forced to allow for distinct modular symmetries in the vertical and horizontal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that (id −Π)(Ω1 𝑃) = 𝑃 · 𝜗 and Π(Ω1 𝑃) = 𝑃 · Ω1 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷,Γ) be a projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (1) A vertical twist for (𝐻, 𝜋, 𝐷,Γ) is a pair (𝑁ver, 𝜈ver), where 𝑁ver is a modular symmetry of (𝐻, 𝜋, 𝐷) commuting with both Γ and 𝜋𝐷(𝜗) and 𝜈ver is a modular automorphism of Ω𝑃, such that 𝑁−1 ver𝜋(·) · 𝑁ver = 𝜋 ◦ 𝜈ver↾𝑃 and 𝑁ver𝜋𝐷(𝜗)𝑁ver ∈ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (2) A horizontal twist for (𝐻, 𝜋, 𝐷,Γ) is a pair (𝑁hor, 𝜈hor), where 𝑁hor is a modular symmetry of (𝐻, 𝜋, 𝐷) commuting with both Γ and 𝜋𝐷(𝜗) and 𝜈hor is a modular automorphism of Ω𝑃, such that 𝑁−1 hor𝜋(·)𝑁hor = 𝜋 ◦ 𝜈hor↾𝑃 and 𝑁hor𝜋𝐷 �Ω1 𝐵 �𝑁hor ⊆ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, if (𝐻, 𝜋, 𝐷,Γ) is a projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π), then any vertical twist (𝑁ver, 𝜈ver) satisfies 𝑁ver𝜋𝐷 �(id −Π)(Ω1 𝑃)�𝑁ver ⊆ LU(1) (𝐻), and any horizontal twist (𝑁hor, 𝜈hor) satisfies 𝑁hor𝜋𝐷 �Π(Ω1 𝑃)�𝑁hor ⊆ LU(1) (𝐻).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now study the existence of vertical and horizontal twists for faithful projectable com- mutator representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Lemmata 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='23 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='59 justify the following convenient definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷,Γ) is faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We define a modular pair for (𝐻, 𝜋, 𝐷,Γ) to be a pair (𝑁, 𝜈), where 𝑁 is a modular symmetry of (𝐻, 𝜋, 𝐷) and 𝜈 is a modular automorphism of Ω𝑃 satisfying the equation 𝑁−1𝜋(·)𝑁 = 𝜋 ◦ 𝜈↾𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In this case, the symbol of (𝑁, 𝜈) is the unique right 1-cocycle 𝜆 : Z → Z>0(𝐵), such that ∀𝑗 ∈ Z, 𝑁↾𝐻𝑗= 𝜋(𝜆(−𝑗))↾𝐻𝑗, 𝜈↾(Ω𝑃)𝑗= (𝜔 ↦→ 𝜔𝜆(𝑗)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We first show that there is a canonical choice of vertical twist, which is unique whenever 𝐵 satisfies Z(𝐵) = C, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=', when 𝐵 is O𝑞(CP1) for 𝑞 ∈ (0, ∞) \\ {1} or 𝐶∞ 𝜃 (T2) for 𝜃 irrational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷,Γ) is a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (Λ𝜅−1/2, Λ𝜅1/2) defines a vertical twist of (𝐻, 𝜋, 𝐷,Γ), which is unique whenever Z(𝐵) = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 76 BRANIMIR ĆAĆIĆ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝑁, 𝜈) is a modular pair for (𝐻, 𝜋, 𝐷,Γ) with symbol 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since the operator 𝜋𝐷(𝜃) satisfies 𝜋𝐷(𝜃)2 = Λ4 𝜅, it follows that (𝑁𝜋𝐷(𝜃)𝑁)2 = Λ2 𝜅𝑁4, so that (𝑁, 𝜈) is a vertical twist for (𝐻, 𝜋, 𝐷,Γ) if and only if sup𝑗∈Z 𝜅−𝑗/2∥𝜋(𝜆(−𝑗)) ↾𝐻𝑗 ∥ < +∞, which is certainly satisfied by 𝜆 � (𝑗 ↦→ 𝜅−𝑗/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Moreover, if Z(𝐵) = C, then 𝜆 = (𝑗 ↦→ 𝑡𝑗) for unique real 𝑡 > 0, so that (𝑁, 𝜈) is a vertical twist for (𝐻, 𝜋, 𝐷,Γ) if and only if sup𝑗∈Z(𝜅1/2𝑡)−𝑗 < +∞, if and only if 𝑡 = 𝜅−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ To characterize existence of horizontal twists, we shall need the following broad gener- alisation of a definition from the literature on spectral triples for crossed products due to Bellissard–Marcolli–Reihani [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Γ be a group, and let ˆ𝐹 : Γ → DPic(𝐵) be a homomorphism, so that the right DPic(𝐵)-action on Z>0(𝐵) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10) pulls back via ˆΦ ◦ 𝜋0( ˆ𝐹) to a right Γ-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For each 𝛾 ∈ Z, let (𝐹(𝛾), 𝜎𝛾, ∇𝛾) � ˆ𝐹(𝛾), and equip 𝐹(𝛾) ⊗𝐵 𝐻 with the inner product defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, for each 𝛽 ∈ Ω1 𝐵, define 𝜌𝛾 [𝛽] : 𝐹(𝛾) ⊗𝐵 𝐻 → 𝐹(𝛾) ⊗𝐵 𝐻 by ∀𝑥 ∈ 𝐹(𝛾), ∀ℎ ∈ 𝐻, 𝜌𝛾 [𝛽](𝑥 ⊗ ℎ) � 𝜎𝛾(𝛽 ⊗ 𝑥) ⟨0⟩ ⊗ 𝜋𝐷 � 𝜎𝛾(𝛽 ⊗ 𝑥) ⟨1⟩ � ℎ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52) and let ∥𝜌𝛾 [𝛽]∥ denote the resulting operator norm of 𝜌𝛾 [𝛽], which we set to equal +∞ whenever 𝜌𝛾 [𝛽] is not bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given a 1-cocycle 𝜆 : Γ → Z>0(𝐵), we say that ˆ𝐹 is 𝜆- metrically equicontinuous with respect to (𝐻, 𝜋, 𝐷) whenever ∀𝛽 ∈ Ω1 𝐵, sup 𝛾∈Γ ��𝜌𝛾 � 𝜆(𝛾−1)2𝛽 ��� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='53) Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall the homomorphism ˆ𝐸 : Γ𝜃 → DPic(𝐶∞ 𝜃 (T2)) of Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' de- fine a group homomorphism 𝜆 : Γ𝜃 → R>0 by 𝜆 � �𝑔 ↦→ (𝑔21𝜃 + 𝑔22)−1/2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ˆ𝐸 is 𝜆- metrically equicontinuous with respect to every faithful bounded commutator representation of (𝐶∞ 𝜃 (T2), Ω𝜃(T2), d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let (𝐻, 𝜋, 𝐷) be such a bounded commutator representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that the left 𝐶∞ 𝜃 (T2)-module Ω1 𝜃 (T2) is generated by {𝑒1, 𝑒2} ⊂ Z(Ω𝜃(T2))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Given 𝑖 = 1, 2 and 𝑔 ∈ Γ𝜃, it follows by construction of ˆ𝐸 that 𝜌𝑔[𝑒𝑖] = 1 𝑔21𝜃+𝑔22 id ⊗𝜋𝐷(𝑒𝑖), so that ∥𝜌𝑔[𝜆(𝑔−1)2𝑒𝑖]∥ = ∥id ⊗𝜋𝐷(𝑒𝑖)∥ ≤ ∥id∥∥𝜋𝐷(𝑒𝑖)∥ ≤ ∥𝜋𝐷(𝑒𝑖)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 the homomorphisms E : Z → Pic(O𝑞(CP1)) and ˆE : Z → DPic(O𝑞(CP1)), and define a group homomorphism 𝜆 : Z → R>0 by 𝜆 � (𝑘 ↦→ 𝑞−𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ˆE is 𝜆-metrically equicontinuous with respect to every faithful bounded commutator representation of (O𝑞(CP1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑞(CP1), d𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Indeed, let (𝐻, 𝜋, 𝐷) be such a bounded commutator representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that Ω𝑞(CP1) = E−2·𝑒+⊕ E2·𝑒−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Choose a coba- sis (𝜂∓ 𝑖 )𝑁∓ 𝑖=1 for E∓2, and define 𝜏± : 𝐻 → E±2 ⊗O𝑞(CP1) 𝐻 by 𝜏±(ℎ) � �𝑁∓ 𝑖=1(𝜂∓ 𝑖 )∗ ⊗ 𝜋𝐷(𝜂∓ 𝑖 𝑒±)ℎ for ℎ ∈ 𝐻;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that 𝜏± is bounded and left 𝐵-linear since, for all ℎ, 𝑘 ∈ 𝐻 and 𝑥 ∈ E±2, ⟨𝑥 ⊗ 𝑘, 𝜏±(ℎ)⟩ = � 𝑘, 𝜋𝐷 � 𝑥∗ �∑︁𝑁∓ 𝑖=1(𝜂∓ 𝑖 )∗𝜂∓ 𝑖 � 𝑒±� ℎ � = ⟨𝑘, 𝜋𝐷(𝑥∗𝑒±)ℎ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For all 𝑖, 𝑗 ∈ Z, define a unitary 𝑉𝑖,𝑗 : E𝑖⊗O𝑞(CP1) (E𝑗⊗O𝑞(CP1) 𝐻) → (E𝑖⊗O𝑞(CP1) E𝑗)⊗O𝑞(CP1) 𝐻 by 𝑉𝑖,𝑗 � (𝑥 ⊗ (𝑦 ⊗ ℎ) ↦→ (𝑥 ⊗ 𝑦) ⊗ ℎ) and, for each 𝑝 ∈ E𝑖, the bounded adjointable map 𝜋𝑖,𝑗(𝑝) : E𝑗 ⊗O𝑞(CP1) 𝐻 → E𝑖+𝑗 ⊗O𝑞(CP1) 𝐻 by 𝜋𝑖,𝑗(𝑝) � (𝑦 ⊗ ℎ ↦→ 𝑝 · 𝑦 ⊗ ℎ), which satisfies ∥𝜆𝑖,𝑗(𝑝)∥ ≤ ∥𝑝∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, let 𝑗 ∈ Z and 𝑝 ∈ E∓2 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝑝𝑒±𝑥 = 𝑞−𝑗 �𝑁∓ 𝑖=1 𝑝𝑥(𝜂∓)∗ 𝑖 𝜂∓ 𝑖 𝑒± for all 𝑥 ∈ E𝑗, it follows that 𝜌𝑗[𝜆(−𝑗)2𝑝𝑒±] = 𝜋∓2,𝑗±2(𝑝) ◦ (E(2) 𝑗,±2 ⊗ id) ◦ 𝑉𝑗,±2 ◦ (id ⊗ 𝜏±), and hence ∥𝜌𝑗[𝜆(−𝑗)2𝑝𝑒±]∥ ≤ ∥𝜋∓2,𝑗±2(𝑝)∥∥E(2) 𝑗,±2 ⊗ id∥∥𝑉𝑗,±2∥∥id ⊗ 𝜏±∥ ≤ ∥𝑝∥∥𝜏±∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 77 In light of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='9, one may ask whether our generalised notion of metric equiconti- nuity makes sense at the level of Hermitian line 𝐵-bimodules with connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following proposition answer this question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (𝐻, 𝜋, 𝐷) is a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let Γ be a group, let ˆ𝐹1, ˆ𝐹2 : Γ → DPic(𝐵) be homomorphisms, and suppose that ˆ𝐹1 � ˆ𝐹2 in Hom(Γ, DPic(𝐵)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, let 𝜆 : Γ → Z>0(𝐵) be a right 1-cocycle for the pullback of the DPic(𝐵)-action of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='10) by 𝜋0( ˆ𝐹1) = 𝜋0( ˆ𝐹2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ˆ𝐹1 is 𝜆-metrically equicontinuous if and only if ˆ𝐹2 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Choose a natural isomorphism 𝜂 : ˆ𝐹1 → ˆ𝐹2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝛾 ∈ Γ be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' For 𝑖 = 1, 2, let (𝐹𝑖(𝛾), 𝜎𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾, ∇𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾) � ˆ𝐹𝑖(𝛾)), and define 𝜌𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾 [𝛽] : 𝐹𝑖(𝛾) ⊗𝐵 𝐻 → 𝐹𝑖(𝛾) ⊗𝐵 𝐻 for each 𝛽 ∈ Ω1 𝐵 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since 𝜂𝛾 : (𝐹1(𝛾), 𝜎1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾, ∇1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾) → (𝐹2(𝛾), 𝜎2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾, ∇2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾) is an isomorphism is DPic(𝐵), the map 𝜂𝛾 ⊗ id : 𝐹1(𝛾) ⊗𝐵 𝐻 → 𝐹2(𝛾) ⊗𝐵 𝐻 is a well-defined unitary that satisfies (𝜂𝛾 ⊗ id) ◦ 𝜌1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾 [𝛽] = 𝜌2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝛾 [𝛽] ◦ (𝜂𝛾 ⊗ id) for all 𝛽 ∈ Ω1 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Hence, given a Hermitian line 𝐵-bimodule with connection (𝐸, 𝜎𝐸, ∇𝐸) and a group 1- cocycle 𝜆 : Z → Z>0(𝐵) for the right Z-action generated by ˆΦ−1 𝐸 , we define (𝐸, 𝜎𝐸, ∇𝐸) to be 𝜆- metrically equicontinuous whenever some (and hence every) homomorphism ˆ𝐹 : Z → DPic(𝐵) satisfying ˆ𝐹(1) � (𝐸, 𝜎𝐸, ∇𝐸) is 𝜆-metrically equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The following characterisation of metric equicontinuity in our general sense for crossed products by extended diffeomor- phisms now shows that metric equicontinuity (in our sense) with respect to a trivial 1-cocycle corresponds to the existing definition in the literature on crossed product spectral triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='70 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Bellissard–Marcolli–Reihani [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a bounded commu- tator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝜔, 𝜙) ∈ � Diff(𝐵) and let 𝜆 : Z → Z>0(𝐵) be a right 1-cocycle for the right Z-action generated by 𝜙−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ˆ𝜏(𝜔, 𝜙) is 𝜆-metrically equicontinuous with respect to (𝐻, 𝜋, 𝐷) if and only if ∀𝑏 ∈ 𝐵, sup 𝑘∈Z ��𝜋(𝜆(𝑘)−1) · [𝐷, 𝜋(𝜙−𝑘(𝑏))] · 𝜋(𝜆(𝑘)−1) �� < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='54) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='70, it suffices to check that the homomorphism ˆ𝜏 ◦ (𝑘 ↦→ (𝜔, 𝜙)𝑘) is 𝜆-metrically equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑘 ∈ Z be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a unitary 𝑉𝑘 : 𝐵𝑘 𝜙 ⊗𝐵 𝐻 → 𝐻 by 𝑉 � (𝑏𝑘 𝜙 ⊗ ℎ ↦→ 𝜋(𝜙−𝑘(𝑏))ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By construction of ˆ𝜏((𝜔, 𝜙)𝑘) � (𝐵𝜙, 𝜎𝜙𝑘, ∇(𝜔,𝜙)𝑘), it follows that 𝑉𝑘𝜌𝑘[𝛽]𝑉∗ 𝑘 = 𝜋𝐷 �𝜙−𝑘(𝛽)� for all 𝛽 ∈ Ω1 𝐵, so that 𝑉𝑘𝜌𝑘 � ˆΦ[ˆ𝜏(𝜔,𝜙)](𝜆(−𝑘)2) · d𝐵(𝑏) � 𝑉∗ 𝑘 = 𝜋𝐷 � 𝜆(𝑘)−2d𝐵𝜙−𝑘(𝑏) � = i𝜋(𝜆(𝑘)−1)[𝐷, 𝜋(𝜙−𝑘(𝑏))]𝜋(𝜆(𝑘)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' for every 𝑏 ∈ 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Since d𝐵(𝐵) generates Ω1 𝐵 as a left 𝐵-module, this proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ At last, we characterise horizontal twists among all modular pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝐵, d𝐵), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) to (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃, Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝑁, 𝜈) be a modular pair for ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) with symbol 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝑁, 𝜈) defines a horizontal twist for ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) if and only if ˆL◦ Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) is 𝜆-metrically equicontinuous with respect to (𝐻, 𝜋, 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52, we may assume that ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) = (𝜄𝑃)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (𝐻, 𝜋, 𝐷) without any loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we reprise the notation of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In particular, it follows that 𝜏 ◦ 𝑁 ◦ 𝜏∗ = id ⊗𝜈 ⊗ id, which makes it clear that 𝑁 ˜𝜋(·)𝑁−1 = ˜𝜋 ◦ 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 78 BRANIMIR ĆAĆIĆ Let 𝛽 ∈ Ω1 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Fix 𝑗 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' By the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50, it follows that ˜𝜋 ˜𝐷(𝛽)↾ ˜𝐻𝑗= 𝜏∗ ◦ (𝜎3 ⊗ 𝜌𝑗[𝛽]) ◦ 𝜏↾ ˜𝐻𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, for every 𝑥 ∈ C2, 𝑝 ∈ 𝑃𝑗, and ℎ ∈ 𝐻, we find that 𝜏𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁𝜏∗(𝑥 ⊗ 𝑝 ⊗ ℎ) = (id ⊗𝜈 ⊗ id) � 𝜎3𝑥 ⊗ 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨0⟩ ⊗ 𝜋𝐷 � 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨1⟩𝜆(𝑗) � ℎ � = 𝜎3 ⊗ 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨0⟩𝜆(𝑗) ⊗ 𝜋𝐷 � 𝜎𝑗(𝛽 ⊗ 𝑝) ⟨1⟩𝜆(𝑗) � ℎ = 𝜎3 ⊗ 𝜌𝑗 � 𝜆(−𝑗)2𝛽 � (𝑥 ⊗ 𝑝 ⊗ ℎ), which implies that ∥𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁↾ ˜𝐻𝑗 ∥ = ��𝜎3 ⊗ 𝜌𝑗 � 𝜆(−𝑗)2𝛽 ��� = ��𝜌𝑗 � 𝜆(−𝑗)2𝛽 ��� by unitarity of 𝜎3 ∈ 𝑀2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the operator 𝑁 ˜𝜋 ˜𝐷(𝛽)𝑁 ∈ LU(1) loc (𝐻) is bounded if and only if the set of operator norms ���𝜌𝑗 � 𝜆(−𝑗)2𝛽 ��� �� 𝑗 ∈ Z � is bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='52 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='67, let (𝐻, 𝜋, 𝐷) be a faithful bounded commutator representation of (𝐶∞ 𝜃 (T2), Ω𝜃(T2), d), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) to the real multiplication instanton (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='65, the modular pair (Λ𝜖−1 𝜃 , Λ𝜖𝜃) is the unique vertical twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='67, the homomorphism ˆL ◦ Hor𝜖2 𝜃 (𝑃𝜃, Ω𝑃𝜃, d𝑃𝜃, Π𝑃𝜃) is (𝑚 ↦→ 𝜖−𝑚/2 𝜃 )- equicontinuous with respect to (𝐻, 𝜋, 𝐷), so that (Λ𝜖−1/2 𝜃 , Λ𝜖1/2 𝜃 ) is the unique horizontal twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='71 together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60 applied to (𝜂, 𝑡) = (𝑒1, 𝜖𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (Λ𝜖−1 𝜃 , Λ𝜖𝜃) and (Λ𝜖−1/2 𝜃 , Λ𝜖1/2 𝜃 ) are non-trivial and distinct since 𝜖𝜃 ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55, let (𝐻, 𝜋, 𝐷) be a faithful bounded commu- tator representation of (O𝑞(CP1), Ω𝑞(CP1), d𝑞), and let ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) be a lift of (𝐻, 𝜋, 𝐷) to the 𝑞-monopole (O𝑞(SU(2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑞(SU(2)), d𝑞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='65, the modular pair (Λ𝑞−1, Λ𝑞) is the unique vertical twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='68, the homomorphism ˆE : Z → DPic(O𝑞(CP1)) of Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='26 is (𝑚 ↦→ 𝑞−𝑚/2)-equicontinuous with respect to (𝐻, 𝜋, 𝐷), so that (Λ𝑞−1/2, Λ𝑞1/2) is the unique horizontal twist of ( ˜𝐻, ˜𝜋, ˜𝐷, ˜Γ) by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='71 together with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='60 applied to (𝜂, 𝑡) = (𝑒±, 𝑞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Note that (Λ𝑞−1, Λ𝑞) and (Λ𝑞−1/2, Λ𝑞1/2) are non-trivial and distinct since 𝑞 ≠ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We now show that total Hodge–de Rham commutator representations admit canonical horizontal twists under a mild hypothesis that is vacuous when Z(𝐵) = C or when 𝐵 is commutative and admits polar decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Suppose that (Δver, Δhor, ★, 𝜏) is a total Riemannian geometry on (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜇𝑃 be the symbol of Δhor, and suppose that 𝜇𝑃(1) has a square root in Z>0(𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, let 𝜇1/2 𝑃 : Z → Z>0(𝐵) be the unique right 1-cocycle for the right Z-action of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12) that satisfies 𝜇1/2 𝑃 (·)2 = 𝜇𝑃, and let (𝑁hor, 𝜈hor) be the modular pair with symbol 𝜇1/2 𝑃 for total Hodge–de Rham commutator representation (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ 𝑃, 2Π − id) induced by (Δver, Δhor, ★, 𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then (𝑁hor, 𝜈hor) defines a horizontal twist for (Ω𝑃, 𝜋𝑃, d𝑃 + d∗ 𝑃, 2Π − id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We use the notation of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, it suffices to show that 𝑁hor satisfies 𝑁hor · e(Ω1 𝐵) · 𝑁hor ⊆ LU(1) (Ω𝑃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝛽 ∈ Ω1 𝐵 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Recall that the pre-Hilbert space Ω𝑃 admits the orthogonal decom- postion Ω𝑃 = �∞ 𝑗=−∞ �1 𝑟=0 �𝑁 𝑠=0(Ω𝑟,𝑠 𝑃 )𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, fix (𝑟, 𝑠, 𝑗) ∈ {0, 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑁} ×Z, so that e(𝛽) maps (Ω𝑟,𝑠 𝑃 )𝑗 to (Ω𝑟,𝑠+1 𝑃 )𝑗, and let 𝑇𝑟,𝑠 𝑗 � 𝑁hore(𝛽)𝑁hor↾(Ω𝑟,𝑠 𝑃 )𝑗= e( ˆΦ𝑗 𝑃(𝜇𝑃(𝑗)−1)𝛽)↾(Ω𝑟,𝑠 𝑃 )𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It therefore suffices to bound the operator norm of 𝑇𝑟,𝑠 𝑗 uniformly in 𝑗 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' NONCOMMUTATIVE U(1)-GAUGE THEORY 79 Let 𝐸 � (Ω𝑟,𝑠 𝑃 )𝑗, 𝐹 � (Ω𝑟,𝑠+1 𝑃 )𝑗, 𝑉 � (Ω𝑟,𝑠 𝑃 )U(1), and 𝑊 � (Ω𝑟,𝑠+1 𝑃 )U(1), which we view as orthogonal direct summands of the pre-Hilbert space Ω𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that each of these pre-Hilbert spaces also defines a 𝐵-self-correspondence of finite type by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29, where each pre-Hilbert space norm is bounded from above by the corresponding norm as a 𝐵-self-correspondence of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Write ˆ𝐿 ◦ Hor𝜅(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) � (𝑃𝑗, 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗, ∇𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗), where we conflate the Hermitian line 𝐵-bimodule L(𝑃)(𝑗) with 𝑃𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' recall that Ω1 𝐵 defines a 𝐵-self- correspondence of finite type by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5, so that Ω1 𝐵 ⊗𝐵 𝑃𝑗 and 𝑃𝑗 ⊗𝐵 Ω1 𝐵 both define 𝐵-self-correspondences of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, we can view each of Ω1 𝐵 ⊗𝐵 𝐸, (Ω1 𝐵 ⊗𝐵 𝑃𝑗) ⊗𝐵 𝑉, (𝑃𝑗 ⊗𝐵 Ω1 𝐵) ⊗𝐵 𝑉, 𝑃𝑗 ⊗𝐵 (Ω1 𝐵 ⊗ 𝑉) and 𝑃𝑗 ⊗𝐵 𝑊 as pre-Hilbert spaces with respect to the inner product defined, mutatis mutandis, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, define 𝜏𝑗 : Ω1 𝐵 ⊗𝐵 𝑃𝑗 → 𝑃𝑗 ⊗𝐵 Ω1 𝐵 by ∀𝜂 ∈ Ω1 𝐵 ⊗𝐵 𝑃𝑗, 𝜏𝑗(𝜂) � 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜇𝑃(−𝑗))𝜂) = 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂)𝜇𝑃(𝑗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' It now follows that 𝑇𝑟,𝑠 𝑗 : 𝐸 → 𝐹 factorizes as the composition 𝐸 𝛽⊗− −−−→ Ω1 𝐵 ⊗𝐵 𝐸 �−→ (Ω1 𝐵 ⊗𝐵 𝑃𝑗) ⊗𝐵 𝑉 𝜏𝑗 ⊗id −−−−→ (𝑃𝑗 ⊗𝐵 Ω1 𝐵) ⊗𝐵 𝑉 �−→ 𝑃𝑗 ⊗𝐵 (Ω1 𝐵 ⊗ 𝑉) id ⊗𝑚𝑟,𝑠 −−−−−→ 𝑃𝑗 ⊗𝐵 𝑊 �−→ 𝐹, where the first two arrows denoted by � are the usual (inverse) associators, which are unitary [18, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12], where 𝑃𝑗 ⊗𝐵 𝑊 �−→ 𝐹 is given by multiplication in Ω𝑃 and hence unitary by the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='29, and where 𝑚𝑟,𝑠 : Ω1 𝐵 ⊗ 𝑉 → 𝑊 is given by multiplication in Ω𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let us now look at the non-trivial arrows in this composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, an explicit calculation shows that 𝛽 ⊗ − � (𝜉 ↦→ 𝛽 ⊗ 𝜉) is bounded with operator norm ∥𝛽 ⊗ −∥ ≤ ∥𝛽∥, where ∥𝛽∥ = ∥𝑔𝐵(𝛽, 𝛽)∥1/2 is the norm of 𝛽 as an element of the 𝐵-self-correspondence of finite type Ω𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, since 𝜏𝑗 is right 𝐵-linear map between pre-Hilbert 𝐵-modules of finite type, it is necessarily bounded and adjointable, so that 𝜏𝑗 ⊗ id is bounded as a map between pre-Hilbert spaces with operator norm ∥𝜏𝑗 ⊗id∥ ≤ ∥𝜏𝑗∥∥id∥ = ∥𝜏𝑗∥ by standard results [18, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, since 𝑚𝑟,𝑠 : Ω1 𝐵 ⊗𝐵 𝑉 → 𝑊is a right 𝐵-linear map of pre-Hilbert 𝐵-modules of finite type, it is bounded and adjointable, and hence bounded as a map of pre-Hilbert spaces with operator norm ∥𝑚𝑟,𝑠∥, so that id ⊗𝑚𝑟,𝑠 is also bounded as a map of pre-Hilbert spaces with operator norm ∥id ⊗𝑚𝑟,𝑠∥ ≤ ∥id∥∥𝑚𝑟,𝑠∥ = ∥𝑚𝑟,𝑠∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Thus, the operator norm of 𝑇𝑟,𝑠 𝑗 is bounded from above by ∥𝛽∥∥𝜏𝑗∥∥𝑚𝑟,𝑠∥, so that, at last, it suffices to show that ∥𝜏𝑗∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Finally, let 𝜂 ∈ Ω1 𝐵 ⊗𝐵 𝑃𝑗 be given, so that 𝜂 = �𝑛 𝑖=1 𝛼𝑖 ⊗ 𝑝𝑖 for 𝛼1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝛼𝑛 ∈ Ω1 𝐵 and 𝑝1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' , 𝑝𝑛 ∈ 𝑃𝑗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' hence, let ˜𝜂 � �𝑛 𝑖=1 𝛼𝑖 · 𝑝𝑖 ∈ (Ω0,1 𝑃 )𝑗, so that ★(˜𝜂) = ∑︁ 𝑖 ★(𝛼𝑖𝑝𝑖) = − ∑︁ 𝑖 𝜗 ★𝐵 (𝛼𝑖)𝑝𝑖𝜇𝑃(𝑗)2−𝑁𝜅𝑗 = (−1)𝑁 ∑︁ 𝑖 ★𝐵(𝛼𝑖)𝑝𝑖𝜇𝑃(𝑗)−𝑁𝜗𝜇𝑃(𝑗)2 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, ★𝑃((𝜂, 𝜂)) = ∑︁ 𝑖,𝑗 𝑝∗ 𝑖 𝑔𝐵(𝛼𝑖, 𝛼𝑗)𝑝𝑗𝜗★𝐵(1) = (−1)𝑁∑︁ 𝑖,𝑗 𝑝∗ 𝑖 𝛼𝑖★𝐵(1)𝑝𝑗𝜇𝑃(𝑗)−𝑁𝜗 = ˜𝜂∗★𝑃(˜𝜂)𝜇𝑃(𝑗)−2, while on the other, ★𝑃 �(𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂), 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂))� = ∑︁ 𝑖,𝑗 𝑔𝐵(𝛼𝑖, 𝑝∗ 𝑖 𝑝𝑗𝛼𝑗)𝜗★𝐵 (1) = (−1)𝑁 ∑︁ 𝑖,𝑗 𝛼∗ 𝑖 𝑝∗ 𝑖 𝑝𝑗 ★𝐵 (𝛼𝑗)𝜗 = ˜𝜂∗ ★𝑃 (˜𝜂), so that (𝜏𝑗(𝜂), 𝜏𝑗(𝜂)) = (𝜇𝑃(𝑗)−1)∗(𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂), 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂))𝜇𝑃(𝑗)−1 = (𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂), 𝜎𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='𝑗(𝜂))𝜇𝑃(𝑗)−2 = (𝜂, 𝜂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ 80 BRANIMIR ĆAĆIĆ We conclude with a first step towards relating our constructions to Rieffel’s compact quantum metric spaces [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We show that a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) equipped with vertical and horizontal twists yields a Lipschitz seminorm [58, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1] on the 𝐶∗-algebra completion of 𝑃 that satisfies a twisted Leibniz inequality [58, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' This, in turn, will recover, up to equivalence of seminorm, Kaad–Kyed’s compact quantum metric space on quantum SU(2) for a canonical choice of parameters [58, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='75 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Kaad–Kyed [58, Lemma 48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let (𝐻, 𝜋, 𝐷,Γ) be a faithful projectable commutator representation of (𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Ω𝑃, d𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Π) with vertical twist (𝑁ver, 𝜈ver) and horizontal twist (𝑁hor, 𝜈hor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Define a U(1)-invariant norms ∥ · ∥𝜏 and ∥ · ∥𝜏,tot on 𝑃 and Ω1 𝑃, respectively, by ∀𝑝 ∈ 𝑃, ∥𝑝∥𝜏 � max � ∥𝜈ver(𝑝)∥ + ∥𝜈hor(𝑝)∥, ∥𝜈−1 ver(𝑝)∥ + ∥𝜈−1 hor(𝑝)∥ � , ∀𝜔 ∈ Ω1 𝑃, ∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot � ∥𝑁ver(𝜋𝐷 ◦ (id −Π))(𝜔)𝑁ver + 𝑁hor(𝜋𝐷 ◦ Π)(𝜔)𝑁hor∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then ∥ · ∥𝜏 makes 𝑃 into a normed ∗-algebra, while ∥ · ∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot is invariant under the ∗-operation and satisfies ∀𝑝1, 𝑝2 ∈ 𝑃, ∀𝜔 ∈ Ω1 𝑃, ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot ≤ ∥𝑝1∥𝜏∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot∥𝑝2∥𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55) Hence, the U(1)-invariant seminorm 𝐿𝜏 � ∥d𝑃(·)∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot on 𝑃 annihilates C ⊆ 𝑃, is invariant under the ∗-operation, and satisfies ∀𝑝1, 𝑝2 ∈ 𝑃, 𝐿𝜏(𝑝1𝑝2) ≤ 𝐿𝜏(𝑝1)∥𝑝2∥𝜏 + ∥𝑝1∥𝜏𝐿𝜏(𝑝2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='76 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Kaad–Kyed [58, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Under the hypotheses of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='75, the U(1)-invariant seminorms ∥ · ∥𝜏,ver and ∥ · ∥𝜏,hor on Ω1 𝑃 defined by ∥ · ∥𝜏,ver � ∥𝑁ver(𝜋𝐷 ◦ (id −Π))(·)𝑁ver∥, ∥ · ∥𝜏,hor � ∥𝑁hor(𝜋𝐷 ◦ Π)(·)𝑁hor∥ satisfy the inequality max{∥𝜔∥𝜏,ver, ∥𝜔∥𝜏,hor} ≤ ∥𝜔∥𝜏,tot for all 𝜔 ∈ Ω1 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝜔 ∈ Ω1 𝑃 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' let 𝜔hor � Π(𝜔), and write (id −Π)(𝜔) = 𝑝𝜗 for unique 𝑝 ∈ 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑐 � 𝜋𝐷(𝜗)Λ−1 𝜅 , which is a U(1)-invariant self-adjoint unitary by definition of a projectable commutator representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, 𝑐 manifestly commutes with the operator 𝜋𝐷(𝑝𝜗) = 𝜋(𝑝)𝑐Λ𝜅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other hand, since Ω1 𝑃,hor = 𝑃 · d𝐵(𝐵) and [𝐷, 𝜋(𝑏)] = [𝜋𝐷(𝜗)𝜕𝜅 + 𝐷hor, 𝜋(𝑏)] = [𝐷hor, 𝜋(𝑏)] for all 𝑏 ∈ 𝐵, it follows that 𝑐 anticommutes with 𝜋𝐷(𝜔hor) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Setting 𝐸± � 1 2 (id ±𝑐), we may now decompose 𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver + 𝑁hor𝜋𝐷(𝜔hor)𝑁hor with respect to the orthogonal direct sum decomposition 𝐻 = 𝐸+(𝐻) ⊕ 𝐸−(𝐻) as 𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver+𝑁hor𝜋𝐷(𝜔hor)𝑁hor = � 𝐸+𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸+ 𝐸+𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸− 𝐸−𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸+ 𝐸−𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸− � , so that ∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ver = ���� �𝐸+𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸+ 0 0 𝐸−𝑁ver𝜋𝐷(𝑝𝜗)𝑁ver𝐸− ����� ≤ ∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot, ∥𝜔∥𝜏,hor = ���� � 0 𝐸+𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸− 𝐸−𝑁hor𝜋𝐷(𝜔hor)𝑁hor𝐸+ 0 ����� ≤ ∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' In what follows, we use the notation of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' The only non- trivial points are positive-definiteness of ∥ · ∥𝜏,tot, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' note that ∥ · ∥𝜏,𝜏 is positive- definite by the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='76, while (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55) implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56) by the usual Leibniz rule for NONCOMMUTATIVE U(1)-GAUGE THEORY 81 d𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Let 𝑝1, 𝑝2 ∈ 𝑃 and 𝜔 ∈ Ω1 𝑃 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' set 𝜔ver � (id −Π)(𝜔) and 𝜔hor � Π(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Then, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='76, ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ver = ∥𝑁ver𝜋𝐷(𝑝1𝜔ver𝑝2)𝑁ver∥ ≤ ∥𝜈−1 ver(𝑝)∥∥𝑁ver𝜋𝐷(𝜔ver)∥∥𝜈ver(𝑝)∥ ≤ ∥𝜈−1 ver(𝑝)∥∥𝜔∥𝜏,tot∥𝜈ver(𝑝)∥, and similarly ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='hor ≤ ∥𝜈−1 hor(𝑝)∥∥𝜔∥𝜏,tot∥𝜈hor(𝑝)∥, so that, in turn, ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot ≤ ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='ver + ∥𝑝1𝜔𝑝2∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='hor ≤ ∥𝜈−1 ver(𝑝)∥∥𝜔∥𝜏,tot∥𝜈ver(𝑝)∥ + ∥𝜈−1 hor(𝑝)∥∥𝜔∥𝜏,tot∥𝜈hor(𝑝)∥ ≤ ∥𝑝∥𝜏∥𝜔∥𝜏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='tot∥𝑝∥𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Continuing from Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='73, we may apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='75 to (/𝑆𝑞(SU(2)), ˜𝜋, ˜/𝐷𝑞,Γ𝑞) equipped with its unique vertical twist (Λ𝑞−1, Λ𝑞) and unique horizontal twist (Λ𝑞−1/2, Λ𝑞1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' We claim that the resulting seminorm 𝐿𝜏 is equivalent to 𝐿𝑞2,𝑞, where (𝐿𝑡,𝑞)𝑡∈(0,∞) is the family of Lipschitz seminorms on O𝑞(SU(2)) with which Kaad–Kyed make 𝐶𝑞(SU(2)) into a compact quantum metric space [58, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='6 & Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' First, note that ∀𝑝 ∈ O𝑞(SU(2)), ∥𝑝∥𝜏 = max � ∥Λ𝑞(𝑝)∥ + ∥Λ𝑞1/2 (𝑝)∥, ∥Λ−1 𝑞 (𝑝)∥ + ∥Λ−1 𝑞1/2 (𝑝)∥ � = ∥𝑝∥𝑞2,𝑞, where (∥ · ∥𝑡,𝑞)𝑡∈(0,∞) is the family of norms on O𝑞(SU(2)) of [58, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5], so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='56) for 𝐿𝜏 is identical to the inequality of [58, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='8] for 𝐿𝑞2,𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Next, using the explicit construction of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='55 together with the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='50, we may now write 𝐿𝜏 = ∥𝜕tot(·)∥, where 𝜕tot : O𝑞(SU(2)) → LU(1) (/𝑆𝑞(SU(2)) is given by 𝜕tot � Λ𝑞−1i[ ˜/𝐷𝑞,ver, ˜𝜋(·)]Λ𝑞−1 + Λ𝑞−1i[ ˜/𝐷𝑞,hor, ˜𝜋(·)]Λ𝑞−1 = 𝜎2 ⊗ �Λ𝑞 ◦ 𝜕𝑞2 0 0 Λ𝑞 ◦ 𝜕𝑞2 � + 𝜎3 ⊗ � 0 Λ𝑞−1/2 ◦ 𝜕+ Λ𝑞−1/2 ◦ 𝜕− 0 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' here, by abuse of notation, we identify 𝑀2(O𝑞(SU(2))) � 𝑀2(C) ⊗ O𝑞(SU(2)) with its image in L(C2 ⊗ O𝑞(SU(2))) via left multiplication of O𝑞(SU(2)) on itself, while, for 𝑡 ∈ (0, ∞), we define 𝜕𝑡 : O𝑞(SU(2)) → O𝑞(SU(2)) by 𝜕𝑡 � � 𝑗∈Z 2𝜋i[𝑗]𝑡 idO𝑞(SU(2))𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' At last, we relate 𝐿𝜏 to 𝐿𝑞2,𝑞 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the one hand, note that if 𝐻 is a Z/2Z-graded pre-Hilbert space with an odd self-adjoint unitary 𝑐 and 𝑆 : 𝐻 → 𝐻 is an odd bounded operator supercommuting with 𝑐, then ∥𝑆∥ = ∥𝑆0∥ for 𝑆0 � −i𝑐 ◦ 𝑆↾𝐻even= i𝑆 ◦ 𝑐↾𝐻even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' On the other, we may construct unitary 𝑈 : O𝑞(SU(2))2 → /𝑆𝑞(SU(2))even by 𝑈 � �� 𝑝1 𝑝2 � ↦→ � 1 0 � ⊗ � 1 0 � ⊗ 𝑝1 + � 0 1 � ⊗ � 0 1 � ⊗ 𝑝2 �, Applying these considerations to 𝑐 = 𝜎1 ⊗ id ⊗ id and 𝑆 = 𝜕tot(𝑝) for 𝑝 ∈ O𝑞(SU(2)) shows that 𝐿𝜏 = ∥𝜕′ tot(·)∥, where 𝜕′ tot : O𝑞(SU(2)) → 𝑀2(O𝑞(SU(2))) is given by 𝜕′ tot � � Λ𝑞 ◦ 𝜕𝑞2 −Λ𝑞−1/2 ◦ 𝜕+ Λ𝑞−1/2 ◦ 𝜕− −Λ𝑞 ◦ 𝜕𝑞2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' But now, for each 𝑡 ∈ (0, ∞), a careful comparison with Kaad–Kyed’s notations [58, §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content='1] shows that 𝐿𝑡,𝑞 = ∥𝜕𝑡,𝑞(·)∥, where 𝜕𝑡,𝑞 : O𝑞(SU(2)) → 𝑀2(O𝑞(SU(2))) is given by 𝜕𝑡,𝑞 = �−i𝐾𝑡Λ𝑡1/2 ◦ 𝜕𝑡 −Λ𝑞−1/2 ◦ 𝜕+ −Λ𝑞−1/2 ◦ 𝜕− i𝐾𝑡Λ𝑡1/2 ◦ 𝜕𝑡 � , 𝐾𝑡 � 1 2𝜋(1 + 𝑡−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Hence, an elementary comparison of 𝜕′ tot with 𝜕𝑞2,𝑞 implies that ∀𝑝 ∈ O𝑞(SU(2)), 1 1 + 𝐾−1 𝑞2 𝐿𝜏(𝑝) ≤ 𝐿𝑞2,𝑞(𝑝) ≤ (1 + 𝐾𝑞2)𝐿𝜏(𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' 82 BRANIMIR ĆAĆIĆ References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Abadie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Eilers, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Exel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Morita equivalence for crossed products by Hilbert 𝐶∗-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfyP4s/content/2301.01749v1.pdf'} +page_content=' Math.' metadata={'source': 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b/LdFOT4oBgHgl3EQf0DSF/content/tmp_files/2301.12934v1.pdf.txt @@ -0,0 +1,1228 @@ +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023 +1 +Coarse-to-fine Hybrid 3D Mapping System with +Co-calibrated Omnidirectional Camera and +Non-repetitive LiDAR +Ziliang Miao1, Buwei He1, Wenya Xie1, Wenquan Zhao1, Xiao Huang1, Jian Bai2, and Xiaoping Hong1 +Abstract—This paper presents a novel 3D mapping robot with +an omnidirectional field-of-view (FoV) sensor suite composed of a +non-repetitive LiDAR and an omnidirectional camera. Thanks to +the non-repetitive scanning nature of the LiDAR, an automatic +targetless co-calibration method is proposed to simultaneously +calibrate the intrinsic parameters for the omnidirectional camera +and the extrinsic parameters for the camera and LiDAR, which +is crucial for the required step in bringing color and texture +information to the point clouds in surveying and mapping +tasks. Comparisons and analyses are made to target-based +intrinsic calibration and mutual information (MI)-based extrinsic +calibration, respectively. With this co-calibrated sensor suite, +the hybrid mapping robot integrates both the odometry-based +mapping mode and stationary mapping mode. Meanwhile, we +proposed a new workflow to achieve coarse-to-fine mapping, +including efficient and coarse mapping in a global environment +with odometry-based mapping mode; planning for viewpoints in +the region-of-interest (ROI) based on the coarse map (relies on the +previous work [1]); navigating to each viewpoint and performing +finer and more precise stationary scanning and mapping of the +ROI. The fine map is stitched with the global coarse map, +which provides a more efficient and precise result than the +conventional stationary approaches and the emerging odometry- +based approaches, respectively. +Index Terms—Mapping, Robotic Systems, Omnidirectional +Vision, Calibration and Identification, SLAM. +I. INTRODUCTION +T +HREE-DIMENSIONAL scanning (obtain the raw points) +and mapping (register or stitch the points into a +point cloud map) are becoming increasingly important in +robotics [2], digital construction [3], and virtual reality [4], +where digitization of the physical 3D space could provide +tremendous insights in modeling, planning, management, +optimization, and quality assurance. Photogrammetry has been +developed to capture the 3D world. However, its application +has been limited in aviation settings where accurate GPS +Manuscript received: Nov. 21, 2022; Revised Jan. 19, 2023; Accepted Jan. +28, 2023. +This paper was recommended for publication by Editor Javier Civera +upon +evaluation +of +the +Associate +Editor +and +Reviewers’ +comments. +This work was supported by Shenzhen Science and Technology Project +(JSGG20211029095803004, +JSGG20201103100401004) +and +SUSTech +startup fund. (Ziliang Miao and Buwei He contributed equally to this work; +Corresponding author: Xiaoping Hong) +1These authors are with School of System Design and Intelligent +Manufacturing (SDIM), Southern University of Science and Technology +(SUSTech), +China +miaozl2019@mail.sustech.edu.cn, +hebw2019@mail.sustech.edu.cn, hongxp@sustech.edu.cn +2Jian Bai is with State Key Laboratory of Modern Optical Instrumentation, +Zhejiang University, China +Digital Object Identifier (DOI): see top of this page. +RTK signals are required. Recently, the need for large-scale +mapping of building environments has been rising, mainly +due to the requirements from Building Information Modeling +(BIM) systems. Thanks to the availability of emerging 3D +robotic LiDAR sensors [5], [6], Mobile Laser Scanner (MLS) +systems are increasingly adopted [7] (Fig. 1a, #3 and #4), +where point clouds from these sensors could be registered +to the global frame through sensor motion estimation (i.e., +odometry) at each instance. However, due to the movement +nature, such approaches largely depend on estimations of +temporal characteristics such as translation and rotation, or +spatial characteristics such as sensor FoV and landmark +coverages. The results vary from scan to scan with no +guarantee of precision. Hence, a more robust and precise +method is desired. +On +the +other +hand, +the +traditional +Terrestrial +Laser +Scanner (TLS) has been employed in many precision-stringent +applications (Fig. 1a, #1 and #2). The TLS-based stationary +mapping is usually inefficient (due to the accurate but slow +laser rotation) but could provide precise results. Viewpoints +(also known as stationary scanning locations) need to be +carefully planned to ensure the spatial coverage and enough +overlapping regions of adjacent viewpoints to make accurate +point cloud stitching [8], but on the other hand, as fewer as +possible to reduce scanning time and cost. The planning for +viewpoints largely relies on the overall layout of the scene, +which has been done by human experience so far [9]. +#1 +#2 +#3 +#4 +(a) +Omnidirectional +camera +Livox Mid-360 LiDAR +(with integrated IMU) +Gimbal mount +Mobile platform +(synchronized) +(b) +Fig. 1. 3D mapping systems: (a) the current TLS (#1 FARO Focus Premium, +#2 LEICA BLK360) and MLS (#3 LEICA BLK2GO, #4 NavVis VLX) +systems; (b) the proposed hybrid mapping robotic system. +Combining the strength from both worlds would be ideal in +large-scale 3D mapping applications. As shown in Fig. 1b, +arXiv:2301.12934v1 [cs.RO] 30 Jan 2023 + +OISEE +SCOUT2 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023 +the proposed hybrid mapping robot is developed carrying +a gimbal mount and a novel sensor suite consisting of an +omnidirectional non-repetitive Livox Mid-360 LiDAR1 and +an omnidirectional camera. The sensors’ FoV and the non- +repetitive scanning nature are shown in Fig. 2a. In the +odometry-based mapping mode, the sensor suite is kept +horizontal by fixing the gimbal mount to coarsely and +efficiently map the entire space with the mobile platform. +Based on the coarse map, a few viewpoints are planned for the +stationary mapping of targeted ROIs. In the stationary mapping +mode, the robot will navigate and stay still at each viewpoint, +performing 360°×300° scanning by traversing the vertical FoV +through the gimbal mount. These precise scans are registered +with each other and then stitched with the pre-generated coarse +map forming a global map with fine ROIs. +The main contributions of this work are as follows: +1) The +first +hybrid +3D +mapping +robot +system +that +integrates odometry-based and stationary mapping modes +is proposed. The consistency of point clouds in two +modes can be guaranteed with the single omnidirectional +non-repetitive Livox Mid-360 LiDAR. +2) An omnidirectional camera is introduced in the proposed +system to complement the omnidirectional LiDAR. +A +novel +automatic +targetless +co-calibration +method +is proposed to simultaneously calibrate the intrinsic +parameters and the extrinsic parameters. +3) An automated coarse-to-fine hybrid mapping workflow is +demonstrated, including odometry-based coarse mapping +in the global environment, planning for the viewpoints +in the ROIs, and finer stationary mapping at viewpoints. +The entire project is open-sourced on GitHub2 to aid the +development of this emerging field. +II. RELATED WORKS +A. Mapping Solutions +3D mapping solutions are of great interest in many +emerging fields [3]. TLS-based and MLS-based approaches +are commonly adopted. +The traditional TLS-based approach uses a heavy-duty +single-laser scanner and traverses the entire FoV through +step-wise rotations about the horizontal and vertical axes. +It provides sufficiently dense points with good precision. +However, this method is slow and laborious. It has to be +repeated on many viewpoints, which need to be chosen wisely +because a lack of viewpoints will cause missing information +in the desired ROI, while the excess of viewpoints will lead +to longer scanning hours and poorer efficiency. Currently, +viewpoints planning relies on human intuition or experiences, +making it challenging to plan effectively in large and complex +working environments like the construction scenes [9]. +On the contrary, the MLS-based approach provides real- +time scanning and mapping results as the LiDAR moves. +The current MLS devices are classified by their usage +configurations, such as handheld (Fig. 1a, #3), backpack +1The authors gratefully acknowledge Livox Technology for the equipment +support. +2https://github.com/ZiliangMiao/Hybrid Mapping Cocalibration.git +(Fig. 1a, #4), and trolley. Most of these mobile systems rely on +conventional LiDARs (16, 32, or 64 lines) and construct the +3D map by registering the point cloud with LiDAR odometry +or LiDAR-IMU odometry. Such mobile systems greatly speed +up the mapping process without planning for viewpoints. +However, it cannot replace the TLS-based approaches due to +insufficient mapping precision and sparse point clouds [3]. The +repetitive scanning nature of mechanical LiDAR is unsuitable +for stationary scanning due to limited FoV coverage (20% +coverage for 32-line LiDAR). Therefore, the indispensable +motion for more coverage will cause errors in pose estimation, +which are accumulated throughout the process, limiting the +usage in high-precision applications. +Both TLS-based and MLS-based approaches have their +unique advantages and drawbacks. It is desired to devise +a mechanism to combine both modes. For example, a +combination of TLS and MLS is used to solve the registration +problem between non-overlapping spaces [8] or use TLS scans +as references to MLS mapping registration to achieve low +mapping errors [10]. Moreover, MLS is also used to provide a +3D map to solve the viewpoints planning problem of TLS [9]. +However, all these methods are based on heterogeneous +sensors for different modes, with different synchronization, +data structure, and protocols, which are difficult to construct +a one-stop mapping robot with a streamlined and automated +workflow. +The unique non-repetitive scanning nature of the Livox +LiDAR provides a combination of an instantaneous high +density at a short time interval for odometry (with effective +point density as 32-line LiDAR within 0.1 seconds) and an +image-level resolution at relatively long time intervals for +scanning (within 3 seconds, as shown in Fig. 2b), which makes +it surprisingly suitable for such hybrid working mechanism. +The +feature +provides +sufficiently +good +performance +in +odometry scenarios [11] and a dense FoV coverage for image- +like feature processing [6], [12], [13]. In this paper, the two +working modes are integrated into the same robot, ensuring +overall mapping efficiency and precision with an automated +coarse-to-fine hybrid mapping workflow. +B. Calibration Methods +In addition to LiDAR, Cameras are usually required +in +3D +mapping +systems +to +give +an +overview +of +the +mapped environment [14]. Cameras could provide high-quality +geometric, color, and texture information [15], which enables +further modeling and rendering [16] of the point clouds +and permits tasks in object detection, segmentation, and +classification [17]. Meanwhile, for autonomous navigation, the +camera is also vital to visual-LiDAR odometry through sensor +fusion [4]. All these functions would rely on the accurate +calibration of the intrinsic parameters of the camera and +extrinsic parameters between the cameras and LiDAR [15]. +Traditionally, multiple cameras are usually required to +be complementary to the omnidirectional FoV of LiDAR. +This work employs an omnidirectional camera over the +traditional multi-camera vision to avoid bulky construction, +high cost, shutter synchronization, and cascaded extrinsic + +MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR +3 +calibrations. The intrinsic and extrinsic parameters of this +novel omnidirectional sensor suite are essentially needed. +The intrinsic parameters of the omnidirectional camera +must be well calibrated since these types usually possess +much larger and more complex distortions than pin-hole +cameras [18]. In [18]–[20], higher-order polynomial-based +intrinsic +models +are +introduced +with +many +degrees +of +freedom to obtain satisfactory results. A popular OcamCalib +toolbox based on the checkerboard is provided [19]. These +methods could be susceptible to over-fitting with high-order +polynomials and often require evenly distributed artificial +targets and dense features across the entire space. Typically, +these calibration processes are manual and could lead to +tedious procedures with a large margin of error. Additionally, +the omnidirectional camera in our work is constructed with +a refractive-reflective geometry to capture a ring-like FoV +beyond 180°. This construction makes intrinsic calibration +even more difficult. An accurate, automatic, and targetless +calibration method is desired. +The +extrinsic +calibration +method +between +the +omnidirectional camera and LiDAR has only been explored +in [21] using edge correspondence to match point clouds +and images. The bearing angle images highlight the edge +features, which are manually positioned. Targetless extrinsic +calibration methods for monocular cameras and LiDAR have +been developed recently. With the non-repetitive LiDARs, +CamVox [12] could project the image-like LiDAR point +clouds onto the camera image plane and extract edge pixels +using the grayscale images based on reflectivity and depth. +The method proposed in [13] uses voxels to extract the edge +points in 3D space and classifies the edges based on depth +continuity. Both methods work well with conventional pin-hole +cameras and need to be extended toward the omnidirectional +cameras with significantly larger distortions. An additional +targetless extrinsic calibration method employing mutual +information (MI) is also developed [22], which maximizes +the intensity correlations of LiDAR and camera. However, +the misrepresented information caused by lighting conditions, +surface +reflection +properties, +and +spectral +reflectance +disagreement could result in worse calibration than the +edge-based methods. +In the proposed targetless co-calibration method, the high- +resolution dense point cloud of the non-repetitive scanning +LiDAR gives abundant and ground-truth-level features, which +eliminates the artificial targets and manual involvement and +reduces the error caused by insufficient coverage and sparse +features of the targets. With the co-calibration method, the +intrinsic and extrinsic parameters are obtained simultaneously +and can be re-calibrated fast and reliably in work scenes. +III. PROPOSED SYSTEM +A. Co-calibrated Omnidirectional Sensor Suite +The Livox Mid-360 LiDAR has a 360° × 55° FoV and +features a non-repetitive scanning pattern, with increasingly +denser points over time (the coverage of FoV approaches +100%), as shown in Fig. 2b. The unique feature specifically +benefits both odometry-based and stationary mapping modes. +The omnidirectional camera provides color information of +the surroundings and has a corresponding 360° × 70° FoV +(Fig. 2a). Both sensors are synchronized and are mounted +on a two-axis gimbal (Fig. 1) to extend the scanning FoV +to 360° × 300°. +-7° +-10° ++60° ++52° +-10° ++60° +-7° ++52° +Omnidirectional Camera +Livox Mid-360 LiDAR +(a) +T = 0.1s +T = 0.5s +T = 3.0s +(b) +Fig. 2. Configuration of the sensors: (a) omnidirectional camera and Livox +Mid-360 LiDAR, both on the gimbal mount; (b) point cloud accumulation +over time due to the non-repetitive scanning nature of the Livox LiDAR. +（Color represents reflectivity of LiDAR points) +#1 +#2 +#3 +... +0 +5.1E-3 +Probability density +0 +5.1E-3 +Probability density +Fig. 3. Proposed co-calibration process. * The grayscale value indicates the +average reflectivity of the projected LiDAR points within a pixel. +The co-calibration simultaneously obtains the intrinsic +(camera) and extrinsic (camera-LiDAR) parameters, defined +respectively as Θ ≜ [u0, v0, c, d, e, a0, . . . , an]T and ∆ ≜ +[α, β, γ, tx, ty, tz]T, which will be introduced later. With + +HO +DJP2002NOOEORXDHZDZDIHZp)120,△=argma +0,1 +ax +nn +Cf( +-14 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023 +the unique benefit of the non-repetitive scanning LiDAR, +an extremely dense point cloud is always available, which +provides a 3D ground truth of the environment. This high- +resolution point cloud could be projected onto the 2D image +plane with pixel values from LiDAR reflectivity, from which +clear edge features could be extracted. To align the edges +from LiDAR and the camera, the co-calibration iteratively +maximizes the correspondence of projected LiDAR edge +points with the omnidirectional camera edge pixels. Kernel +Density Estimation (KDE) is employed to estimate the camera +edge distribution with different distribution smoothness (by +varying bandwidth coefficient) to obtain global optimum. +The entire process of co-calibration can be divided into the +following two steps (Fig. 3): +1) Edge Extraction: Edge extractions are performed for +both camera and LiDAR. For the camera, exposure fusion [23] +is adopted to enhance the dynamic range of images to capture +more details for low and high-brightness objects. Canny edge +extraction [24] is performed on the enhanced image, with +edge points Q = [q1, q2 . . . , qn]. For LiDAR, since the +FoV is smaller, point clouds scanned from different pitch +angles are stitched together. The stitching is performed by the +generalized iterative closest point (GICP) algorithm [25] with +the initial transformation given by the state of the gimbal. +The stitched point cloud with reflectivity is then projected +to an image plane with the azimuthal angle and elevation +angle as the coordinates, generating a grayscale image by +taking the average reflectivity of the projected LiDAR points +within each pixel. The Canny edge extraction is performed +on this grayscale image. Uniform sampling is performed in +each stage to remove the non-uniform point distribution. The +edge pixels are then identified in the original 3D point cloud +P = [LP1, LP2 . . . , LPm]. +2) Iterative Optimization: +The iterative optimization is +performed in the omnidirectional image space. The LiDAR +edge points are projected to the image coordinates through +the following equations: +CP = C +LT(LP; ∆) = C +LR · LP + C +Lt, LP ∈ P, +(1) +p = Π(CP; Θ) = +�c +d +e +1 +� �r cos φ − u0 +r sin φ − v0 +� +, +(2) +r = F(θ; a0, . . . , an) = a0 + a1θ1 + . . . + anθn, +(3) +θ = arccos( +z +� +x2 + y2 + z2 ), +(4) +φ = arccos( +x +� +x2 + y2 ), +(5) +where CP and LP denote the 3D point coordinates in camera +and LiDAR coordinate systems, respectively, and they are +related through the extrinsic transformation C +LT(LP; ∆), i.e., +rotation C +LR and translation C +Lt with the extrinsic parameters +∆. The symbol p denotes the location of the point in the +camera image space, and Π(CP; Θ) expresses the intrinsic +transformation from +CP += +[x, y, z]T (3D point) to p +(2D point), with the distortion correction matrix +� +c +d +e +1 +� +. +The pixel radius r from the image center [u0, v0]T is +transformed from the elevation angles θ by a polynomial +function F(θ; a0, . . . , an) in the camera model; θ and φ are the +elevation and azimuth angle of CP (Note the omnidirectional +camera features a ring image). +To facilitate the alignment between the camera edges +and the LiDAR edges, the camera edge distribution with +nonparametric probability density function is constructed +with the Gaussian Kernel by Kernel Density Estimation +(KDE) [26]. The optimization is based on maximizing the +probabilities of the projected LiDAR edge points onto the +camera edge distribution: +ˆΘ, ˆ∆ = arg max +Θ, ∆ +1 +n +m +� +i=1 +|| ˆf(pi; h, Q)||2, +(6) +ˆf(pi; h, Q) = 1 +nh +n +� +j=1 +K +�pi − qj +h +� +, +(7) +K(x) = +1 +√ +2π det(Σ)e− 1 +2 (x−µ)TΣ−1(x−µ), +(8) +µ = [0, 0]T, Σ = I2×2, +(9) +where h denotes the bandwidth of the KDE. +Several rounds of iterative optimization with reducing +bandwidth are carried out to approach the correct calibration +values smoothly. At the start of the process, the bandwidth +is set at a large number to get a continuous and smooth +cost function, which allows the optimization to approach +the optimal region quickly without many local optima. +Then the bandwidth is reduced gradually to increase the +gradient, ensuring a sensitive optimization around the optimum +(optimization of the x-axis translation is shown in Fig. 4). +0.20 0.25 + 0.30 +0.35 +Bandwidth=16 +Bandwidth=4 +Bandwidth=1 +1 +0 +Normalized cost +Translation in the x-axis (m) +(a) +0.265 0.275 0.285 0.295 +Translation in the x-axis (m) +1 +2 +4 +3 +(b) +Fig. 4. +Iterative optimization with the reducing KDE bandwidth: (a) the +normalized cost w.r.t. the translation in the x-axis under the different values +of bandwidth; (b) zoom in to a sub-region of (a) to demonstrate the iterative +process. +The +optimization +uses +the +Levenberg-Marquardt +method +implemented +in +Ceres-solver +[27]. +For +computational +efficiency, +the +parabolic +Epanechnikov +kernel K(x) = +3 +4(1 − xTx) can be substituted for the +Gaussian kernel. +B. Coarse-to-fine Hybrid Mapping +The coarse-to-fine hybrid mapping workflow is outlined in +Fig. 5. With the co-calibration and synchronization, all the +obtained LiDAR points are represented in both coordinates +and color. Odometry/SLAM methods are used as a backbone +to provide localization in both coarse and fine mapping. We +used FAST-LIO (LiDAR-Inertial odometry [11]) in our current + +MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR +5 +Fig. +5. +Proposed +coarse-to-fine +hybrid +mapping +workflow. +The +odometry/SLAM serves as a backbone to provide localization results. +system, but the choice is not limited; other odometry/SLAM +methods could be utilized as well. At the coarse mapping +stage, the robot obtains the localization and motion results +from the odometry, from which the scanned points are +converted and registered to the global map. Based on the +coarse map, a few viewpoints for stationary mapping are +planned for the targeted ROIs, which is well developed in +previous work by considering the constraints such as range, +grazing angle, FoV, and overlap [1]. The robot then navigates +to the generated viewpoints one-by-one through the backbone +odometry/SLAM and performs the fine mapping, respectively. +At each viewpoint, stationary scans are performed at several +gimbal states, with overlapping FoV regions between the +adjacent two states, and cover a large overall FoV (360° × +300°). These point clouds will be pre-registered based on the +gimbal angles (as initial angles) at each viewpoint. The scans +from all the viewpoints are then combined with the global +coarse map based on robot localization (again provided by the +LiDAR-Inertial odometry) as the initial state for optimization. +Finally, the GICP [25] algorithm is used to optimize all the +localization results and gimbal states and refine all stationary +scans and the coarse map to form the fine map. Notably, +we could choose either odometry or SLAM methods in +the localization backbone. Although SLAM has more loop- +closure functions than odometry, the final GICP optimization +is accurate enough to yield a much better localization result. +IV. EXPERIMENTS AND RESULTS +A. Co-calibration Results +The effectiveness of the proposed co-calibration method is +demonstrated in three natural scenes, as shown in Fig. 6. The +projection error (in pixels) is defined as: +e = 1 +n +n +� +i=1 +d(pi; Q), +(10) +where d is to calculate the distance from the LiDAR projected +point pi to the nearest point in target set Q. Note that the +largest 10% of the distances are considered outliers with no +correspondences and are eliminated. Overall, the co-calibration +works well in all scenes with projection errors on the order of +3 pixels or less. The colorized point clouds after co-calibration +also show much better consistency, as seen in Fig. 6b. +(a) +(b) +Fig. 6. Co-calibration results in three scenes: (a) aligned LiDAR edge points +(red) on camera images; (b) comparison of colorized point clouds before and +after co-calibration with the average projection errors in pixels. +We +further +compare +our +co-calibration +results +with +the classical target-based intrinsic calibration [19], [28], +and the state-of-the-art MI-based extrinsic calibration [22], +respectively, as shown below. +1) Analysis of the Intrinsic Results: As a comparison, the +target-based intrinsic calibration for omnidirectional cameras +is performed [19]. Thirty checkerboards are manually selected +as a reference set (Fig. 7a). As the number and position +of the targets affect the calibration profoundly, we evaluate +the calibration result as a function of the targets’ number +and randomly select a specific number of checkerboards +from the reference set for calibration (repeated 100 times +independently). The mean reprojection error is used to +represent the calibration accuracy. The results in Fig. 7b +show that as the number of checkerboards increases, the +calibration is more accurate and converged. It is likely that +more checkerboards would increase the FoV coverage and +feature points density and improve the effectiveness of the +target-based method. However, it is labor-intensive to place +many checkerboards uniformly and densely around the sensor +and manually select the appropriate ones, which may be +impossible in the field. The co-calibration method, on the +contrary, employs dense LiDAR points as abundant, well- +covered, and accurate features; and the elimination of artificial +targets and human involvement enables an accurate, efficient, +and field-friendly approach. Our co-calibration result yields +a significantly improved performance on the same reference +set, compared with the conventional method (orange and blue +boxplot in Fig. 7b, respectively). +2) Analysis +of +the +Extrinsic +Results: +The +mutual +information +(MI)-based +extrinsic +calibration +method +utilizes the fact that the reflectivity of LiDAR points +and corresponding grayscale intensity values of camera +pixels are correlated since both of them capture the spectral +response of the object at light frequencies (LiDAR 905 nm, +camera 400-800 nm), which are usually similar. These values + +Projection Error: 7.15 →3.17 +Projection Error: 7.36 →2.85 +皖A·35 +天国310 +Proiection Error: 7.08 → 2.63Projection Error: 7.15 →3.17 +Projection Error: 7.36 →2.85 +皖A·35 +天国310 +Proiection Error: 7.08 → 2.636 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023 +-0.4 +-0.3 +-0.2 +-0.1 +0.6 +0.5 +0 +X +0.4 +Z +0.2 +0 +0 -0.2 -0.4 +-0.5 +Y +x-axis (m) +y-axis (m) +z-axis (m) +(a) +Proposed +0 +10 +20 +30 +Avg. +14 +15 +16 +13 +12 +11 +10 +9 +8 +7 +6 +5 +Number of checkerboards +Projection error (px) +3.19 +5.89 +5.74 +5.93 +6.64 +6.97 +7.86 +7.71 +8.43 +9.28 +9.64 +10.3 +11.9 +40 +(b) +Fig. 7. Comparison with the target-based intrinsic calibration: (a) the poses +of the thirty checkerboards; (b) boxplots of projection errors of target-based +calibration (blue) and the proposed co-calibration (orange). +are then used to calibrate the extrinsic parameters between +the camera and LiDAR by maximizing the MI of the two +distributions [22]. Fig. 8 shows the comparisons of the two +optimization methods demonstrating the normalized costs on +different extrinsic parameters. The proposed co-calibration +method shows a much more sensitive and reliable gradient +in the cost function near the optimum than the MI-based +method. +0 +1 +Normalized cost +MI-based +Proposed +Rotation in the x-axis (rad) +-0.4 -0.2 +0 +0.2 +0.4 +-0.4 +-0.2 +0 +0.2 +0.4 +Rotation in the y-axis (rad) +0 +1 +Normalized cost +MI-based +Proposed +MI-based +Proposed +Rotation in the z-axis (rad) +1.2 +1.4 +1.6 +1.8 +2 +0 +1 +Normalized cost +Translation in the x-axis (m) +-0.5 +0 +0.5 +1 +0 +1 +Normalized cost +MI-based +Proposed +Translation in the y-axis (m) +0 +1 +Normalized cost +-1 +-0.5 +0 +0.5 +1 +MI-based +Proposed +0 +1 +Normalized cost +Translation in the z-axis (m) +-0.5 +0 +0.5 +1 +MI-based +Proposed +Fig. 8. Comparisons of the normalized cost function between the proposed +method and the MI-based method. The optimal values should lie in the gray +areas estimated based on manufacturing. +The inaccurate calibration result of the MI-based method +could be attributed mainly to three reasons: the lighting +conditions, the surface reflection properties, and the spectral +reflectance disagreement. The camera’s light source Ii is the +external ambient lighting which does not change with the +camera pose. On the contrary, LiDAR uses an active laser from +the sensor and therefore differs significantly from the camera, +as shown in Fig. 10a. Besides the lighting, the surfaces of the +objects are important. The detected intensity could be modeled +as follows: +Ir = Kd · Ii · f(θ), +(11) +where Ir and Ii indicate the reflection intensity and incident +intensity, respectively, Kd +is the reflectance, and f(θ) +describes the surface properties of the object with respect to +incident angle θ. For most objects, the surface is Lambertian +(diffusive), and in that case, f(θ) = cos θ. However, many +surfaces do not follow this property, and it could be a specular +reflection that the LiDAR does not collect any signal; or the +retroreflection that the majority of the energy will be directed +back toward the LiDAR itself and gives a strong intensity, such +as those on traffic signs and warning stickers, which show a +contrast difference in the LiDAR intensities from the camera +intensities shown in the red boxes in Fig. 9b. Additionally, +the spectral reflectance of objects at various light wavelengths +could be different. For instance, materials composed of plant +fibers show a large reflectance at around 905 nm, even those +dyed in black colors. As a result, no contrast could be seen in +LiDAR intensities of materials with different colors, as shown +in green boxes in Fig. 9b. All three factors mentioned above +could cause significant differences in intensity response from +the LiDAR and the camera and reduce the applicability of the +MI-based method. +LiDAR +Camera +Ambient Light +Retro +A +B +A +B +Spread +Lamber�an +(Lamber�an+Retro) +(a) +0 +255 +Intensity and reflectivity +(in grayscale) +Camera +LiDAR +(b) +Fig. 9. +Analysis of the MI-based extrinsic calibration: (a) the types of +reflection of the LiDAR and camera w.r.t. the rough surface and the +retroreflective surface; (b) the inconsistent intensity cases between LiDAR and +camera, including retroreflection cases (red boxes), and the special spectral +reflectance cases (green boxes). +B. Coarse-to-fine Hybrid Mapping Results +The proposed coarse-to-fine hybrid mapping method is +demonstrated in an academic building on the SUSTech +campus. The global coarse map is generated by Fast-LIO in +ten minutes, and the ROI is selected based on this global +coarse map (Fig. 10a). In this case, five viewpoints are +properly planned in this ROI (Fig. 10b), and perform stationary +scanning for three minutes in each (Fig. 10c). +Plane thickness could be used as a quantitative metric for +precision evaluation and comparison between coarse and fine +mapping. Local planes with a small third eigenvalue λ3 are +selected by diagonalizing the covariance matrix. Assuming +the points along the plane’s normal direction follow the +Gaussian distribution (corresponding to the third eigenvalue +λ3 with the normal direction of the plane defined by its +eigenvector), we could set the thickness of the plane as 4√λ3. + +MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR +7 +(a) +(b) +(c) +Fig. 10. +Coarse-to-fine hybrid mapping: (a) odometry-based global coarse +mapping; (b) coarse map of the selected ROI, with markers indicating the +planned viewpoints; (c) fine map of the ROI, the color illustrates the scans +from respective viewpoints. +TABLE I +SPECS COMPARISON OF CURRENT MAPPING SYSTEMS +Proposed +#1 FARO Focus +Premium 150 +Type +Hybrid Mapping +TLS +FoV +360° × 300° +360° × 300° +Range +0.1-40 m +0.5-150 m +PPS +200,000 pts/s +2,000,000 pts/s +Precision +∼ 40 mm (coarse) +∼ 20 mm (fine) +∼ 1mm [29] +Accuracy +∼ 10 mm (coarse) +∼ 2 mm (fine) +∼ 1mm [29] +Registration +Odometry+Optimization +Optimization +Work Manner +Mobile Robot +Manual (tripod) +Viewpoints Planning +Coarse map-based +Intuition-based +Vision +1-omni camera +1-camera +#2 LEICA +BLK360 +#3 LEICA +BLK2GO +#4 NavVis +VLX +TLS +MLS +MLS +360° × 300° +360° × 270° +360° × 30°(×2) +0.5-45 m +0.5-25 m +0.9-100 m +680,000 pts/s +420,000 pts/s +300,000 pts/s (×2) +∼ 20 mm [30] +∼ 20 mm [30] +15-50 mm(walls, 80.5%) [31] +∼ 1 mm [30] +∼ 30 mm [30] +15-50 mm(beams, 98.2%) [31] +Optimization +Odometry/SLAM +Odometry/SLAM +Manual (tripod) +Manual (handheld) +Manual (backpack) +Intuition-based +No need +No need +3-camera +3-camera +4-camera +The coarse and fine maps of the three different scenes are +shown in Fig. 11a, whereas the zoomed views show the point +cloud quality with the top view of the selected planes to +demonstrate the mapping quality. The quantitative evaluations +of the plane thickness (the mapping precision) in these scenes +are summarized in Fig. 11b. Besides precision (spread of data), +accuracy (correctness) is also important to examine. Fig. 11c +illustrates the measurement accuracy (compared to results +from a TLS system, which we regard as ground truth). It is +evident that both the precision and accuracy of fine mapping +outperform coarse mapping. Although odometry-based coarse +mapping has good performances in best-case scenarios, it +could be significantly improved by fine mapping in the average +values and worse-case scenarios, which are the main concerns +of the surveying and mapping industry. +With the accurate co-calibration results, LiDAR points +(a) +#3 +#2 +#1 +Scenes +0 +10 +20 +30 +40 +50 +Mapping precision (mm) +Coarse Mapping +Fine Mapping +60 +(b) +#3 +#2 +#1 +Scenes +-20 +0 +20 +40 +60 +80 +Mapping accuracy (mm) +Coarse Mapping +Fine Mapping +100 +(c) +(d) +(e) +Fig. 11. Comparison of coarse and fine mapping: (a) coarse and fine maps +in three scenes (scene #1 is from Fig. 10b, scene #2 and #3 are new). The +left column shows the large-scale coarse map, and the right column shows +the zoomed-in coarse and fine map in top view (to visualize wall thickness) +and third person view (to visualize scene); (b) mapping precision from the +three scenes; (c) mapping accuracy from the three scenes; (d) top view of the +colorized fine map; (e) third-person view of the colorized ROI. +can be colorized from the image information through the +transformation in Eqn. 1 and Eqn. 2. Fig. 11d shows the +colorized hybrid mapping, and Fig. 11e illustrates the fine +mapping of the zoomed-in ROI. The coarse-to-fine map with +great precision and accurate colorization pave the way for +higher precision with a single unified setup and workflow. It +benefits industries requiring both efficiency and accuracy, such +as construction automation and building inspection. +Lastly, a detailed comparison of the proposed system +with the current widely used TLS and MLS systems +(shown in Fig. 1a) is made in Table I, where several key + +0 +2m +ROI +10m +2m0 +2m +ROI +10m +2m0 +2m +ROI +10m +2mCoarse Mapping +Fine +Scene #1 +Top view +Third-person +view +Scene #2 +Top view +Third-person +view +Scene #3MappingTop vie +Third-person +vlewX +取消ROI8 +IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. JANUARY, 2023 +parameters are listed. The most crucial difference is that the +proposed system integrates two working modes in a single +streamlined workflow, ensuring overall mapping efficiency +and precision/accuracy. All other systems are either TLS +which only works in stationary mode, or MLS in mobile +mode. Due to this capability, it is the first robotic system +that allows automatic viewpoint planning instead of human +intuition-based viewpoints selection. In addition, the mobile +robot could navigate itself with overall good localization and +provide good initial states for fine map optimization. The +mapping precision and accuracy of the proposed system are +also compared with these systems [29]–[31]. The proposed +system achieves performance close to the LEICA TLS but +allows mobility as MLS, agreeing with the purpose of the +system. +V. CONCLUSION +This paper proposed a coarse-to-fine hybrid 3D mapping +robotic system based on an omnidirectional camera and a non- +repetitive Livox LiDAR. A hybrid mapping approach with both +odometry-based and stationary mapping modes is integrated +into one mobile mapping robot, achieving a streamlined +and automated mapping workflow with the assurance of +efficiency and mapping precision and accuracy. Meanwhile, +the proposed automatic and targetless co-calibration method +provides accurate parameters to generate colorized mapping. +Specifically, the calibration is based on edges extracted +from camera images and LiDAR reflectivity, and the result +is compared with the mutual-information-based calibration +method, which was under-performing possibly due to varied +reflection nature in light sources, surface reflection properties, +and the spectral reflectance disagreement in the MI-based +method. In future work, more complicated planning strategies +could be developed to further optimize both the objectives +of scanning time and spatial coverage. We believe this new +automated mapping robot will open up a new horizon for +surveying and inspection robotics. +REFERENCES +[1] P. S. Blaer and P. K. Allen, “View planning and automated data +acquisition for three-dimensional modeling of complex sites,” Journal +of Field Robotics, vol. 26, no. 11-12, pp. 865–891, 2009. +[2] C. 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Pavelka, “Comparison of leica +blk360 and leica blk2go on chosen test objects.” ISPRS Annals of +Photogrammetry, Remote Sensing & Spatial Information Sciences, 2022. +[31] S. De Geyter, J. Vermandere, H. De Winter, M. Bassier, and +M. Vergauwen, “Point cloud validation: On the impact of laser scanning +technologies on the semantic segmentation for bim modeling and +evaluation,” Remote Sensing, vol. 14, no. 3, p. 582, 2022. + diff --git a/LdFOT4oBgHgl3EQf0DSF/content/tmp_files/load_file.txt b/LdFOT4oBgHgl3EQf0DSF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c9584f70fd5bd14ebbc6f6b2d36f42831dff049 --- /dev/null +++ b/LdFOT4oBgHgl3EQf0DSF/content/tmp_files/load_file.txt @@ -0,0 +1,655 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf,len=654 +page_content='IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' JANUARY, 2023 1 Coarse-to-fine Hybrid 3D Mapping System with Co-calibrated Omnidirectional Camera and Non-repetitive LiDAR Ziliang Miao1, Buwei He1, Wenya Xie1, Wenquan Zhao1, Xiao Huang1, Jian Bai2, and Xiaoping Hong1 Abstract—This paper presents a novel 3D mapping robot with an omnidirectional field-of-view (FoV) sensor suite composed of a non-repetitive LiDAR and an omnidirectional camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Thanks to the non-repetitive scanning nature of the LiDAR, an automatic targetless co-calibration method is proposed to simultaneously calibrate the intrinsic parameters for the omnidirectional camera and the extrinsic parameters for the camera and LiDAR, which is crucial for the required step in bringing color and texture information to the point clouds in surveying and mapping tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Comparisons and analyses are made to target-based intrinsic calibration and mutual information (MI)-based extrinsic calibration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With this co-calibrated sensor suite, the hybrid mapping robot integrates both the odometry-based mapping mode and stationary mapping mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Meanwhile, we proposed a new workflow to achieve coarse-to-fine mapping, including efficient and coarse mapping in a global environment with odometry-based mapping mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' planning for viewpoints in the region-of-interest (ROI) based on the coarse map (relies on the previous work [1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' navigating to each viewpoint and performing finer and more precise stationary scanning and mapping of the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The fine map is stitched with the global coarse map, which provides a more efficient and precise result than the conventional stationary approaches and the emerging odometry- based approaches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Index Terms—Mapping, Robotic Systems, Omnidirectional Vision, Calibration and Identification, SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' INTRODUCTION T HREE-DIMENSIONAL scanning (obtain the raw points) and mapping (register or stitch the points into a point cloud map) are becoming increasingly important in robotics [2], digital construction [3], and virtual reality [4], where digitization of the physical 3D space could provide tremendous insights in modeling, planning, management, optimization, and quality assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Photogrammetry has been developed to capture the 3D world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, its application has been limited in aviation settings where accurate GPS Manuscript received: Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 21, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Revised Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 19, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Accepted Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 28, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' This paper was recommended for publication by Editor Javier Civera upon evaluation of the Associate Editor and Reviewers’ comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' This work was supported by Shenzhen Science and Technology Project (JSGG20211029095803004, JSGG20201103100401004) and SUSTech startup fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (Ziliang Miao and Buwei He contributed equally to this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Corresponding author: Xiaoping Hong) 1These authors are with School of System Design and Intelligent Manufacturing (SDIM), Southern University of Science and Technology (SUSTech), China miaozl2019@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='cn, hebw2019@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='cn, hongxp@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='cn 2Jian Bai is with State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, China Digital Object Identifier (DOI): see top of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' RTK signals are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Recently, the need for large-scale mapping of building environments has been rising, mainly due to the requirements from Building Information Modeling (BIM) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Thanks to the availability of emerging 3D robotic LiDAR sensors [5], [6], Mobile Laser Scanner (MLS) systems are increasingly adopted [7] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1a, #3 and #4), where point clouds from these sensors could be registered to the global frame through sensor motion estimation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=', odometry) at each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, due to the movement nature, such approaches largely depend on estimations of temporal characteristics such as translation and rotation, or spatial characteristics such as sensor FoV and landmark coverages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The results vary from scan to scan with no guarantee of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Hence, a more robust and precise method is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' On the other hand, the traditional Terrestrial Laser Scanner (TLS) has been employed in many precision-stringent applications (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1a, #1 and #2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The TLS-based stationary mapping is usually inefficient (due to the accurate but slow laser rotation) but could provide precise results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Viewpoints (also known as stationary scanning locations) need to be carefully planned to ensure the spatial coverage and enough overlapping regions of adjacent viewpoints to make accurate point cloud stitching [8], but on the other hand, as fewer as possible to reduce scanning time and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The planning for viewpoints largely relies on the overall layout of the scene, which has been done by human experience so far [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' #1 #2 #3 #4 (a) Omnidirectional camera Livox Mid-360 LiDAR (with integrated IMU) Gimbal mount Mobile platform (synchronized) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 3D mapping systems: (a) the current TLS (#1 FARO Focus Premium, #2 LEICA BLK360) and MLS (#3 LEICA BLK2GO, #4 NavVis VLX) systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) the proposed hybrid mapping robotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Combining the strength from both worlds would be ideal in large-scale 3D mapping applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1b, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='12934v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='RO] 30 Jan 2023 OISEE SCOUT2 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' JANUARY, 2023 the proposed hybrid mapping robot is developed carrying a gimbal mount and a novel sensor suite consisting of an omnidirectional non-repetitive Livox Mid-360 LiDAR1 and an omnidirectional camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The sensors’ FoV and the non- repetitive scanning nature are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In the odometry-based mapping mode, the sensor suite is kept horizontal by fixing the gimbal mount to coarsely and efficiently map the entire space with the mobile platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Based on the coarse map, a few viewpoints are planned for the stationary mapping of targeted ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In the stationary mapping mode, the robot will navigate and stay still at each viewpoint, performing 360°×300° scanning by traversing the vertical FoV through the gimbal mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' These precise scans are registered with each other and then stitched with the pre-generated coarse map forming a global map with fine ROIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The main contributions of this work are as follows: 1) The first hybrid 3D mapping robot system that integrates odometry-based and stationary mapping modes is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The consistency of point clouds in two modes can be guaranteed with the single omnidirectional non-repetitive Livox Mid-360 LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2) An omnidirectional camera is introduced in the proposed system to complement the omnidirectional LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' A novel automatic targetless co-calibration method is proposed to simultaneously calibrate the intrinsic parameters and the extrinsic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 3) An automated coarse-to-fine hybrid mapping workflow is demonstrated, including odometry-based coarse mapping in the global environment, planning for the viewpoints in the ROIs, and finer stationary mapping at viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The entire project is open-sourced on GitHub2 to aid the development of this emerging field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' RELATED WORKS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Mapping Solutions 3D mapping solutions are of great interest in many emerging fields [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' TLS-based and MLS-based approaches are commonly adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The traditional TLS-based approach uses a heavy-duty single-laser scanner and traverses the entire FoV through step-wise rotations about the horizontal and vertical axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It provides sufficiently dense points with good precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, this method is slow and laborious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It has to be repeated on many viewpoints, which need to be chosen wisely because a lack of viewpoints will cause missing information in the desired ROI, while the excess of viewpoints will lead to longer scanning hours and poorer efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Currently, viewpoints planning relies on human intuition or experiences, making it challenging to plan effectively in large and complex working environments like the construction scenes [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' On the contrary, the MLS-based approach provides real- time scanning and mapping results as the LiDAR moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The current MLS devices are classified by their usage configurations, such as handheld (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1a, #3), backpack 1The authors gratefully acknowledge Livox Technology for the equipment support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='com/ZiliangMiao/Hybrid Mapping Cocalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='git (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1a, #4), and trolley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Most of these mobile systems rely on conventional LiDARs (16, 32, or 64 lines) and construct the 3D map by registering the point cloud with LiDAR odometry or LiDAR-IMU odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Such mobile systems greatly speed up the mapping process without planning for viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, it cannot replace the TLS-based approaches due to insufficient mapping precision and sparse point clouds [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The repetitive scanning nature of mechanical LiDAR is unsuitable for stationary scanning due to limited FoV coverage (20% coverage for 32-line LiDAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Therefore, the indispensable motion for more coverage will cause errors in pose estimation, which are accumulated throughout the process, limiting the usage in high-precision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Both TLS-based and MLS-based approaches have their unique advantages and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It is desired to devise a mechanism to combine both modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For example, a combination of TLS and MLS is used to solve the registration problem between non-overlapping spaces [8] or use TLS scans as references to MLS mapping registration to achieve low mapping errors [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Moreover, MLS is also used to provide a 3D map to solve the viewpoints planning problem of TLS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, all these methods are based on heterogeneous sensors for different modes, with different synchronization, data structure, and protocols, which are difficult to construct a one-stop mapping robot with a streamlined and automated workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The unique non-repetitive scanning nature of the Livox LiDAR provides a combination of an instantaneous high density at a short time interval for odometry (with effective point density as 32-line LiDAR within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1 seconds) and an image-level resolution at relatively long time intervals for scanning (within 3 seconds, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2b), which makes it surprisingly suitable for such hybrid working mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The feature provides sufficiently good performance in odometry scenarios [11] and a dense FoV coverage for image- like feature processing [6], [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In this paper, the two working modes are integrated into the same robot, ensuring overall mapping efficiency and precision with an automated coarse-to-fine hybrid mapping workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Calibration Methods In addition to LiDAR, Cameras are usually required in 3D mapping systems to give an overview of the mapped environment [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Cameras could provide high-quality geometric, color, and texture information [15], which enables further modeling and rendering [16] of the point clouds and permits tasks in object detection, segmentation, and classification [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Meanwhile, for autonomous navigation, the camera is also vital to visual-LiDAR odometry through sensor fusion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' All these functions would rely on the accurate calibration of the intrinsic parameters of the camera and extrinsic parameters between the cameras and LiDAR [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Traditionally, multiple cameras are usually required to be complementary to the omnidirectional FoV of LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' This work employs an omnidirectional camera over the traditional multi-camera vision to avoid bulky construction, high cost, shutter synchronization, and cascaded extrinsic MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR 3 calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The intrinsic and extrinsic parameters of this novel omnidirectional sensor suite are essentially needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The intrinsic parameters of the omnidirectional camera must be well calibrated since these types usually possess much larger and more complex distortions than pin-hole cameras [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In [18]–[20], higher-order polynomial-based intrinsic models are introduced with many degrees of freedom to obtain satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' A popular OcamCalib toolbox based on the checkerboard is provided [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' These methods could be susceptible to over-fitting with high-order polynomials and often require evenly distributed artificial targets and dense features across the entire space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Typically, these calibration processes are manual and could lead to tedious procedures with a large margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Additionally, the omnidirectional camera in our work is constructed with a refractive-reflective geometry to capture a ring-like FoV beyond 180°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' This construction makes intrinsic calibration even more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' An accurate, automatic, and targetless calibration method is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The extrinsic calibration method between the omnidirectional camera and LiDAR has only been explored in [21] using edge correspondence to match point clouds and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The bearing angle images highlight the edge features, which are manually positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Targetless extrinsic calibration methods for monocular cameras and LiDAR have been developed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With the non-repetitive LiDARs, CamVox [12] could project the image-like LiDAR point clouds onto the camera image plane and extract edge pixels using the grayscale images based on reflectivity and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The method proposed in [13] uses voxels to extract the edge points in 3D space and classifies the edges based on depth continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Both methods work well with conventional pin-hole cameras and need to be extended toward the omnidirectional cameras with significantly larger distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' An additional targetless extrinsic calibration method employing mutual information (MI) is also developed [22], which maximizes the intensity correlations of LiDAR and camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, the misrepresented information caused by lighting conditions, surface reflection properties, and spectral reflectance disagreement could result in worse calibration than the edge-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In the proposed targetless co-calibration method, the high- resolution dense point cloud of the non-repetitive scanning LiDAR gives abundant and ground-truth-level features, which eliminates the artificial targets and manual involvement and reduces the error caused by insufficient coverage and sparse features of the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With the co-calibration method, the intrinsic and extrinsic parameters are obtained simultaneously and can be re-calibrated fast and reliably in work scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PROPOSED SYSTEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Co-calibrated Omnidirectional Sensor Suite The Livox Mid-360 LiDAR has a 360° × 55° FoV and features a non-repetitive scanning pattern, with increasingly denser points over time (the coverage of FoV approaches 100%), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The unique feature specifically benefits both odometry-based and stationary mapping modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The omnidirectional camera provides color information of the surroundings and has a corresponding 360° × 70° FoV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Both sensors are synchronized and are mounted on a two-axis gimbal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1) to extend the scanning FoV to 360° × 300°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 7° 10° +60° +52° 10° +60° 7° +52° Omnidirectional Camera Livox Mid-360 LiDAR (a) T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1s T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5s T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='0s (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Configuration of the sensors: (a) omnidirectional camera and Livox Mid-360 LiDAR, both on the gimbal mount;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) point cloud accumulation over time due to the non-repetitive scanning nature of the Livox LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' （Color represents reflectivity of LiDAR points) #1 #2 #3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1E-3 Probability density 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1E-3 Probability density Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Proposed co-calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' * The grayscale value indicates the average reflectivity of the projected LiDAR points within a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The co-calibration simultaneously obtains the intrinsic (camera) and extrinsic (camera-LiDAR) parameters, defined respectively as Θ ≜ [u0, v0, c, d, e, a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' , an]T and ∆ ≜ [α, β, γ, tx, ty, tz]T, which will be introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With HO DJP2002NOOEORXDHZDZDIHZp)120,△=argma 0,1 ax nn Cf( 14 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' JANUARY, 2023 the unique benefit of the non-repetitive scanning LiDAR, an extremely dense point cloud is always available, which provides a 3D ground truth of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' This high- resolution point cloud could be projected onto the 2D image plane with pixel values from LiDAR reflectivity, from which clear edge features could be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' To align the edges from LiDAR and the camera, the co-calibration iteratively maximizes the correspondence of projected LiDAR edge points with the omnidirectional camera edge pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Kernel Density Estimation (KDE) is employed to estimate the camera edge distribution with different distribution smoothness (by varying bandwidth coefficient) to obtain global optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The entire process of co-calibration can be divided into the following two steps (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 3): 1) Edge Extraction: Edge extractions are performed for both camera and LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For the camera, exposure fusion [23] is adopted to enhance the dynamic range of images to capture more details for low and high-brightness objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Canny edge extraction [24] is performed on the enhanced image, with edge points Q = [q1, q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' , qn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For LiDAR, since the FoV is smaller, point clouds scanned from different pitch angles are stitched together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The stitching is performed by the generalized iterative closest point (GICP) algorithm [25] with the initial transformation given by the state of the gimbal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The stitched point cloud with reflectivity is then projected to an image plane with the azimuthal angle and elevation angle as the coordinates, generating a grayscale image by taking the average reflectivity of the projected LiDAR points within each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The Canny edge extraction is performed on this grayscale image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Uniform sampling is performed in each stage to remove the non-uniform point distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The edge pixels are then identified in the original 3D point cloud P = [LP1, LP2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' , LPm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2) Iterative Optimization: The iterative optimization is performed in the omnidirectional image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The LiDAR edge points are projected to the image coordinates through the following equations: CP = C LT(LP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' ∆) = C LR · LP + C Lt, LP ∈ P, (1) p = Π(CP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Θ) = �c d e 1 � �r cos φ − u0 r sin φ − v0 � , (2) r = F(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' , an) = a0 + a1θ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' + anθn, (3) θ = arccos( z � x2 + y2 + z2 ), (4) φ = arccos( x � x2 + y2 ), (5) where CP and LP denote the 3D point coordinates in camera and LiDAR coordinate systems, respectively, and they are related through the extrinsic transformation C LT(LP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' ∆), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=', rotation C LR and translation C Lt with the extrinsic parameters ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The symbol p denotes the location of the point in the camera image space, and Π(CP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Θ) expresses the intrinsic transformation from CP = [x, y, z]T (3D point) to p (2D point), with the distortion correction matrix � c d e 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The pixel radius r from the image center [u0, v0]T is transformed from the elevation angles θ by a polynomial function F(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' , an) in the camera model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' θ and φ are the elevation and azimuth angle of CP (Note the omnidirectional camera features a ring image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' To facilitate the alignment between the camera edges and the LiDAR edges, the camera edge distribution with nonparametric probability density function is constructed with the Gaussian Kernel by Kernel Density Estimation (KDE) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The optimization is based on maximizing the probabilities of the projected LiDAR edge points onto the camera edge distribution: ˆΘ, ˆ∆ = arg max Θ, ∆ 1 n m � i=1 || ˆf(pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' h, Q)||2, (6) ˆf(pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' h, Q) = 1 nh n � j=1 K �pi − qj h � , (7) K(x) = 1 √ 2π det(Σ)e− 1 2 (x−µ)TΣ−1(x−µ), (8) µ = [0, 0]T, Σ = I2×2, (9) where h denotes the bandwidth of the KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Several rounds of iterative optimization with reducing bandwidth are carried out to approach the correct calibration values smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' At the start of the process, the bandwidth is set at a large number to get a continuous and smooth cost function, which allows the optimization to approach the optimal region quickly without many local optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Then the bandwidth is reduced gradually to increase the gradient, ensuring a sensitive optimization around the optimum (optimization of the x-axis translation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='35 Bandwidth=16 Bandwidth=4 Bandwidth=1 1 0 Normalized cost Translation in the x-axis (m) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='295 Translation in the x-axis (m) 1 2 4 3 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Iterative optimization with the reducing KDE bandwidth: (a) the normalized cost w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' the translation in the x-axis under the different values of bandwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) zoom in to a sub-region of (a) to demonstrate the iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The optimization uses the Levenberg-Marquardt method implemented in Ceres-solver [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For computational efficiency, the parabolic Epanechnikov kernel K(x) = 3 4(1 − xTx) can be substituted for the Gaussian kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Coarse-to-fine Hybrid Mapping The coarse-to-fine hybrid mapping workflow is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With the co-calibration and synchronization, all the obtained LiDAR points are represented in both coordinates and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Odometry/SLAM methods are used as a backbone to provide localization in both coarse and fine mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' We used FAST-LIO (LiDAR-Inertial odometry [11]) in our current MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Proposed coarse-to-fine hybrid mapping workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The odometry/SLAM serves as a backbone to provide localization results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' system, but the choice is not limited;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' other odometry/SLAM methods could be utilized as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' At the coarse mapping stage, the robot obtains the localization and motion results from the odometry, from which the scanned points are converted and registered to the global map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Based on the coarse map, a few viewpoints for stationary mapping are planned for the targeted ROIs, which is well developed in previous work by considering the constraints such as range, grazing angle, FoV, and overlap [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The robot then navigates to the generated viewpoints one-by-one through the backbone odometry/SLAM and performs the fine mapping, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' At each viewpoint, stationary scans are performed at several gimbal states, with overlapping FoV regions between the adjacent two states, and cover a large overall FoV (360° × 300°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' These point clouds will be pre-registered based on the gimbal angles (as initial angles) at each viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The scans from all the viewpoints are then combined with the global coarse map based on robot localization (again provided by the LiDAR-Inertial odometry) as the initial state for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Finally, the GICP [25] algorithm is used to optimize all the localization results and gimbal states and refine all stationary scans and the coarse map to form the fine map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Notably, we could choose either odometry or SLAM methods in the localization backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Although SLAM has more loop- closure functions than odometry, the final GICP optimization is accurate enough to yield a much better localization result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Co-calibration Results The effectiveness of the proposed co-calibration method is demonstrated in three natural scenes, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The projection error (in pixels) is defined as: e = 1 n n � i=1 d(pi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Q), (10) where d is to calculate the distance from the LiDAR projected point pi to the nearest point in target set Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Note that the largest 10% of the distances are considered outliers with no correspondences and are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Overall, the co-calibration works well in all scenes with projection errors on the order of 3 pixels or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The colorized point clouds after co-calibration also show much better consistency, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Co-calibration results in three scenes: (a) aligned LiDAR edge points (red) on camera images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) comparison of colorized point clouds before and after co-calibration with the average projection errors in pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' We further compare our co-calibration results with the classical target-based intrinsic calibration [19], [28], and the state-of-the-art MI-based extrinsic calibration [22], respectively, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1) Analysis of the Intrinsic Results: As a comparison, the target-based intrinsic calibration for omnidirectional cameras is performed [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Thirty checkerboards are manually selected as a reference set (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' As the number and position of the targets affect the calibration profoundly, we evaluate the calibration result as a function of the targets’ number and randomly select a specific number of checkerboards from the reference set for calibration (repeated 100 times independently).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The mean reprojection error is used to represent the calibration accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 7b show that as the number of checkerboards increases, the calibration is more accurate and converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It is likely that more checkerboards would increase the FoV coverage and feature points density and improve the effectiveness of the target-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, it is labor-intensive to place many checkerboards uniformly and densely around the sensor and manually select the appropriate ones, which may be impossible in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The co-calibration method, on the contrary, employs dense LiDAR points as abundant, well- covered, and accurate features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' and the elimination of artificial targets and human involvement enables an accurate, efficient, and field-friendly approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Our co-calibration result yields a significantly improved performance on the same reference set, compared with the conventional method (orange and blue boxplot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 7b, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2) Analysis of the Extrinsic Results: The mutual information (MI)-based extrinsic calibration method utilizes the fact that the reflectivity of LiDAR points and corresponding grayscale intensity values of camera pixels are correlated since both of them capture the spectral response of the object at light frequencies (LiDAR 905 nm, camera 400-800 nm), which are usually similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' These values Projection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='15 →3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='17 Projection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='36 →2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='85 皖A·35 天国310 Proiection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='08 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='63Projection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='15 →3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='17 Projection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='36 →2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='85 皖A·35 天国310 Proiection Error: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='08 → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='636 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' JANUARY, 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 0 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0 0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 Y x-axis (m) y-axis (m) z-axis (m) (a) Proposed 0 10 20 30 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 14 15 16 13 12 11 10 9 8 7 6 5 Number of checkerboards Projection error (px) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='89 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='97 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='86 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='71 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='43 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='28 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='64 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='9 40 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Comparison with the target-based intrinsic calibration: (a) the poses of the thirty checkerboards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) boxplots of projection errors of target-based calibration (blue) and the proposed co-calibration (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' are then used to calibrate the extrinsic parameters between the camera and LiDAR by maximizing the MI of the two distributions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 8 shows the comparisons of the two optimization methods demonstrating the normalized costs on different extrinsic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The proposed co-calibration method shows a much more sensitive and reliable gradient in the cost function near the optimum than the MI-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 0 1 Normalized cost MI-based Proposed Rotation in the x-axis (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 Rotation in the y-axis (rad) 0 1 Normalized cost MI-based Proposed MI-based Proposed Rotation in the z-axis (rad) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='8 2 0 1 Normalized cost Translation in the x-axis (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 1 0 1 Normalized cost MI-based Proposed Translation in the y-axis (m) 0 1 Normalized cost 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 1 MI-based Proposed 0 1 Normalized cost Translation in the z-axis (m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5 1 MI-based Proposed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Comparisons of the normalized cost function between the proposed method and the MI-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The optimal values should lie in the gray areas estimated based on manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The inaccurate calibration result of the MI-based method could be attributed mainly to three reasons: the lighting conditions, the surface reflection properties, and the spectral reflectance disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The camera’s light source Ii is the external ambient lighting which does not change with the camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' On the contrary, LiDAR uses an active laser from the sensor and therefore differs significantly from the camera, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Besides the lighting, the surfaces of the objects are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The detected intensity could be modeled as follows: Ir = Kd · Ii · f(θ), (11) where Ir and Ii indicate the reflection intensity and incident intensity, respectively, Kd is the reflectance, and f(θ) describes the surface properties of the object with respect to incident angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For most objects, the surface is Lambertian (diffusive), and in that case, f(θ) = cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' However, many surfaces do not follow this property, and it could be a specular reflection that the LiDAR does not collect any signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' or the retroreflection that the majority of the energy will be directed back toward the LiDAR itself and gives a strong intensity, such as those on traffic signs and warning stickers, which show a contrast difference in the LiDAR intensities from the camera intensities shown in the red boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Additionally, the spectral reflectance of objects at various light wavelengths could be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' For instance, materials composed of plant fibers show a large reflectance at around 905 nm, even those dyed in black colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' As a result, no contrast could be seen in LiDAR intensities of materials with different colors, as shown in green boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' All three factors mentioned above could cause significant differences in intensity response from the LiDAR and the camera and reduce the applicability of the MI-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' LiDAR Camera Ambient Light Retro A B A B Spread Lamber�an (Lamber�an+Retro) (a) 0 255 Intensity and reflectivity (in grayscale) Camera LiDAR (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Analysis of the MI-based extrinsic calibration: (a) the types of reflection of the LiDAR and camera w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' the rough surface and the retroreflective surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) the inconsistent intensity cases between LiDAR and camera, including retroreflection cases (red boxes), and the special spectral reflectance cases (green boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Coarse-to-fine Hybrid Mapping Results The proposed coarse-to-fine hybrid mapping method is demonstrated in an academic building on the SUSTech campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The global coarse map is generated by Fast-LIO in ten minutes, and the ROI is selected based on this global coarse map (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In this case, five viewpoints are properly planned in this ROI (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10b), and perform stationary scanning for three minutes in each (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Plane thickness could be used as a quantitative metric for precision evaluation and comparison between coarse and fine mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Local planes with a small third eigenvalue λ3 are selected by diagonalizing the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Assuming the points along the plane’s normal direction follow the Gaussian distribution (corresponding to the third eigenvalue λ3 with the normal direction of the plane defined by its eigenvector), we could set the thickness of the plane as 4√λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' MIAO et al: COARSE-TO-FINE HYBRID 3D MAPPING SYSTEM WITH CO-CALIBRATED OMNIDIRECTIONAL CAMERA AND NON-REPETITIVE LIDAR 7 (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Coarse-to-fine hybrid mapping: (a) odometry-based global coarse mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) coarse map of the selected ROI, with markers indicating the planned viewpoints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (c) fine map of the ROI, the color illustrates the scans from respective viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' TABLE I SPECS COMPARISON OF CURRENT MAPPING SYSTEMS Proposed #1 FARO Focus Premium 150 Type Hybrid Mapping TLS FoV 360° × 300° 360° × 300° Range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='1-40 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5-150 m PPS 200,000 pts/s 2,000,000 pts/s Precision ∼ 40 mm (coarse) ∼ 20 mm (fine) ∼ 1mm [29] Accuracy ∼ 10 mm (coarse) ∼ 2 mm (fine) ∼ 1mm [29] Registration Odometry+Optimization Optimization Work Manner Mobile Robot Manual (tripod) Viewpoints Planning Coarse map-based Intuition-based Vision 1-omni camera 1-camera #2 LEICA BLK360 #3 LEICA BLK2GO #4 NavVis VLX TLS MLS MLS 360° × 300° 360° × 270° 360° × 30°(×2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5-45 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5-25 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='9-100 m 680,000 pts/s 420,000 pts/s 300,000 pts/s (×2) ∼ 20 mm [30] ∼ 20 mm [30] 15-50 mm(walls, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='5%) [31] ∼ 1 mm [30] ∼ 30 mm [30] 15-50 mm(beams, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content='2%) [31] Optimization Odometry/SLAM Odometry/SLAM Manual (tripod) Manual (handheld) Manual (backpack) Intuition-based No need No need 3-camera 3-camera 4-camera The coarse and fine maps of the three different scenes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11a, whereas the zoomed views show the point cloud quality with the top view of the selected planes to demonstrate the mapping quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The quantitative evaluations of the plane thickness (the mapping precision) in these scenes are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Besides precision (spread of data), accuracy (correctness) is also important to examine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11c illustrates the measurement accuracy (compared to results from a TLS system, which we regard as ground truth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It is evident that both the precision and accuracy of fine mapping outperform coarse mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Although odometry-based coarse mapping has good performances in best-case scenarios, it could be significantly improved by fine mapping in the average values and worse-case scenarios, which are the main concerns of the surveying and mapping industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' With the accurate co-calibration results, LiDAR points (a) #3 #2 #1 Scenes 0 10 20 30 40 50 Mapping precision (mm) Coarse Mapping Fine Mapping 60 (b) #3 #2 #1 Scenes 20 0 20 40 60 80 Mapping accuracy (mm) Coarse Mapping Fine Mapping 100 (c) (d) (e) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Comparison of coarse and fine mapping: (a) coarse and fine maps in three scenes (scene #1 is from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 10b, scene #2 and #3 are new).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The left column shows the large-scale coarse map, and the right column shows the zoomed-in coarse and fine map in top view (to visualize wall thickness) and third person view (to visualize scene);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (b) mapping precision from the three scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (c) mapping accuracy from the three scenes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (d) top view of the colorized fine map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' (e) third-person view of the colorized ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' can be colorized from the image information through the transformation in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1 and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11d shows the colorized hybrid mapping, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 11e illustrates the fine mapping of the zoomed-in ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The coarse-to-fine map with great precision and accurate colorization pave the way for higher precision with a single unified setup and workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' It benefits industries requiring both efficiency and accuracy, such as construction automation and building inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Lastly, a detailed comparison of the proposed system with the current widely used TLS and MLS systems (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' 1a) is made in Table I, where several key 0 2m ROI 10m 2m0 2m ROI 10m 2m0 2m ROI 10m 2mCoarse Mapping Fine Scene #1 Top view Third-person view Scene #2 Top view Third-person view Scene #3MappingTop vie Third-person vlewX 取消ROI8 IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' JANUARY, 2023 parameters are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The most crucial difference is that the proposed system integrates two working modes in a single streamlined workflow, ensuring overall mapping efficiency and precision/accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' All other systems are either TLS which only works in stationary mode, or MLS in mobile mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Due to this capability, it is the first robotic system that allows automatic viewpoint planning instead of human intuition-based viewpoints selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In addition, the mobile robot could navigate itself with overall good localization and provide good initial states for fine map optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The mapping precision and accuracy of the proposed system are also compared with these systems [29]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' The proposed system achieves performance close to the LEICA TLS but allows mobility as MLS, agreeing with the purpose of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' CONCLUSION This paper proposed a coarse-to-fine hybrid 3D mapping robotic system based on an omnidirectional camera and a non- repetitive Livox LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' A hybrid mapping approach with both odometry-based and stationary mapping modes is integrated into one mobile mapping robot, achieving a streamlined and automated mapping workflow with the assurance of efficiency and mapping precision and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Meanwhile, the proposed automatic and targetless co-calibration method provides accurate parameters to generate colorized mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Specifically, the calibration is based on edges extracted from camera images and LiDAR reflectivity, and the result is compared with the mutual-information-based calibration method, which was under-performing possibly due to varied reflection nature in light sources, surface reflection properties, and the spectral reflectance disagreement in the MI-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' In future work, more complicated planning strategies could be developed to further optimize both the objectives of scanning time and spatial coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' We believe this new automated mapping robot will open up a new horizon for surveying and inspection robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LdFOT4oBgHgl3EQf0DSF/content/2301.12934v1.pdf'} +page_content=' Blaer and P.' metadata={'source': 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“connected things” +to “connected intelligence”, featured by ultra high density, large- +scale, dynamic heterogeneity, diversified functional requirements +and machine learning capabilities, which leads to a growing +need for highly efficient intelligent algorithms. The classic +optimization-based algorithms usually require highly precise +mathematical model of data links and suffer from poor per- +formance with high computational cost in realistic 6G appli- +cations. Based on domain knowledge (e.g., optimization models +and theoretical tools), machine learning (ML) stands out as a +promising and viable methodology for many complex large-scale +optimization problems in 6G, due to its superior performance, +generalizability, computational efficiency and robustness. In this +paper, we systematically review the most representative “learning +to optimize” techniques in diverse domains of 6G wireless +networks by identifying the inherent feature of the underlying +optimization problem and investigating the specifically designed +ML frameworks from the perspective of optimization. In partic- +ular, we will cover algorithm unrolling, learning to branch-and- +bound, graph neural network for structured optimization, deep +reinforcement learning for stochastic optimization, as well as end- +to-end learning for semantic optimization, for solving challenging +large-scale optimization problems arising from various important +wireless applications. To enable ML implementation in dis- +tributed wireless networks across massive number of end devices, +federated learning for distributed optimization will further be +presented. Through the in-depth discussion, we shed light on +the excellent performance of ML-based optimization algorithms +with respect to the classical methods, and provide insightful +guidance to develop advanced ML techniques in 6G networks. +Neural network design, theoretical tools of different ML methods, +Yandong Shi is with the China Telecom Research Institute, Guangzhou +510660, China (e-mail: shiyd2@chinatelecom.cn). +Lixiang Lian, Yuanming Shi, and Yong Zhou are with the School of Infor- +mation Science and Technology, ShanghaiTech University, Shanghai 201210, +China +(e-mail: +lianlx@shanghaitech.edu.cn; +shiym@shanghaitech.edu.cn; +zhouyong@shanghaitech.edu.cn). +Zixin Wang is with the School of Information Science and Technol- +ogy, ShanghaiTech University, Shanghai 201210, China, also with the +Shanghai Institute of Microsystem and Information Technology, Chinese +Academy of Sciences, Shanghai 200050, China, and also with the Uni- +versity of Chinese Academy of Sciences, Beijing 100049, China (e-mail: +wangzx2@shanghaitech.edu.cn). +Liqun Fu is with the School of Informatics and the Key Laboratory of +Underwater Acoustic Communication and Marine Information Technology +(Ministry of Education), Xiamen University, Xiamen 361005, China (e-mail: +liqun@xmu.edu.cn). +Lin Bai is with the School of Cyber Science and Technology, Beihang +University, Beijing 100191, China (e-mail: l.bai@buaa.edu.cn). +Jun Zhang is with the Department of Electronic and Computer Engineering, +Hong Kong University of Science and Technology, Hong Kong (e-mail: +eejzhang@ust.hk). +Wei Zhang is with the School of Electrical Engineering and Telecommuni- +cations, The University of New South Wales, Sydney, NSW 2052, Australia +(e-mail: w.zhang@unsw.edu.au). +implementation issues, as well as challenges and future research +directions are also discussed to support the practical use of ML +model in wireless applications. +Index Terms—Large-scale optimization, machine learning, +deep neural network, 6G, large-scale networks, wireless com- +munications, learning to optimize, non-convex optimization. +I. INTRODUCTION +The sixth generation (6G) wireless systems have re- +cently attracted considerable attention from both industry and +academia, whose visions are towards ubiquitous 3D coverage +(space-air-ground-sea integrated network) [1], the intelligent +and green networks [2], Internet of everything (IoE) [3], etc. +Compared with the previous generations, 6G can provide +services with more stringent requirements, such as higher +throughput, lower latency, higher reliability, denser connec- +tion, higher energy efficiency, as well as connected intelligence +with machine learning capability [3]. Driven by the new +industrial and technological revolution, 6G can also support +new services/applications beyond 5G such as immersive cloud +extended reality (XR), holographic communications, sensory +interconnection, digital twins and so on [4], which may +demand new performance metrics to facilitate diversified and +personalized user services. The requirements of 6G system +have made the fine-grained optimization of radio resources and +effective learning of network-related information urgent neces- +sities. Due to the large-scale, high density, heterogeneous qual- +ities of services, and integrated multi-functional cross-layer +design, the optimization problems in 6G can be extremely +time-sensitive and complex, which pose great challenges +for efficient optimization algorithm design. Machine learning +(ML) has been recently leveraged as a disruptive technology +to solve the challenging optimization problems in 6G, as well +as support ubiquitous artificial intelligence (AI) services and +IoE applications [4]–[6] including synaesthesia internet, digital +twins, smart industry, smart agriculture, super traffic, precision +medicine and blockchain economy. In this section, we first +discuss the properties of optimization problems in 6G wireless +networks and summarize the advantages and disadvantages +of classic optimization-based methods. Then we introduce +the motivation for ML-based optimization frameworks and +summarize the existing design paradigms to solve different +classes of optimization problems. Table I summarizes the main +notations used throughout this paper. +arXiv:2301.03377v1 [eess.SP] 3 Jan 2023 + +2 +A. Large-Scale Optimization for 6G +The performance of 6G wireless networks can be enhanced +by adopting various optimization algorithms, which solve +the practical engineering problem through the mathematical +tools. Constructing and solving optimization problems can +effectively handle the technical issues in engineering and guide +the performance-related policy development. The properties of +optimization problems in 6G networks lay in the following +three aspects: (1) The objective functions of 6G optimization +problems can be complicated to meet personalized services +of heterogeneous networks and highly non-convex due to the +enabling of integrated functions in 6G, such as joint sensing, +communication, computing and control [7]. Other factors such +as diversified services 6G facilitates (e.g. distributed edge +training and inference [8]) and integrated cross-layer designs +will further complicate the objective function. For example, +the mutual information is adopted as the objective function +in semantic communications [9] to optimize the efficiency- +accuracy trade-off, which is intractable. (2) Optimization vari- +ables and model parameters of 6G optimization problems can +be of high dimension due to the massive devices, large-scale +antennas in wireless networks and large amounts of data in +various 6G technologies and services [3]. The feasible region +of optimization variables can be highly stringent to satisfy +practical network conditions under resource constraints and +provide robust and reliable services for ubiquitous networking. +For example, the spectral-efficiency maximization problem +in heterogeneous networks involves lots of non-convex con- +straints to support different transmission types (i.e., uplink and +downlink transmission constraints) [10]. (3) The optimization +problems in 6G wireless networks usually involve real-time +network-dependent parameters, such as the network structure, +channel state information (CSI), traffic condition, etc. There- +fore, the near-optimal performance of various optimization- +based algorithms (OAs) should be achieved in real time, which +is a fundamental challenge. +Highly effective algorithms based on classical optimization +theory have been extensively developed for various classes +of optimization problems in 6G. For example, many iterative +algorithms (e.g., approximate message passing (AMP), orthog- +onal matching pursuit (OMP) and alternating direction method +of multipliers (ADMM)) are designed for signal recovery +(e.g., signal detection [11] and channel estimation [12]). The +semidefinite relaxation (SDR) and successive convex approx- +imation (SCA) techniques are widely applied to solve non- +convex optimization problems (e.g., non-convex quadratically +constrained quadratic programming problems [13]) in wireless +communications. Despite that some of these algorithms can +achieve good performance through theoretical analysis and +numerical simulations, classic optimization-based algorithms +(COAs) in realistic 6G applications face many challenges. +1) Optimization Performance: +Due to the highly non- +convexity, COAs can be intractable, suboptimal or heuristic +without performance guarantees. Besides, the tuning of free- +parameters in COAs relies on prior knowledge or model +assumption, whose setting can greatly affect the achievable +performance of COAs. For example, iterative soft thresholding +algorithm (ISTA) for sparse recovering suffers from an inher- +ent trade-off between estimation performance and convergence +rate, which is controlled by the choice of regularization +parameter [14]. +2) Computational Cost: Most of the COAs are iterative +in nature, which typically induces high computational cost to +obtain optimal solutions. For example, for solving mixed com- +binatorial optimization problems, the complexity of branch- +and-bound (BB) algorithm grows exponentially with the scale +of problem [15]. However, most signal processing techniques +in 6G have stringent latency requirement. Furthermore, the +COAs depend on the real-time environmental parameters (e.g., +network topology, channel conditions). When the environment +changes, the iterative COAs need to be executed repeatedly to +accommodate to the dynamic environment, which is unford- +able for time-sensitive applications. +3) Tractability of System Modeling: The design of COAs +highly depends on the availability and accuracy of system +modeling. However, it is hard to precisely capture the network +architecture, communication environment and transmission +data links using mathematical models due to the time-varying, +heterogeneity, complexity and nonlinearity in 6G wireless +networks. The imperfect and mismatched system model can +greatly deteriorate the performance of COAs when applied in +practical systems. +Therefore, traditional COAs are computational inefficient +and scale poorly for large-scale optimization problems in 6G +systems. The reliance on perfect mathematical models and +possible intractability of optimal solutions make the COAs +entail serious performance gap between theoretical design and +real-time application [16]. Motivated by the disadvantages of +COAs, ML has surged as a powerful technique to solve the +challenging optimization problems in wireless networks. +B. Machine Learning in 6G +The goal of ML-based optimization algorithm (MOA) de- +sign is to achieve near-optimal performance with high com- +putational efficiency for challenging large-scale optimization +problems in 6G wireless networks, enabling a paradigm shift +from classic optimization theory-based approaches to employ- +ing more promising deep learning (DL) architectures [3]. ML +for large-scale optimization features the following advantages +[17], [18]. +1) Superior Performance: MOAs entail near-optimal or +superior learning performance compared with COAs due to +the data-driven feature as well as the sophisticated design of +neural networks (NNs) and learning strategies. For example, +algorithm unrolling methods enjoy a superior performance for +signal detection [19], channel estimation [20] and precoding +design [21] compared with their corresponding COA counter- +parts and other traditional algorithms. Graph neural network +(GNN) enjoys better performance for resource allocation prob- +lems with fewer iterations compared with traditional weighted +minimum mean-square error (WMMSE) algorithm [22]. +2) Scalability and Generalizability: With enhanced learn- +ing capacity, MOAs can be used to solve large-scale and +complex optimization problems. Incorporating the properties + +3 +TABLE I +LIST OF ACRONYMS IN ALPHABETICAL ORDER +Acronym +Explanation +6G +6th Generation +ADMM +Alternating Direction Method of Multipliers +AE +Auto-Encoder +AI +Artificial Intelligence +AMP +Approximate Message Passing +BB +Branch-and-Bound +BS +Base Station +CMCP +Complex Modulus Constrained Problem +CNN +Convolutional Neural Network +COA +Classic Optimization-Based Algorithm +CS +Compressive Sensing +CSI +Channel State Information +D2D +Device to Device Communication +DL +Deep Learning +DNN/NN +Deep Neural Network/Neural Network +DQN/DQL +Deep Q-Network/Deep Q-Learning +DRL +Deep Reinforcement Learning +FL +Federated Learning +GCN +Graph Convolutional Network +GNN +Graph Neural Network +IB +Information Bottleneck +IoT +Internet of Things +ISTA +Iterative Soft Thresholding Algorithm +JADCE +Joint Activity Detection and Channel Estimation +JSCC +Joint Source-Channel Coding +LBB +Learning to Branch-and-Bound +MAS/MADRL +Multi-Agent System/Multi-Agent DRL +MDP +Markov Decision Process +MIMO +Multi-Input Multi-Output +MINLP +Mixed Integer Nonlinear Problem +MIP +Mixed Integer Programming +ML +Machine Learning +MLP +Multi-Layer Perceptron +MOA +Machine Learning-Based Optimization Algorithm +MSE +Mean Square Error +OA +Optimization-Based Algorithm +OMP +Orthogonal Matching Pursuit +QoS +Quality of Service +RL +Reinforcement Learning +RNN +Recurrent Neural Network +RIS +Reconfigurable Intelligent Surfaces +SNR +Signal to Noise Ratio +SDR +Semidefinite Relaxation +SCA +Successive Convex Approximation +VAE +Variational Auto-Encoder +WMMSE +Weighted Minimum Mean-Square Error +of target task into the NN architectures further improves +the scalability and generalizability of MOAs. For example, +message passing-based GNNs can be safely generalized to +solve large-scale problems even when trained on small-scale +samples [22], thereby leading to reduced training cost. The +decentralized nature of some specialized MOAs enables the +efficient training in large-scale networks, such as the multi- +agent reinforcement learning (MARL) [23]. Advanced ML +techniques, such as transfer learning [15] and meta-learning +[24] can tackle the task mismatch problems with fewer training +samples, thereby improving the generalizability of MOAs. +3) Computational Efficiency: ML inference only requires +a small number of simple operations and can be realized +in real time. By shifting the computations from online to +offline, ML is highly attractive for computational intensive +optimization tasks in 6G [16]. The online deployment of well- +trained MOAs can effectively reduce the system delay and +improve the overall performance. +4) Robustness: ML-based approaches shall be robust to the +imperfect model assumptions and dynamic wireless environ- +ment due to the data-driven nature. Through learning from ex- +periences, MOAs work well even under unknown environment +when the mathematical model is unavailable. For example, +continual learning [25] is robust to the dynamic environments +by sequentially handling different wireless parameters (e.g., +different CSI distributions). +Despite of the recent successful tries of MOAs, the impor- +tant questions in this context are what kind of optimization +problems can be effectively solved by ML techniques and +which ML techniques would provide reliable, timely and ef- +fective solutions for these optimization problems. In this paper, +we will try to answer these questions by focusing mainly on +some exemplary ML design frameworks applied to solve vari- +ous large-scale optimization problems in 6G networks. Within +each framework, we highlight the motivations, the NN design +principles, type of optimization problems, toy examples of its +applications in 6G wireless networks, the theoretical analysis, +the related research challenges as well as the summary of +advantages and disadvantages. Specifically, the algorithm un- +rolling [14], learning to branch-and-bound (LBB) [15], GNN +for structured optimization [22], deep reinforcement learning +(DRL) for stochastic optimization [26], end-to-end learning +for semantic optimization [27] as well as federated learning +(FL) for distributed optimization [28] will be covered in the +following sections to shed light on the excellent performance +of ML compared with the conventional optimization algorithm +in a variety of practical domains and provide guidance on +the usage of ML techniques in 6G networks, followed by the +summary of network design philosophies, theoretical tools, +implementation issues and discussion of future directions to +drive forward the research in this area. Before the detailed +elaborations of specific MOA designs, we firstly summarize +the existing design paradigms of MOAs from different per- +spectives in the following subsections. +C. Category for ML-Based Optimization Algorithms +1) Learning Principle: From the perspective of learning +principle, learning to optimize can be divided into supervised + +4 +Machine Learning for Large-Scale Optimization in 6G Wireless Networks +Algorithm +Unrolling +Signal Recovery Problems +Signal +Detection +MIMO +Channel +Estimation +Joint +Activity +Detection +and +Channel +Estimation +Nonconvex Sum-Utility +Maximization Problems +Precoding +Design +Power +Control +Learning to +Branch-and-Bound +Mixed Integer Nonlinear +Problems +Resource +Allocation +Non-Convex Complex +Modulus Constrained +Problems +MIMO +Detection +Beamfor +ming +Graph Neural Network for +Structured Optimization +Cellular/Cell-Free +Networks +Beamfor +ming +Power +Allocation +D2D Networks +Power +Allocation +Link +Scheduling +Distributed Systems +Decentral +ized +Networks +Heteroge +neous +Networks +Other Signal Processing +Applications +Mobile +Traffic +Prediction +Channel +Tracking +Deep Reinforcement Learning +for Stochastic Optimization +Stochastic Integer +Programming Problems +Intelligent +Traffic +Discrete- +valued Power +Control +Device +Scheduling +Stochastic Mixed-integer +Programming Problems +Mobile Edge +Network +Optimization +Space- +Air- +Ground- +Integrated +Network +Distributed Constraint +Optimization +Scalable +Radio +Resource +Allocation +End-to-End Learning for +Semantic Optimization +Semantic Meaning +Extraction and +Interpretation +Text +Task +Image +Task +Speech +Task +Federated Learning for +Distributed Optimization +Cross-Device FL +Hybrid +Beamforming +Channel +Estimation +DRL- +Assisted +FL +Cross-Silo FL +Power +Control +Discussions and Future +Research Directions +NN Design for Wireless +Communications +Theoretical Tools +Implementation Issues and Software +Platforms +Challenges and Future Research +Directions +Fig. 1. An overview of the main topics of ML-based optimization algorithms, related problems and the wireless applications. +and unsupervised learning. With the labeled training data, +supervised learning directly learns the nonlinear mapping +between problem parameters and optimal solutions of opti- +mization problems through generic NNs (e.g., multi-layer per- +ceptron (MLP) for channel estimation [29]) or specialized NNs +(e.g., algorithm unrolling for joint active device and channel +estimation (JADCE) [14]). The universal approximation theo- +rem states that feed-forward NN with one single hidden layer +can approximate continuous functions to arbitrary precision +[30], which provides theoretical support for supervised MOA +designs by treating NN as a universal function approximator. +With the unlabeled training data, unsupervised approach learns +the optimal solutions by adopting the objective function of +original optimization problem as the training loss function. +Then the NN is evaluated and updated by means of any +gradient-based optimizer (e.g., GNN takes minus weighted +sum rate as loss function for scalable radio resource man- +agement [22] and DRL aims to maximize expected cumula- +tive discounted rewards for resource allocation in vehicle-to- +vehicle communications [31]). +Compared with the COAs, supervised MOAs enjoy similar +or superior optimization performance by leveraging the strong +representation power of NN. By constructing connections with +COAs, the supervised MOAs show better interpretability and +enables performance analysis for certain network architectures. +However, the original optimization problem needs to be solved +repeatedly with varying problem parameters to collect the +training labels, which can induce huge computational costs +in data acquisition. Besides, in supervised MOA, the NN is +trained as a universal approximator of an existing optimization +algorithm, and thus the performance greatly depends on that +of existing algorithms. Instead, the unsupervised MOAs can +greatly simplify the process of data acquisition and can be +effectively adopted to solve non-convex or NP-hard problem, +where no tractable COAs can be found, thereby greatly +eliminating the dependence on the COAs and facilitating +great flexibility to search for optimal solutions. However, its +performance is primarily restricted by the type of optimization +problems. For highly non-convex problem suffering from the +“curse” of local minima, unsupervised MOA may be trapped +into a spurious solution with poor performance. In addition, +the unsupervised MOA is usually computational complex and +requires a larger training set to produce expected outcomes. +2) NN Architecture: From the perspective of NN architec- +ture, there are mainly two design principles as summarized +below. +• Generic NN: Most of the MOAs adopt generic NNs, such +as MLP, convolutional neural network (CNN), autoen- +coder (AE), etc, which are not tailored for specific tasks. +The generic NNs are not specific to a particular task or +dataset, but can be applied for general tasks and datasets +to achieve an acceptable performance. For example, with +tremendous training data and large NN architecture, MLP +is usually adopted as a benchmark algorithm for perfor- +mance comparison in multi-input multi-output (MIMO) +detection [32], power control [33], semantic communi- +cation [34], etc. However, the flexibility comes with the +cost of poor data efficiency (high training overhead), poor +robustness and poor generalization ability. +• Task-Specific NN: Another design principle of NN ar- + +5 +chitecture is customized implementation by incorporating +the structure of target task, datasets and domain-specific +knowledge into the NN architecture. The specialized NN +demonstrates some unique advantages empirically and +theoretically, such as robustness to model uncertainties, +scalability and generalizability in large-scale problem, +and high training efficiency. Some of specialized NNs +can handle tricky constraints in the optimization prob- +lems (e.g., integer or constant envelope constraints) and +enable performance guarantees under certain conditions. +Nonetheless, the task-specific NN is highly customized +and different problems require separate NN designs. +Besides, for complex problems, it can be hard to con- +struct a specific NN. The design of specific NN highly +depends on the structure of problem itself, or the property +of existing algorithms that can be used to solve the +problem. For example, the NN architecture of algorithm +unrolling comes from parameterizing the original iterative +algorithm [35]. Therefore, the key to designing a spe- +cialized NN is to identify the special characteristics of +problem/datasets/classic algorithms and sophisticatedly +integrate them into the design of the NN architecture and +training algorithm. +3) Theoretical Analysis: From the perspective of theoretical +analysis, most of the MOAs treat the NN as a black box +without interpretations (e.g., MLP, CNN and AE), whose +performance can only be demonstrated numerically. However, +in wireless communication systems, transparentness and relia- +bility are of pivotal importance for a practical algorithm design +[36]. Therefore, it is paramount to understand the learning +mechanism of NN and its applicable conditions. Inspired by +the pertinent COAs, some of the MOAs in the “supervised +learning” category enable theoretical analysis by constructing a +relationship of performance between these two types of meth- +ods. If equivalence can be proved, the performance analysis +of MOAs can be developed based on that of COAs (e.g., the +performance analysis of algorithm unrolling approaches [14], +LBB approach [15], GNN approach [33], etc.). +D. Related Works and Our Contributions +There exist several survey papers on ML for wireless com- +munications [37]–[41]. All these studies provide visions of ar- +tificial intelligence-based wireless network designs, enumerat- +ing on the applicable cases and scenarios in wireless networks +where the ML can make a viable impact. Particularly, the vast +majority of surveys [37]–[41] started with the introduction +of the fundamentals of ML, including the ML theory, the +framework of generic deep networks as well as its update rules +and training methods, followed by envisioning the wireless +applications of different ML techniques in future wireless +networks from different aspects. Among them, Zappone et +al. [38] provided an in-depth quantitative analysis for each +use-case of DL-based wireless network design. Mao et al. +[40] focused on the discussion of DL applications in different +wireless communication layers (i.e., physical layer, data link +layer, network layer and upper layer), while Zhang et al.. +[39] considered the mobile, sensor networks and their related +applications. However, none of existing works provides a +thorough survey of ML/DL techniques for large-scale wireless +networks from the perspective of optimization. In light of +the underlying different optimization problems involved in a +variety of practical fields, various customized ML paradigms +are extensively reviewed in this paper. By identifying the +task-specific structures of large-scale optimization problems +and classifying the existing ML frameworks accordingly, this +paper bridges the elusive ML algorithms and well-grounded +optimization theory to improve the interpretability and trans- +parency of deep neural networks (DNNs), and inspires a game- +changing new perspective for solving large-scale optimization +problems in wireless networks. The major contributions are +summarized as follows: +• The challenges of wireless network optimizations in +6G systems, as well as the restrictions of traditional +optimization algorithms are discussed to motivate the +ML-based wireless network optimizations. The existing +design paradigms of MOAs are systematically elaborated +from different perspectives, such as learning principles, +NN architectures and theoretical foundations, which are +presented in Section I. +• The connections between large-scale optimization prob- +lems and specialized DL algorithms are thoroughly dis- +cussed to reveal the potential of MOA for improving +the system performance and provide insightful guidance +on the use of ML in 6G networks. Specifically, some +promising learning to optimize frameworks (i.e., algo- +rithm unrolling, LBB, GNN and DRL) are thoroughly dis- +cussed from Section II to Section V, detailing properties +and classifications of large-scale wireless optimization +problems, NN design patterns catering to the features +of optimization problems and the use-cases in various +wireless applications. +• Semantic communication is elaborated for end-to-end +optimization of communication systems in Section VI, +which provides a classic semantic communication archi- +tecture to endow the NNs with the ability of semantic +information extraction and recovery to optimize the trans- +mission efficiency and reliability trade-offs. +• Federated learning, as a prominent distributed ML +scheme to effectively solve the model optimization prob- +lem in ML over hyper-scale wireless networks, is de- +scribed in Section VII, where the issues of network/data +heterogeneity and instability, as well as the deployment in +different wireless communication systems are illustrated. +• The guidelines for MOA designs in terms of the choice +of loss functions, network architectures, training methods, +and the ways to handle optimization constraints are given +in Section VIII. The issues pertaining to the theoretical +progress of various MOAs, implementations as well as +challenges and future research directions are also pre- +sented in Section VIII. +We summarize the main topics of ML-based optimization +algorithms, related problems and the wireless applications in +Fig. 1. + +6 +II. ALGORITHM UNROLLING +In this section, we introduce one widely adopted “learn +to optimize” algorithm design framework, termed algorithm +unrolling, which constructs a layered network to mimic each +iteration of a classic iterative algorithm. We start from the +motivation of algorithm unrolling and its design framework, +which provides a guidance on how to unroll an iterative algo- +rithm, followed by the case studies in wireless communication +networks. The advantages and disadvantages of algorithm +unrolling are summarized at the end of this section. +A. Motivations and Design Frameworks +The success of “end-to-end learning” framework requires +huge training datasets and significant computational cost due +to a large number of training parameters of generic NNs while +serving as a universal function approximator. The low training +efficiency of generic NN hinders its application for dynamic +and large-scale wireless networks. Moreover, in future 6G +wireless networks with scenarios where abundant high-quality +training samples such as CSI are unavailable, the performance +of general DNN may significantly degrade and even underper- +form traditional algorithms. Furthermore, the generic NN is +treated as a “black-box”, where the functionality of each layer +and the performance guarantees of NN are hard to obtain. The +lack of interpretability of black-box DNNs can be a serious +limitation in contrast with optimization-based approaches with +theoretical guarantees in wireless networks, where reliability +and predictability are of vital importance. To address these +limitations, algorithm unrolling is an emerging method that +provides a concrete and systematic connection between classic +iterative algorithms and deep neural networks. By unfolding +an iterative algorithm with algorithm parameters transferred +to training parameters of NN, the unrolled NN enables inter- +pretability of each layer and even theoretical guarantees are +possible. Due to the potential in developing efficient, high- +performance and theoretical guaranteed NN using reasonably +sized training sets, it is a pleasant surprise that algorithm +unrolling rapidly grows in both theoretic investigations and +practical applications [35]. +Algorithm unrolling was first introduced by Gregor and +LeCun [42] to accelerate the ISTA for improving the com- +putational efficiency of sparse coding. The basic idea is to +map each ISTA iteration to a neural network layer and then +stack the layers together, which can be viewed as executing an +ISTA iteration multiple times by a layer-wise neural network. +The same techniques can be further applied to general iterative +algorithms, in which the update form is given by +xt+1 = g(xt; θt), +t = 0, 1, 2, · · · , T. +(1) +Here, xt ∈ Rn, t = 1, · · · , T are the iterative variable +vector (e.g., the signal to be recovered or the variable to be +optimized), g(·; ·) : Rn → Rn is the iterative function of +a specific iteration algorithm, and θt ∈ Rm, t = 1, · · · , T +are trainable parameters (including model parameters and +regularization coefficients) of the algorithm. The overall prin- +ciple of algorithm unrolling is to unroll a specific iterative +algorithm into a deep network by mapping each iteration +function g(·; ·) into a single network layer and stacking a +finite number of layers together. The forward procedure of +NN is equivalent to the execution of the iterative algorithm. +The unrolled network architecture thus depends on the original +iterative algorithm (e.g., the unrolled ISTA turns to be an +recurrent neural network (RNN) [42]), as the single network +layer shares the same structure of iteration function g(·; ·). The +details for the algorithm unrolling are illustrated in Figure. +2. The trainable parameters θt ∈ Rm, t = 1, · · · , T can be +learned through end-to-end approach: +minimize +ΘT +L +� +xT +1(ΘT ) +� +, +(2) +where L(·) is the loss function for training, ΘT = {θt}T +t=0 +is the entire trainable parameters of total T-layer network and +xT +1(·) is the output function of the unrolled network. Due to +the customized structure of NN, the end-to-end training may +suffer from spurious local minima and gradient explosion or +vanishment during the training. Instead of directly solving (2), +the common adopted training strategy for unrolled networks +is layer-wise method [43], which can achieve more efficient +training due to the better parameter initialization. That is, the +whole training process can be divided into T sequential sub- +training processes. For the t-th sub-training process, we aim to +refine the trainable parameters Θt, where a two-stage method +is used. The first stage is dedicated to optimize parameter +θt individually, while the latter learns the whole Θt jointly +by fixing the learned θt as initialization. In the testing stage, +feeding the data forward through the unrolled network with +learned parameters is equivalent to executing the parameter- +optimized iterative algorithm for a finite number of iterations. +In recent years, algorithm unrolling approaches have been +extensively applied to solve various signal processing prob- +lems, such as sparse and low rank regression, probabilistic +graphical model, differential equations and quadratic opti- +mization [44], and have been applied to a wide range of +applications including but not limited to compressive sensing +(CS) [45], deconvolution [46], and image processing [47]. +The superior performance and high computational efficiency +of algorithm unrolling have been proved in various domains. +At a high level, algorithm unrolling methods take advantages +of both optimization-based priors and data-driven learning +ability of NN. Compared with generic black-box NNs, un- +rolled NNs have much fewer parameters due to the inheri- +tance of the structure and domain knowledge from a specific +iterative algorithm, allowing it to benefit in terms of the +computational efficiency, interpretability, generalizability and +reliability. Theoretical tools from the traditional optimization +theory can be explored to describe the convergence behavior +and performance guarantees of unrolled NN. Compared with +iterative counterparts, algorithm unrolling enables improved +performance due to its expanded representation capability by +tuning trainable parameters of model-based algorithm through +data-drive learning. +Due to the stringent requirements of large-scale 6G wireless +networks on efficiency and reliability, algorithm unrolling has +recently attracted much attention in wireless communications +for solving sparse optimization problems and some non- + +7 +… +for +do +end for +Stacking +One Layer Network +Mapping +Trainable Parameters +Unrolled Network +… +Input +Layer +Hidden +Layer +Output +Layer +Fig. 2. The general framework of algorithm unrolling method. +convex optimization problems. In the following subsections, +we review several representative cases of algorithm unrolling +in wireless communication applications. Table II summarizes +the types of problems we will cover, the corresponding appli- +cation topics, the underlying iterative algorithms and the type +of trainable parameters of the unrolled methods. +B. Application 1: Signal Recovery Problems +With the emergence of large-scale signal processing tech- +niques in 6G systems, such as big data, massive IoT, massive +access network, large-scale antennas, etc., signal recovery, es- +pecially sparse signal recovery, has been a widely encountered +optimization problem in diverse wireless applications [55], +[56]. Consider the following widely accepted model for signal +recovery in wireless networks +y = Ax + w, +(3) +where y is the received vector or matrix at the BS, A is +the measurement matrix (e.g., the channel matrix in signal +detection, beam selection matrix in beamspace channel esti- +mation and pilot matrix in JADCE), w is Gaussian noise and +x is the unknown signal to be recovered (e.g., symbols in +signal detection, channels in channel estimation and channels +of active devices in JADCE). +When the x in (3) is a sparse signal, the well known ISTA +has the following simple iterative expression: xt+1 = ηλ(xt+ +1 +C AT (y − Axt)), where η, λ and C are threshold operator, +regulation parameter and step size, respectively. However, +traditional ISTA suffers from high computation complexity for +recovering sparse signals. By a simple variable substitution to +separate the inputs and output of network (i.e., W t +1 = 1 +C AT , +W t +2 = I − 1 +C AT A, and θt = λ) and parameterizing them +as trainable factors, the algorithm unrolling approach maps +the original ISTA into an unrolled RNN as follows with +t = 0, 1, . . . , T − 1, +xt+1 = ηθt � +W t +1y + W t +2xt� +. +(4) +This approach inherits the structure and domain knowledge +of the ISTA-based algorithm, also improves the convergence +rate and computational efficiency of original algorithm through +end-to-end training, which can be extended for recovering +signals with different sparse patterns (e.g. group sparse pattern +[14]). +In summary, the imperfectness of practical data links and +inherent various sparsity structures of practical signals render +many conventional (sparse) signal recovery algorithms, such +as ISTA, AMP, OMP, etc., suffering from the unsatisfactory +performance, slow convergence and high complexity when +employed in practical large-scale wireless networks. Due to the +superior performance, algorithm unrolling methods have been +applied to solve signal recovery problems in the applications +of signal detection [11], [19], [32], [48], channel estimation +[12], [20], [49] and JADCE [14], [50], [51]. +1) Data Detection: Data detection at the receiver has been +a challenging task due to the wireless channel fading. Con- +ventional data detection techniques, such as linear detectors +[57], SDR [58], sphere decoding [59], AMP [60], etc., depend +on the mathematical models of wireless channels and usually +assume perfect CSI at the receiver which is replaced by its es- +timate in the practical system. The channel dependence of data +detection undermines the detector performance in practical +wireless systems where the channels can be highly complex, +poorly understood, hard to be modeled or with estimation +errors. Moreover, the traditional detection techniques suffer +from significant complexity-reliability trade-off, which cannot +be efficiently implemented at the scale required by 6G massive +MIMO systems. To overcome these limitations, ML-based re- +ceivers have been extensively studied, which learn the mapping +from the channel outputs to the transmitted symbols in a data- +driven manner, and several algorithm unrolling approaches +for MIMO detection including DetNet [32], OAMPNet [19], +MMNet [11] and CMDNet [48] were proposed. +When assuming perfect CSI at receiver, Samuel et al. [32] +unfolded the projected gradient descent algorithm called Det- +Net by treating the gradient step sizes as learned parameters, +followed by common MLPs for improving expressive power. +However, the architecture of DetNet does not contain the +properties of iterative methods, leading to high complexity and +poor interpretability. To further improve detection performance +with imperfect CSI, a model-driven unrolled orthogonal AMP +(OAMP) network called OAMPNet [19] was proposed to +jointly estimate channel and detect signal by taking channel +statistics and channel estimation errors into consideration, + +8 +TABLE II +ALGORITHM UNROLLING METHODS FOR WIRELESS APPLICATIONS +Target Problems +Applications +Unrolled Networks +Original +Algorithms +Type of Trained Parameters +Signal Recovery +Problems +MIMO Detection +DetNet [32] +PGD +DNN weight and gradient step-size +OAMPNet [19] +OAMP +Step-size and nonlinear estimator factor +MMNet [11] +ISTA +Scale factor +CMDNet [48] +CMD +Step-size and gradient scale +Channel Estimation +GM-LAMP [20] +AMP +Linear transform coefficient and shrinkage +parameter +mpNet [49] +MP +Linear transform coefficient +ADMM-OGChannelNet [12] +ADMM +Linear transform coefficient and step-size +JCADE +DADMM [50] +ADMM +Step-size, shrinkage parameter and auxiliary +variable +FAT-DL [51] +AMP +Denoiser factor and scale factor +LISTA-GS [14] +ISTA-GS +Linear transform coefficient, step-size and +shrinkage parameter +Non-convex Sum-Utility +Maximization Problems +Precoding Design +IAIDNN [21] +WMMSE +Linear transform coefficient and trainable +offset +PDD-TJAPB [52] +WMMSE +Long-term variable +UPGDNet [53] +PGD +DNN weight and scale factor +Power Control +PDD-SSCA [54] +WMMSE +Short-term variable +which has only a few trainable parameters and can be trained +with a much smaller data set compared with DetNet. DetNet +and OAMPNet are both trained offline with simple model +assumptions (e.g., i.i.d. Gaussian channels, low-order mod- +ulation schemes) and can suffer a large performance gap +for realistic channels [11]. To overcome these limitations, +MMNet [11] was proposed by unrolling ISTA, which enables +online training by exploiting the locality of realistic channels +in both frequency and time domains. In particular, MMNet +parameterizes the linear transformation in ISTA, followed +by the estimation of noise variance for different transmitted +symbols in the nonlinear denoiser. To further support fast +online training, an unrolled concrete maximum a-posteriori +detection network (CMDNet) was proposed by [48], which +is theoretically revealed to be able to learn the approximate +probabilities of the individual optimal detector. +2) Massive MIMO Channel Estimation: In order to opti- +mize the data rate and energy consumption trade-off, channel +estimation is crucial in communication systems. As there +are only a few dominant propagation paths despite the high +dimension of the channel, massive MIMO channel estimation +problems turn out to be sparse signal recovery problems. +Classical CS-based algorithms suffer from high computational +complexity and poor estimation accuracy in low signal to noise +ratio (SNR) regions, especially for large-scale antenna arrays +in wide-band system with complex sparse structures. Recent +years witness an emergence of a number of unrolled channel +estimation networks benefiting from both domain knowledge +and DL, such as GM-LAMP [20], mpNet [49] and ADMM- +OGChannelNet [12] for MIMO channel estimation problems. +In millimeter wave (mmWave) MIMO systems, GM-LAMP +[20] unrolls AMP by deriving a new shrinkage function based +on the Gaussian mixture prior information of beamspace +channels to improve the estimation accuracy of AMP. To +address the basis mismatch issue in off-grid mmWave channel +estimation problem, deep unrolled network architecture ADM- +MOGChannelNet [12] was proposed by mapping the data flow +to the iterative procedures of ADMMOG algorithm, which is +computationally more efficient with performance guarantees. +However, due to the intrinsic supervised nature, these meth- +ods all require collecting a database of clean channels for +offline training, which may hinder their practical applicability. +When ground-truth channel data are unavailable, an unrolled +matching pursuit mpNet [49] was designed for MIMO channel +estimation in an unsupervised way, which can automatically +correct its channel estimation algorithm based on incoming +data with the advantage of training online due to its simple +network structure. +3) Joint Activity Detection and Channel Estimation: The +JADCE problem in grant-free massive access scenario [14] is +a typical sparse optimization problem in wireless networks, as +the sporadic transmission leads the joint activity and channel +matrix to exhibit group-sparse pattern. On the other hand, solv- +ing JADCE problem also becomes more challenging for large- +scale networks with large-scale antenna arrays and massive +number of IoT devices. Considering a multi-antenna BS and +a large number of devices, the signal model in JADCE can be +written as Y = AX + W , where A denotes the pilot matrix +and X = ΛH is the device state matrix with diagonal activity +matrix Λ and channel matrix H. To recover the group-sparse +device state matrix X with improved estimation accuracy and +low computational complexity, extended unrolled versions of + +9 +ADMM-based [50], AMP-based [51] and ISTA-based [14] +frameworks were proposed, respectively, for JADCE problems +in massive access scenarios. +Assuming the device state matrix enjoys Bernoulli-Gaussian +mixture distribution, an AMP-based unrolled network with +dimension reduction module was proposed by [51] to reduce +the length of pilot sequences and computational complexity +for JADCE problems. To directly exploit group sparsity, other +studies [14], [50] focused on solving the ℓ2,1 regularized +group least absolute shrinkage and selection operator (LASSO) +problem in a model-driven DL approach, which does not +depend on any prior distribution. Specifically, Mao et al. +[50] unrolled ADMM to solve the group LASSO, where the +network parameters are optimally learned using the stochas- +tic gradient descent algorithm. With the advantages of fast +convergence rate, high robustness and theoretical guarantees, +ISTA-based [14] algorithm unrolling framework was proposed +by extending LASSO-based decoder to group LASSO to +circumvent the high computational cost of classic ISTA and +poor algorithm robustness of AMP simultaneously. +C. Application +2: +Non-Convex +Sum-Utility +Maximization +Problems +A wide variety of resource management problems are di- +rectly or indirectly reliant on the sum-utility maximization +problems, which aim at maximizing the system sum-utility +(e.g. sum-rate) subjecting to the transmit power constraint. +The general formulation with K devices and one BS can be +expressed as +maximize +V +U (R1(V ), · · · , RK(V )) +(5a) +subject to Q(V ) ≤ P, +(5b) +where V denotes the resource (e.g., precoding matrix at the +BS) to be optimized, U(·) is the network utility function +(e.g., weighted-sum function), Rk(·) represents the achievable +rate for device k and Q(·) ≤ P is the power constraint. +Unfortunately, most of the sum-utility maximization problems +are non-convex and very difficult to solve. +In addition to the direct parameterization of the iterative +algorithm into a neural network as in Section II-B, the al- +gorithm unrolling methods can also use trainable parameters +to approximate the complex operators (e.g., matrix inversion) +in the iterative algorithm to achieve the purpose of reducing +the computational complexity for wireless applications such as +precoding design [21], [52], [53] and power control [54]. The +iterative WMMSE algorithm is one of the most representative +algorithms for sum-rate maximization problems [61], which is +guaranteed to converge to a stationary point. The general form +of WMMSE is given by, +U t = Ft(V t−1), W t = Gt(U t, V t−1), V t = Jt(U t, W t). +(6) +U, W are introduced auxiliary variables and Ft(·), Gt(·), +Jt(·) are the iterative mapping functions at the t-th WMMSE +iteration, where Ft(·) and Jt(·) contain computationally inten- +sive matrix inversion operations. By using trainable parameters +to approximate the matrix inversion and matrix multiplication +functions, unrolled WMMSE enjoys low computational com- +plexity, whose t-th layer network can be written as follows, +U t = ˜Ft(V t−1, Xu,t, Y u,t, Zu,t) + Ou,t +(7a) +W t = ˜Gt(U t, V t−1, Xw,t, Y w,t, Zw,t) +(7b) +V t = ˜Jt(U t, W t, Xv,t, Y v,t, Zv,t) + Ov,t, +(7c) +where +trainable +parameters +{Xu,t, Y u,t, Zu,t}, +{Xw,t, Y w,t, Zw,t} and {Xv,t, Y v,t, Zv,t} are used to +approximate the corresponding inversed matrix in original +Ft(·), Gt(·) and Jt(·) with Ou,t and Ov,t as the trainable +offsets. +A deep-unfolding framework IAIDNN was proposed in +[21], where a number of trainable parameters are introduced +to replace the high-complexity matrix inversion operations in +classic WMMSE algorithm. Beneficial from both optimal per- +formance of WMMSE algorithm and extensive representation +power of DNNs, IAIDNN achieves the performance of the +iterative WMMSE algorithm with much lower computational +complexity and a smaller number of training samples. In [52], +an unrolled WMMSE approach was also proposed to solve a +short-term sub-problem decomposed by original non-convex +stochastic problem with low complexity for precoding design +in reconfigurable intelligent surfaces (RIS)-aided communi- +cation systems. Such unrolled WMMSE network was also +adopted in part of [54] to approximate the iterative WMMSE +algorithm with low training complexity and reduced memory +overhead, which is adopted as a short-term sub-algorithm +in a two-stage stochastic optimization problem (e.g., power +minimization for two-timescale hybrid beamforming). The +above unrolled networks are all trained under a supervised way +due to the complex structure of WMMSE. An unsupervised +deep unrolling framework based on projection gradient descent +called UPGDNet was proposed in [53] to solve the sum-rate +maximization problems in the scenario of multiuser ultra- +reliable low-latency communications (URLLC) with finite +block length transmission, which demonstrates a satisfactory +generalization ability and low complexity. +D. Advantages and Disadvantages of Algorithm Unrolling +The advantages of algorithm unrolling methods are summa- +rized as follows. +1) Higher Computational Efficiency: Compared with the +end-to-end learning based on generic NN, reduced number +of training parameters in unrolled NN can significantly boost +the computational efficiency of NN. Besides, due to the +incorporation of domain-specific knowledge through unrolling, +the training of unrolled NN is faster and requires fewer training +data. Algorithm unrolling methods can significantly improve +the convergence rate of original iterative algorithm (e.g., +LISTA-GS [14] converges less than 10 iterations while ISTA +needs more than hundreds iterations), and also can reduce the +complexity of one-step iterative process (e.g., IAIDNN [21] re- +duces the computational complexity of WMMSE from O(n3) +to O(n2.73) with n denoting number of system parameters). + +10 +2) Better Learning Performance: By extending the itera- +tive counterparts and training using datasets, the algorithm +unrolling can achieve superior performance compared with the +conventional iterative algorithm. For example, mpNet [49] and +LISTA-GS [14] achieve more accurate estimation performance +(more than 5dB normalized mean-squared error enhancement) +compared with ISTA-based algorithms. +3) Interpretability and Theoretical Analysis: Inherited from +the traditional iterative algorithm, the behavious of each +unrolled network layer is interpretable. Algorithm unrolling +methods, to some extent, build a bridge between deep learning +and iterative formulations, where the optimization tools can be +used to define the optimal learned parameters that leading to +fastest convergence rate (e.g., the optimal parameters defined +in LISTA-GS guarantee the linear convergence rate for recov- +ering group-sparse matrices [14]). +However, there are some disadvantages of these methods. +First, for complex iterative algorithms when highly nonlinear +or non-smooth operations are involved, it is hard to develop +efficient NN to unfold the complicated iterative operations. +Second, for iterative algorithms with slow convergence, the +depth of unrolled NN will be large, which can easily suffer +from gradient explosion or vanishment during the training +stage. Third, even if the extended representation ability of +algorithm unrolling, its convergence is hard to be guaranteed +for complex iterative algorithms (e.g. unrolled WMMSE and +unrolled ADMM). Moreover, its performance is restricted by +the iterative algorithms. Analyzing the impact of trainable +parameters on the convergence and learning accuracy is also +challenging. +III. LEARNING TO BRANCH-AND-BOUND +Many important applications in wireless networks involve +complicated combinatorial problems, whose optimal solutions +are hard to obtain efficiently. A learning-based BB algorithm, +namely LBB, is introduced in this section to tackle the +combinatorial problem with low computational complexity. +Instead of end-to-end learning, the LBB replaces the complex +pruning step of traditional BB with NN to accelerate searching +for optimal solutions in feasible region. We first introduce the +traditional globally-optimal BB algorithm to solve a general +combinatorial problem. Then we introduce a learned BB to +learn the optimal pruning policy of BB in a data-driven +manner, which will be followed by the case studies in wireless +communication systems. The advantages and disadvantages of +LBB are summarized at the end of this section. +A. Global Optimal Branch-and-Bound +BB algorithm can find global optimal solutions for non- +convex combinatorial problems, e.g., discrete and mixed com- +binatorial optimization problems. It implicitly enumerates all +possible solutions by dividing the original problem into a se- +ries of sub-problems (branch step) and systematically discards +the non-promising sub-problems based on lower bounds or +upper bounds (bound step). Specifically, the branch step is to +partition the feasible region into smaller subregions in a tree +structure, where the root node is the original problem and the +leaf node represents the subproblem over the corresponding +subregions. The bound step uses the features obtained at each +node to prune off the subregions that do not contain the +optimal solution, until BB eventually converges on an exact +result [62]. To guide and accelerate the searching process, the +branch step involves node and variable selection determining +which node to explore and the fractional variables to branch +on in next iteration, whereas the bound step consists of two +main policies: evaluating the bounds of selected nodes by +solving the subproblems and pruning policy which determines +whether to explore the subregion corresponding to the node. +The pruning policy at bound step depends on the optimal +solution and optimal objective value of relaxed subproblem at +each node, where the constraints for the undetermined discrete +variables are relaxed into convex continuous constraints and +then various convex solvers can be applied to solve the relaxed +convex subproblem, e.g., linear programs (LPs), second order +cone programs (SOCPs) or semidefinite programs (SDPs). For +example, binary constraints can be relaxed into box constraints +and the non-convex complex modulus constraints can be +relaxed to their convex envelopes [15]. +By learning the time-consuming components of BB al- +gorithm using NN, the learning-based BB can significantly +reduce computational complexity with near optimal perfor- +mance. Authors in [63] provided a survey of learning-based +BB techniques regarding to the key decisions in BB, such +as learning-based branching and learning-based pruning, and +summarized the merits and flaws of different learning methods. +In original branch step of BB algorithm, various branching +rules (e.g., strong branching, hybrid branching) are used in +branch variable selection by calculating the score of can- +didate variables to indicate their qualities and then picking +the variable with the highest score. In this way, enormous +branch decisions are required while a single bad one could +sharply increase the size of search tree without improvement +in the learning performance. To accelerate the branching rules, +imitation learning was adopted in [64], [65] to learn a fast +auxiliary branching policy by approximating the traditional +branching rules using expert experience, which can outperform +the initial expert with carefully designed features. In addition, +such approach leads to a simpler learning task with smaller +Vapnik-Chervonenkis dimension, which only needs a few +training samples, and thereby speeding up the training process +consequently. To overcome the complex feature calculation +at each selected node, authors in [66] encoded branching +policies into a graph convolutional network (GCN), where +features on the graph can be efficiently extracted by vari- +ous message passing approaches. To solve large-scale mixed +integer programming (MIP), the authors in [67] constructed +two corresponding GCN components (i.e. neural diving and +neural branching) to learn a branching policy for enhancing +the performance. +Even with the learned branching policy, the computational +complexity of BB algorithm is generally exponential w.r.t. the +number of optimize variables due to the inefficient pruning +policy. Based on this observation, other studies [68], [69] +focused on the supervised learning of optimal pruning policy +to solve large-scale MIP efficiently. In the next subsection, + +11 +we shall introduce the motivations and design frameworks of +pruning policy learning-based LBB in large-scale 6G wireless +networks. +B. Motivations and Design Frameworks of LBB +In 6G wireless networks, typical resource allocation prob- +lems, such as subcarrier allocation in orthogonal frequency +division multiple access (OFDMA) [70], user association [71] +and access point selection in could-radio access network (C- +RAN) [72], can be formulated into mixed combinatorial opti- +mization problems. With the rapid growth of wireless network +scales (e.g., the massive number of IoT devices and the +massive number of antennas), the dimension of optimization +variable becomes very large, which makes learning-based BB +a promising approach to tackle the exponential complexity +of traditional BB while maintaining the global optimality. In +this subsection, we highlight a promising learning strategy for +node pruning in BB algorithm proposed in [68], [69], termed +LBB, which has been shown to achieve low computational +complexity with near-optimal performance. +The main idea of LBB is to model the tree search process +as a sequential decision problem because whether or not +exploring the subregion of the tree node corresponds to the +preserve decision or prune decision. Such problem can be +efficiently solved by a binary classifier with problem features +as the input and decision states as the output. The procedure +of LBB includes training data generation, feature design, +binary classifier learning and searching space controlling, as +described below. +1) Training Data Generation and Feature Design: The +original BB algorithm is directly applied to generate the +training dataset. Specifically, the problem parameters of each +randomly generated wireless network are collected. For each +set of problem parameters, the original BB algorithm is +adopted to obtain the optimal solution and the features of all +explored nodes are recorded. Then the nodes whose feasible +sets contain the optimal solution are labeled as preserve while +the others are labeled as prune. The input features of the NN +are also of vital importance for training a good classifier in +LBB. Features are divided into tow categories, i.e., problem- +independent features and problem-dependent features [15], +[73]. Problem-independent features correspond to the structure +of the binary tree generated by the BB algorithm, which +includes node features computed from the current node (e.g., +the depth of current node), branching features computed from +the branching variable of the current node (e.g., the branching +variable’s value at the current node) and tree features computed +from the BB search tree (e.g., global upper and lower bounds) +[68]. While problem-dependent features correspond to the +domain knowledge of different specific problems. Typically +in wireless communications, the CSI feature and some de- +scriptions of the radio resources (e.g. power feature) are +usually considered as problem-dependent features to exploit +the domain knowledge for efficient policy learning. By feeding +the problem dependent features into the classifier learning, the +LBB can avoid solving relaxed problems at each branching +node, which can further reduce the computational burden. +2) Binary Classifier Learning: The binary classifier de- +termines whether to preserve a node or not and a good +classifier can prune as many non-optimal nodes as possible to +minimize the search time. Standard classifiers, such as logistic +regression [74] and support vector machine (SVM) [75], are +inefficient for high-dimensional data classification with tangled +mapping between input and output [76]. To effectively capture +the complicated relationship between the input features and +output classification labels, MLP can be employed for binary +classifier learning in LBB due to its powerful expression +ability [15]. In each layer of MLP, the input is multiplied +with a learned weight matrix, followed by a rectified linear +unit function (Relu) as activation function. The output layer +employs the soft-max function to calculate the probability of +each class, while the weighted cross entropy is adopted as loss +function. +Target Problems: +MINLPs,CMCPs +Optimal Nodes +Nonoptimal Nodes +Pruned Nodes +Pruning +Fig. 3. Learning to branch-and-bound method. +3) Training Sample Unbalance and Searching Space Con- +trolling: +It is observed that the number of non-optimal +(pruned) nodes is much larger than the number of optimal +(preserved) nodes and the mistakes at early stages can have +greater impact on the performance during BB searching pro- +cess. Thus, to enable efficient network training and achieve a +good classification performance, weighted cross entropy was +adopted in [73], where the higher weights are assigned to the +nodes labeled as preserve and the nodes with small depth in the +training dataset to raise the priority of these training samples +in the learning of NN [15]. Since too aggressive pruning policy +may lead to no feasible solutions and too moderate pruning +policy may lead to abundant preserved nodes, a threshold +was adopted to control the searching space dynamically. For +instance, a higher threshold for the class pruning will preserve +more nodes than that with a lower threshold, leading to a larger +searching space and better performance. The searching space +controlling can guarantee the feasibility of LBB with reduced +computational complexity. +The overall frameworks of BB and LBB are illustrated in +Figure 3. With the advantages of near-optimality and low com- +putational complexity for challenging non-convex and combi- +natorial problems, LBB methods have gain much traction in +the context of large-scale wireless networks to solve various +resource allocation problems. In the following subsections, +we review several representatives of non-convex combinatorial +optimization problem with extensive applications in wireless + +12 +networks to fully reveal the power of LBB for efficient and +high-quality resource management. +C. Application 1: Mixed Integer Nonlinear Problems +In wireless networks, typical resource management prob- +lems, such as user selection, user association, spectrum al- +location, computational offloading, interference and power +management, can be formulated into a mixed integer nonlinear +problem (MINLP), which is general NP-hard. The generic +formulation is given by [15] +MINLP : minimize +a,w +f(a, w) +(8a) +subject to Q(a, w) ≤ 0, +(8b) +ai ∈ N, wi ∈ C, ∀i, +(8c) +where f(·, ·) is the objective function (usually non-linear), +such as power consumption, sum rate, communication delay, +etc, discrete-valued variable ai and continuous-valued variable +wi are the elements of a and w, such as sub-carrier index, user +index, device index for discrete variable and beamforming, +power for continuous variable, and Q(·, ·) denotes certain +constraints, e.g., the quality of service (QoS) and power +constraints. +A typical example of MINLP is network power mini- +mization problem in C-RAN, which consists of binary vari- +ables (i.e., the selection of remote radio heads and front- +haul links), continuous variables (i.e., downlink beamform- +ing coefficients), QoS constraints and power constraints. To +get the near-optimal performance with affordable complexity, +LBB via imitation learning was proposed in [15], where the +depth-first-search was adopted as the branch variable selection +rule which always choses the first undetermined node during +variable selection process. In feature design, besides problem- +independent features such as the depth of node, the local upper +bound of node, current global lower bound and so on, the CSI +feature and power feature are designed as problem-dependent +features. LBB algorithm was shown to achieve success us- +ing only tens to hundreds of training samples for solving +MINLPs in the applications of resource allocation problem in +device-to-device (D2D) communications [73] and mobile edge +computing [77]. Specifically, in [77], a learning strategy for +node pruning in BB algorithm was proposed for the offloading +resource assignment in mobile edge computing, where DNN +was applied to approximate the unknown mapping between +the attributes of BB tree nodes and the pruning decisions in +the BB tree search. To further reduce the sample complexity, +imitation learning was adopted in [73] to solve MINLPs of +resource allocation in D2D systems, where DAgger algorithm +was employed to correct the mistakes the learned policy makes +by iteratively collecting data. Specifically, instead of using +the training data only once, the pruning policy πt learned in +imitation learning shall explore all the tree nodes and generate +their corresponding features at iteration t. Then based on +these datasets, a new prune policy πt+1 can be learned at +next iteration to correct the mistakes made by πt. To adapt +to time-varying network settings, a transfer learning via self- +imitation method was adopted in [15] to quickly adapt the +learned pruning policy in LBB to the new task with reduced +training samples. +D. Application 2: Non-Convex Complex Modulus Constrained +Problems +Many resource management problems in wireless networks +can be formulated as non-convex complex modulus con- +strained problems (CMCPs), which is general NP-hard. The +generic formulation of CMCP is given by [78], +CMCP : minimize +a,w +g(w) +(9a) +subject to |bH +i w + ci| ≥ 1, ∀i, +(9b) +arg (bH +i w + ci) ∈ Ai, ∀i, +(9c) +where g(·) : CN +→ R is a convex objective function +(e.g., power consumption), constraint (9b) denotes the non- +convex minimum complex modulus constraints on N linear +transformations of the optimization variable w ∈ CN (e.g., +the transmitted symbol vector and the beamformimg vector) +with coefficients bi ∈ CN and ci ∈ C, and constraint (9c) +represents the corresponding argument constraints where each +argument set Ai can be continuous or discrete (e.g., box +constraints or discrete phase sets). +One toy example of CMCPs in wireless networks is the +QoS-constrained multi-cast beamforming optimization prob- +lem in multiple-input single-output (MISO) downlink trans- +mission, which minimizes the total transmit power at the BS +(i.e., minw ∥w∥2 +2) subject to individual SNR constraints for +all devices (i.e., |hH +k w| ≥ 1, ∀k). The formulated problem +corresponds to the standard CMCP in (9) by letting bk = hk, +ck = 0 and Ak = [0, 2π], ∀k. Other applications of CMCPs +include MIMO detection and passive beamforming in RIS- +assisted systems [78]. Unlike the BB algorithms for solving +MINLPs with discrete branching variables, the branching of +BB algorithm for solving CMCP is based on argument sets +to deal with the continuous variables [79]. As a result, the +continuous argument set can lead to unbounded extension of +tree search in BB algorithm and tremendous number of binary +classification tasks in the searching process. Therefore, it is +difficult to find the optimal solution of the CMCP using only +one classifier [15], [73]. To address this challenge, ensemble +learning was applied in [78] to train multiple classifiers in +LBB, which are further combined to achieve better perfor- +mance. To facilitate the multi-classifier learning and address +the data unbalance in LBB, [78] devised the under-sampling +training and ensemble testing. In under-sampling training, indi- +vidual classifier is trained on each of the data subsets sampled +both from the majority set and minority set. In ensemble +testing, LBB is executed multiple times using learned multiple +classifiers in parallel to choose the best solution from all +the results. Regarding to the feature design, besides problem- +independent features such as local node features (e.g., the +argument set A) and global tree features (e.g., the global lower +bound), the CSI, the SNR, the received signal and the complex +modulus constraint can be designed as problem-dependent +features. + +13 +E. Advantages and Disadvantages +LBB algorithm was shown to achieve near-optimal per- +formance and meanwhile substantially reduce computational +complexity using only tens to hundreds of training sam- +ples, due to the inheritance of the structure of traditional +BB algorithm and the exploitation of DL techniques. If the +parameters and features are properly chosen, the number of +relaxed problems to be solved can be reduced from O(2L) +in BB to O(L) in LBB for MINLPs [15], where L denotes +the number of integer variables. However, the training of LBB +depends on the labels generated from traditional BB, which +can induce great computational burden to generate the training +set. Meanwhile, feature design in LBB is not supported by +theory and hence different designs can lead to different results +of the algorithm. +IV. GRAPH NEURAL NETWORK FOR STRUCTURED +OPTIMIZATION +The graph-structured topology of wireless network enables +the successful usage of GNN to solve a broad range of design +problems over the wireless networks [80]. As a specialized NN +for graph-structured data, GNN can exploit the domain knowl- +edge of various applications to achieve near-optimal learning +performance with good scalability and generalizability. In +this section, we commence with the framework of GNN for +structured optimization. Then we illustrate the graph modeling +of optimization problems in wireless networks, after which, +several applications of GNN are discussed. The advantages and +disadvantages of GNN-based solution in wireless networks are +summarized at the end of this section. +A. Principles of Graph Neural Network +Recently, a growing number of applications have emerged +possessing graph-structured data, e.g., social networks and +transportation networks, with high dimensional features, active +interactions between graph nodes and potentially time-varying +structures. The emerging GNN can effectively incorporate the +graph structure into the architecture of NN to model the node +properties and the relationships between nodes to explore the +hidden features in graph-structured data. Hence, GNN not +only features for good scalability to large-scale graphs and +good generalizability to dynamic graph structures, but also can +achieve near-optimal performance with more efficient training. +Traditionally, a graph-structured data can be mathematically +represented as a pair G = (V, E), where V is the set of nodes +and E is the set of edges. For a node, its (1-hop) neighborhood +is defined as the set of all nodes with edges connected to +it. These edges can be represented by an adjacency matrix +A, where Ai,j = 1 if and only if edges (i, j) ∈ E for all +nodes i, j ∈ V . The graph is undirected if A is symmetric, +otherwise it is directed. The properties associated to the nodes +and edges are important for learning over graph. Denote zi +as the node features associated with node i ∈ V , ei,j as +the edge features associated with edge (i, j) ∈ E. Since the +graph structure may be changed by the permutation of nodes, +a key desideratum for designing GNNs is that the devised +GNN should satisfy permutation invariance or permutation +equivariance [81], which can precisely capture the internal +structure of graph data. As illustrated in Figure 4, permutation +invariance means that the output of GNN does not depend on +the node order used to encode adjacency matrix. Permutation +equivariance means that the output of GNN is permuted +according to the same permutation as the input node order. +Such properties enable faster training, less training samples, +better scalability and generalizability of GNN. For example, +compared with MLP, the parameters of GNN can be shared for +graphs with varying sizes and permuted input, thereby achiev- +ing good generalizability, while an MLP has to be retrained +when the topology is changed. More precisely, authors in [33] +theoretically showed that the GNN’s generalization error and +required number of training samples are O(n) and O(n2) +lower than those of MLPs, where n is the number of nodes +in the graph. +GNN implements the learning over graph by exacting the +neighbor information to enrich the feature of each node and +spreading these features over the graph according to the graph +structure. The framework of GNN is illustrated in Figure 4. +Generally in a vanilla GNN with multiple hidden layers, the +output of last GNN layer contains the processed information +of all nodes and edges, which can be used for classification +or prediction. In each GNN layer, each node aggregates +the information of its neighbors to update its hidden state. +Specifically, let d(ℓ) +k +be the hidden state of the k-th node at +the ℓ-th GNN layer. Each GNN layer contains the aggregation +and combination step as follows. +1) Aggregation Step: The aggregation step aims to update +the node’s hidden state with its neighbors’ information. Typi- +cally, the k-th node uses an NN to aggregate its neighbors’ out- +puts of the previous layer (the (ℓ−1)-th layer), followed by a +pooling function (e.g., element-wise max-pooling or element- +wise mean-pooling) that is invariant to the permutation of the +inputs. The update is given by +a(ℓ) +k += PLj∈N (k) +� +f (ℓ) +1 (d(ℓ−1) +k +, d(ℓ−1) +j +, ej,k|Θ(ℓ) +1 ) +� +, +(10) +where PLj∈N (k)(·) denotes the pooling function used for +aggregating the outputs of the nodes in N(k), N(k) denotes +the set of neighboring nodes of node k, f (ℓ) +1 (·|Θ(ℓ) +1 ) is the +aggregation function implemented using MLP with model +parameters Θ(ℓ) +1 +at the ℓ-th layer and ej,k is the feature of +edge (j, k). +2) Combination Step: After obtaining the aggregated in- +formation, another combination function is applied to process +information and update the hidden state at each node. Specifi- +cally, the aggregated information is combined with the node’s +own information as follows: +d(ℓ) +k += f (ℓ) +2 +� +d(ℓ−1) +k +, a(ℓ) +k , zk|Θ(ℓ) +2 +� +, +(11) +where f (ℓ) +2 (·|Θ(ℓ) +2 ) represents parameterized combination +function with model parameters Θ(ℓ) +2 +and zk is the feature +of node k. +To improve the learning ability of GNN over different +graphs, the aggregation function f1, the combination function +f2 and the pooling function PL should be carefully designed. +Authors in [33] gave general guidelines for designing these + +14 +Hidden Layer +Original Order +Output Layer +Input Layer +Permutation +Permutation +Fig. 4. Illustration of permutation equivariance and permutation invari- +ance. Different colors of nodes represent different orders of nodes. +Devices +BSs +Cell-free Network +D2D Network +Fig. 5. +Examples of the wireless communication graph modeling (i.e. +cell-free networks and D2D networks). +functions. First, the message aggregation and combination +should be simplified to reduce the complexity of calculations. +Second, different pooling functions are suitable for different +graphs and optimization problems. For example, the max- +pooling function is suitable when the neighbors’ influence is +sparse or the problem parameters are noisy, because it focuses +on the neighbors that are most influential. While the sum- +pooling function gives summary statistics of the neighbors, +which works well when we aim to obtain a summary of the +neighbors. Third, the aggregation and combination function +should be properly designed to satisfy the permutation invari- +ance or permutation equivariance property. Last, the input em- +bedding is of vital importance for information representation. +Therefore, it is often preferable to employ an input embedding +neural network to lift or compress the features into a proper +dimension to search for a balance between training complexity +and learning performance. +B. Graph Optimization Problems in Wireless Networks +In wireless networks, the relationship between users, BSs +and even antennas spontaneously yields to a wireless com- +munication graph. Specifically, a wireless network can be +modeled as a directed/undirected graph with node and edge +features, where communication devices (users and BSs) can +be treated as nodes, channels between devices can be treated +as directed edges. An edge (i, j) exists if there are interdepen- +dencies, e.g., communication link, interference link or other +connectivity patterns, from node i to node j. The features +associated with the node represent the properties of devices, +e.g., user’s weight in weighted sum rate maximization, while +the features associated with the edge stand for the link proper- +ties, e.g., channel gains. Mathematically, the general form of +optimization problem on wireless communication graph can +be expressed as [22] +minimize +X +f(X, Z, A) +(12a) +subject to Q(X, Z, A) ≤ 0, +(12b) +where f(·) is the objective function (usually non-convex), X +denotes the collection of optimization variables assigned to all +nodes, Z denotes the node feature matrix which can model the +heterogeneity of node, and A is the edge feature tensor, which +can incorporate more comprehensive link properties, Q(·) is +the constraint. The graph optimization problem is general +enough to cover most of the resource management problems in +wireless networks, e.g., power control, beamforming design, +link scheduling, etc. +The graph-structured optimization problem (12) defined on +wireless communication graph satisfies the permutation equiv- +ariance and permutation invariance properties [82]. Namely, +the objective function f and constraint function Q are invariant +to the permutation of the device indices, while the output +of GNN, i.e., the optimal variable X would be permuted in +the same way as the permutation of device orders. To solve +the graph optimization problem effectively, the GNNs which +satisfy the permutation equivariance/invariance properties are +favorable. To fully reveal the advantage of GNN-based solu- +tion for (12), authors in [33] bridged the GNN with a class of +COAs, i.e., distributed message passing, to justify that for any +graph optimization problem, there exists a GNN that can solve +it from the algorithmic perspective. Compared with the COAs, +the data driven GNN can achieve near-optimal performance +with reduced computational complexity. Compared with MLP, +the superior performance of GNN in solving graph optimiza- +tion problem in wireless networks is theoretically proven in +[33] in terms of the optimization performance and sample +efficiency. +Table III summarizes the wireless applications of GNN. +Based on the topology of graph, the applications of GNN may +cover resource management problems in D2D/cellular/cell- +free/distributed network and other signal processing fields. In +the following subsections, we introduce several application +examples of GNN, detailing the graph modeling of different +problems and the GNN architecture developed to solve the +respective optimization problems. + +12:2812:2812:2815 +TABLE III +WIRELESS APPLICATIONS OF GRAPH NEURAL NETWORK +Applications +Wireless Issues +Ref +Node +Edge +Node Feature +Edge Feature +Cellular/Cell-free +Networks +Beamforming +Optimizations +[83] +Antenna / User +Communication link +Beam feature +Channel coefficient +[84] +User / RIS +Link between nodes +Received pilots / Mean of all user features +\ +Power Allocation +[85] +BS / User +Communication link +State information +Channel coefficient +D2D Networks +Power Allocation +[22], [82] +Transceiver pair +Directed interference link +The state of the direct channel and +the weight of transceiver pair +The states of the +interference channel +Link Scheduling +[86] +Transceiver pair +Directed interference link +Channel gains or distance +Channel gains or distance +[87], [88] +Communication link +Interference link +Queue length and link rate +\ +Distributed Systems +Decentralized +Networks +[89] +Transmitter +Communication link +Application state +Channel coefficient +[90] +Receiver +Direct link / Interference link +Proportional-fairness ratio +\ +[91] +Device +Communication link +Binary internal characteristics +\ +Heterogeneous +Networks +[10] +Different device +Meta-path +Subchannel gain and power budget +Distance between nodes +[92] +Antenna or user +Communication link +State information +Channel coefficient +Other Signal Processing +Applications +Mobile Traffic +Prediction +[93]–[95] +Region +Road connecting the regions +Historical traffic +\ +Channel Tracking +[96] +Element of channel vector +Link between channel element +Channel coefficient +Channel spatial correlation +C. Application 1: Graph Neural Networks in Cellular/Cell- +Free Networks +In cellular or cell free networks, we can treat the different +communication devices, including antennas, user equipments +(UEs), RISs, and BSs, as graph nodes and their interde- +pendencies as graph edges, as illustrated in Fig. 5. In the +following, we will provide some examples of GNN-based +resource allocation in cellular or cell-free networks. +1) Beamforming Optimization: The multi-antenna beam- +forming optimization problems were considered in [83], where +a bipartite GNN framework consisting of antenna/user vertices +and channel edges was proposed to improve the scalability and +the generalization ability of DNN approaches. When consid- +ering the RIS-assisted cellular networks, GNN is proposed in +[84] to directly map the received pilots to the beamformers at +the BS and reflective patterns at the RIS by solving sum-rate +maximization problem. To model it as a graph optimization +problem, the RIS and users are taken as graph nodes and +the communication links between users and RIS are taken +as the graph edges. The input features of user nodes are the +received pilots and the input feature of RIS node is the mean +of all user features. After updating the input features through +multiple GNN layers, the output features of user nodes will +be mapped to the beamforming matrix at BS and the output +feature of RIS node will be mapped to the reflective pattern +of RIS. To capitalize on the special properties of GNN, the +GNN layers are designed such that the beamforming vectors +corresponding to different users are permutation equivariant +and the reflective pattern at RIS is permutation invariant for +the change of the order of user channels, thereby enjoying the +beneficial performance of GNN. +2) Power Allocation: +Consider a cellular/cell-free net- +work consisting of one/multiple BSs and multiple users. The +cellular/cell-free system can be modeled as a fully connected +bipartite graph, where the BSs and users are viewed as +nodes, the communication links including direct links and +interference links between BSs and users are regarded as +edges. In the power allocation problem, the edge features +are typically the channel coefficients and node features are +the state information of user demand (e.g., the data arrival +rate for traffic demand [85]). After updating through GNN +layers, the hidden states at the user nodes are mapped to the +power values through learnable MLPs. Many existing works +[85] further developed advanced GNN algorithms to handle +various adverse factors in wireless systems. For example, a +random edge GNN (REGNN) was proposed in [85] to handle +the fast fading channels in power allocation problem, where +the underlying graph structure is a random variable drawn from +a specific distribution. The authors in [85] further proposed +an unsupervised model-free primal-dual learning method to +address the general utility-constrained (e.g., binary power +constraint) wireless resource allocation problems (e.g., the +sum-rate maximization problems). +D. Application 2: Graph Neural Networks in D2D Networks +In D2D networks, supposing there are multiple transceiver +pairs, each transceiver pair usually can be treated as one node, +where the node features include the direct link CSI, the weight +of each transmission pair and other related information. The +interference link between transmission pairs can be treated +as the edge, where the edge features include interference +CSIs. The underlying graph, as shown in Fig. 5, can be +directed or undirected according to the definition of edge +features. The GNNs can be effectively adopted to solve various +resource allocation problems in D2D networks, such as power +allocation [22], [82], [97], link scheduling [86]–[88] and so +on. +1) Power Allocation: Based on a standard D2D graph, +IGCNet [82] and MPGNN [22] were proposed to apply +GNN over the D2D graph for the power control problem, +where the wireless graph modeling are designed to match the +permutation equivariance property of interference channels, +and the aggregation and combination functions are realized +using MLPs. To inherit the advantages of classic algorithm, +an unrolled WMMSE algorithm was parameterized in [97] +to design the GNN architecture for solving power allocation +problem in a single-hop ad hoc interference network. + +16 +2) Link Scheduling: By exploiting and incorporating the +underlying topology of wireless networks to learning al- +gorithms, GNNs are well-suited for efficient scheduling of +transmission links in wireless communications [86]–[88]. To +eliminate the expensive channel estimation stage, Lee et al. +[86] firstly constructed a graph embedding process for link +scheduling in D2D networks, where each D2D pair is designed +as a node while interference links among D2D pairs are +the edges for efficient graph design. Based on Structure2Vec +architecture, each node in the graph is represented by a low- +dimensional feature vector, where the link classifier can be +trained with high efficiency to decide whether a D2D pair +should be activated. Other studies [87], [88] focused on link +scheduling in wireless multi-hop networks, where the stream +of packets from a source user (one node) to a destination +user (the other node) may pass through multiple edges on +the designed graph. In [87], a trainable graph convolutional +network (GCN) module was proposed to improve the opti- +mality gap with the traditional greedy approaches for the NP- +hard maximum weight independent set (MWIS) problem in +link scheduling. To reduce the delay of scheduling, a delay- +oriented distributed scheduler based on GCNs was proposed in +[88], in which multi-step lookahead backlogs and the network +topology were fully captured by the node embedding layers. +E. Application 3: Graph Neural Networks in Distributed Sys- +tems +The typical characteristics of distributed systems are de- +centralized setting (i.e., the transceivers only have knowledge +of their local radio environment and make local decisions +based on this information) and heterogeneous nature (i.e., there +are different types of nodes with different node features in a +communication graph), where GNNs also have shown their +superior performance and scalability for these systems. +1) Decentralized Networks: To address the intrinsic infor- +mation delay and asynchrony between devices in decentralized +cooperative wireless systems, a primal-dual learning based +aggregation GNNs (Agg-GNNs) was proposed in [89] to de- +sign localized resource policies with delay and asynchronous +constraints. This can be achieved by adding multiple network +layers to process delayed information after signal aggregations +in Agg-GNNs, which gather spatial and temporal-correlated +information of the global wireless networks. Similar primal- +dual learning method was adopted in [90] to learn resilient +radio resource management (RRM) policies with adaptive per- +user minimum-capacity constraints, which can adapt to the +current network conditions via optimized slack variables. By +parameterizing the RRM policies using scalable GNNs based +on the graph topology of wireless networks, negligible duality +gap can be proved and superior trade-off between average rate +and user fairness can be achieved. The parallel implementation +of GNNs was discussed in [83] to facilitate the deployment of +decentralized GNN in distributed MIMO configurations, e.g., +cell-free MIMO and fog radio access networks. Furthermore, +authors in [91] analyzed the robustness of a decentralized +GNN-based binary classifier for inference considering the +imperfect fading channels and wireless noises in the exchange +of local information between neighboring nodes, where a novel +retransmission mechanism to enhance the prediction robust- +ness was proposed under different communication systems. +2) Heterogeneous Networks: In heterogeneous networks, +different types of communication subjectives (e.g., users, ac- +cess points, mobile stations, etc.) can be modeled as different +types of nodes connected by different types of edges (e.g., +transmission edges and inference edges) to indicate a more +complex wireless communication network. A heterogeneous +graph neural network (HGNN) was firstly proposed in [10] to +characterize two different types of nodes (access points (APs) +and mobile stations (MSs)) and various types of edges between +nodes (e.g., uplink/downlink transmission path and inter- +AP/inter-MS interference path) in distributed cell-free massive +systems. To address the impact of heterogeneous nodes, an +adaptive node embedding layer was proposed, where the node +features of AP and MS are transformed by two embedding +matrices to handle the varying input feature dimensions before +the process of GNN layers. Similar HGNN was considered +in [98] for D2D resource allocation, which sets individual +aggregation/update functions according to different relations +between nodes to address network heterogeneity. Different +from the traditional GNN-based power allocation, whose out- +puts are permutation equivariant to arbitrary permutations of +users, author in [92] constructed a permutation equivariant +heterogeneous GNN (PGNN) to learn the optimal power +allocation policy in cellular networks whose outputs are only +equivariant to some permutations of user nodes to precisely +match the identified properties in the power allocation policy. +It shows that the PGNN achieves better learning performance +in terms of sample efficiency, computational complexity and +performance optimality due to the exploitation of the well- +matched policy properties and the heterogeneous design of +GNN. +F. Other Signal Processing Applications +1) Mobile Traffic Prediction: Graph-based methods can +also be applied to address large-scale mobile traffic prediction +[93], [94], where the challenge is to exploit the time-evolving +nature of mobile movements and the spatial relations of mobile +traffic demand. Typically, a graph is constructed to characterize +the spatial structure of the traffic data in different geometric +regions by dividing the area into discrete grids, where node +represents the region and edge represents the road connecting +the regions. To capture the temporal correlations of traffic +data, RNN-based [93], [95] and CNN-based [94] methods +can be employed. For example, Guo et al. in [95] proposed +graph convolutional RNN for traffic prediction, where GCN +and gated recurrent unit were used to exploit the spatial +and temporal structure of the traffic data. He et al. in [93] +constructed a spatial relation graph of traffic data to capture +the near-far spatial correlation, and utilized an attention- +based structural RNN to capture the temporal dependency +and spatial relationship simultaneously. Besides the spatial- +temporal structures of traffic sequences, the user’s mobility +patterns have also been exploited in [94] by a graph-based +temporal convolutional network for accurate traffic prediction, + +17 +where each node denotes a wireless AP, the directed edges +indicate the movements of mobile users during the time steps +of interest, and the temporal convolutional network layers +further model the temporal trend of mobile data traffic on each +AP. +2) Channel Tracking: The massive MIMO channels can +be modeled as a graph, where GNN can extract the spatial +correlations within the large-scale channels for efficient chan- +nel estimation. Specially, different antennas are considered +as nodes with their channel coefficients being node features, +while the spatial relationships are modeled as edges with +the channel spatial correlations being edge features for graph +modeling. In [96], a GNN-based channel tracking framework +was designed, which contains an encoder fed with historical +channel samples, a core network performing GNN updates and +a decoder to decode the node and edge attributes. It shows that +the graph-structure captured GNN significantly outperforms +feed-forward NNs for channel tracking. +G. Advantages and Disadvantages +The advantages of GNNs are summarized as follows. +1) High Scalability and Generalizability: +GNNs enjoy +promising scalability and generalization ability for two rea- +sons. First, the permutation invariance and permutation equiv- +ariance properties of GNNs enable the learned NN to adapt to +large-scale and dynamic scenarios by exploiting the analogies +or equivalent patterns between the training network topology +and dynamic testing conditions automatically. Second, GNNs +leverage the distributed message passing architecture to learn +local relationships among graph nodes and combinatorial gen- +eralization over graphs [80]. As a result, GNNs can generalize +to large-scale communication networks with varying sizes and +permuted structures (e.g., more users, antennas, BSs, etc.). +2) Good Learning Performance and High Computational +Efficiency: Compared with generic NNs, GNNs are more suit- +able for graph-structured data and distributed systems, which +can exploit the task-specific knowledge for better learning +performance with less training samples [22]. +However, GNNs also suffer from explanation, generalization +and representation limitations. For example, GNNs cannot +compute some important graph properties such as the longest +or shortest cycle, diameter, or certain motifs [99], which are +crucial for the theoretical performance analysis over graph. +GNNs also show a poor learning performance when aggre- +gating messages across a long path, and this situation cannot +be improved by increasing the number of aggregation network +layers in practice [100]. The theoretic understandings of GNNs +are still in an early stage. +V. DEEP REINFORCEMENT LEARNING FOR STOCHASTIC +OPTIMIZATION +The long-term performance optimization in dynamic and +uncertain wireless networks can be cast as a stochastic op- +timization problem, where the network entities need to learn +the optimal policies over time under system uncertainties. DRL +has emerged as an efficient and powerful ML tool in address- +ing the sequential decision-making problems in dynamic and +Fig. 6. Reinforcement learning method in wireless communications. +large-scale networks without explicit models of transmission +environment, which allows the agents to update the decision +policies through interactions with the unknown environment. +In this section, we start with the motivations of DRL. Then +we present the basic concepts of reinforcement learning (RL) +and the categories of DRL techniques, followed by the case +studies of DRL in wireless communication networks to fully +reveal the power of DRL in solving stochastic optimization +problems in dynamic large-scale wireless networks. The pros +and cons are summarized at the end of this section. +A. Motivations of DRL +With the emergence of diversified application scenarios and +the ever-growing density of wireless devices in modern net- +works, such as IoT, unmanned aerial vehicle (UAV), and mas- +sive machine-type communication systems, the performance +optimization in these networks becomes extremely compli- +cated due to the dynamic and uncertain environment. The +general trends towards intelligent, autonomous, self-adaptive, +and decentralized networks have been indispensable, where +the agents learn the optimal decision policies automatically +based on local or minimal exchanged information to maximize +the network performance over time in dynamic and large- +scale modern emerging networks. This kind of problem can +be modeled as a Markov decision process (MDP) and various +techniques, such as dynamic programming [101] and RL can +be employed to solve the MDP problem. However, in stochas- +tic and dynamic environments, it is infeasible to build an +explicit mathematical model to fully capture the characteristics +of the time-varying environments, which renders many model- +based methods impractical. RL is a machine learning technique +that aims at maximizing the accumulated discounted reward of +an MDP with collected experiences in a data-driven manner. +Specifically, through trial-and-error interactions between the +agent and the environment, the RL enables the agent to +establish a general long-term optimal control policy while +keeping track of the real-time environmental dynamics for +sequential optimization problems. +Even though RL can effectively solve the MDP without +explicit state transformation models, the exploration of an +unknown environment consumes lots of time, especially for +a highly dynamic and large-scale network. Motivated by +the strong learning ability of DNN as a universal function +approximator, the combination of RL and DNN, namely + +Observations and Rewards +IndustrialIoT +Intelligent RL agents +Wireless Environment +Resource Allocations18 +DRL, demonstrates great success in complex and dynamic +wireless networks. The DRL enjoys the fast learning ability of +DNN to speedup the learning process of RL and the transfer +ability of DNN to be scalable to large-scale networks. On +the other hand, DRL benefits from the RL techniques to be +adaptive to the dynamic condition and amenable to distributed +implementation. The DRL technique is illustrated in Figure +6. Before discussing the technical details of DRL, we will +first introduce the basic knowledge of RL in the following +subsection. +B. RL and Categories of DRL +MDP can be described with ⟨S, A, P, R, γ⟩, where S and +A denote the state space and action space, respectively, P +denotes the state transition probability, R and γ denote the +reward function and the discount factor over the future reward. +The Markov property of state transition is expressed as +P [s′ = s(t + 1) | s(1), . . . , s(t), a(1), . . . , a(t)] +=P [s′ = s(t + 1) | s(t), a(t)] . +(13) +The intelligent agent determines its next move a(t) ∈ A +according to its current state s(t) ∈ S as well as its learned +policy π, which involves in the state transition of the environ- +ment, i.e., s(t + 1) ∼ P(s′ | s(t), a(t)). At the end of each +step, the agent receives a reward r(t) = R(s(t), a(t)) from +the environment, stores the tuple ⟨s(t), a(t), r(t), s(t + 1)⟩ +and updates its policy according to the corresponding state- +action pair. Note that P and R are generally unknown due +to the randomness and the complexity of the environment, +therefore we shall learn the optimal control policy with model- +free RL. The objective of RL is to find an optimal policy to +maximize the expected accumulated discounted reward, i.e., +maxπ Eπ +��∞ +j=0 γjr(t + j) +� +, where π is the policy denoting +the mapping from a state to an action, r(t) = R(s(t), π(s(t)) +is the instantaneous reward obtained after the action a(t) = +π(s(t)) is performed under state s(t). In particular, a good +policy shall balance the instant reward in the current round and +the potential dividend in the future rounds to achieve long-term +optimality in sequential decision-making. +In each round of the decision process, the intelligent agent +evaluates the value of the state under a policy π, which is +defined as Vπ(s) = Eπ +��∞ +j=0 γjr(t + j) | s ∈ S +� +. The state +value function Vπ(s) represents the discounted reward that +could be achieved at initial state s following the policy π. Sim- +ilarly, the state-action function can be defined as Qπ(s, a) = +Eπ +��∞ +j=0 γjr(t + j) | s ∈ S, a ∈ A +� +, +which +characterizes +the expected discounted reward when starting from initial state +s and action a, then following the policy π thereafter. In par- +ticular, there is a conversion relationship between state value +function Vπ(s) = � +a π(a|s)Qπ(s, a) and state-action func- +tion Qπ(s, a) = � +s′ P(s′|s, a) (R(s, a) + γVπ(s′)), where +π(a|s) denotes the conditional probability of each action +a under given state s following policy π. By substituting +Qπ(s, a) into Vπ(s), we have the Bellman expectation equa- +tion Vπ(s) = � +a π(a|s) (R(s, a) + γ � +s′ P(s′|s, a)Vπ(s′)) . +Given the optimal policy π∗, we have the following Bellman +optimality equation for V∗ and Q∗, respectively, i.e., +Vπ∗(s) = max +a +� +R(s, a) + γ +� +s′ +P(s′|s, a)V (s′) +� +, +(14) +Qπ∗(s, a) = max +a +� +R(s, a) + γ +� +s′ +P(s′|s, a)Vπ∗(s′) +� +, +(15) +where the optimal action in each round shall be found to +achieve the Bellman optimality equation. +When the dimension of state and action space is small, Q- +learning [102] is a widely used algorithm in RL to find the +optimal action sequences by visiting each action-state pair. +However, in complex and large-scale networks, the state and +action space can be very large. To accelerate the learning +process, DNN can be embedded into the RL framework to +approximate computational-expensive quantities with parame- +terized functions. Based on the types of quantities fitted by the +DNN, the DRL techniques can be categorized into value-based +and policy-based algorithms. +1) Value-Based DRL: The value-based DRL can be applied +to solve MDP with discrete state/action space, where the +value functions are approximated using DNN. The widely used +method is deep Q-learning (DQL) [102], which implements +a deep Q-network to fit the value of Q⋆ +π(s, a) in (15). In +particular, DQL accepts the features of states as input and +outputs the fitted action-state values for all actions, where +the action with the largest state-action value is adopted, i.e. +a(t) = arg max +a∈A +Q(s, a). To train the NN, a loss function +defined as the mean square error (MSE) between a target +output, calculated using target network parameters θ′, and +the actual output, characterized by parameters θ, is utilized. +To reduce the oscillations, the target network θ′ is updated +much slower than the actual network θ. With the learned Q +values, the actions chosen at each time slot follow a widely +used ϵ-greedy strategy, where the agent chooses the action +randomly with probability ϵ and the action that maximizes the +current action-state value with probability 1−ϵ to balance the +exploration and exploitation in the training process. +2) Policy-Based DRL: The agents in the policy-based al- +gorithms learn the policy mapping from the current state to +the optimal action or the probability of each action directly +through the DNN technique instead of evaluating the state- +action values, which can be applied to both continuous and +discrete action spaces. In particular, the neural networks are +optimized by minimizing the following loss function, +ρ(θ) = lim +T →∞ +1 +T +T +� +t=1 +rt = +� +s∈S +dθ(s) +� +a∈A +πθ(a|s)R(s, a), (16) +where dθ(s) = limT →∞ P(st = s|s0) denotes the steady-state +probability distribution under policy π, which is parameterized +by θ. To facilitate the model parameter update, actor-critic +framework, as in deep deterministic policy gradient (DDPG) +[103], can be employed. In particular, the critic learns a +parameterized state-action function Qω +π(s, a) by using the + +19 +Bellman equation as in DQL, where a copy of the actual +critic network can be adopted to calculate the target values +with slowly updated weights to improve the learning stability. +On the other hand, the actor learns a parameterized function +πθ(a | s) specifying the current policy through mapping +the states to a specified action, which can be obtained by +maximizing the learned value function of the critic. +TABLE IV +DRL-ENABLED STOCHASTIC OPTIMIZATION IN WIRELESS NETWORKS +Prolem Types +DRL Methods +Application Scenarios +Refs +IP +Value-Based +Intelligent Traffic +[104]–[112] +Discrete-Valued Power Control +[31], [113]–[116] +Device Scheduling +[117]–[119] +Stochastic +MINLP +Policy-Based +Mobile Edge Network Optimization +[120]–[127] +Space-Air-Ground-Integrated Network +[128]–[132] +DCOP +MADRL +Scalable Radio Resource Allocation +[26], [133]–[136] +Recently, DRL-based approaches have attracted attentions +of the wireless community to solve different types of wireless +resource allocation problems, such as integer programming, +mixed-integer linear programming and sequential optimization +problem, in a wide range of applications including but not +limited to multi-access scheduling, power control, beamform- +ing designs, and bandwidth allocation. In general, DRL-based +approaches inherit the advantages of conventional control +theorem with theoretical guarantees for convergence and opti- +mality [102], [103], [137]–[139] in the long-term performance +optimization, and the advantages of data-driven DL with high +computational efficiency and excellent learning performance in +complex, dynamic, and large-scale wireless networks. In the +following subsections, we shall review several representative +cases of the application of DRL in wireless resource allocation +according to the types of optimization problems. The covered +cases are summarized in Table IV based on the optimization +problem types, application scenarios, DRL methods, MDP +elements, etc. +C. Application 1: Stochastic Integer Programming Problems +The stochastic integer programming (IP) problems involving +discrete variables in dynamic scenarios have been quite com- +mon in wireless resource allocation, which have the following +general forms: +maximize +x1,...,xT +T +� +t=1 +ctft(xt; yt), +(17a) +subject to gt(xt) ≤ bt, +(17b) +yt ∼ P(y | yt−1, xt−1), +(17c) +xt ∈ Zn, +(17d) +where the objective function ft(·) can be the utility function +of wireless networks, gt(·) can be the performance constraint +functions, and the discrete-valued variables xt can be resource +allocation variables, such as device scheduling, yt stands for +the dynamic state of the environment with Markov evolutions. +As a non-convex problem with NP-hardness, the COAs for +solving IPs suffer from unintended performance degradation, +high computational complexity and high requirements for +datasets in the actual implementation. DL-based algorithms, +such as LBB in Section III, can be employed to solve the IP +problem. However, LBB and DRL focus on different applica- +tion scenarios. For example, LBB aims at solving one-shot IP +problems, which cannot handle long-term constraints. Instead, +DRL is specifically used to cope with long-term constrained +stochastic optimization problems by adaptively interacting +with the environment to learn the long-term optimal policies. +Besides, LBB requires explicit models to achieve a good +learning performance. Instead, DRL approaches are model- +free, which can successfully perform for unknown dynamics +through autonomous exploration. In conjunction with the good +robustness, DRL provides great potentials of scalability in han- +dling high-dimensional problems and flexibility in embedding +diverse task-specific features for decision making compared to +the LBB-based algorithms. +Therefore, the value-based DRL algorithms can be effec- +tively adopted to solve the challenging stochastic IP problem +by transforming the original IP problem into an MDP with +discrete action spaces. Specifically, the objective function or +the metric strongly correlated with the optimization objective +can be regarded as a reward function in MDP. The state should +include the features relevant to the decision policies while the +action can be the discrete optimization variables. Then the +value-based DRL algorithms can be employed to search for the +optimal actions in discrete action space. We introduce several +applications of value-based DRL in solving the IP problem. +1) Intelligent Traffic: +With a growing increase of au- +tonomous vehicles and intelligent roadside units, establishing +an intelligent transportation system (ITS) is becoming im- +portant to improve transportation efficiency in the pursuit of +smart cities, which has been attracting considerable attentions +from researchers, and plenty of DRL-based methods have +been proposed thereafter. Some of the hottest applications for +DRL-assisted ITS are adaptive traffic signal control (ATSC), +autonomous driving, and on-board energy management, the +details of which, including DRL algorithms and MDP model- +ing, are summarized in the table V. +For example, Choe et al. [104] developed a deep Q network +(DQN)-based approach to maximize transportation efficiency +by minimizing local accumulated waiting time according to +handcraft features of local traffic. Further, Jin et al. [105] +took traffic flow into account to minimize the waiting rate of +vehicles for higher transportation efficiency. Wei et al. [106] +integrated multi-source factors into the design of the reward +function to improve the overall performance in reducing the +network latency. Beyond optimizations over traffic signals, +the need for autonomous driving such as lane changing and +auto-breaking rises with the rapid development of intelligent +vehicles and the Internet of Vehicles (IoV). For instance, Hoel +et al. [108] jointly optimized lane change policy and speed +change policy of autonomous driving vehicles according to +their locations and speeds to minimize their travel time. Ye +et al. [110] further extended the state from abstract features +to real-time traffic flow scenes and enabled autonomous ve- +hicles to optimize car following and land-changing policies +for higher travel efficiency. The joint control of roadside + +20 +TABLE V +DRL-ASSISTED INTELLIGENT TRANSPORTATION SYSTEMS +Application Scenarios +DRL Methods +Refs +State Space +Action Space +Reward Function +ATSC +DQN +[104] +Handcraft features of local traffic +Traffic signal +Accumulated waiting time +LSTM+DQN +[105] +Number of vehicles traffic flow +Duration of traffic signal +Waiting rate of vehicles +CNN+DQN +[106] +Local transportation state +Traffic signal +Mixed function of delay, +queue length and waiting time +CNN+DQN +[107] +Local transportation state, +number of intelligent vehicles +Traffic signal, +vehicle detouring +Mixed function of waiting +time and system influence +Autonomous Driving +DQN +[108] +Current speed, land index +and road information +Lane change, speed change +Self-defined indicator +DQN +[109] +Current pedestrain status +Pedestrain detection +Self-defined indicator +DQN +[110] +Real-time traffic flow scenes +Car following, lane change +Self-defined indicator +On-Board Management +DQN +[111] +State of charge, power demand +and generator speed +Throttle of the engine +Mixed function of state of charge +and instantaneous fuel consumption rate +DQN +[112] +State of charge +Operation rate parameter +of energy source +Deviation between +SOC of all units +units and vehicles was firstly investigated to improve traffic +efficiency in [107] by jointly controlling the traffic signal of an +intelligent traffic light and the detouring behavior of intelligent +vehicles connected to the IoV according to the real-time traffic +flow information. Through the unique design of the MDP +modeling, the agent learned to maximize the accumulated +reduced waiting time while preventing the side roads from high +congestion as well, thus maximizing the traffic efficiency of +the whole traffic network. The numerical simulations in [107] +show that the DQN-based algorithm proposed therein can +achieve better performance than that of conventional strategies +as well as the DRL methods that only control the traffic signals +with affordable computational consumption for highly time- +sensitive real-time traffic control scenarios. +2) Discrete-Valued Power Control: To maximize network +utility, the discrete-valued power control as a classic IP prob- +lem can be effectively solved by DQN as proposed in [31], +[113]–[116]. For example, Xu et al. [113] firstly adopted DQN +to dynamically allocate discrete-valued transmit powers of BSs +in C-RANs, which manifests superior performance in terms +of energy efficiency and the adaptability to highly dynamic +environments. Ye et al. [31] further extended the DQN to +address the joint allocation of sub-band and transmit power +in V2V networks, where the proposed DQN-based scheduling +strategy can be executed in distributed systems to meet the +stringent latency requirements. The authors in [114] applied +the DQN in the mobile edge computing scenarios to jointly +optimize the transmit power and device scheduling, which +shows significant enhancement in aspects of network delay and +resource consumption. Chu et al. [115] considered an energy- +harvesting assisted communication system without any prior +knowledge assumed of the energy dynamics and developed a +two-stage strategy to solve the joint optimization problem of +battery prediction and sum-rate maximization, where a long +short-term memory (LSTM)-based network is used to improve +the prediction accuracy of battery status in the first stage and +the DRL is employed to optimize real-time transmit power +and access policy for sum-rate maximization. +3) Device Scheduling: In wireless networks, as each device +can only be in the state of scheduled or non-scheduled, the +device scheduling optimization turns out to be an IP problem +and DQN can be effectively used to address it. For example, +Lee et al. [140] proposed a circumstance-independent DQN- +based scheduler to maximize the network utility under various +conditions and QoS constraints. The unconstrained Lagrangian +function was adopted as a reward function to cope with various +constraints. The DQN-based scheduler was also successfully +used in FL systems and mobile edge computing systems +[117]–[119] to schedule participating devices. For example, +Zhou et al. [117] adopted DQN to schedule training batches +to optimize the quality of query response for a cloud-enabled +DNN inference system. Young et al. [118] further considered +the cloud-edge hybrid inference system and proposed Au- +toScale, an automatic DQN-based scheduler, to dynamically +schedule the inference execution target for improving the pre- +diction accuracy. To tackle the problem of non-i.i.d distribution +of heterogeneous data in the inference of FL system, Young et +al. [141] proposed AutoFL, a heterogeneity-aware DQN-based +scheduler, to schedule the FL participants for overall energy +efficiency enhancement. +D. Application 2: Stochastic Mixed Integer Nonlinear Prob- +lems +Stochastic MINLPs involving both continuous-valued and +discrete-valued variables in dynamic environments have been +widely encountered in the wireless communication systems, + +21 +which have the following general forms: +maximize +{xi}T +1 ,{zi}T +1 +T +� +t=1 +ctft(xt, zt; yt), +(18a) +subject to g(xt) ≤ bt, +(18b) +h(zt) ≤ dt, +(18c) +xt ∈ Zn, +(18d) +zt ∈ Rn, ∀t, +(18e) +yt ∼ P(y | yt−1, xt−1, bmzt−1), +(18f) +where ft(·) denotes the utility function of wireless net- +works, gt(·) and ht(·) can be the performance/resource con- +straint functions, such as the QoS constraints and the max- +imum/average power constraints, xt and zt denote discrete- +valued and continuous-valued network resources, respectively, +yt denotes the dynamic states of the environment with Marko- +vian properties. Obviously, it is computationally expensive and +analytically intractable to solve such an NP-hard stochastic +non-convex problem with COAs. On the other hand, the value- +based DRL algorithms can only deal with the discrete action +space to evaluate the state-action values for each possible +action. Hence, bypassing the evaluation of the state-action +values and directly choosing the action under the currently +learned policy, the policy-based methods can be effectively +employed to solve the challenging stochastic MINLPs. To +model the stochastic MINLP as an MDP, the state can be +defined as the features related to the decision-making, the +discrete-valued and continuous-valued variables yet to be +optimized can be considered as actions of MDP, which are +sampled from the learned policy mapping function. The reward +can be defined as the objective function or the metric strongly +correlated with the optimization objective. Then, the policy- +based methods can be employed to learn the optimal policy +function directly in the continuous action space. Note that +the discrete-valued variables in the stochastic MINLPs can +be obtained by rounding the optimized continuous-valued +variables in the testing. We introduce several applications of +the policy-based DRL in solving the stochastic MINLPs. +1) Mobile Edge Network Optimization: Applications mi- +grated from cloud to edge have been a prevalent trend in +the era of 5G and beyond. By enabling a large number of +access points and integrating the widely distributed comput- +ing, caching, and communication resources, the mobile edge +network (MEN) can significantly reduce the communication +latency and improve the network performance by jointly op- +timizing the heterogeneous resources at the edge. DRL-based +methods [120]–[127] have shown promising performance in +the multi-resource joint optimization in MEN attributed to +their self-adaptation to dynamic environment, the general- +ization power for high-dimensional problem and the real- +time inference in practice. For instance, Ke et al. [120] first +formulated a joint optimization problem for task offloading, +bandwidth allocation, and energy sensing in IoT networks, and +then proposed a DDPG-based joint design scheme to minimize +the transmission delay and energy consumption, which can +well handle the time-varying channel and dynamic bandwidth. +Further, the authors in [121] developed the Wolpertinger +DDPG to eliminate the possible performance degradation of +DDPG induced by rounding discrete variables in a similar +context. Li et al. [123] proposed an LSTM-assisted DRL algo- +rithm to allocate resources across slices under varying service +demands, where the LSTM mechanism was adopted for higher +tracking accuracy of user mobility and the system utility. Xu et +al. [124] further considered the joint optimization of channel +allocation and continuous energy harvesting time while taking +energy consumption and queue length into account, where +the proposed DRL algorithm can achieve higher throughput +with stringent performance constraints. Recently, the block- +chain application, as a computational intensive scenario, has +been widely combined with the MEN to achieve higher +mining efficiency. For example, Du et al. [127] developed +an asynchronous DRL-based algorithm to adaptively allocate +the channel resources and establish the pricing policy by +maximizing the rational profit among all miners while taking +the wireless fading channel into account. Specifically, the +advantageous actor-critic algorithm (A3C) was adopted to +avoid the overestimation or underestimation of the chosen +action, which can achieve better performance than that of the +DDPG as numerically verified. +2) Space-Air-Ground-Integrated +Network: +Space-air- +ground +integrated +(SAGI) +network +provides +ubiquitous +communication and computing services from cloud to edge. +Due to the physical distance among layers (e.g., space, air +and ground), it is essential to develop a joint optimization +framework to coordinate the resources across different layers +and timescales to satisfy stringent QoS constraints, where +various DRL-based approaches have been proposed [128]– +[132]. For instance, Liao et al. [128] developed an actor-critic- +based DRL algorithm to jointly optimize the task offloading +and computational resource allocation by minimizing the +cross-layer queuing delay, where a queuing-aware agent was +developed to balance the instantaneous queuing boosting and +long-term latency constraints. Further, they proposed a block- +chain and semi-distributed learning-based DRL algorithm by +minimizing latency while guaranteeing long-term security +requirements. Wang et al. [131] took the multi-dimensional +resource heterogeneity and network dynamics into account +and proposed a soft actor-critic-based DRL algorithm by +minimizing the energy consumption and the queuing latency +of the offloading tasks. +E. Application 3: Distributed Constraint Optimization Prob- +lems +Distributed constraint optimization problems in multi-agent +systems (MASs) involving multiple nodes are challenging but +very common in wireless systems, which have the following +general forms: +maximize +{xi,t}N +1 +T +� +t=1 +ctft({xi,t}N +1 ; {yi,t}N +1 ), +(19a) +subject to gi,t(xi,t) ≤ bi,t, ∀i, +(19b) +{yi,t+1}N +1 ∼ P({yi,t+1}N +1 |{xi,t}N +1 , {yi,t}N +1 ), +(19c) + +22 +where {xi,t}N +1 and {yi,t}N +1 denote the set of variables and +system parameters of the wireless systems, respectively, ft(·) +denotes the objective function of the MAS, gi,t(·) denotes the +performance constraint function of each variable. Note that +the evolution of system states {yi,t+1}N +1 can be modeled as +a Markov process. Specifically, the agent can be the BS or +edge device in wireless networks, the system parameters can +be wireless fading channels or edge resource status, while the +set of variables can be the corresponding resource allocation +policies. It is worth noting that the system parameters of MAS +at the current time slot depend on the system parameters and +action variables of MAS at the last time slot, therefore the +interactions among agents determine the system performance. +Due to the coupling among the agents and the non-convexity +of the objective function, it is challenging to achieve the +desired performance within acceptable computational delay +using COAs. Fortunately, built on the MAS, the multi-agent +deep RL (MADRL) shows unmatched performance in deal- +ing with the multi-agent co-learning problems, which allows +multiple agents to learn multiple individual policies and one +global policy collaboratively based on the interactions among +agents and environment. +MADRL enables each agent to develop its own decision- +making policy which can be executed decentralized, thereby +improving the scalability of network significantly. In particular, +the agents in MAS can build either competitive relationships or +cooperative relationships. As a result, the objective of MADRL +algorithms can be divided into three categories: maximizing +the global reward by coordinating all agents, maximizing the +reward of each agent by constructing an equilibrium among +agents, or constructing a competitive equilibrium among dif- +ferent groups of cooperative agents. In wireless networks, it +is more general that the MAS operates in cooperative mode, +which is the focus of the following discussion. Note that the +learned policy in MADRL can be unstable due to the fact that +the state of each agent depends not only on its own states and +actions but also on the actions and states of other agents. Such +mutual coupling depends on the communication range of the +network. In some scenarios, the agent can only communicate +with the surrounding agents, thereby leading to a partially +observable MDP. Therefore, besides the consistent trial-and- +error explorations as in a single-agent system, an efficient +communication mechanism among agents is also important to +achieve the long-term optimal policy in dynamic MAS. In the +following, we illustrate the application examples of MADRL +for cooperative MAS design. +To overcome the issue of huge CSI overhead for centralized +RRM design in large-scale wireless networks, Yasir et al. [133] +firstly adopted the MADRL technique to dynamically allocate +transmit power at each BS through mutual coordination for +sum-rate maximization of wireless networks. By modeling +each BS as an intelligent agent, each agent determines its +transmit power individually according to its local and neigh- +boring channel information, achieving the same performance +as that of COA with full CSI. Further, the authors in [134] ex- +tended it to the continuous-valued power control, where three +different RL algorithms were proposed to feature a promising +performance of MADRL. However, the above methods require +additional CSI exchange between the neighboring BSs, which +can downgrade the spectrum efficiency. To address this, DEC- +MAPC proposed in [26] achieved fully decentralized power +control only using local CSI while maximizing the sum- +rate of the network. Specifically, to achieve fully distributed +implementation, DEC-MAPC was proposed to decompose the +global state-action value into a monotonic increasing non- +linear function of all local state-action values. As a result, each +BS only needs to determine its transmit power by maximizing +the local state-action value, then the network utility can be +maximized. To address the continuous power control while +cooperation, an actor-critic framework with a double critic +network was adopted in DEC-MAPC, leading to more accurate +estimation of state-action values and higher network utility. +F. Advantages and Disadvantages +The advantages of DRL methods are summarized as fol- +lows. +1) High Adaptability: The training data for policy updates +are collected from historical interactions with the environment. +As a result, the training of DRL policy can keep track of +real-time wireless dynamics. Compared with generic NNs, +the training data in DRL methods are label-free, making it +unrestricted to the traditional algorithms and allowing more +degrees of freedom to improve the learning performance. +Additionally, the DRL is model-free, thus enabling the ex- +ploration of unknown and complicated environments. +2) Suitable for Long-Term Optimization: The DRL-based +methods can learn a long-term optimal policy that takes the +potential reward in the future and long-term system constraints +into account rather than just considering instantaneous system +reward and one-shot constraint. +However, the DRL methods also have some limitations. +First, it is hard to train a global optimal policy because +the action space is generally too large to be exhaustively +traversed. Besides, to obtain a good policy, numerous training +experiences shall be stored, which however is challenging due +to the limited storage capacity of local devices. Second, while +the DRL methods can be deployed with a relatively simple +network structure, there are lots of hyperparameters involved +in the training and execution, whose values are chosen man- +ually by costly trials. The fine-tuning of hyperparameters is +labor-intensive and time-consuming, and improperly chosen +values can significantly degrade the performance of DRL. +Third, DRL is developed based on the MDP modeling. For +some complicated applications, the definition of MDP can be +challenging. For example, it can be hard to acquire some state +information efficiently in practice. It also can be quite difficult +to define a highly featured state space and choose a good +reward function in some scenarios. +VI. END-TO-END LEARNING FOR SEMANTIC +OPTIMIZATION +The joint optimization of physical layer transmitter and +receiver in wireless communication systems is extremely chal- +lenging due to infinitely large searching space for modular +functions, the complex interactions among modules and the + +23 +highly non-convexity of optimized global performance in +terms of communication rate, transmission reliability, resource +consumption, transmission delays, etc., in conventional block- +based communication systems. DL-enabled end-to-end learn- +ing has been studied to merge the transmission blocks and +jointly design the transmitter and receiver in a data-driven +manner. To boost the system capacity and improve transmis- +sion efficiency and reliability, semantic communication has +been envisioned as a new transmission paradigm by delivering +the semantic meaning rather than bit stream of transmitted +messages. In this section, we identify the motivation of se- +mantic communication and review the classic framework of a +semantic system, followed by the DL-enabled semantic system +design and the overviews of its applications for different types +of transmission tasks. +A. Motivation and Challenges +In conventional wireless communication systems, message +compression and message error correction are achieved by +source coding and channel coding, respectively, aiming for +high transmission efficiency and reliability. In view of this, +conventional communication networks have been designed to +optimize data-oriented performance metrics such as communi- +cation data rate, spectrum/energy efficiency, symbol or bit level +accuracy and latency, while ignoring the semantic meaning be- +hind the transmitted messages [142]. For instance, the bit-error +rate (BER) or symbol-error rate (SER) is usually taken as per- +formance metric in communication systems to measure the bit +or symbol level accuracy and effectiveness of transmit symbols +[143]. With the communication system capacity approaching +Shannon limit and the booming development of ML, it is an +increasing belief in the community that classical Shannon’s +information theory needs to be upgraded for the next evolution +of wireless communication networks. A variety of services +emerged in 6G wireless systems are service/content-centric, +which means they are more concerned about the semantic- +related information instead of physical data symbols, which +sparks a paradigm shift from the symbol transmission to the +semantic meaning transmission in communication systems. +By delivering the semantic meaning of the message relevant +to the transmission task directly rather than its exact copy, +semantic communication is expected to break through the +classic design paradigms of Shannon which is targeting at the +accurate transference of source signals to the destination re- +ceiver. Specifically, the semantic communication can increase +the system capacity by identifying and extracting the seman- +tic meaning and eliminating the irrelevant information from +transmitted messages to realize compression. The semantic +communication can guarantee the reliability of transmission +through exact semantic meaning recovery/interpretation at the +receiver. However, to reap the benefit of semantic communica- +tion, semantic-aware optimization for wireless techniques and +network structures should be activated to accommodate to the +new requirements in semantic communication systems. +There are several challenges for semantic-aware optimiza- +tion, which are summarized as follows. +1) New Metric Design: To enable semantic-aware opti- +mization, it is indispensable to design metrics for both exact +semantic meaning extraction and accurate semantic meaning +transmission. The first metric measures the meaning behind +the transmitted symbols mathematically. The second metric +characterizes the semantic errors between recovered and trans- +mitted semantic meaning to guarantee successful semantic +meaning inference at receiver. +2) Joint Design of Transmitter and Receiver: Instead of the +separate design, the coordinated design of transmitter (source) +and receiver (destination) can achieve high system capacity +and reliable transmission simultaneously by exploiting seman- +tic side information. Such joint design is expected to compress +the transmitted signals maximally while reserving the semantic +meaning at transmitter and recover the semantic meaning at +receiver to combat the channel fading and semantic noise. +3) Mathematical Theories: It is still an ongoing research +direction to develop efficient and elegant mathematical theories +to evaluate the overall performance of a semantic communi- +cation system. +Motivated by recent ML tools, DL-enabled semantic com- +munication system has received considerable attention, where +the transmitter and receiver implemented by DNNs can be +jointly learned targeting at good overall performance [27]. In +the following, we will introduce the architecture of a classic +semantic communication system, after which we will review +the existing techniques to address the challenges for semantic- +aware optimization. +B. Architectures of Semantic Communication Systems +Semantic communication system usually contains three +components including semantic transmitter and receiver, +knowledge base, as well as semantic noise and error [27], +[144]. +1) Semantic Transmitter and Receiver: The desired seman- +tic transmitter and receiver are expected to be agents with in- +telligence (e.g. humans and smart devices), aiming to perform +the functions of semantic communication terminals, e.g., exe- +cuting highly intelligent compression/extraction/interpretation +algorithms, sensing the environment to obtain high-level data +and updating the knowledge bases, etc. Semantic encoders are +typically deployed in the semantic transmitter, which are able +to extract the meaning of the source (e.g. text, speech and +image messages) and encode these features into symbols (bits) +for transmission. The receiver with semantic decoder should +be able to recover the compressed features sent by semantic +transmitter as well as perform various intelligent tasks based +on the inferred semantic information (e.g., automatic speech +recognition when transmitting speech signals). +2) Knowledge Base: The semantic transmitter and receiver +contain certain knowledge bases (KBs) to capture the meaning +of the knowledge entities and their complex relationships. The +KBs at transmitter and receiver are expected to be constantly +updated by self-learning and both contain the knowledge ele- +ments involved in the current communication, which constitute +the core of a semantic communication system [144]. The KBs +are the knowledge models that the transmitter and receiver + +24 +Semantic +Encoder +Source Message +Semantic +Information +Corrupted +Information +Noise +Wireless Channel +Semantic +Receiver +Semantic +Transmitter +Semantic +Decoder +Target Message +Transmitter KB +Receiver KB +Fig. 7. Semantic communication system. +observed previously and can be shared through communi- +cations. With the transmitter KB, the semantic transmitter +extracts the features of the transmitting messages and then +the semantic receiver can interpret and infer the meanings +of them based on the receiver KB. Based on different types +of source messages such as text, image or audio, the KBs +could be different for various applications. In addition, the +KBs at the semantic transmitter and receiver may also be +different due to the different abilities for understanding (e.g., +the transmitter is Chinese language system while the receiver +only uses English). +3) Semantic Noise and Error: Semantic noise that inter- +feres with the interpretation of the semantic information dur- +ing encoding, data transportation, and decoding processes is +introduced as one of the semantic communication components +[27]. Semantic communication system contains two kinds of +noises, namely channel noise and semantic noise. In addition +to physical channel noise such as additive white Gaussian +noise (AWGN), fading channels, and multi-path effect, which +are introduced by channel impairments and can cause the +signal attenuation and distortion, the semantic noise is defined +as a type of disturbance in message interpretation processes +due to the ambiguity in words, sentences or symbols used +in the message transmission [34]. Semantic noise can lead to +semantic errors in the receiver and misunderstanding of the +received message. The semantic noise may occur when KBs +between the semantic transmitter and receiver are mismatched. +These two kinds of noises will eventually lead to semantic +understanding errors at the receiver and it is hard to distinguish +which factor causes the errors. +C. Semantic Meaning Extraction and Interpretation +In wireless networks, the core issue of semantic communi- +cation is how to extract the semantic information and then +perfectly recover it after data transportation, which can be +expressed mathematically under information-theoretic view as +follows. For the source signal x and its corresponding received +signal y, which belong to a pair of random variables (X, Y ), +the probability model involved in semantic communication is +represented as a Markov chain Y ↔ X ↔ Z ↔ ˆZ. The +random variable Z and ˆZ represent the semantic information +after semantic encoder and the received semantic information +at semantic decoder, respectively. Assuming a wireless fading +channel from the semantic encoder to the semantic decoder, +then we have ˆZ = HZ + W , where the random variables +H and W represent the channel model and the channel noise +model, respectively. The semantic encoder Cθ(·) can be rep- +resented by parameter θ, while the semantic decoder Cφ(·) is +parameterized by φ at the receiver side. Then the communica- +tion probabilistic model satisfies p(y|x) = pφ(y|ˆz) · pθ(ˆz|x), +where pθ(ˆz|x) = pc(ˆz|z) · pθ(z|x) denotes the transmitter +and channel probabilistic encoder and pθ, pc, pφ denote the +transition probabilities of the semantic encoder, the wireless +channel, and the semantic decoder, respectively. The Markov +chain of semantic communication thus reduces to Y ↔ X ↔ +ˆZ. Since the goal of semantic communication is to maximize +expected faithfulness in representing observed messages (that +is to say to minimize the semantic errors) and minimize the +amount of data to be transmitted [9]. Then the general form +of optimization problem of semantic communication can be +expressed as +minimize +P ˆ +Z|X +� +f(Y , X, ˆZ), g(X, ˆZ) +� +(20a) +subject to (X, Y ) ∈ K, +(20b) +where P ˆ +Z|X denotes a statistical mapping of source infor- +mation to received semantic information, function f(·, ·, ·) +measures the semantic error, function g(·, ·) characterizes the +number of symbols to be transmitted in semantic communica- +tion, and K denotes the background knowledge. +The key of semantic communication is to define the +semantic-aware optimization metrics g to quantify the se- +mantic information and f to measure the semantic error in +(20a) and jointly design the semantic encoder and decoder +to obtain the optimal mapping P ˆ +Z|X in a task-oriented sense. +Inspired by the powerful representation ability of DNN and its +successful employment in natural language processing (NLP), +DL-based end-to-end semantic system has gain much traction, +in which the semantic encoder and decoder are implemented +by DNNs to represent and interpret the semantic meaning, +and are jointly trained in an end-to-end manner to achieve +the global optimality. In the following, we will detail an +information-theoretic framework for semantic communication +system design by theoretically characterizing the trade-off +between compression of semantic feature extraction and dis- +tortion of semantic meaning transfer. +An Information Bottleneck Optimal Semantic System: The +IB framework was proposed in [145], [146] as a principle +approach to characterize the trade-off between information +compression and target signal reconstruction. Therefore, IB +can be used to provide theoretical guidance for the semantic +system design. +From the perspective of reliable transmission, as long as +the encoding information entropy remains unchanged, that is, +when I( ˆZ; Y ) = I(X; Y ), the semantic communication can +recover the target information Y completely and losslessly +through the semantic decoder theoretically, where the mutual +information I(X; Y ) = � +y∈Y +� +x∈X p(x, y) log( p(x,y) +p(x)p(y)) + +25 +obtained from the joint probability distribution p(x, y) and the +marginal probability distribution p(x), p(y) is a measure of the +mutual dependence between two random variables. However, +the loss of information is inevitable in the practice due to the +signal compression and system noise, therefore it is natural to +maximize I( ˆZ; Y ) while restricting the information flow from +source signal X to compressed feature ˆZ in semantic commu- +nication. As a result, the IB-based optimization problem can +be formulated as: +maximize +P ˆ +Z|X +I( ˆZ; Y ) +(21a) +subject to I( ˆZ; X) ≤ α. +(21b) +Specifically, given samples of P ˆ +Z|X, the objective function +of the IB optimization problem is to maximize the mutual +information between received semantic information and target +signal to minimize the information loss of semantic interpre- +tation for reliable transmission, while the constraint keeps the +mutual information between the source signal and the seman- +tic information within a certain range to guarantee a target +compression ratio of semantic extraction for improved system +efficiency. By solving (21), we can learn the parameterized +optimal semantic encoder Cθ(·). +To solve the above IB optimization problem, a Lagrangian +operator β ≥ 0 can be introduced to maximize its Lagrangian +dual equation LIB(θ) = I( ˆZ; Y )−βI( ˆZ; X). The Lagrangian +operator β controls the trade-off between the compression +ratio of the received semantic information ˆZ with respect to +the source information X and the amount of semantic infor- +mation transferred from the transmitter to the receiver. When +β = 0, the objective LIB aims at minimal distortion (maximal +semantic information transfer), whereas for β → ∞, data rate +is minimized (compression ratio is maximized). To overcome +the intractability of mutual information in IB optimization, the +authors of [147] constructed a lower bound for LIB(θ) using +some variational distribution qψ(ˆz), which can be easily opti- +mized. Specifically, exploiting elementary properties of mutual +information, entropy and Kullback–Leibler divergence (KLD), +the LIB(θ) is lower bounded by Epθ(y,ˆz)[log pφ(y|ˆz)] − +βEp(x)[DKL(pθ(ˆz|x)||qψ(ˆz))]. Using the re-parametrization +trick with conditions that hold for variational auto-encoder +(VAE) [148], this lower bound enables the optimization of +the parameters of semantic encoder θ, decoder φ and the +variational parameters ψ via gradient-based methods. By +parameterizing the semantic encoder and decoder as NNs, +the IB optimization problem can be effectively solved by ML +techniques. +D. Applications of Semantic Communications +In Table VI, we summarize the different DL-enabled se- +mantic systems based on the different transmission tasks (text +[34], [149], [150], image [151], [152], speech signals [153], +[154] and general signals [155], [156]), from the perspective +of performance metrics, semantic quantity module, semantic +error module, loss function, KBs as well as the adaptation to +dynamic environment for task-specific applications. +1) Text Signals: A DL-based semantic communication sys- +tem, namely DeepSC, was proposed in [34] for text trans- +mission, which was the first work clarifying the concept of +semantic information and semantic error at the sentence level. +A new metric, namely sentence similarity, was proposed to +reflect the learning performance of semantic system for text +transmission. To jointly train the deep encoder and decoder, +cross-entropy and lower bound of mutual information (LBMI) +constitute the loss function, where the first term measures the +semantic errors between transmitted message and recovered +message and the second term measures the system capacity. +By minimizing the loss function, the DeepSC can be trained to +maximize the system capacity while minimizing the semantic +errors. A transformer-based DNN in DeepSC further makes it +applicable to varying communication conditions. Considering +the capacity-limited devices in a more practical scenario, au- +thors in [149] proposed a distributed semantic communication +system for IoT networks, called L-DeepSC, where bilingual +evaluation understudy (BLEU) score and cross-entropy are +adopted as performance metric and loss function, respectively. +To reduce the model sharing cost on IoT devices, the semantic +models are compressed through network sparsification and +quantization. A refined CSI estimation scheme based on deep +denoising network was proposed to eliminate the impact of +fading channels on the semantic model training. +2) Image Signals: When aiming at wireless image trans- +mission, a joint source-channel coding (JSCC) was proposed +by [151], which can be regarded as an early semantic com- +munication system. In JSCC, two CNNs were considered as +autoencoder for image feature extraction at transmitter and +autodecoder for image reconstruction at receiver, respectively, +where a non-trainable layer in the middle represents the noisy +communication channel. Peak signal-to-noise ratio (PSNR) +metric was used to measure the learning performance of +semantic system for image transmission and the MSE loss +function was used to minimize the average distortion between +the original input images and its reconstruction. To exploit +the feedback channel information, Kurka et al. [152] further +extended deep JSCC to DeepJSCC-f by deploying layered +autoencoders, where the transmission of each image signal +is divided into multiple layers. +3) Speech Signals: Other series of works focused on learn- +ing semantic information directly from the raw speech signals. +Weng et al. [153] firstly proposed DL-enabled semantic com- +munication system called DeepSC-S for speech signals, where +attention-based SE-ResNets semantic encoder and decoder +were proposed to identify the essential speech information +and recover signals. The MSE was used as loss function for +training DeepSC-S, and the signal-to-distortion ration (SDR) +and the perceptual evaluation of speech distortion (PESQ) +score were adopted to measure the performance. Besides, a +novel semantic-oriented speech to text transmission system +was considered by [154], where an attention-based network +is utilized to identify the semantic representation of the input +speech signals and a semantic decoder implemented using +MLPs is employed to transform the received speech features +to the text form for speech recognition. + +26 +TABLE VI +DL-ENABLED SEMANTIC SYSTEMS FOR DIFFERENT TRANSMISSION SIGNALS +Transmission +Signals +Refs +Performance Metrics +Semantic Quantity +Module (SQM) +Semantic Error +Module (SEM) +Loss Function +KBs +Methods for Dynamic +Environment +Text +[34] +Sentence similarity +Transformer +Transformer +Cross-entropy - λ LBMI +Datasets of words +Transfer learning +[149] +BLEU score +MLPs +MLPs +Cross-entropy +Datasets of words +\ +[150] +Edge-based similarity, +word2vec, hybrid-based similarity, +METEOR and BLEU score +Part-of-speech +strategy +Context-based +strategy +Log-softmax +Datasets of +words and speech +Context-based +dynamic programming +algorithm +Image +[151], [152] +PSNR metric +Encoder CNN +Decoder CNN +MSE +Datasets of images +\ +Speech +[153] +SDR and PESQ score +SE-ResNet module +Multiple SE-ResNet +modules +MSE +Datasets of speeches +\ +[154] +WER and semantic +similarity score +Attention-based +NNs +MLPs +Cross-entropy +Datasets of +words and speeches +\ +General +[155] +Recall@1 for image retrieval, +BLEU score for machine translation, +answer accuracy for VQA +Transformer +Transformer +Cross-entropy and MSE +Different dataset +\ +[156] +PSNR and text accuracy +Dob +Dpr +ESD +Library data at receiver +Data adaptation +4) General Signals: In addition to the semantic communi- +cation systems for specific signals, general signal transmission +was considered in [155], [156]. A transformer-based semantic +communication system was proposed in [155] for transmitting +multimodal data considering three different tasks, including +image retrieval, machine translation and visual question an- +swering. However, the training algorithms, such as the loss +functions, for different tasks were designed separately. To +address this, Zhang et al. [156] firstly designed a semantic- +distortion-based universal loss function adapted to general sig- +nals, which consists a hyper-parameter-based linear combina- +tion of distortion measure functions for observable information +Dob and for pragmatic output Dpr. These functions can be +designed to be the KLD, cross entropy, MSE, etc., to adapt to +the different transmission tasks. +5) Adaptation to Dynamic Environment: When the com- +munication environment is dynamic or the transmission task +is changed, it will pose great challenges for the design of DL- +enabled semantic communication systems. First, it requires +different KBs to extract and interpret the transmitted messages +as the varying of communication environment or transmission +tasks. For example, if the transmitted message is unseen at +the transmitter and receiver, the KBs need to be updated +and expanded based on the empirical semantic information, +leading to formidable computational costs for the training of +semantic encoder and decoder. The KBs matching the dynamic +transmission conditions can be learned from the perceived +environments/empirical information and can be shared be- +tween transmitter and receiver via communication to minimize +the semantic inference errors, which however are complex +and long-term processes. Existing semantic communication +designs [149], [151], [152] assumed the shared and fixed +KBs, leading to limited scalability and poor generalizability +in dynamic environments. Second, for the newly updated KBs +and the dynamic changing environment, the semantic coding +strategies should quickly adapt to the new environment and +KBs with minimal training cost. Transfer learning [34], [153] +was adopted to accelerate the DL model training in dynamic +environment by synergizing the past learning experiences to +assist the new problem solving. However, the re-training of +NNs still requires extra communication and computational +cost. A receiver-leading dynamic semantic communication +system with non-shareable KBs at receiver was proposed +in [156], where an individual data adaptation network was +configured at the semantic transmitter to tackle the dynamic +data transmission environment leaving the semantic encoder +adaptive to dynamic environment without retraining. +E. Advantages and Disadvantages +Semantic communications are expected to improve the +communication efficiency and reliability, enhance the quality +of experience for human-oriented services as well as support +a more robust and upgrade/evolution-friendly communication +systems [27]. However, both theoretical and practical imple- +mentation of semantic communication are in an early stage, +which will spark an explosion of research interests in both +academia and industry. +VII. FEDERATED LEARNING FOR DISTRIBUTED +OPTIMIZATION +When optimizing a large-scale model with training data +scattered across massive number of edge devices, distributed +ML has emerged as a key enabling technology to reduce the +communication cost and preserve data privacy in resource +limited wireless networks. FL [157], [158] has been proposed +as a prominent distributed ML scheme to effectively solve the +model optimization problem in ML over large-scale wireless +networks, where each edge device participates in the train- +ing process by exchanging model parameters with data kept +locally. In this section, we firstly review the FL framework, +followed by its applications in wireless communication sys- +tems based on different network structures and the summary +of its pros and cons. +A. Federated Learning Framework +Considering more practical scenarios in large-scale wireless +networks, where most of training data accounting for a global + +27 +model learning are generated locally at end devices, in the +aforementioned centralized learning framework, edge devices +are required to send these data to a central server (e.g. a BS), +triggering high communication costs and data privacy issues. +To mitigate these problems, FL, a distributed framework to +train a global statistical model, was proposed [159]. Under +the coordination of a dedicated central server, FL allows +multiple devices to participate in the global model training +through local model update and model parameters exchange +without sharing their raw data [160], thereby preserving the +data privacy and saving the communication cost. The unique +challenges of FL compared with centralized learning frame- +works include system heterogeneity with various end-device +features (e.g., transmission environment, communication re- +sources, computational powers, etc.), data heterogeneity with +non-identical local data distributions and unbalanced local +datasets, and dynamic wireless environment with uncertain +wireless channels and access links [8], [161]–[163], which +have stimulated a growing research interest in FL. +The goal of FL is to minimize a global loss or empirical risk +function LFL, i.e., minθ +� +i∈S wiLi(θ; Di), where θ represents +the model weights, Li denotes the local loss function of device +i over local dataset Di, S is the set of participating end-devices +and wi denotes the weight for each local loss function with +wi ≥ 0 and � wi = 1. In a typical cross-device FL training +process, as illustrated in Figure 8, a central server orchestrates +and repeats the following steps (referred as federated averaging +algorithm) until the convergence of global model. +1) Device Selection: The central server selects a subset of +devices meeting certain eligibility requirements to participate +in each training round. Typical device selection schemes in +wireless networks include random scheduling, round robin, +proportional fairness as well as incentive mechanism based +on auction game [164], [165]. +2) Global Model Broadcast: The central server broadcasts +the current model set to the selected devices that participate +in the training process. +3) Local Model Training: Based on the received global +model, each selected device takes a batch of samples from its +local dataset and utilizes a local model update algorithm (e.g., +stochastic gradient descent algorithm) to obtain the updated +local model. +4) Model Aggregation: All the local model updates are +aggregated at the central server by computing the model +aggregation function (e.g. weighted average function) to obtain +the updated global model. +To implement federated learning in wireless networks with +limited resources and unreliable communication links, the +efficiency of information transmission becomes the core issue. +In wireless FL, the local model shall be transmitted from end +devices to central server, followed by the calculation of the +aggregation function at the central server. Given the struc- +ture of model aggregation function, the wireless transmission +of local model can be divided into orthogonal transmission +and non-orthogonal transmission [166]–[169]. The practical +orthogonal frequency division multiple access (OFDMA) and +FDMA techniques are adopted to support interference-free +uplink local model transmission and downlink global model +transmission in [166], [167] through orthogonalized frequency +allocation, which could increase the communication latency +instead. In order to improve the transmission efficiency, non- +orthogonal transmission schemes leveraging the principle of +over-the-air computation (AirComp) have been studied in +[168]–[171]. Unlike the orthogonal transmission which re- +quires the decoding of each local model separately, AirComp +allows the central server to receive a desired aggregation +function of local models via their concurrent transmissions on +same resource block [168], [169] by exploiting the waveform +superposition nature of wireless multiple access channels, +therefore the communication latency and the consumed com- +munication resources will not increase with the number of +devices, facilitating its usage for large-scale networks. +Training data is essential for ML model learning. The cen- +tralized ML algorithms assume the training data is independent +and identical distribution (i.i.d.) distributed. In wireless FL, as +the training datasets are generated distributed by end devices, +the heterogeneous nature of large-scale wireless networks +leads to non-independent and non-identical distribution (non- +i.i.d.) datasets at different devices. The dynamic network envi- +ronments, such as the mobility of devices and the randomness +of link connections, further make the sensory data not only +heterogeneous but also non-stationary. As a result, the non- +i.i.d. and non-stationary feature of on-device data becomes +one of the key bottlenecks to accomplish efficient and accurate +distributed ML tasks in hyper-scale wireless networks. In the +following subsections, we overview the existing wireless FL +frameworks to resolve the issue of network/data heterogeneity +and instability. +B. Application 1: Cross-Device Federated Learning +Cross-device FL involves enormous number of IoT devices +or agents, where the data is generated locally and remains +typically decentralized [159]. In order to deploy DNN models +on a larger-scale wireless network, avoid huge communication +overhead for training data collection in centralized learning as +well as protect data privacy, there are already some studies +that leverage FL for wireless communication applications +including hybrid beamforming [172] and channel estimation +[173], [174]. Specifically, in these wireless applications, the +training data across devices are typically non-i.i.d. due to +the heterogeneity of the transmission environment where de- +vices are located. The non-i.i.d. data can slow down the +convergence and deteriorate the learning accuracy of FL. To +improve the overall performance of FL in dealing with various +wireless communication problems with non-i.i.d. training data, +the deeper and wider CNN models were used in FL-based +systems [172], [173] to provide better feature extraction and +representation capability. In addition to the data heterogene- +ity, system parameters, such as SNR, antenna numbers, and +channel statistics of participated devices are also dynamically +changing. As shown in [172], [173], the decentralized FL- +based wireless communication systems are more robust to +the channel imperfections and corruptions compared with the +centralized learning. To cope with the varying system condi- +tions, a federated dynamic detection network was proposed + +28 +Central Server +Global Model Aggregation +Broadcast +Global Model +Upload +Local Models +Devices +Fig. 8. Illustration of cross-device federated learning. +Broadcast +Global Model +Cellular Networks +Upload +Local Models +Global Model Aggregation +Silo Servers +Central Server +Fig. 9. Illustration of cross-silo federated learning. +in [175] to perform the dynamic MIMO detection, where +two independent detection networks were built leveraging the +algorithm unrolling approach, and a specifically designed route +network was built to adaptively select a better detector for +every sample under different conditions. However, the multi- +network design can induce significant training cost and has +the limited adaptability to the changing environment. +The dynamic and uncertain wireless environment poses +great challenges for efficient wireless resource allocation in +FL. The impact of resource allocation policies on the learning +performance of wireless FL, e.g., the test accuracy and training +efficiency, is generally implicit and non-analytic. For instance, +in the training of the CNN-based wireless FL system, it is hard +to obtain an analytical expression of the FL testing accuracy +with respect to the resource allocation parameters (e.g., device +selection, power allocation, computational resource allocation, +etc.). As a result, the conventional convex optimization based +resource allocation algorithms are infeasible for such scenar- +ios. Fortunately, the dynamic resource allocation problem in +the FL system can be formulated as a stochastic optimization +problem by modeling the total training process of FL as +an MDP with the resource status at each training round +being the state, the resource allocation policies being the +action, and FL learning performances (e.g., training latency, +learning accuracy, energy consumption, etc.) being the reward. +Therefore, the model-free DRL can be applied to solve the +dynamic resource allocation problems in FL while regarding +the FL performance changes as a black box [176]–[179], +given the diverse and dynamic wireless conditions for FL +participants. Specifically, in each training round, the resource +allocation policies shall be optimized based on the real-time +resource states (i.e., CSI, calculation resources and available +bandwidth), leading to the change of the FL performance of +DNN model, which is considered as the immediate reward +of the current policy. Then the state evolves based on the +action performed. Through trial-and-error method, the long- +term optimal resource allocation policy can be learned to +maximize the total amount of reward received over time. +We summarize the representative works of DRL-assisted +FL in Table VII, detailing the DRL algorithms, state space, +action space and reward function, respectively. For example, +the authors in [182] proposed an experience-based scheduling +framework for client selection in each communication round +to cope with the non-i.i.d. data distribution, where DQN was +adopted to learn the optimal client selection policy aiming for +higher test accuracy and fewer communication rounds under +dynamic wireless conditions. DQN-based algorithm was also +adopted in [185] to optimize the user access in open radio +access network for long-term throughput maximization and +efficient FL. Further, the authors in [183] developed a double +DQN-based algorithm to optimize the amount of data, energy +and computational resources allocated to each end device for +FL training by minimizing the energy consumption and system +delay. The joint optimization of radio resource allocation and +device scheduling were also investigated in [186], where an +actor-critic based DRL approach was proposed to optimize +the transmit power at the BS and the computing frequencies +at the local devices when participating in the FL training, +such that the FL learning performance and the fairness of +users can be maximized while reducing the energy and time +consumption. In [178], a value-based DRL method across +computing, communication and caching was proposed for FL +optimization in a 5G ultra-dense edge computing networks, +where DQN was adopted to solve the complex optimization +objectives integrating the QoS metric and communication +delays. To reduce the back-haul traffic congestion in the large- +scale IoT network, the authors in [179] adopted a policy-based +algorithm to allocate the back-haul data flow by maximizing +the network utility. In [187], the authors exploited the DRL +technique to optimally allocate the available energy and data +units at each device and design the block generation rate at +miner in a blockchain-assisted FL system. To summarize, by +transforming the implicit optimization target related to the FL +learning accuracy and efficiency into a numerical reward, such +DRL-based algorithms show better fitness in wireless FL for +dynamic resource allocation compared with the COAs under +dynamic environment. +C. Application 2: Cross-Silo Federated Learning +In contrast with cross-device FL, where a large number +of IoT devices participate in FL to complete same learning +task, the clients in cross-silo setting are silo servers, including + +12:2829 +TABLE VII +DRL-ENABLED FEDERATED LEARNING FRAMEWORKS +Algorithm Type +DRL Algorithms +Refs +State Space +Action Space +Reward Function +DQN +[178] +Task queue state and the available resource state +Application partitioning, subcarrier allocation strategy, +and service caching placement +Negative summation of normalized execution time +Value-Based +[180] +Selected action in previous time slot, +local information, interferers’ information +and interfered neighbors’ information +Transmit power, beamformer +Achievable rate minus the sum of +the achievable rate losses of the interfered links +[181] +Number of available channels, number of power level, +length of the binary representation of the feedback signal, +and indicator of no transmission +Channel selection policy, +power allocation policy +Network utility +DDQN +[177] +Channel conditions, resource allocation actions +Channel selection, transmit power +Sum-rate of all cells minus QoS based penalty +[182] +Compressed model weights +Device scheduling +Exponential function of +achieved test accuracy +[183] +Available CPU, energy unit, and wireless bandwidth +Device scheduling +Resource utilization of MEC system. +Policy-Based +A2C +[179] +Statistics of data flow +Available assignment options +Mixed function of delay and PLR +DDPG +[184] +The selected beamformer indexes, +the achievable rate and signal power at all users, +interferer links, and interfered links, respectively +Codeword in the beamformer codebook +Achievable rate minus the sum of +the achievable rate losses of the interfered links +organizations (e.g. medical or financial), geo-distributed data- +centers, etc., each of which has an identity or name that +allows the system to access it specifically [159]. In cross-silo +setting, a number of organizations can share incentive (e.g. +the payoff-sharing scheme in financial FL system [188]) rather +than data directly to train an ML model due to confidentiality +or law issues. In cross-device FL, the training data are usually +partitioned by examples, while in cross-silo FL, in addition to +be partitioned by examples, the training data can be partitioned +by features, e.g., different organizations keep the data of differ- +ent features corresponding to same batch of customers [159]. +The feature partitioned FL requires multi-clients to train the +model collaboratively by exchanging intermediate parameters +rather than model parameters to deal with the missing features. +Lots of works have been done to address the security and +privacy challenges [189] and communication bottlenecks [190] +in feature-partitioned FL. Another source of heterogeneity in +cross-silo FL is the non-i.i.d. and non-stationary siloed data. +Typically in large-scale wireless networks, the clients in cross- +silo setting can be clusters of cellular or D2D networks that +contain non-i.i.d. and non-stationary heterogeneous data. +To overcome the model divergence and staleness in the +training stage and poor accuracy in the inference stage induced +by non-i.i.d. and non-stationary training data of different +clients in cross-silo FL, an adaptive federated multi-task +learning (FMTL) framework was proposed by [191], which +preserves multiple models at the server and the clients through +adaptive model updating and cluster splitting to deal with non- +stationary environments and multi-task learning. Specifically, +in model update scheme, model dichotomy [192] was adopted +to find the geometric centers of local models in two virtual sub- +clusters, whose model update directions were compounded to +determine the update direction of global model. The recursive +cluster splitting, through decreasing the sub-clusters’ distances +of updating directions, was also proposed to avoid the poor +performance induced by the mutation of data distributions. +Furthermore, a binary tree-based low-complexity model se- +lection scheme was proposed to choose the best model for +fitting the current data in both training and testing stages. +Such adaptive FMTL has been shown to accelerate the model +training convergence and reduce the computation complexity +while ensuring model accuracy when it is applied to solve the +GNN-based power control problem in cross-silo FL system +consisting of D2D networks. +D. Advantages and Disadvantages +FL can not only embed the training capabilities of DNN for +hyper-scale wireless networks, but also build a unified multi- +source data application system with privacy preserving among +multiple devices. Besides, FL can realize data sharing and inte- +gration across silo servers for supporting high-precision model +construction [159]. However, FL still faces many challenges in +terms of privacy protection, theoretical analysis and wireless +deployment. For example, the privacy preserving techniques +usually sacrifice the learning performance and induce addi- +tional computational cost, which are undesirable for efficient +FL. The convergence analysis of FL considering the trade- +off between learning performance and resource consumption +is difficult due to the highly non-convexity and intractability +of optimization problem. Moreover, the deployment of FL +in large-scale wireless networks should jointly consider the +issues of dynamic fading channel, communication overhead, +low power constraint of end devices, as well as the availability +and the willingness of participants, which can be challenging +for efficient algorithm design. +VIII. DISCUSSIONS AND FUTURE RESEARCH DIRECTIONS +In light of the appealing benefit of ML for large-scale +optimization in 6G wireless networks, significant efforts are +still needed to upgrade the existing ML techniques or develop +new ones considering the constraints of practical wireless +communication systems. To further pave the path for its more +comprehensive applications in future wireless communication +systems, in this section, we summarize the existing DNN +design principles to accommodate to different kinds of op- +timization problems in wireless networks, after which, we +summarize the existing theoretical tools to characterize the +performance of MOAs. Subsequently, we discuss the software + +30 +platforms and implementation issues for MOAs in 6G wireless +networks and some potential research directions. +A. Neural Network Design for Wireless Communications +The design of NNs (e.g. loss function design, network +architecture design and training algorithm design) for solving +complex optimization problems in various wireless commu- +nication applications requires careful consideration. In the +following, we summarize the design principles of DNNs from +different aspects when applied to solve large-scale optimiza- +tion problems in wireless networks. +1) Loss Functions: The design of loss function is generally +dependent on the communication problem being solved and +the availability of training labels. When training labels are +available, the DNN can be effectively trained in a supervised +manner by constructing the loss functions using the training +labels. For example, regression-based losses (e.g., MSE or +weighted MSE) are usually optimized for estimation problems, +such as channel estimation [29] and MIMO detection [19], and +for resource allocation problems, such as power allocation [82] +and beamforming design [194], when the labels of targeting +signals are available. Cross-entropy losses are adopted for clas- +sification problems, such as codebook-based precoder design +[195], user scheduling [73], and sub-channel selection [196], +when the class labels are available. In some specific applica- +tions, information theory-based losses can help to fulfill the +goals of optimization task, which can be effectively calculated +using the empirical data (including training labels), such as +the mutual information-based loss in semantic communication +[9], min-max generative adversarial net (GAN) loss in channel +estimation [197], information-bottleneck-based loss in edge +inference system [198], etc. Alternatively, when training labels +are unavailable, the average performance measurements are +adopted to formulate the loss function for model training +in an unsupervised manner [199]. For example, in resource +management problem, the average performance measurement +E[f(p(h), h)], e.g., (weighted) sum-rate [84], [200], energy +efficiency [201], communication delay [202], secrecy capacity +[203], etc., are utilized to guide the model training. +2) Network Architectures: For network architecture de- +sign, MLP is usually adopted as a benchmark algorithm for +comparison [19], [33], while for the problems with special +structures (e.g., data structure, algorithm structure and problem +structure), specialized NN architectures are preferred to better +serve the underlying purpose of the task. For data with graph +structure (e.g., network data [200]), GNN is more suitable for +exploiting the inherent graph structure of the problem. For data +with spatio-temporal correlations (e.g. traffic prediction [204]), +CNN or RNN is specialized in capturing the correlations +within data. For data with low dimensional structure, the +generative model can be utilized to capture the fundamental +sparse structure within data (e.g., the underlying probability +distribution of spatial channel can be learned by generative +network [205]). For algorithms with iterative nature, NN can +be designed to imitate the forward operation of iterative +algorithms (e.g., deep unrolling NN inherits the structure +of original iterative algorithm [35]). For problems with the +distributed data, FL framework can be exploited to meet +the distributed requirement [8]. For problems with complex +dynamic environment, DRL constitutes a viable technology +to address the stochastic optimization problem through agents +and environment interactions. +To cope with the more stringent requirements in 6G system +and support diverse applications for future wireless networks, +hybrid DL-based optimizations have attracted increasing atten- +tion to fully integrate the intrinsic diverse features of different +tasks into the design of customized NNs and make the most of +the advantages of various DL techniques to improve the algo- +rithm efficiency. The core of hybrid DL is to match different +features of the problem with appropriate learning technology. +For example, DRL-enabled FL system has been discussed in +Section VII to solve dynamic resource allocation problems in +FL. Besides, the algorithm unrolling approaches can be easily +combined with not only conventional optimization [52], [54] +but also some DL techniques such as GNN [97] to speed +up the iterative algorithms. GNN integrated with DRL has +been discussed in [206] for algorithmic and methodologi- +cal improvements, where DRL and GNN complement each +other for better utility or application-specific enhancements. +In particular, the versatility of DRL and the flexible encoding +capability of GNNs can be combined to address challenging +optimization problems in different applications. Moreover, +contrastive learning, a self-supervised learning approach, has +been leveraged in GNN to address the challenge of data +heterogeneity in graphs [207], [208], where the node features +are learned in an unsupervised way. +3) Training Algorithms: When the network structure is +fixed with reasonable loss function, Adam [209], the most +popular back propagation algorithm, is usually applied for +stable training. To further depict the recurrence relation of +gradients between two adjacent layers, the generalized chain +rule was proposed in [21] to perform back propagation in +unrolling-based DNN algorithms. To obtain better learning +performance for algorithm unrolling methods, the layer-wise +training approach is widely adopted [14]. End-to-end training +of DNNs based on a large number of channel samples can +bypass the explicit channel modeling procedure and poten- +tially provide system-level performance gains compared to the +COAs when solving the optimization problems in wireless +communication systems [16]. Furthermore, to accommodate +the NN to the dynamic changing environment in wireless +communication systems, several techniques can be applied. +For example, transfer learning has been adopted in [15] to +tackle the task mismatch issue (e.g., the network setting in +training is different from that in testing) in LBB algorithm via +self-imitation. Similarly, transfer learning in semantic commu- +nication in [34], [153] enables the trained NN to adapt to the +dynamic communication environment quickly with reduced +number of training data. Meta learning is another technique +to improve the generalizability of DNN in dynamic settings. +For instance, model-agnostic meta-learning is used in [210] to +learn a good model initialization by alternating inner-task and +across-task updates, such that the learned model parameters +can adapt to a new environment with a small number of labeled +data. + +31 +Even though transfer learning and meta learning can signif- +icantly speed up the learning process of NN in new environ- +ment, they still require batch-sized training data to be available +before the learning. In many scenarios of wireless networks, +the training data arrives sequentially in a stream whose inher- +ent features can be drifted due to the dynamic environment. +In this case, a promising solution is to learn the models on +the fly, which can improve the generalizability and scalability +of model sufficiently and save the memory of system. Online +deep learning [211], to learn the DNNs from the sequentially +received data in an online manner, has been investigated in the +various contexts of wireless networks. For example, in [212], +an online DNN framework was proposed to solve the general +optimization problems in wireless communication, where the +self-defined layers rather than convolutional or full-connected +layers were adopted to estimate optimization variables for +each data sample. The CNN-based [213] and GNN-based +[214] online learning algorithms were further proposed, where +the online module is retrained based on the observed testing +samples to overcome the mismatches between training and +testing data induced by the dynamic environment. Besides the +network generalization issues, the complex and unpredictable +environment may also lead to implicit system performance +functions in resource allocation problem, which hinders the +gradient calculation in the training process. To tackle this, a +model-free approximation approach was proposed in [215], in +which the gradients are approximated by their zeroth-ordered +updates through sampling the model functions. +4) Optimization Constraints: +Optimizations in wireless +communications usually need to deal with various complex +constraints to meet specific requirements, which brings ad- +ditional difficulties to the design of ML algorithm. Normal- +ization layer as the output layer of DNN provides a simple +and effective way to deal with modulus constraint (e.g. power +constraint [84]). The proper activation functions can help to +restrain the network output within a feasible region. For exam- +ple, sigmoid can keep the output between 0 and 1. For discrete +constraints, e.g., quantization, various smooth functions can be +constructed to approximate the discontinuous function, or we +can directly set the gradient to be 1 at the back propagation to +avoid the gradient vanishing and explosion [216]. For other +more complex constraints, primal-dual learning [85], [217] +plays an important role, where DNNs are designed to solve the +corresponding unconstrained Lagrangian dual problem. The +authors in [90] showed that the duality gap of radio resource +management optimization problem in wireless networks is +negligible if the parameterized learning through NN is near +universal. +B. Theoretical Tools +ML technology, represented by DNN, has made great +achievements in the fields of computer vision, natural language +processing and communications in recent years with the great +improvement of computing power and the great enrichment of +data. However, due to the “black box” nature of DNN, the lack +of interpretability and theoretical guarantee of the DL-based +framework is a critical issue that needs to be addressed for +applications requiring highly transparent and reliable technolo- +gies (e.g. wireless communications, healthcare and automatic +driving). DNN is a model function characterizing complex +relationships among data, and its “black box” nature is mainly +manifested in the fact that there is a huge unknown gap +between the design of the model and its final performance on +specific tasks. In other words, it is impossible to accurately +predict and control DNN performance when designing the +model, to clearly understand the reasons for its good or bad +performance, and to systematically improve its performance, +but only to rely on some lucky model design and training +tricks. From ML perspective, this is due to the fact that the +ML task (including training data) and DNN theory aspects +(e.g., loss quantities and the generalization performance of the +model) have not been thoroughly understood. Especially when +dealing with high-dimensional real data (e.g. images with +the millions of dimensions), many of the existing statistical +quantities and information-theoretic concepts such as entropy, +mutual information, maximum likelihood and KLD suffer +from the curse of dimensionality for computation, the ill- +posedness for degenerate distributions, as well as the lack +of guarantees for finite samples [218]. To avoid these issues, +the principled formulations are replaced with approximate +bounds, simplified assumptions, heuristics and special tricks +in practice, resulting in a serious performance gap between +theory and practice. +In wireless communications, the specialized MOAs poten- +tially enable rigorous analytical results within certain per- +formance limits [219] due to their task-specific features and +the model-inspired property. In the following, we summarize +the existing research results and progresses on the theoretical +aspects of the MOAs. +1) Algorithm Unrolling: Inherited from the traditional it- +erative algorithm, the behavior of each layer of unrolled NN +is interpretable. The unrolled iterative hard threshold (IHT), +used to solve ℓ0 norm constrained sparse recovery problem, +has been theoretically analyzed in [220], which provided the +optimality condition for the exact sparse recovery of the +unrolled NN and proved the linear convergence rate of the +unrolled IHT. The theoretical studies for the unrolled ISTA +can be found in [221], [222], where the linear convergence +rate of unrolled ISTA has been established and the structure +of optimal network parameters for unrolled ISTA has been +analyzed as well to guide the network structure design and +the training parameters downsizing. The extension of unrolled +ISTA to solve the group-sparse matrix estimation problem has +been studied in [14] and applied to JADCE problem in wireless +networks, which established the linear convergence rate of +unrolled ISTA with group sparsity structure. While some +progresses have been achieved to establish the performance +guarantees of algorithm unrolling, the underlying mechanism +and the impact of learned parameters on the convergence and +learning accuracy are still to be further discovered. +2) Learning to Branch-and-Bound: The complexity of LBB +for solving MINLPs is analyzed in [15] following the common +assumptions and analysis of imitation learning [223], which +shows the expected number of nodes explored and the number +of relaxed problems solved are O(L2) and O(L) with L + +32 +integer variables under proper parameter settings. Therefore, +the computational complexity of LBB is much lower than +that of the traditional BB algorithm especially for large-scale +network with large L. However, the theoretical analysis of +learning performance is still missing. +3) Graph Neural Network for Structured Optimization: The +graph optimization problem for large-scale wireless networks +exhibits the property of permutation invariance or permutation +equivariance depending on the output features. The classic +GNN frameworks enjoy the same properties, i.e., permutation +invariance or equivariance, which guarantees advantageous +performance of GNN when applied to solve the graph- +structured optimization problems [22]. Furthermore, Shen et +al. [33] firstly provided the generalization analysis of GNNs +to theoretically verify the advantages of GNNs over MLPs in +solving wireless communication problems in terms of the gen- +eralization error and the required number of training samples. +Based on the probably approximately correct (PAC) learning +framework, they showed that the GNNs’ generalization error +and required number of training samples are O(n) and O(n2) +lower than those of MLPs, where n is the number of nodes in +the graph. Therefore, the GNNs can be theoretically proved +to solve the graph optimization problem with near-optimal +performance and much fewer training samples than generic +NNs. +4) Deep Reinforcement Learning for Stochastic Optimiza- +tion: The existing theoretical framework of DRL is established +based on the classic control theory and the theoretical results +of conventional RL. Besides the optimization performance +in terms of the expected accumulated reward, another main +concern of RL community is the convergence performance +in terms of the sample efficiency over collected experience +data. Xie et al. [224] proposed a pessimism-based approach +that guarantees the convergence with O(d) sample complexity +while only requiring Bellman closedness as standard in the +exploratory setting. Furthermore, Zhan et al. [225] proposed a +novel theoretical analysis framework for DRL according to the +primal-dual formulation of MDPs. By relaxing the stringent +requirement on the all-policy concentrability and Bellman- +completeness, the framework proposed therein enables poly- +nomial sample complexity under single-policy concentrability. +5) End-to-End Learning for Semantic Optimization: The +theory of semantic communication has caught extensive atten- +tions in the past several years with some preliminary research +results [36], [226]. In [9], the IB theory was used to design +the semantic communication systems by taking the meaning of +semantic information and the compression ratio into consider- +ation. However, the IB formulation is task and label dependent, +that is, the measurement quantity changes as tasks and labels +change. Besides, the IB provides an information-theoretic +guidance for semantic information extraction and transfer, +where the implementation of each functionality relies on the +DNNs. Accordingly, a theoretical understanding of DNN can +facilitate the theoretical analysis of semantic communication +system, which hinges on the development of ML theories. +6) Federated Leaning for Distributed Optimization: The +theoretical development of FL depends on the research +progress of the underlying DL theories. The existing theoret- +ical research of FL focused on the convergence performance +analysis with simple ML models, such as convex loss functions +in [157]. The generic performance analysis for DL model in +FL is still missing. Furthermore, the theoretical analysis of FL +shall be developed in view of various challenging practical +issues including expensive communication, system and data +heterogeneity as well as privacy and security concerns [157]. +7) End-to-End Learning for General Non-convex Optimiza- +tion: The DL theories also enable the theoretical analysis of +deep generative networks [205], ReLU-based DNNs [227], +continual learning [25], etc., for end-to-end learning frame- +works. The error bound of generative model for CS was ana- +lyzed in [228] for ReLU-based generative NN and L-Lipschitz +generative models, which can guide the high dimensional +channel estimation applications in wireless communications +[205]. The ReLU-based DNN was proved mathematically +equivalent to a piecewise linear function under some mild +conditions, which can be applied to theoretically prove that the +end-to-end DNNs can be utilized as a universal approximator +of the MMSE channel estimator to supply theoretical support +for DNN-based channel estimation algorithm design [227]. +Moreover, the convergence analysis of continual learning +framework [25] and model-free online DNNs [229] were +further established for end-to-end wireless applications. As the +development of theoretical understandings on various learning +techniques, more solid and in-depth description of theoretical +analysis of MOA designs can be obtained to guide their usage +in practical 6G wireless networks. +C. Implementation Issues and Software Platforms +1) Implementation Issues: +In academic, most existing +learning to optimize methods are trained and tested on an +offline simulator with designed dataset (e.g. the DeepMIMO +[230] and Raymobtime [231] for collecting realistic training +data) or generated dataset from simulator (e.g., the data +generated from BB algorithm for training LBB models). The +software platforms for offline training simulator of ML-based +model shall be discussed next. For distributed ML to be +implemented on massive low-power end devices [8], FL, +decentralized learning, model-split learning, distributed RL +as well as trustworthy learning are considered as promising +edge learning algorithms which are amenable to distributed +implementations in large-scale networks. The edge inference +implementation issues can be categorized as horizontal edge +inference and vertical edge inference based on different collab- +orative computing mechanisms, which are also well discussed +in literatures [232]–[234] to realize real-time inference in +practical distributed systems. +The standardizations of AI/ML for communication project +have just been established and discussed in December 2021. +The first technical standard for AI/ML was approved in the +3rd generation partnership project (3GPP) Release 18 [235] +to investigate the implementation-related issues of AI/ML +in physical layer. In order to ensure that the AI/ML model +can be stably applied to the communication system, the +general AI/ML architecture, collaboration between user and +network, life cycle management of AI/ML models, model + +33 +activation/deactivation, model monitoring, and model switch- +ing/updating shall be further discussed and designed in 3GPP +[236]–[238], which can evaluate whether to update the AI +model or go back to the traditional algorithm according to +the monitoring results. +2) Software Platforms: For offline training of ML models, +there are a rapidly growing body of software platforms for +simulations and productization of ML algorithms and models, +which can greatly simplify the construction of the compli- +cated NNs, including the forward operation, gradient back +propagation as well as the parallel computing, and provide +potentially feasible platforms for the implementation of MOAs +in practical wireless communication systems. MATLAB Neu- +ral Network Toolbox, TensorFlow [239] and PyTorch [240] +have provided excellent open-source software frameworks, +which are compatible with common operating systems and +can be installed in various communication devices, e.g., cloud +server, BS, AP or a terminal with certain computational power, +for real-time/offline data collection, model training, updating +and inference. For example, Pytorch can be used to build +a DNN easily in network environment and the library is +well optimized for graphic processing units (GPU). Tensor- +Flow is more suitable for building advanced and large-scale +NNs. As a production-oriented DL framework, Caffe2 [241] +has been developed in Facebook to train NNs on multiple +GPUs in distributed setting for supporting mobile operating +systems (e.g. iOS and Android). In addition, there are many +other software platforms such as Blocks [242], CNTK [243] +and Lasagne [244] that can also support mobile systems +for commercial-grade distributed DL implementation. From +the perspective of promoting scientific research, Open-L2O +[44], a research-oriented open software package, has recently +been established to support both model-free and model-based +“learning to optimize” approaches, which facilitates a fair +performance comparison of different algorithms in a simulated +environment and a fully automatic algorithm design of various +kinds of optimization problems. The advances in computing +capacities and data storage techniques further fertilize the +development of more sophisticated and advanced MOAs. +For example, the GPU can be utilized to execute the DL +algorithms much faster than traditional processors. Besides +general-purpose GPU, customizable field-programmable gate +array (FPGA) and dedicated application specific integrated +circuit (ASIC) also encourage the research progress of DL +in big data processing. +D. Challenges and Future Research Directions +Even though great progresses have been made in the field of +MOA designs, a large amount of data are still required to train +the DNNs to achieve near-optimal performance, which leads +to several challenges for the practical deployment of MOAs. +The acquisition of training data in practice can be difficult +due to the hard-measurable environment, high storage cost and +dynamic nature of wireless networks. To address this, the task- +oriented MOAs as introduced in this paper can significantly +improve the sample efficiency and the generalizability of NNs +by incorporating the prior knowledge and task-specific features +into the DL design. To avoid the transmission cost of data +acquisition, local dataset can be exploited to train the DL +models locally. However, how to address the heterogeneity of +the distributed computing nodes and guarantee a satisfactory +overall performance is another issue to be carefully looked at. +Besides the quantify of training data, the quality of training +data can also greatly affect the learning performance. Robust +MOA design, which is robust to data errors, measurement +noises, hardware faults and mismatched training/testing con- +ditions, is critical to generate reliable and trustworthy results +in the practical applications. In addition to the challenges +related to the data acquisition, we suggest following possible +directions for future research in this area. +1) Theoretical Analysis of MOAs: A huge amount of pa- +rameters need to be optimized when using ML-based al- +gorithm to solve large-scale wireless optimization problems. +Hence, the research progress of formal and rigorous ML +theories shall help us to understand the optimal parameters +to be learned, which can significantly reduce the training +time of ML-based algorithm and guide the design of MOAs. +For instance, the good model parameters were analyzed in +[14] and can be obtained by solving a simple convex opti- +mization problem rather than through computational intensive +back propagation algorithm, which significantly accelerates the +training process of large-scale DL models for JADCE problem. +However, the theoretical understandings of MOAs for wireless +communication applications are still in the initial stage. +2) Ultra-Lite Neural Network Design: Most of the existing +MOAs are highly demanding on computational power and +storage space, which renders their deployment on small size +and low computational power end devices. As motivated by +the edge computing, the DL should be implemented in a +distributed manner based on the local datasets. The straight- +forward network sparsification or pruning can alleviate the +storage and computational burden, while it can deteriorate the +learning performance significantly. Therefore, a light-weight +and low-complexity MOA achieving on-par performance with +the traditional algorithm is attracting increasing attention in +edge computing systems, which allows on-device model train- +ing and computing with high accuracy, small model size and +low computational complexity. +3) Advanced Methodologies and Extended Applications of +MOAs: A trend of MOA is to exploit more sophisticated un- +derlying features of specific optimization problems and design +more advanced and task-specific MOAs to embed expert prior +knowledge into DL techniques to speed up the convergence. In +particular, the model-based approaches can be utilized to in- +spire/assist the design of MOAs and provide theoretical insight +for the designed networks. In addition to the development of +the methodologies, another trend is to explore new applications +in wireless networks. Besides the optimization problems men- +tioned before, learning to optimize techniques are expected to +solve other problems involved in wireless communication for +future research directions, including multi-object optimization +problems [245], bi-level optimization problems [246], conic +programming [247], maximum-likelihood estimation problems +[248], etc, in various emerging applications. + +34 +IX. CONCLUSION +Integrating high-performance intelligent algorithm into the +wireless networks has been an inevitable trend and disruptive +shift for supporting highly transparent, reliable and large-scale +6G communication systems. In this paper, we investigated +some of the most groundbreaking ML technologies applied +for solving challenging large-scale optimization problems in +6G wireless networks, including algorithm unrolling, learning +to branch-and-bound, graph neural network for structured +optimization, deep reinforcement learning for stochastic opti- +mization, end-to-end learning for semantic-aware optimization +as well as the federated wireless learning for distributed +optimization. In each section, the general algorithm was first +introduced, followed by its case studies of the formulated +optimization problems arising from wireless applications as +well as the summary of its advantages and disadvantages. In +the last part, the neural network design for wireless communi- +cations, theoretical tools, implementation issues together with +the future research directions were also discussed to implement +ML algorithms in wireless communications from theory to +practice. 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Wang, “Faster activity and data +detection in massive random access: A multiarmed bandit approach,” +IEEE Internet of Things J., vol. 9, pp. 13664–13678, Aug. 2022. + diff --git a/NNE1T4oBgHgl3EQftgVE/content/tmp_files/load_file.txt b/NNE1T4oBgHgl3EQftgVE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..508c129a26de21e3df0422b8c16d0032f1b4aea0 --- /dev/null +++ b/NNE1T4oBgHgl3EQftgVE/content/tmp_files/load_file.txt @@ -0,0 +1,4338 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf,len=4337 +page_content='1 Machine Learning for Large-Scale Optimization in 6G Wireless Networks Yandong Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' 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Zixin Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Yong Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Liqun Fu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Lin Bai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Senior Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Jun Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and Wei Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IEEE Abstract—The sixth generation (6G) wireless systems are envisioned to enable the paradigm shift from “connected things” to “connected intelligence”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' featured by ultra high density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' large- scale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' dynamic heterogeneity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' diversified functional requirements and machine learning capabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' which leads to a growing need for highly efficient intelligent algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The classic optimization-based algorithms usually require highly precise mathematical model of data links and suffer from poor per- formance with high computational cost in realistic 6G appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on domain knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', optimization models and theoretical tools), machine learning (ML) stands out as a promising and viable methodology for many complex large-scale optimization problems in 6G, due to its superior performance, generalizability, computational efficiency and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this paper, we systematically review the most representative “learning to optimize” techniques in diverse domains of 6G wireless networks by identifying the inherent feature of the underlying optimization problem and investigating the specifically designed ML frameworks from the perspective of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In partic- ular, we will cover algorithm unrolling, learning to branch-and- bound, graph neural network for structured optimization, deep reinforcement learning for stochastic optimization, as well as end- to-end learning for semantic optimization, for solving challenging large-scale optimization problems arising from various important wireless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To enable ML implementation in dis- tributed wireless networks across massive number of end devices, federated learning for distributed optimization will further be presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Through the in-depth discussion, we shed light on the excellent performance of ML-based optimization algorithms with respect to the classical methods, and provide insightful guidance to develop advanced ML techniques in 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Neural network design, theoretical tools of different ML methods, Yandong Shi is with the China Telecom Research Institute, Guangzhou 510660, China (e-mail: shiyd2@chinatelecom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Lixiang Lian, Yuanming Shi, and Yong Zhou are with the School of Infor- mation Science and Technology, ShanghaiTech University, Shanghai 201210, China (e-mail: lianlx@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' shiym@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' zhouyong@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zixin Wang is with the School of Information Science and Technol- ogy, ShanghaiTech University, Shanghai 201210, China, also with the Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China, and also with the Uni- versity of Chinese Academy of Sciences, Beijing 100049, China (e-mail: wangzx2@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Liqun Fu is with the School of Informatics and the Key Laboratory of Underwater Acoustic Communication and Marine Information Technology (Ministry of Education), Xiamen University, Xiamen 361005, China (e-mail: liqun@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Lin Bai is with the School of Cyber Science and Technology, Beihang University, Beijing 100191, China (e-mail: l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='bai@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Jun Zhang is with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong (e-mail: eejzhang@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Wei Zhang is with the School of Electrical Engineering and Telecommuni- cations, The University of New South Wales, Sydney, NSW 2052, Australia (e-mail: w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='zhang@unsw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' implementation issues, as well as challenges and future research directions are also discussed to support the practical use of ML model in wireless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Index Terms—Large-scale optimization, machine learning, deep neural network, 6G, large-scale networks, wireless com- munications, learning to optimize, non-convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' INTRODUCTION The sixth generation (6G) wireless systems have re- cently attracted considerable attention from both industry and academia, whose visions are towards ubiquitous 3D coverage (space-air-ground-sea integrated network) [1], the intelligent and green networks [2], Internet of everything (IoE) [3], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with the previous generations, 6G can provide services with more stringent requirements, such as higher throughput, lower latency, higher reliability, denser connec- tion, higher energy efficiency, as well as connected intelligence with machine learning capability [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Driven by the new industrial and technological revolution, 6G can also support new services/applications beyond 5G such as immersive cloud extended reality (XR), holographic communications, sensory interconnection, digital twins and so on [4], which may demand new performance metrics to facilitate diversified and personalized user services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The requirements of 6G system have made the fine-grained optimization of radio resources and effective learning of network-related information urgent neces- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the large-scale, high density, heterogeneous qual- ities of services, and integrated multi-functional cross-layer design, the optimization problems in 6G can be extremely time-sensitive and complex, which pose great challenges for efficient optimization algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Machine learning (ML) has been recently leveraged as a disruptive technology to solve the challenging optimization problems in 6G, as well as support ubiquitous artificial intelligence (AI) services and IoE applications [4]–[6] including synaesthesia internet, digital twins, smart industry, smart agriculture, super traffic, precision medicine and blockchain economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this section, we first discuss the properties of optimization problems in 6G wireless networks and summarize the advantages and disadvantages of classic optimization-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then we introduce the motivation for ML-based optimization frameworks and summarize the existing design paradigms to solve different classes of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Table I summarizes the main notations used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='03377v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='SP] 3 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Large-Scale Optimization for 6G The performance of 6G wireless networks can be enhanced by adopting various optimization algorithms, which solve the practical engineering problem through the mathematical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Constructing and solving optimization problems can effectively handle the technical issues in engineering and guide the performance-related policy development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The properties of optimization problems in 6G networks lay in the following three aspects: (1) The objective functions of 6G optimization problems can be complicated to meet personalized services of heterogeneous networks and highly non-convex due to the enabling of integrated functions in 6G, such as joint sensing, communication, computing and control [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Other factors such as diversified services 6G facilitates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' distributed edge training and inference [8]) and integrated cross-layer designs will further complicate the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the mutual information is adopted as the objective function in semantic communications [9] to optimize the efficiency- accuracy trade-off, which is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (2) Optimization vari- ables and model parameters of 6G optimization problems can be of high dimension due to the massive devices, large-scale antennas in wireless networks and large amounts of data in various 6G technologies and services [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The feasible region of optimization variables can be highly stringent to satisfy practical network conditions under resource constraints and provide robust and reliable services for ubiquitous networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the spectral-efficiency maximization problem in heterogeneous networks involves lots of non-convex con- straints to support different transmission types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', uplink and downlink transmission constraints) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (3) The optimization problems in 6G wireless networks usually involve real-time network-dependent parameters, such as the network structure, channel state information (CSI), traffic condition, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' There- fore, the near-optimal performance of various optimization- based algorithms (OAs) should be achieved in real time, which is a fundamental challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Highly effective algorithms based on classical optimization theory have been extensively developed for various classes of optimization problems in 6G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, many iterative algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', approximate message passing (AMP), orthog- onal matching pursuit (OMP) and alternating direction method of multipliers (ADMM)) are designed for signal recovery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', signal detection [11] and channel estimation [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The semidefinite relaxation (SDR) and successive convex approx- imation (SCA) techniques are widely applied to solve non- convex optimization problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', non-convex quadratically constrained quadratic programming problems [13]) in wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Despite that some of these algorithms can achieve good performance through theoretical analysis and numerical simulations, classic optimization-based algorithms (COAs) in realistic 6G applications face many challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Optimization Performance: Due to the highly non- convexity, COAs can be intractable, suboptimal or heuristic without performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, the tuning of free- parameters in COAs relies on prior knowledge or model assumption, whose setting can greatly affect the achievable performance of COAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, iterative soft thresholding algorithm (ISTA) for sparse recovering suffers from an inher- ent trade-off between estimation performance and convergence rate, which is controlled by the choice of regularization parameter [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Computational Cost: Most of the COAs are iterative in nature, which typically induces high computational cost to obtain optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, for solving mixed com- binatorial optimization problems, the complexity of branch- and-bound (BB) algorithm grows exponentially with the scale of problem [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, most signal processing techniques in 6G have stringent latency requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, the COAs depend on the real-time environmental parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', network topology, channel conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When the environment changes, the iterative COAs need to be executed repeatedly to accommodate to the dynamic environment, which is unford- able for time-sensitive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Tractability of System Modeling: The design of COAs highly depends on the availability and accuracy of system modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, it is hard to precisely capture the network architecture, communication environment and transmission data links using mathematical models due to the time-varying, heterogeneity, complexity and nonlinearity in 6G wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The imperfect and mismatched system model can greatly deteriorate the performance of COAs when applied in practical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, traditional COAs are computational inefficient and scale poorly for large-scale optimization problems in 6G systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The reliance on perfect mathematical models and possible intractability of optimal solutions make the COAs entail serious performance gap between theoretical design and real-time application [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivated by the disadvantages of COAs, ML has surged as a powerful technique to solve the challenging optimization problems in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Machine Learning in 6G The goal of ML-based optimization algorithm (MOA) de- sign is to achieve near-optimal performance with high com- putational efficiency for challenging large-scale optimization problems in 6G wireless networks, enabling a paradigm shift from classic optimization theory-based approaches to employ- ing more promising deep learning (DL) architectures [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ML for large-scale optimization features the following advantages [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Superior Performance: MOAs entail near-optimal or superior learning performance compared with COAs due to the data-driven feature as well as the sophisticated design of neural networks (NNs) and learning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, algorithm unrolling methods enjoy a superior performance for signal detection [19], channel estimation [20] and precoding design [21] compared with their corresponding COA counter- parts and other traditional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Graph neural network (GNN) enjoys better performance for resource allocation prob- lems with fewer iterations compared with traditional weighted minimum mean-square error (WMMSE) algorithm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Scalability and Generalizability: With enhanced learn- ing capacity, MOAs can be used to solve large-scale and complex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Incorporating the properties ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='LIST OF ACRONYMS IN ALPHABETICAL ORDER ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Acronym ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Explanation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='6G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='6th Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ADMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Alternating Direction Method of Multipliers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='AE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Auto-Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='AI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Artificial Intelligence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='AMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Approximate Message Passing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='BB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Branch-and-Bound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Base Station ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CMCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Complex Modulus Constrained Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Convolutional Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='COA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Classic Optimization-Based Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Compressive Sensing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel State Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='D2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Device to Device Communication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Deep Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DNN/NN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Deep Neural Network/Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DQN/DQL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Deep Q-Network/Deep Q-Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DRL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Deep Reinforcement Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='FL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Federated Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='GCN ' metadata={'source': 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+page_content='Internet of Things ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ISTA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Iterative Soft Thresholding Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='JADCE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Joint Activity Detection and Channel Estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='JSCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Joint Source-Channel Coding ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MIMO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Multi-Input Multi-Output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MINLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Mixed Integer Nonlinear Problem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MIP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Mixed Integer Programming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ML ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Machine Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Multi-Layer Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MOA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Machine Learning-Based Optimization Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Mean Square Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='OA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Optimization-Based Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='OMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Orthogonal Matching Pursuit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='QoS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Quality of Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='RL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Reinforcement Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='RNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Recurrent Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='RIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Reconfigurable Intelligent Surfaces ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='SNR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Signal to Noise Ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='SDR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Semidefinite Relaxation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='SCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Successive Convex Approximation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='VAE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Variational Auto-Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='WMMSE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Weighted Minimum Mean-Square Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='of target task into the NN architectures further improves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='the scalability and generalizability of MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, message passing-based GNNs can be safely generalized to solve large-scale problems even when trained on small-scale samples [22], thereby leading to reduced training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The decentralized nature of some specialized MOAs enables the efficient training in large-scale networks, such as the multi- agent reinforcement learning (MARL) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advanced ML techniques, such as transfer learning [15] and meta-learning [24] can tackle the task mismatch problems with fewer training samples, thereby improving the generalizability of MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Computational Efficiency: ML inference only requires a small number of simple operations and can be realized in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By shifting the computations from online to offline, ML is highly attractive for computational intensive optimization tasks in 6G [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The online deployment of well- trained MOAs can effectively reduce the system delay and improve the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 4) Robustness: ML-based approaches shall be robust to the imperfect model assumptions and dynamic wireless environ- ment due to the data-driven nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Through learning from ex- periences, MOAs work well even under unknown environment when the mathematical model is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, continual learning [25] is robust to the dynamic environments by sequentially handling different wireless parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', different CSI distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Despite of the recent successful tries of MOAs, the impor- tant questions in this context are what kind of optimization problems can be effectively solved by ML techniques and which ML techniques would provide reliable, timely and ef- fective solutions for these optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this paper, we will try to answer these questions by focusing mainly on some exemplary ML design frameworks applied to solve vari- ous large-scale optimization problems in 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Within each framework, we highlight the motivations, the NN design principles, type of optimization problems, toy examples of its applications in 6G wireless networks, the theoretical analysis, the related research challenges as well as the summary of advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the algorithm un- rolling [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' learning to branch-and-bound (LBB) [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' GNN for structured optimization [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' deep reinforcement learning (DRL) for stochastic optimization [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' end-to-end learning for semantic optimization [27] as well as federated learning (FL) for distributed optimization [28] will be covered in the following sections to shed light on the excellent performance of ML compared with the conventional optimization algorithm in a variety of practical domains and provide guidance on the usage of ML techniques in 6G networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' followed by the summary of network design philosophies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' theoretical tools,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' implementation issues and discussion of future directions to drive forward the research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Before the detailed elaborations of specific MOA designs, we firstly summarize the existing design paradigms of MOAs from different per- spectives in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Category for ML-Based Optimization Algorithms 1) Learning Principle: From the perspective of learning principle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' learning to optimize can be divided into supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Machine Learning for Large-Scale Optimization in 6G Wireless Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Algorithm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Unrolling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Signal Recovery Problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MIMO ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Directions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' An overview of the main topics of ML-based optimization algorithms, related problems and the wireless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the labeled training data, supervised learning directly learns the nonlinear mapping between problem parameters and optimal solutions of opti- mization problems through generic NNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', multi-layer per- ceptron (MLP) for channel estimation [29]) or specialized NNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', algorithm unrolling for joint active device and channel estimation (JADCE) [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The universal approximation theo- rem states that feed-forward NN with one single hidden layer can approximate continuous functions to arbitrary precision [30], which provides theoretical support for supervised MOA designs by treating NN as a universal function approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the unlabeled training data, unsupervised approach learns the optimal solutions by adopting the objective function of original optimization problem as the training loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the NN is evaluated and updated by means of any gradient-based optimizer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', GNN takes minus weighted sum rate as loss function for scalable radio resource man- agement [22] and DRL aims to maximize expected cumula- tive discounted rewards for resource allocation in vehicle-to- vehicle communications [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with the COAs, supervised MOAs enjoy similar or superior optimization performance by leveraging the strong representation power of NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By constructing connections with COAs, the supervised MOAs show better interpretability and enables performance analysis for certain network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the original optimization problem needs to be solved repeatedly with varying problem parameters to collect the training labels, which can induce huge computational costs in data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, in supervised MOA, the NN is trained as a universal approximator of an existing optimization algorithm, and thus the performance greatly depends on that of existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Instead, the unsupervised MOAs can greatly simplify the process of data acquisition and can be effectively adopted to solve non-convex or NP-hard problem, where no tractable COAs can be found, thereby greatly eliminating the dependence on the COAs and facilitating great flexibility to search for optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, its performance is primarily restricted by the type of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For highly non-convex problem suffering from the “curse” of local minima, unsupervised MOA may be trapped into a spurious solution with poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition, the unsupervised MOA is usually computational complex and requires a larger training set to produce expected outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) NN Architecture: From the perspective of NN architec- ture, there are mainly two design principles as summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Generic NN: Most of the MOAs adopt generic NNs, such as MLP, convolutional neural network (CNN), autoen- coder (AE), etc, which are not tailored for specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The generic NNs are not specific to a particular task or dataset, but can be applied for general tasks and datasets to achieve an acceptable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, with tremendous training data and large NN architecture, MLP is usually adopted as a benchmark algorithm for perfor- mance comparison in multi-input multi-output (MIMO) detection [32], power control [33], semantic communi- cation [34], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the flexibility comes with the cost of poor data efficiency (high training overhead), poor robustness and poor generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Task-Specific NN: Another design principle of NN ar- 5 chitecture is customized implementation by incorporating the structure of target task, datasets and domain-specific knowledge into the NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The specialized NN demonstrates some unique advantages empirically and theoretically, such as robustness to model uncertainties, scalability and generalizability in large-scale problem, and high training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Some of specialized NNs can handle tricky constraints in the optimization prob- lems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', integer or constant envelope constraints) and enable performance guarantees under certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Nonetheless, the task-specific NN is highly customized and different problems require separate NN designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, for complex problems, it can be hard to con- struct a specific NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The design of specific NN highly depends on the structure of problem itself, or the property of existing algorithms that can be used to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the NN architecture of algorithm unrolling comes from parameterizing the original iterative algorithm [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, the key to designing a spe- cialized NN is to identify the special characteristics of problem/datasets/classic algorithms and sophisticatedly integrate them into the design of the NN architecture and training algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Theoretical Analysis: From the perspective of theoretical analysis, most of the MOAs treat the NN as a black box without interpretations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', MLP, CNN and AE), whose performance can only be demonstrated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, in wireless communication systems, transparentness and relia- bility are of pivotal importance for a practical algorithm design [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, it is paramount to understand the learning mechanism of NN and its applicable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Inspired by the pertinent COAs, some of the MOAs in the “supervised learning” category enable theoretical analysis by constructing a relationship of performance between these two types of meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' If equivalence can be proved, the performance analysis of MOAs can be developed based on that of COAs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the performance analysis of algorithm unrolling approaches [14], LBB approach [15], GNN approach [33], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Related Works and Our Contributions There exist several survey papers on ML for wireless com- munications [37]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' All these studies provide visions of ar- tificial intelligence-based wireless network designs, enumerat- ing on the applicable cases and scenarios in wireless networks where the ML can make a viable impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Particularly, the vast majority of surveys [37]–[41] started with the introduction of the fundamentals of ML, including the ML theory, the framework of generic deep networks as well as its update rules and training methods, followed by envisioning the wireless applications of different ML techniques in future wireless networks from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Among them, Zappone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [38] provided an in-depth quantitative analysis for each use-case of DL-based wireless network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [40] focused on the discussion of DL applications in different wireless communication layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', physical layer, data link layer, network layer and upper layer), while Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='. [39] considered the mobile, sensor networks and their related applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, none of existing works provides a thorough survey of ML/DL techniques for large-scale wireless networks from the perspective of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In light of the underlying different optimization problems involved in a variety of practical fields, various customized ML paradigms are extensively reviewed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By identifying the task-specific structures of large-scale optimization problems and classifying the existing ML frameworks accordingly, this paper bridges the elusive ML algorithms and well-grounded optimization theory to improve the interpretability and trans- parency of deep neural networks (DNNs), and inspires a game- changing new perspective for solving large-scale optimization problems in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The major contributions are summarized as follows: The challenges of wireless network optimizations in 6G systems, as well as the restrictions of traditional optimization algorithms are discussed to motivate the ML-based wireless network optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The existing design paradigms of MOAs are systematically elaborated from different perspectives, such as learning principles, NN architectures and theoretical foundations, which are presented in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The connections between large-scale optimization prob- lems and specialized DL algorithms are thoroughly dis- cussed to reveal the potential of MOA for improving the system performance and provide insightful guidance on the use of ML in 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, some promising learning to optimize frameworks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', algo- rithm unrolling, LBB, GNN and DRL) are thoroughly dis- cussed from Section II to Section V, detailing properties and classifications of large-scale wireless optimization problems, NN design patterns catering to the features of optimization problems and the use-cases in various wireless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic communication is elaborated for end-to-end optimization of communication systems in Section VI, which provides a classic semantic communication archi- tecture to endow the NNs with the ability of semantic information extraction and recovery to optimize the trans- mission efficiency and reliability trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Federated learning, as a prominent distributed ML scheme to effectively solve the model optimization prob- lem in ML over hyper-scale wireless networks, is de- scribed in Section VII, where the issues of network/data heterogeneity and instability, as well as the deployment in different wireless communication systems are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The guidelines for MOA designs in terms of the choice of loss functions, network architectures, training methods, and the ways to handle optimization constraints are given in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The issues pertaining to the theoretical progress of various MOAs, implementations as well as challenges and future research directions are also pre- sented in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We summarize the main topics of ML-based optimization algorithms, related problems and the wireless applications in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 6 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ALGORITHM UNROLLING In this section, we introduce one widely adopted “learn to optimize” algorithm design framework, termed algorithm unrolling, which constructs a layered network to mimic each iteration of a classic iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We start from the motivation of algorithm unrolling and its design framework, which provides a guidance on how to unroll an iterative algo- rithm, followed by the case studies in wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The advantages and disadvantages of algorithm unrolling are summarized at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivations and Design Frameworks The success of “end-to-end learning” framework requires huge training datasets and significant computational cost due to a large number of training parameters of generic NNs while serving as a universal function approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The low training efficiency of generic NN hinders its application for dynamic and large-scale wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, in future 6G wireless networks with scenarios where abundant high-quality training samples such as CSI are unavailable, the performance of general DNN may significantly degrade and even underper- form traditional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, the generic NN is treated as a “black-box”, where the functionality of each layer and the performance guarantees of NN are hard to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The lack of interpretability of black-box DNNs can be a serious limitation in contrast with optimization-based approaches with theoretical guarantees in wireless networks, where reliability and predictability are of vital importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address these limitations, algorithm unrolling is an emerging method that provides a concrete and systematic connection between classic iterative algorithms and deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By unfolding an iterative algorithm with algorithm parameters transferred to training parameters of NN, the unrolled NN enables inter- pretability of each layer and even theoretical guarantees are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the potential in developing efficient, high- performance and theoretical guaranteed NN using reasonably sized training sets, it is a pleasant surprise that algorithm unrolling rapidly grows in both theoretic investigations and practical applications [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Algorithm unrolling was first introduced by Gregor and LeCun [42] to accelerate the ISTA for improving the com- putational efficiency of sparse coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The basic idea is to map each ISTA iteration to a neural network layer and then stack the layers together, which can be viewed as executing an ISTA iteration multiple times by a layer-wise neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The same techniques can be further applied to general iterative algorithms, in which the update form is given by xt+1 = g(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' θt), t = 0, 1, 2, · · · , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (1) Here, xt ∈ Rn, t = 1, · · · , T are the iterative variable vector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the signal to be recovered or the variable to be optimized), g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ·) : Rn → Rn is the iterative function of a specific iteration algorithm, and θt ∈ Rm, t = 1, · · · , T are trainable parameters (including model parameters and regularization coefficients) of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The overall prin- ciple of algorithm unrolling is to unroll a specific iterative algorithm into a deep network by mapping each iteration function g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ·) into a single network layer and stacking a finite number of layers together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The forward procedure of NN is equivalent to the execution of the iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The unrolled network architecture thus depends on the original iterative algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the unrolled ISTA turns to be an recurrent neural network (RNN) [42]), as the single network layer shares the same structure of iteration function g(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The details for the algorithm unrolling are illustrated in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The trainable parameters θt ∈ Rm, t = 1, · · · , T can be learned through end-to-end approach: minimize ΘT L � xT +1(ΘT ) � , (2) where L(·) is the loss function for training, ΘT = {θt}T t=0 is the entire trainable parameters of total T-layer network and xT +1(·) is the output function of the unrolled network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the customized structure of NN, the end-to-end training may suffer from spurious local minima and gradient explosion or vanishment during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Instead of directly solving (2), the common adopted training strategy for unrolled networks is layer-wise method [43], which can achieve more efficient training due to the better parameter initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' That is, the whole training process can be divided into T sequential sub- training processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For the t-th sub-training process, we aim to refine the trainable parameters Θt, where a two-stage method is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The first stage is dedicated to optimize parameter θt individually, while the latter learns the whole Θt jointly by fixing the learned θt as initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the testing stage, feeding the data forward through the unrolled network with learned parameters is equivalent to executing the parameter- optimized iterative algorithm for a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In recent years, algorithm unrolling approaches have been extensively applied to solve various signal processing prob- lems, such as sparse and low rank regression, probabilistic graphical model, differential equations and quadratic opti- mization [44], and have been applied to a wide range of applications including but not limited to compressive sensing (CS) [45], deconvolution [46], and image processing [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The superior performance and high computational efficiency of algorithm unrolling have been proved in various domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' At a high level, algorithm unrolling methods take advantages of both optimization-based priors and data-driven learning ability of NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with generic black-box NNs, un- rolled NNs have much fewer parameters due to the inheri- tance of the structure and domain knowledge from a specific iterative algorithm, allowing it to benefit in terms of the computational efficiency, interpretability, generalizability and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Theoretical tools from the traditional optimization theory can be explored to describe the convergence behavior and performance guarantees of unrolled NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with iterative counterparts, algorithm unrolling enables improved performance due to its expanded representation capability by tuning trainable parameters of model-based algorithm through data-drive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the stringent requirements of large-scale 6G wireless networks on efficiency and reliability, algorithm unrolling has recently attracted much attention in wireless communications for solving sparse optimization problems and some non- 7 … for do end for Stacking One Layer Network Mapping Trainable Parameters Unrolled Network … Input Layer Hidden Layer Output Layer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The general framework of algorithm unrolling method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' convex optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following subsections, we review several representative cases of algorithm unrolling in wireless communication applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Table II summarizes the types of problems we will cover, the corresponding appli- cation topics, the underlying iterative algorithms and the type of trainable parameters of the unrolled methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 1: Signal Recovery Problems With the emergence of large-scale signal processing tech- niques in 6G systems, such as big data, massive IoT, massive access network, large-scale antennas, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', signal recovery, es- pecially sparse signal recovery, has been a widely encountered optimization problem in diverse wireless applications [55], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Consider the following widely accepted model for signal recovery in wireless networks y = Ax + w, (3) where y is the received vector or matrix at the BS, A is the measurement matrix (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the channel matrix in signal detection, beam selection matrix in beamspace channel esti- mation and pilot matrix in JADCE), w is Gaussian noise and x is the unknown signal to be recovered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', symbols in signal detection, channels in channel estimation and channels of active devices in JADCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When the x in (3) is a sparse signal, the well known ISTA has the following simple iterative expression: xt+1 = ηλ(xt+ 1 C AT (y − Axt)), where η, λ and C are threshold operator, regulation parameter and step size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, traditional ISTA suffers from high computation complexity for recovering sparse signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By a simple variable substitution to separate the inputs and output of network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', W t 1 = 1 C AT , W t 2 = I − 1 C AT A, and θt = λ) and parameterizing them as trainable factors, the algorithm unrolling approach maps the original ISTA into an unrolled RNN as follows with t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' , T − 1, xt+1 = ηθt � W t 1y + W t 2xt� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (4) This approach inherits the structure and domain knowledge of the ISTA-based algorithm, also improves the convergence rate and computational efficiency of original algorithm through end-to-end training, which can be extended for recovering signals with different sparse patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' group sparse pattern [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In summary, the imperfectness of practical data links and inherent various sparsity structures of practical signals render many conventional (sparse) signal recovery algorithms, such as ISTA, AMP, OMP, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', suffering from the unsatisfactory performance, slow convergence and high complexity when employed in practical large-scale wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the superior performance, algorithm unrolling methods have been applied to solve signal recovery problems in the applications of signal detection [11], [19], [32], [48], channel estimation [12], [20], [49] and JADCE [14], [50], [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Data Detection: Data detection at the receiver has been a challenging task due to the wireless channel fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Con- ventional data detection techniques, such as linear detectors [57], SDR [58], sphere decoding [59], AMP [60], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', depend on the mathematical models of wireless channels and usually assume perfect CSI at the receiver which is replaced by its es- timate in the practical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The channel dependence of data detection undermines the detector performance in practical wireless systems where the channels can be highly complex, poorly understood, hard to be modeled or with estimation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, the traditional detection techniques suffer from significant complexity-reliability trade-off, which cannot be efficiently implemented at the scale required by 6G massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome these limitations, ML-based re- ceivers have been extensively studied, which learn the mapping from the channel outputs to the transmitted symbols in a data- driven manner, and several algorithm unrolling approaches for MIMO detection including DetNet [32], OAMPNet [19], MMNet [11] and CMDNet [48] were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When assuming perfect CSI at receiver, Samuel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [32] unfolded the projected gradient descent algorithm called Det- Net by treating the gradient step sizes as learned parameters, followed by common MLPs for improving expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the architecture of DetNet does not contain the properties of iterative methods, leading to high complexity and poor interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To further improve detection performance with imperfect CSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' a model-driven unrolled orthogonal AMP (OAMP) network called OAMPNet [19] was proposed to jointly estimate channel and detect signal by taking channel statistics and channel estimation errors into consideration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='TABLE II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ALGORITHM UNROLLING METHODS FOR WIRELESS APPLICATIONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Target Problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Unrolled Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Algorithms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Type of Trained Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Signal Recovery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Problems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MIMO Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DetNet [32] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='PGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DNN weight and gradient step-size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='OAMPNet [19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='OAMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Step-size and nonlinear estimator factor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MMNet [11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ISTA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Scale factor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CMDNet [48] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='CMD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Step-size and gradient scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel Estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='GM-LAMP [20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='AMP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Linear transform coefficient and shrinkage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='mpNet [49] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='MP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Linear transform coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ADMM-OGChannelNet [12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ADMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Linear transform coefficient and step-size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='JCADE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DADMM [50] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='ADMM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Step-size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' shrinkage parameter and auxiliary variable FAT-DL [51] AMP Denoiser factor and scale factor LISTA-GS [14] ISTA-GS Linear transform coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' step-size and shrinkage parameter Non-convex Sum-Utility Maximization Problems Precoding Design IAIDNN [21] WMMSE Linear transform coefficient and trainable offset PDD-TJAPB [52] WMMSE Long-term variable UPGDNet [53] PGD DNN weight and scale factor Power Control PDD-SSCA [54] WMMSE Short-term variable which has only a few trainable parameters and can be trained with a much smaller data set compared with DetNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DetNet and OAMPNet are both trained offline with simple model assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Gaussian channels, low-order mod- ulation schemes) and can suffer a large performance gap for realistic channels [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome these limitations, MMNet [11] was proposed by unrolling ISTA, which enables online training by exploiting the locality of realistic channels in both frequency and time domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, MMNet parameterizes the linear transformation in ISTA, followed by the estimation of noise variance for different transmitted symbols in the nonlinear denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To further support fast online training, an unrolled concrete maximum a-posteriori detection network (CMDNet) was proposed by [48], which is theoretically revealed to be able to learn the approximate probabilities of the individual optimal detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Massive MIMO Channel Estimation: In order to opti- mize the data rate and energy consumption trade-off, channel estimation is crucial in communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As there are only a few dominant propagation paths despite the high dimension of the channel, massive MIMO channel estimation problems turn out to be sparse signal recovery problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Classical CS-based algorithms suffer from high computational complexity and poor estimation accuracy in low signal to noise ratio (SNR) regions, especially for large-scale antenna arrays in wide-band system with complex sparse structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Recent years witness an emergence of a number of unrolled channel estimation networks benefiting from both domain knowledge and DL, such as GM-LAMP [20], mpNet [49] and ADMM- OGChannelNet [12] for MIMO channel estimation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In millimeter wave (mmWave) MIMO systems, GM-LAMP [20] unrolls AMP by deriving a new shrinkage function based on the Gaussian mixture prior information of beamspace channels to improve the estimation accuracy of AMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address the basis mismatch issue in off-grid mmWave channel estimation problem, deep unrolled network architecture ADM- MOGChannelNet [12] was proposed by mapping the data flow to the iterative procedures of ADMMOG algorithm, which is computationally more efficient with performance guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, due to the intrinsic supervised nature, these meth- ods all require collecting a database of clean channels for offline training, which may hinder their practical applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When ground-truth channel data are unavailable, an unrolled matching pursuit mpNet [49] was designed for MIMO channel estimation in an unsupervised way, which can automatically correct its channel estimation algorithm based on incoming data with the advantage of training online due to its simple network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Joint Activity Detection and Channel Estimation: The JADCE problem in grant-free massive access scenario [14] is a typical sparse optimization problem in wireless networks, as the sporadic transmission leads the joint activity and channel matrix to exhibit group-sparse pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' On the other hand, solv- ing JADCE problem also becomes more challenging for large- scale networks with large-scale antenna arrays and massive number of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Considering a multi-antenna BS and a large number of devices, the signal model in JADCE can be written as Y = AX + W , where A denotes the pilot matrix and X = ΛH is the device state matrix with diagonal activity matrix Λ and channel matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To recover the group-sparse device state matrix X with improved estimation accuracy and low computational complexity, extended unrolled versions of 9 ADMM-based [50], AMP-based [51] and ISTA-based [14] frameworks were proposed, respectively, for JADCE problems in massive access scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Assuming the device state matrix enjoys Bernoulli-Gaussian mixture distribution, an AMP-based unrolled network with dimension reduction module was proposed by [51] to reduce the length of pilot sequences and computational complexity for JADCE problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To directly exploit group sparsity, other studies [14], [50] focused on solving the ℓ2,1 regularized group least absolute shrinkage and selection operator (LASSO) problem in a model-driven DL approach, which does not depend on any prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [50] unrolled ADMM to solve the group LASSO, where the network parameters are optimally learned using the stochas- tic gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the advantages of fast convergence rate, high robustness and theoretical guarantees, ISTA-based [14] algorithm unrolling framework was proposed by extending LASSO-based decoder to group LASSO to circumvent the high computational cost of classic ISTA and poor algorithm robustness of AMP simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 2: Non-Convex Sum-Utility Maximization Problems A wide variety of resource management problems are di- rectly or indirectly reliant on the sum-utility maximization problems, which aim at maximizing the system sum-utility (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' sum-rate) subjecting to the transmit power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The general formulation with K devices and one BS can be expressed as maximize V U (R1(V ), · · · , RK(V )) (5a) subject to Q(V ) ≤ P, (5b) where V denotes the resource (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', precoding matrix at the BS) to be optimized, U(·) is the network utility function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', weighted-sum function), Rk(·) represents the achievable rate for device k and Q(·) ≤ P is the power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Unfortunately, most of the sum-utility maximization problems are non-convex and very difficult to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition to the direct parameterization of the iterative algorithm into a neural network as in Section II-B, the al- gorithm unrolling methods can also use trainable parameters to approximate the complex operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', matrix inversion) in the iterative algorithm to achieve the purpose of reducing the computational complexity for wireless applications such as precoding design [21], [52], [53] and power control [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The iterative WMMSE algorithm is one of the most representative algorithms for sum-rate maximization problems [61], which is guaranteed to converge to a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The general form of WMMSE is given by, U t = Ft(V t−1), W t = Gt(U t, V t−1), V t = Jt(U t, W t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (6) U, W are introduced auxiliary variables and Ft(·), Gt(·), Jt(·) are the iterative mapping functions at the t-th WMMSE iteration, where Ft(·) and Jt(·) contain computationally inten- sive matrix inversion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By using trainable parameters to approximate the matrix inversion and matrix multiplication functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' unrolled WMMSE enjoys low computational com- plexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' whose t-th layer network can be written as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' U t = ˜Ft(V t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t) + Ou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t (7a) W t = ˜Gt(U t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' V t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Xw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t) (7b) V t = ˜Jt(U t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t) + Ov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (7c) where trainable parameters {Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' {Xw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t} and {Xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t} are used to approximate the corresponding inversed matrix in original Ft(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Gt(·) and Jt(·) with Ou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t and Ov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t as the trainable offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A deep-unfolding framework IAIDNN was proposed in [21], where a number of trainable parameters are introduced to replace the high-complexity matrix inversion operations in classic WMMSE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Beneficial from both optimal per- formance of WMMSE algorithm and extensive representation power of DNNs, IAIDNN achieves the performance of the iterative WMMSE algorithm with much lower computational complexity and a smaller number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [52], an unrolled WMMSE approach was also proposed to solve a short-term sub-problem decomposed by original non-convex stochastic problem with low complexity for precoding design in reconfigurable intelligent surfaces (RIS)-aided communi- cation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such unrolled WMMSE network was also adopted in part of [54] to approximate the iterative WMMSE algorithm with low training complexity and reduced memory overhead, which is adopted as a short-term sub-algorithm in a two-stage stochastic optimization problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', power minimization for two-timescale hybrid beamforming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The above unrolled networks are all trained under a supervised way due to the complex structure of WMMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' An unsupervised deep unrolling framework based on projection gradient descent called UPGDNet was proposed in [53] to solve the sum-rate maximization problems in the scenario of multiuser ultra- reliable low-latency communications (URLLC) with finite block length transmission, which demonstrates a satisfactory generalization ability and low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages of Algorithm Unrolling The advantages of algorithm unrolling methods are summa- rized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Higher Computational Efficiency: Compared with the end-to-end learning based on generic NN, reduced number of training parameters in unrolled NN can significantly boost the computational efficiency of NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, due to the incorporation of domain-specific knowledge through unrolling, the training of unrolled NN is faster and requires fewer training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Algorithm unrolling methods can significantly improve the convergence rate of original iterative algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', LISTA-GS [14] converges less than 10 iterations while ISTA needs more than hundreds iterations), and also can reduce the complexity of one-step iterative process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', IAIDNN [21] re- duces the computational complexity of WMMSE from O(n3) to O(n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='73) with n denoting number of system parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 10 2) Better Learning Performance: By extending the itera- tive counterparts and training using datasets, the algorithm unrolling can achieve superior performance compared with the conventional iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, mpNet [49] and LISTA-GS [14] achieve more accurate estimation performance (more than 5dB normalized mean-squared error enhancement) compared with ISTA-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Interpretability and Theoretical Analysis: Inherited from the traditional iterative algorithm, the behavious of each unrolled network layer is interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Algorithm unrolling methods, to some extent, build a bridge between deep learning and iterative formulations, where the optimization tools can be used to define the optimal learned parameters that leading to fastest convergence rate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the optimal parameters defined in LISTA-GS guarantee the linear convergence rate for recov- ering group-sparse matrices [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, there are some disadvantages of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' First, for complex iterative algorithms when highly nonlinear or non-smooth operations are involved, it is hard to develop efficient NN to unfold the complicated iterative operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Second, for iterative algorithms with slow convergence, the depth of unrolled NN will be large, which can easily suffer from gradient explosion or vanishment during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Third, even if the extended representation ability of algorithm unrolling, its convergence is hard to be guaranteed for complex iterative algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' unrolled WMMSE and unrolled ADMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, its performance is restricted by the iterative algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Analyzing the impact of trainable parameters on the convergence and learning accuracy is also challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' LEARNING TO BRANCH-AND-BOUND Many important applications in wireless networks involve complicated combinatorial problems, whose optimal solutions are hard to obtain efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A learning-based BB algorithm, namely LBB, is introduced in this section to tackle the combinatorial problem with low computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Instead of end-to-end learning, the LBB replaces the complex pruning step of traditional BB with NN to accelerate searching for optimal solutions in feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We first introduce the traditional globally-optimal BB algorithm to solve a general combinatorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then we introduce a learned BB to learn the optimal pruning policy of BB in a data-driven manner, which will be followed by the case studies in wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The advantages and disadvantages of LBB are summarized at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Global Optimal Branch-and-Bound BB algorithm can find global optimal solutions for non- convex combinatorial problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', discrete and mixed com- binatorial optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' It implicitly enumerates all possible solutions by dividing the original problem into a se- ries of sub-problems (branch step) and systematically discards the non-promising sub-problems based on lower bounds or upper bounds (bound step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the branch step is to partition the feasible region into smaller subregions in a tree structure, where the root node is the original problem and the leaf node represents the subproblem over the corresponding subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The bound step uses the features obtained at each node to prune off the subregions that do not contain the optimal solution, until BB eventually converges on an exact result [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To guide and accelerate the searching process, the branch step involves node and variable selection determining which node to explore and the fractional variables to branch on in next iteration, whereas the bound step consists of two main policies: evaluating the bounds of selected nodes by solving the subproblems and pruning policy which determines whether to explore the subregion corresponding to the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The pruning policy at bound step depends on the optimal solution and optimal objective value of relaxed subproblem at each node, where the constraints for the undetermined discrete variables are relaxed into convex continuous constraints and then various convex solvers can be applied to solve the relaxed convex subproblem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', linear programs (LPs), second order cone programs (SOCPs) or semidefinite programs (SDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, binary constraints can be relaxed into box constraints and the non-convex complex modulus constraints can be relaxed to their convex envelopes [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By learning the time-consuming components of BB al- gorithm using NN, the learning-based BB can significantly reduce computational complexity with near optimal perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Authors in [63] provided a survey of learning-based BB techniques regarding to the key decisions in BB, such as learning-based branching and learning-based pruning, and summarized the merits and flaws of different learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In original branch step of BB algorithm, various branching rules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', strong branching, hybrid branching) are used in branch variable selection by calculating the score of can- didate variables to indicate their qualities and then picking the variable with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this way, enormous branch decisions are required while a single bad one could sharply increase the size of search tree without improvement in the learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To accelerate the branching rules, imitation learning was adopted in [64], [65] to learn a fast auxiliary branching policy by approximating the traditional branching rules using expert experience, which can outperform the initial expert with carefully designed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition, such approach leads to a simpler learning task with smaller Vapnik-Chervonenkis dimension, which only needs a few training samples, and thereby speeding up the training process consequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome the complex feature calculation at each selected node, authors in [66] encoded branching policies into a graph convolutional network (GCN), where features on the graph can be efficiently extracted by vari- ous message passing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To solve large-scale mixed integer programming (MIP), the authors in [67] constructed two corresponding GCN components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' neural diving and neural branching) to learn a branching policy for enhancing the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Even with the learned branching policy, the computational complexity of BB algorithm is generally exponential w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the number of optimize variables due to the inefficient pruning policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on this observation, other studies [68], [69] focused on the supervised learning of optimal pruning policy to solve large-scale MIP efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the next subsection, 11 we shall introduce the motivations and design frameworks of pruning policy learning-based LBB in large-scale 6G wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivations and Design Frameworks of LBB In 6G wireless networks, typical resource allocation prob- lems, such as subcarrier allocation in orthogonal frequency division multiple access (OFDMA) [70], user association [71] and access point selection in could-radio access network (C- RAN) [72], can be formulated into mixed combinatorial opti- mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the rapid growth of wireless network scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the massive number of IoT devices and the massive number of antennas), the dimension of optimization variable becomes very large, which makes learning-based BB a promising approach to tackle the exponential complexity of traditional BB while maintaining the global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this subsection, we highlight a promising learning strategy for node pruning in BB algorithm proposed in [68], [69], termed LBB, which has been shown to achieve low computational complexity with near-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The main idea of LBB is to model the tree search process as a sequential decision problem because whether or not exploring the subregion of the tree node corresponds to the preserve decision or prune decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such problem can be efficiently solved by a binary classifier with problem features as the input and decision states as the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The procedure of LBB includes training data generation, feature design, binary classifier learning and searching space controlling, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Training Data Generation and Feature Design: The original BB algorithm is directly applied to generate the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the problem parameters of each randomly generated wireless network are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For each set of problem parameters, the original BB algorithm is adopted to obtain the optimal solution and the features of all explored nodes are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the nodes whose feasible sets contain the optimal solution are labeled as preserve while the others are labeled as prune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The input features of the NN are also of vital importance for training a good classifier in LBB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Features are divided into tow categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', problem- independent features and problem-dependent features [15], [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Problem-independent features correspond to the structure of the binary tree generated by the BB algorithm, which includes node features computed from the current node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the depth of current node), branching features computed from the branching variable of the current node (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the branching variable’s value at the current node) and tree features computed from the BB search tree (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', global upper and lower bounds) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' While problem-dependent features correspond to the domain knowledge of different specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Typically in wireless communications, the CSI feature and some de- scriptions of the radio resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power feature) are usually considered as problem-dependent features to exploit the domain knowledge for efficient policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By feeding the problem dependent features into the classifier learning, the LBB can avoid solving relaxed problems at each branching node, which can further reduce the computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Binary Classifier Learning: The binary classifier de- termines whether to preserve a node or not and a good classifier can prune as many non-optimal nodes as possible to minimize the search time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Standard classifiers, such as logistic regression [74] and support vector machine (SVM) [75], are inefficient for high-dimensional data classification with tangled mapping between input and output [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To effectively capture the complicated relationship between the input features and output classification labels, MLP can be employed for binary classifier learning in LBB due to its powerful expression ability [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In each layer of MLP, the input is multiplied with a learned weight matrix, followed by a rectified linear unit function (Relu) as activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The output layer employs the soft-max function to calculate the probability of each class, while the weighted cross entropy is adopted as loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Target Problems: MINLPs,CMCPs Optimal Nodes Nonoptimal Nodes Pruned Nodes Pruning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Learning to branch-and-bound method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Training Sample Unbalance and Searching Space Con- trolling: It is observed that the number of non-optimal (pruned) nodes is much larger than the number of optimal (preserved) nodes and the mistakes at early stages can have greater impact on the performance during BB searching pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Thus, to enable efficient network training and achieve a good classification performance, weighted cross entropy was adopted in [73], where the higher weights are assigned to the nodes labeled as preserve and the nodes with small depth in the training dataset to raise the priority of these training samples in the learning of NN [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Since too aggressive pruning policy may lead to no feasible solutions and too moderate pruning policy may lead to abundant preserved nodes, a threshold was adopted to control the searching space dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, a higher threshold for the class pruning will preserve more nodes than that with a lower threshold, leading to a larger searching space and better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The searching space controlling can guarantee the feasibility of LBB with reduced computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The overall frameworks of BB and LBB are illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the advantages of near-optimality and low com- putational complexity for challenging non-convex and combi- natorial problems, LBB methods have gain much traction in the context of large-scale wireless networks to solve various resource allocation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following subsections, we review several representatives of non-convex combinatorial optimization problem with extensive applications in wireless 12 networks to fully reveal the power of LBB for efficient and high-quality resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 1: Mixed Integer Nonlinear Problems In wireless networks, typical resource management prob- lems, such as user selection, user association, spectrum al- location, computational offloading, interference and power management, can be formulated into a mixed integer nonlinear problem (MINLP), which is general NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The generic formulation is given by [15] MINLP : minimize a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='w f(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' w) (8a) subject to Q(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' w) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (8b) ai ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' wi ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (8c) where f(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ·) is the objective function (usually non-linear),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' such as power consumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' sum rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' communication delay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' etc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' discrete-valued variable ai and continuous-valued variable wi are the elements of a and w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' such as sub-carrier index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' user index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' device index for discrete variable and beamforming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power for continuous variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and Q(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ·) denotes certain constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the quality of service (QoS) and power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A typical example of MINLP is network power mini- mization problem in C-RAN, which consists of binary vari- ables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the selection of remote radio heads and front- haul links), continuous variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', downlink beamform- ing coefficients), QoS constraints and power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To get the near-optimal performance with affordable complexity, LBB via imitation learning was proposed in [15], where the depth-first-search was adopted as the branch variable selection rule which always choses the first undetermined node during variable selection process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In feature design, besides problem- independent features such as the depth of node, the local upper bound of node, current global lower bound and so on, the CSI feature and power feature are designed as problem-dependent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' LBB algorithm was shown to achieve success us- ing only tens to hundreds of training samples for solving MINLPs in the applications of resource allocation problem in device-to-device (D2D) communications [73] and mobile edge computing [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, in [77], a learning strategy for node pruning in BB algorithm was proposed for the offloading resource assignment in mobile edge computing, where DNN was applied to approximate the unknown mapping between the attributes of BB tree nodes and the pruning decisions in the BB tree search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To further reduce the sample complexity, imitation learning was adopted in [73] to solve MINLPs of resource allocation in D2D systems, where DAgger algorithm was employed to correct the mistakes the learned policy makes by iteratively collecting data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, instead of using the training data only once, the pruning policy πt learned in imitation learning shall explore all the tree nodes and generate their corresponding features at iteration t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then based on these datasets, a new prune policy πt+1 can be learned at next iteration to correct the mistakes made by πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To adapt to time-varying network settings, a transfer learning via self- imitation method was adopted in [15] to quickly adapt the learned pruning policy in LBB to the new task with reduced training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 2: Non-Convex Complex Modulus Constrained Problems Many resource management problems in wireless networks can be formulated as non-convex complex modulus con- strained problems (CMCPs), which is general NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The generic formulation of CMCP is given by [78], CMCP : minimize a,w g(w) (9a) subject to |bH i w + ci| ≥ 1, ∀i, (9b) arg (bH i w + ci) ∈ Ai, ∀i, (9c) where g(·) : CN → R is a convex objective function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', power consumption), constraint (9b) denotes the non- convex minimum complex modulus constraints on N linear transformations of the optimization variable w ∈ CN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the transmitted symbol vector and the beamformimg vector) with coefficients bi ∈ CN and ci ∈ C, and constraint (9c) represents the corresponding argument constraints where each argument set Ai can be continuous or discrete (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', box constraints or discrete phase sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' One toy example of CMCPs in wireless networks is the QoS-constrained multi-cast beamforming optimization prob- lem in multiple-input single-output (MISO) downlink trans- mission, which minimizes the total transmit power at the BS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', minw ∥w∥2 2) subject to individual SNR constraints for all devices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', |hH k w| ≥ 1, ∀k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The formulated problem corresponds to the standard CMCP in (9) by letting bk = hk, ck = 0 and Ak = [0, 2π], ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Other applications of CMCPs include MIMO detection and passive beamforming in RIS- assisted systems [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Unlike the BB algorithms for solving MINLPs with discrete branching variables, the branching of BB algorithm for solving CMCP is based on argument sets to deal with the continuous variables [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the continuous argument set can lead to unbounded extension of tree search in BB algorithm and tremendous number of binary classification tasks in the searching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, it is difficult to find the optimal solution of the CMCP using only one classifier [15], [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address this challenge, ensemble learning was applied in [78] to train multiple classifiers in LBB, which are further combined to achieve better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To facilitate the multi-classifier learning and address the data unbalance in LBB, [78] devised the under-sampling training and ensemble testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In under-sampling training, indi- vidual classifier is trained on each of the data subsets sampled both from the majority set and minority set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In ensemble testing, LBB is executed multiple times using learned multiple classifiers in parallel to choose the best solution from all the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Regarding to the feature design, besides problem- independent features such as local node features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the argument set A) and global tree features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the global lower bound), the CSI, the SNR, the received signal and the complex modulus constraint can be designed as problem-dependent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 13 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages LBB algorithm was shown to achieve near-optimal per- formance and meanwhile substantially reduce computational complexity using only tens to hundreds of training sam- ples, due to the inheritance of the structure of traditional BB algorithm and the exploitation of DL techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' If the parameters and features are properly chosen, the number of relaxed problems to be solved can be reduced from O(2L) in BB to O(L) in LBB for MINLPs [15], where L denotes the number of integer variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the training of LBB depends on the labels generated from traditional BB, which can induce great computational burden to generate the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Meanwhile, feature design in LBB is not supported by theory and hence different designs can lead to different results of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' GRAPH NEURAL NETWORK FOR STRUCTURED OPTIMIZATION The graph-structured topology of wireless network enables the successful usage of GNN to solve a broad range of design problems over the wireless networks [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a specialized NN for graph-structured data, GNN can exploit the domain knowl- edge of various applications to achieve near-optimal learning performance with good scalability and generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this section, we commence with the framework of GNN for structured optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then we illustrate the graph modeling of optimization problems in wireless networks, after which, several applications of GNN are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The advantages and disadvantages of GNN-based solution in wireless networks are summarized at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Principles of Graph Neural Network Recently, a growing number of applications have emerged possessing graph-structured data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', social networks and transportation networks, with high dimensional features, active interactions between graph nodes and potentially time-varying structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The emerging GNN can effectively incorporate the graph structure into the architecture of NN to model the node properties and the relationships between nodes to explore the hidden features in graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Hence, GNN not only features for good scalability to large-scale graphs and good generalizability to dynamic graph structures, but also can achieve near-optimal performance with more efficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Traditionally, a graph-structured data can be mathematically represented as a pair G = (V, E), where V is the set of nodes and E is the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For a node, its (1-hop) neighborhood is defined as the set of all nodes with edges connected to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' These edges can be represented by an adjacency matrix A, where Ai,j = 1 if and only if edges (i, j) ∈ E for all nodes i, j ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The graph is undirected if A is symmetric, otherwise it is directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The properties associated to the nodes and edges are important for learning over graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Denote zi as the node features associated with node i ∈ V , ei,j as the edge features associated with edge (i, j) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Since the graph structure may be changed by the permutation of nodes, a key desideratum for designing GNNs is that the devised GNN should satisfy permutation invariance or permutation equivariance [81], which can precisely capture the internal structure of graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As illustrated in Figure 4, permutation invariance means that the output of GNN does not depend on the node order used to encode adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Permutation equivariance means that the output of GNN is permuted according to the same permutation as the input node order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such properties enable faster training, less training samples, better scalability and generalizability of GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, compared with MLP, the parameters of GNN can be shared for graphs with varying sizes and permuted input, thereby achiev- ing good generalizability, while an MLP has to be retrained when the topology is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' More precisely, authors in [33] theoretically showed that the GNN’s generalization error and required number of training samples are O(n) and O(n2) lower than those of MLPs, where n is the number of nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' GNN implements the learning over graph by exacting the neighbor information to enrich the feature of each node and spreading these features over the graph according to the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The framework of GNN is illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Generally in a vanilla GNN with multiple hidden layers, the output of last GNN layer contains the processed information of all nodes and edges, which can be used for classification or prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In each GNN layer, each node aggregates the information of its neighbors to update its hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, let d(ℓ) k be the hidden state of the k-th node at the ℓ-th GNN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Each GNN layer contains the aggregation and combination step as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Aggregation Step: The aggregation step aims to update the node’s hidden state with its neighbors’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Typi- cally, the k-th node uses an NN to aggregate its neighbors’ out- puts of the previous layer (the (ℓ−1)-th layer), followed by a pooling function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', element-wise max-pooling or element- wise mean-pooling) that is invariant to the permutation of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The update is given by a(ℓ) k = PLj∈N (k) � f (ℓ) 1 (d(ℓ−1) k , d(ℓ−1) j , ej,k|Θ(ℓ) 1 ) � , (10) where PLj∈N (k)(·) denotes the pooling function used for aggregating the outputs of the nodes in N(k), N(k) denotes the set of neighboring nodes of node k, f (ℓ) 1 (·|Θ(ℓ) 1 ) is the aggregation function implemented using MLP with model parameters Θ(ℓ) 1 at the ℓ-th layer and ej,k is the feature of edge (j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Combination Step: After obtaining the aggregated in- formation, another combination function is applied to process information and update the hidden state at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifi- cally, the aggregated information is combined with the node’s own information as follows: d(ℓ) k = f (ℓ) 2 � d(ℓ−1) k , a(ℓ) k , zk|Θ(ℓ) 2 � , (11) where f (ℓ) 2 (·|Θ(ℓ) 2 ) represents parameterized combination function with model parameters Θ(ℓ) 2 and zk is the feature of node k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To improve the learning ability of GNN over different graphs, the aggregation function f1, the combination function f2 and the pooling function PL should be carefully designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Authors in [33] gave general guidelines for designing these 14 Hidden Layer Original Order Output Layer Input Layer Permutation Permutation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Illustration of permutation equivariance and permutation invari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Different colors of nodes represent different orders of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Devices BSs Cell-free Network D2D Network Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Examples of the wireless communication graph modeling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' cell-free networks and D2D networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' First, the message aggregation and combination should be simplified to reduce the complexity of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Second, different pooling functions are suitable for different graphs and optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the max- pooling function is suitable when the neighbors’ influence is sparse or the problem parameters are noisy, because it focuses on the neighbors that are most influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' While the sum- pooling function gives summary statistics of the neighbors, which works well when we aim to obtain a summary of the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Third, the aggregation and combination function should be properly designed to satisfy the permutation invari- ance or permutation equivariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Last, the input em- bedding is of vital importance for information representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, it is often preferable to employ an input embedding neural network to lift or compress the features into a proper dimension to search for a balance between training complexity and learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Graph Optimization Problems in Wireless Networks In wireless networks, the relationship between users, BSs and even antennas spontaneously yields to a wireless com- munication graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, a wireless network can be modeled as a directed/undirected graph with node and edge features, where communication devices (users and BSs) can be treated as nodes, channels between devices can be treated as directed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' An edge (i, j) exists if there are interdepen- dencies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', communication link, interference link or other connectivity patterns, from node i to node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The features associated with the node represent the properties of devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', user’s weight in weighted sum rate maximization, while the features associated with the edge stand for the link proper- ties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', channel gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Mathematically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the general form of optimization problem on wireless communication graph can be expressed as [22] minimize X f(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A) (12a) subject to Q(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (12b) where f(·) is the objective function (usually non-convex),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' X denotes the collection of optimization variables assigned to all nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Z denotes the node feature matrix which can model the heterogeneity of node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and A is the edge feature tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' which can incorporate more comprehensive link properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Q(·) is the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The graph optimization problem is general enough to cover most of the resource management problems in wireless networks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', power control, beamforming design, link scheduling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The graph-structured optimization problem (12) defined on wireless communication graph satisfies the permutation equiv- ariance and permutation invariance properties [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Namely, the objective function f and constraint function Q are invariant to the permutation of the device indices, while the output of GNN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the optimal variable X would be permuted in the same way as the permutation of device orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To solve the graph optimization problem effectively, the GNNs which satisfy the permutation equivariance/invariance properties are favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To fully reveal the advantage of GNN-based solu- tion for (12), authors in [33] bridged the GNN with a class of COAs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', distributed message passing, to justify that for any graph optimization problem, there exists a GNN that can solve it from the algorithmic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with the COAs, the data driven GNN can achieve near-optimal performance with reduced computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with MLP, the superior performance of GNN in solving graph optimiza- tion problem in wireless networks is theoretically proven in [33] in terms of the optimization performance and sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Table III summarizes the wireless applications of GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on the topology of graph, the applications of GNN may cover resource management problems in D2D/cellular/cell- free/distributed network and other signal processing fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following subsections, we introduce several application examples of GNN, detailing the graph modeling of different problems and the GNN architecture developed to solve the respective optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='12:2812:2812:2815 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='TABLE III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='WIRELESS APPLICATIONS OF GRAPH NEURAL NETWORK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Wireless Issues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Ref ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Node Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Edge Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Cellular/Cell-free ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Beamforming ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Optimizations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[83] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Antenna / User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Beam feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[84] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='User / RIS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Link between nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Received pilots / Mean of all user features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='\\ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Power Allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[85] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='BS / User ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='State information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='D2D Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Power Allocation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [82] Transceiver pair Directed interference link The state of the direct channel and the weight of transceiver pair The states of the interference channel Link Scheduling [86] Transceiver pair Directed interference link Channel gains or distance Channel gains or distance [87],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [88] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Interference link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Queue length and link rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='\\ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Distributed Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Decentralized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[89] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Transmitter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Application state ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[90] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Receiver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Direct link / Interference link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Proportional-fairness ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='\\ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[91] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Binary internal characteristics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='\\ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Heterogeneous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Different device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Meta-path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Subchannel gain and power budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Distance between nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[92] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Antenna or user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Communication link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='State information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Other Signal Processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Applications ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Mobile Traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[93]–[95] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Road connecting the regions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Historical traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='\\ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel Tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[96] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Element of channel vector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Link between channel element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel coefficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Channel spatial correlation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 1: Graph Neural Networks in Cellular/Cell- Free Networks In cellular or cell free networks, we can treat the different communication devices, including antennas, user equipments (UEs), RISs, and BSs, as graph nodes and their interde- pendencies as graph edges, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we will provide some examples of GNN-based resource allocation in cellular or cell-free networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Beamforming Optimization: The multi-antenna beam- forming optimization problems were considered in [83], where a bipartite GNN framework consisting of antenna/user vertices and channel edges was proposed to improve the scalability and the generalization ability of DNN approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When consid- ering the RIS-assisted cellular networks, GNN is proposed in [84] to directly map the received pilots to the beamformers at the BS and reflective patterns at the RIS by solving sum-rate maximization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To model it as a graph optimization problem, the RIS and users are taken as graph nodes and the communication links between users and RIS are taken as the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The input features of user nodes are the received pilots and the input feature of RIS node is the mean of all user features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' After updating the input features through multiple GNN layers, the output features of user nodes will be mapped to the beamforming matrix at BS and the output feature of RIS node will be mapped to the reflective pattern of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To capitalize on the special properties of GNN, the GNN layers are designed such that the beamforming vectors corresponding to different users are permutation equivariant and the reflective pattern at RIS is permutation invariant for the change of the order of user channels, thereby enjoying the beneficial performance of GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Power Allocation: Consider a cellular/cell-free net- work consisting of one/multiple BSs and multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The cellular/cell-free system can be modeled as a fully connected bipartite graph, where the BSs and users are viewed as nodes, the communication links including direct links and interference links between BSs and users are regarded as edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the power allocation problem, the edge features are typically the channel coefficients and node features are the state information of user demand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the data arrival rate for traffic demand [85]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' After updating through GNN layers, the hidden states at the user nodes are mapped to the power values through learnable MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Many existing works [85] further developed advanced GNN algorithms to handle various adverse factors in wireless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, a random edge GNN (REGNN) was proposed in [85] to handle the fast fading channels in power allocation problem, where the underlying graph structure is a random variable drawn from a specific distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The authors in [85] further proposed an unsupervised model-free primal-dual learning method to address the general utility-constrained (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', binary power constraint) wireless resource allocation problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the sum-rate maximization problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 2: Graph Neural Networks in D2D Networks In D2D networks, supposing there are multiple transceiver pairs, each transceiver pair usually can be treated as one node, where the node features include the direct link CSI, the weight of each transmission pair and other related information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The interference link between transmission pairs can be treated as the edge, where the edge features include interference CSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The underlying graph, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 5, can be directed or undirected according to the definition of edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The GNNs can be effectively adopted to solve various resource allocation problems in D2D networks, such as power allocation [22], [82], [97], link scheduling [86]–[88] and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Power Allocation: Based on a standard D2D graph, IGCNet [82] and MPGNN [22] were proposed to apply GNN over the D2D graph for the power control problem, where the wireless graph modeling are designed to match the permutation equivariance property of interference channels, and the aggregation and combination functions are realized using MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To inherit the advantages of classic algorithm, an unrolled WMMSE algorithm was parameterized in [97] to design the GNN architecture for solving power allocation problem in a single-hop ad hoc interference network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 16 2) Link Scheduling: By exploiting and incorporating the underlying topology of wireless networks to learning al- gorithms, GNNs are well-suited for efficient scheduling of transmission links in wireless communications [86]–[88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To eliminate the expensive channel estimation stage, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [86] firstly constructed a graph embedding process for link scheduling in D2D networks, where each D2D pair is designed as a node while interference links among D2D pairs are the edges for efficient graph design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on Structure2Vec architecture, each node in the graph is represented by a low- dimensional feature vector, where the link classifier can be trained with high efficiency to decide whether a D2D pair should be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Other studies [87], [88] focused on link scheduling in wireless multi-hop networks, where the stream of packets from a source user (one node) to a destination user (the other node) may pass through multiple edges on the designed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [87], a trainable graph convolutional network (GCN) module was proposed to improve the opti- mality gap with the traditional greedy approaches for the NP- hard maximum weight independent set (MWIS) problem in link scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To reduce the delay of scheduling, a delay- oriented distributed scheduler based on GCNs was proposed in [88], in which multi-step lookahead backlogs and the network topology were fully captured by the node embedding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 3: Graph Neural Networks in Distributed Sys- tems The typical characteristics of distributed systems are de- centralized setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the transceivers only have knowledge of their local radio environment and make local decisions based on this information) and heterogeneous nature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', there are different types of nodes with different node features in a communication graph), where GNNs also have shown their superior performance and scalability for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Decentralized Networks: To address the intrinsic infor- mation delay and asynchrony between devices in decentralized cooperative wireless systems, a primal-dual learning based aggregation GNNs (Agg-GNNs) was proposed in [89] to de- sign localized resource policies with delay and asynchronous constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' This can be achieved by adding multiple network layers to process delayed information after signal aggregations in Agg-GNNs, which gather spatial and temporal-correlated information of the global wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Similar primal- dual learning method was adopted in [90] to learn resilient radio resource management (RRM) policies with adaptive per- user minimum-capacity constraints, which can adapt to the current network conditions via optimized slack variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By parameterizing the RRM policies using scalable GNNs based on the graph topology of wireless networks, negligible duality gap can be proved and superior trade-off between average rate and user fairness can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The parallel implementation of GNNs was discussed in [83] to facilitate the deployment of decentralized GNN in distributed MIMO configurations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', cell-free MIMO and fog radio access networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, authors in [91] analyzed the robustness of a decentralized GNN-based binary classifier for inference considering the imperfect fading channels and wireless noises in the exchange of local information between neighboring nodes, where a novel retransmission mechanism to enhance the prediction robust- ness was proposed under different communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Heterogeneous Networks: In heterogeneous networks, different types of communication subjectives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', users, ac- cess points, mobile stations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=') can be modeled as different types of nodes connected by different types of edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', transmission edges and inference edges) to indicate a more complex wireless communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A heterogeneous graph neural network (HGNN) was firstly proposed in [10] to characterize two different types of nodes (access points (APs) and mobile stations (MSs)) and various types of edges between nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', uplink/downlink transmission path and inter- AP/inter-MS interference path) in distributed cell-free massive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address the impact of heterogeneous nodes, an adaptive node embedding layer was proposed, where the node features of AP and MS are transformed by two embedding matrices to handle the varying input feature dimensions before the process of GNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Similar HGNN was considered in [98] for D2D resource allocation, which sets individual aggregation/update functions according to different relations between nodes to address network heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Different from the traditional GNN-based power allocation, whose out- puts are permutation equivariant to arbitrary permutations of users, author in [92] constructed a permutation equivariant heterogeneous GNN (PGNN) to learn the optimal power allocation policy in cellular networks whose outputs are only equivariant to some permutations of user nodes to precisely match the identified properties in the power allocation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' It shows that the PGNN achieves better learning performance in terms of sample efficiency, computational complexity and performance optimality due to the exploitation of the well- matched policy properties and the heterogeneous design of GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Other Signal Processing Applications 1) Mobile Traffic Prediction: Graph-based methods can also be applied to address large-scale mobile traffic prediction [93], [94], where the challenge is to exploit the time-evolving nature of mobile movements and the spatial relations of mobile traffic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Typically, a graph is constructed to characterize the spatial structure of the traffic data in different geometric regions by dividing the area into discrete grids, where node represents the region and edge represents the road connecting the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To capture the temporal correlations of traffic data, RNN-based [93], [95] and CNN-based [94] methods can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' in [95] proposed graph convolutional RNN for traffic prediction, where GCN and gated recurrent unit were used to exploit the spatial and temporal structure of the traffic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' in [93] constructed a spatial relation graph of traffic data to capture the near-far spatial correlation, and utilized an attention- based structural RNN to capture the temporal dependency and spatial relationship simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides the spatial- temporal structures of traffic sequences, the user’s mobility patterns have also been exploited in [94] by a graph-based temporal convolutional network for accurate traffic prediction, 17 where each node denotes a wireless AP, the directed edges indicate the movements of mobile users during the time steps of interest, and the temporal convolutional network layers further model the temporal trend of mobile data traffic on each AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Channel Tracking: The massive MIMO channels can be modeled as a graph, where GNN can extract the spatial correlations within the large-scale channels for efficient chan- nel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specially, different antennas are considered as nodes with their channel coefficients being node features, while the spatial relationships are modeled as edges with the channel spatial correlations being edge features for graph modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [96], a GNN-based channel tracking framework was designed, which contains an encoder fed with historical channel samples, a core network performing GNN updates and a decoder to decode the node and edge attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' It shows that the graph-structure captured GNN significantly outperforms feed-forward NNs for channel tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages The advantages of GNNs are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) High Scalability and Generalizability: GNNs enjoy promising scalability and generalization ability for two rea- sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' First, the permutation invariance and permutation equiv- ariance properties of GNNs enable the learned NN to adapt to large-scale and dynamic scenarios by exploiting the analogies or equivalent patterns between the training network topology and dynamic testing conditions automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Second, GNNs leverage the distributed message passing architecture to learn local relationships among graph nodes and combinatorial gen- eralization over graphs [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, GNNs can generalize to large-scale communication networks with varying sizes and permuted structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', more users, antennas, BSs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Good Learning Performance and High Computational Efficiency: Compared with generic NNs, GNNs are more suit- able for graph-structured data and distributed systems, which can exploit the task-specific knowledge for better learning performance with less training samples [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, GNNs also suffer from explanation, generalization and representation limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, GNNs cannot compute some important graph properties such as the longest or shortest cycle, diameter, or certain motifs [99], which are crucial for the theoretical performance analysis over graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' GNNs also show a poor learning performance when aggre- gating messages across a long path, and this situation cannot be improved by increasing the number of aggregation network layers in practice [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The theoretic understandings of GNNs are still in an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DEEP REINFORCEMENT LEARNING FOR STOCHASTIC OPTIMIZATION The long-term performance optimization in dynamic and uncertain wireless networks can be cast as a stochastic op- timization problem, where the network entities need to learn the optimal policies over time under system uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DRL has emerged as an efficient and powerful ML tool in address- ing the sequential decision-making problems in dynamic and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Reinforcement learning method in wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' large-scale networks without explicit models of transmission environment, which allows the agents to update the decision policies through interactions with the unknown environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this section, we start with the motivations of DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then we present the basic concepts of reinforcement learning (RL) and the categories of DRL techniques, followed by the case studies of DRL in wireless communication networks to fully reveal the power of DRL in solving stochastic optimization problems in dynamic large-scale wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The pros and cons are summarized at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivations of DRL With the emergence of diversified application scenarios and the ever-growing density of wireless devices in modern net- works, such as IoT, unmanned aerial vehicle (UAV), and mas- sive machine-type communication systems, the performance optimization in these networks becomes extremely compli- cated due to the dynamic and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The general trends towards intelligent, autonomous, self-adaptive, and decentralized networks have been indispensable, where the agents learn the optimal decision policies automatically based on local or minimal exchanged information to maximize the network performance over time in dynamic and large- scale modern emerging networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' This kind of problem can be modeled as a Markov decision process (MDP) and various techniques, such as dynamic programming [101] and RL can be employed to solve the MDP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, in stochas- tic and dynamic environments, it is infeasible to build an explicit mathematical model to fully capture the characteristics of the time-varying environments, which renders many model- based methods impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' RL is a machine learning technique that aims at maximizing the accumulated discounted reward of an MDP with collected experiences in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, through trial-and-error interactions between the agent and the environment, the RL enables the agent to establish a general long-term optimal control policy while keeping track of the real-time environmental dynamics for sequential optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Even though RL can effectively solve the MDP without explicit state transformation models, the exploration of an unknown environment consumes lots of time, especially for a highly dynamic and large-scale network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivated by the strong learning ability of DNN as a universal function approximator, the combination of RL and DNN, namely Observations and Rewards IndustrialIoT Intelligent RL agents Wireless Environment Resource Allocations18 DRL, demonstrates great success in complex and dynamic wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The DRL enjoys the fast learning ability of DNN to speedup the learning process of RL and the transfer ability of DNN to be scalable to large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' On the other hand, DRL benefits from the RL techniques to be adaptive to the dynamic condition and amenable to distributed implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The DRL technique is illustrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Before discussing the technical details of DRL, we will first introduce the basic knowledge of RL in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' RL and Categories of DRL MDP can be described with ⟨S, A, P, R, γ⟩, where S and A denote the state space and action space, respectively, P denotes the state transition probability, R and γ denote the reward function and the discount factor over the future reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The Markov property of state transition is expressed as P [s′ = s(t + 1) | s(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' , s(t), a(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' , a(t)] =P [s′ = s(t + 1) | s(t), a(t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (13) The intelligent agent determines its next move a(t) ∈ A according to its current state s(t) ∈ S as well as its learned policy π, which involves in the state transition of the environ- ment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', s(t + 1) ∼ P(s′ | s(t), a(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' At the end of each step, the agent receives a reward r(t) = R(s(t), a(t)) from the environment, stores the tuple ⟨s(t), a(t), r(t), s(t + 1)⟩ and updates its policy according to the corresponding state- action pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Note that P and R are generally unknown due to the randomness and the complexity of the environment, therefore we shall learn the optimal control policy with model- free RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The objective of RL is to find an optimal policy to maximize the expected accumulated discounted reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', maxπ Eπ ��∞ j=0 γjr(t + j) � , where π is the policy denoting the mapping from a state to an action, r(t) = R(s(t), π(s(t)) is the instantaneous reward obtained after the action a(t) = π(s(t)) is performed under state s(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, a good policy shall balance the instant reward in the current round and the potential dividend in the future rounds to achieve long-term optimality in sequential decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In each round of the decision process, the intelligent agent evaluates the value of the state under a policy π, which is defined as Vπ(s) = Eπ ��∞ j=0 γjr(t + j) | s ∈ S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The state value function Vπ(s) represents the discounted reward that could be achieved at initial state s following the policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Sim- ilarly, the state-action function can be defined as Qπ(s, a) = Eπ ��∞ j=0 γjr(t + j) | s ∈ S, a ∈ A � , which characterizes the expected discounted reward when starting from initial state s and action a, then following the policy π thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In par- ticular, there is a conversion relationship between state value function Vπ(s) = � a π(a|s)Qπ(s, a) and state-action func- tion Qπ(s, a) = � s′ P(s′|s, a) (R(s, a) + γVπ(s′)), where π(a|s) denotes the conditional probability of each action a under given state s following policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By substituting Qπ(s, a) into Vπ(s), we have the Bellman expectation equa- tion Vπ(s) = � a π(a|s) (R(s, a) + γ � s′ P(s′|s, a)Vπ(s′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Given the optimal policy π∗, we have the following Bellman optimality equation for V∗ and Q∗, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', Vπ∗(s) = max a � R(s, a) + γ � s′ P(s′|s, a)V (s′) � , (14) Qπ∗(s, a) = max a � R(s, a) + γ � s′ P(s′|s, a)Vπ∗(s′) � , (15) where the optimal action in each round shall be found to achieve the Bellman optimality equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When the dimension of state and action space is small, Q- learning [102] is a widely used algorithm in RL to find the optimal action sequences by visiting each action-state pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, in complex and large-scale networks, the state and action space can be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To accelerate the learning process, DNN can be embedded into the RL framework to approximate computational-expensive quantities with parame- terized functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on the types of quantities fitted by the DNN, the DRL techniques can be categorized into value-based and policy-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Value-Based DRL: The value-based DRL can be applied to solve MDP with discrete state/action space, where the value functions are approximated using DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The widely used method is deep Q-learning (DQL) [102], which implements a deep Q-network to fit the value of Q⋆ π(s, a) in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, DQL accepts the features of states as input and outputs the fitted action-state values for all actions, where the action with the largest state-action value is adopted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' a(t) = arg max a∈A Q(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To train the NN, a loss function defined as the mean square error (MSE) between a target output, calculated using target network parameters θ′, and the actual output, characterized by parameters θ, is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To reduce the oscillations, the target network θ′ is updated much slower than the actual network θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the learned Q values, the actions chosen at each time slot follow a widely used ϵ-greedy strategy, where the agent chooses the action randomly with probability ϵ and the action that maximizes the current action-state value with probability 1−ϵ to balance the exploration and exploitation in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Policy-Based DRL: The agents in the policy-based al- gorithms learn the policy mapping from the current state to the optimal action or the probability of each action directly through the DNN technique instead of evaluating the state- action values, which can be applied to both continuous and discrete action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, the neural networks are optimized by minimizing the following loss function, ρ(θ) = lim T →∞ 1 T T � t=1 rt = � s∈S dθ(s) � a∈A πθ(a|s)R(s, a), (16) where dθ(s) = limT →∞ P(st = s|s0) denotes the steady-state probability distribution under policy π, which is parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To facilitate the model parameter update, actor-critic framework, as in deep deterministic policy gradient (DDPG) [103], can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, the critic learns a parameterized state-action function Qω π(s, a) by using the 19 Bellman equation as in DQL, where a copy of the actual critic network can be adopted to calculate the target values with slowly updated weights to improve the learning stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' On the other hand, the actor learns a parameterized function πθ(a | s) specifying the current policy through mapping the states to a specified action, which can be obtained by maximizing the learned value function of the critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' TABLE IV DRL-ENABLED STOCHASTIC OPTIMIZATION IN WIRELESS NETWORKS Prolem Types DRL Methods Application Scenarios Refs IP Value-Based Intelligent Traffic [104]–[112] Discrete-Valued Power Control [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [113]–[116] Device Scheduling [117]–[119] Stochastic MINLP Policy-Based Mobile Edge Network Optimization [120]–[127] Space-Air-Ground-Integrated Network [128]–[132] DCOP MADRL Scalable Radio Resource Allocation [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [133]–[136] Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DRL-based approaches have attracted attentions of the wireless community to solve different types of wireless resource allocation problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' such as integer programming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' mixed-integer linear programming and sequential optimization problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' in a wide range of applications including but not limited to multi-access scheduling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' beamform- ing designs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and bandwidth allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In general, DRL-based approaches inherit the advantages of conventional control theorem with theoretical guarantees for convergence and opti- mality [102], [103], [137]–[139] in the long-term performance optimization, and the advantages of data-driven DL with high computational efficiency and excellent learning performance in complex, dynamic, and large-scale wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following subsections, we shall review several representative cases of the application of DRL in wireless resource allocation according to the types of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The covered cases are summarized in Table IV based on the optimization problem types, application scenarios, DRL methods, MDP elements, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 1: Stochastic Integer Programming Problems The stochastic integer programming (IP) problems involving discrete variables in dynamic scenarios have been quite com- mon in wireless resource allocation, which have the following general forms: maximize x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=',xT T � t=1 ctft(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' yt), (17a) subject to gt(xt) ≤ bt, (17b) yt ∼ P(y | yt−1, xt−1), (17c) xt ∈ Zn, (17d) where the objective function ft(·) can be the utility function of wireless networks, gt(·) can be the performance constraint functions, and the discrete-valued variables xt can be resource allocation variables, such as device scheduling, yt stands for the dynamic state of the environment with Markov evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a non-convex problem with NP-hardness, the COAs for solving IPs suffer from unintended performance degradation, high computational complexity and high requirements for datasets in the actual implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DL-based algorithms, such as LBB in Section III, can be employed to solve the IP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, LBB and DRL focus on different applica- tion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, LBB aims at solving one-shot IP problems, which cannot handle long-term constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Instead, DRL is specifically used to cope with long-term constrained stochastic optimization problems by adaptively interacting with the environment to learn the long-term optimal policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, LBB requires explicit models to achieve a good learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Instead, DRL approaches are model- free, which can successfully perform for unknown dynamics through autonomous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In conjunction with the good robustness, DRL provides great potentials of scalability in han- dling high-dimensional problems and flexibility in embedding diverse task-specific features for decision making compared to the LBB-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, the value-based DRL algorithms can be effec- tively adopted to solve the challenging stochastic IP problem by transforming the original IP problem into an MDP with discrete action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the objective function or the metric strongly correlated with the optimization objective can be regarded as a reward function in MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The state should include the features relevant to the decision policies while the action can be the discrete optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the value-based DRL algorithms can be employed to search for the optimal actions in discrete action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We introduce several applications of value-based DRL in solving the IP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Intelligent Traffic: With a growing increase of au- tonomous vehicles and intelligent roadside units, establishing an intelligent transportation system (ITS) is becoming im- portant to improve transportation efficiency in the pursuit of smart cities, which has been attracting considerable attentions from researchers, and plenty of DRL-based methods have been proposed thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Some of the hottest applications for DRL-assisted ITS are adaptive traffic signal control (ATSC), autonomous driving, and on-board energy management, the details of which, including DRL algorithms and MDP model- ing, are summarized in the table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Choe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [104] developed a deep Q network (DQN)-based approach to maximize transportation efficiency by minimizing local accumulated waiting time according to handcraft features of local traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Further, Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [105] took traffic flow into account to minimize the waiting rate of vehicles for higher transportation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [106] integrated multi-source factors into the design of the reward function to improve the overall performance in reducing the network latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Beyond optimizations over traffic signals, the need for autonomous driving such as lane changing and auto-breaking rises with the rapid development of intelligent vehicles and the Internet of Vehicles (IoV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, Hoel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [108] jointly optimized lane change policy and speed change policy of autonomous driving vehicles according to their locations and speeds to minimize their travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [110] further extended the state from abstract features to real-time traffic flow scenes and enabled autonomous ve- hicles to optimize car following and land-changing policies for higher travel efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The joint control of roadside 20 TABLE V DRL-ASSISTED INTELLIGENT TRANSPORTATION SYSTEMS Application Scenarios DRL Methods Refs State Space Action Space Reward Function ATSC DQN [104] Handcraft features of local traffic Traffic signal Accumulated waiting time LSTM+DQN [105] Number of vehicles traffic flow Duration of traffic signal Waiting rate of vehicles CNN+DQN [106] Local transportation state Traffic signal Mixed function of delay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' queue length and waiting time CNN+DQN [107] Local transportation state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' number of intelligent vehicles Traffic signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' vehicle detouring Mixed function of waiting time and system influence Autonomous Driving DQN [108] Current speed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' land index and road information Lane change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' speed change Self-defined indicator DQN [109] Current pedestrain status Pedestrain detection Self-defined indicator DQN [110] Real-time traffic flow scenes Car following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' lane change Self-defined indicator On-Board Management DQN [111] State of charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power demand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='and generator speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Throttle of the engine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Mixed function of state of charge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='and instantaneous fuel consumption rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='DQN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='[112] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='State of charge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Operation rate parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='of energy source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='Deviation between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='SOC of all units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='units and vehicles was firstly investigated to improve traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='efficiency in [107] by jointly controlling the traffic signal of an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='intelligent traffic light and the detouring behavior of intelligent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='vehicles connected to the IoV according to the real-time traffic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='flow information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Through the unique design of the MDP modeling, the agent learned to maximize the accumulated reduced waiting time while preventing the side roads from high congestion as well, thus maximizing the traffic efficiency of the whole traffic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The numerical simulations in [107] show that the DQN-based algorithm proposed therein can achieve better performance than that of conventional strategies as well as the DRL methods that only control the traffic signals with affordable computational consumption for highly time- sensitive real-time traffic control scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Discrete-Valued Power Control: To maximize network utility, the discrete-valued power control as a classic IP prob- lem can be effectively solved by DQN as proposed in [31], [113]–[116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [113] firstly adopted DQN to dynamically allocate discrete-valued transmit powers of BSs in C-RANs, which manifests superior performance in terms of energy efficiency and the adaptability to highly dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [31] further extended the DQN to address the joint allocation of sub-band and transmit power in V2V networks, where the proposed DQN-based scheduling strategy can be executed in distributed systems to meet the stringent latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The authors in [114] applied the DQN in the mobile edge computing scenarios to jointly optimize the transmit power and device scheduling, which shows significant enhancement in aspects of network delay and resource consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [115] considered an energy- harvesting assisted communication system without any prior knowledge assumed of the energy dynamics and developed a two-stage strategy to solve the joint optimization problem of battery prediction and sum-rate maximization, where a long short-term memory (LSTM)-based network is used to improve the prediction accuracy of battery status in the first stage and the DRL is employed to optimize real-time transmit power and access policy for sum-rate maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Device Scheduling: In wireless networks, as each device can only be in the state of scheduled or non-scheduled, the device scheduling optimization turns out to be an IP problem and DQN can be effectively used to address it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [140] proposed a circumstance-independent DQN- based scheduler to maximize the network utility under various conditions and QoS constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The unconstrained Lagrangian function was adopted as a reward function to cope with various constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The DQN-based scheduler was also successfully used in FL systems and mobile edge computing systems [117]–[119] to schedule participating devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [117] adopted DQN to schedule training batches to optimize the quality of query response for a cloud-enabled DNN inference system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [118] further considered the cloud-edge hybrid inference system and proposed Au- toScale, an automatic DQN-based scheduler, to dynamically schedule the inference execution target for improving the pre- diction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To tackle the problem of non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d distribution of heterogeneous data in the inference of FL system, Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [141] proposed AutoFL, a heterogeneity-aware DQN-based scheduler, to schedule the FL participants for overall energy efficiency enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 2: Stochastic Mixed Integer Nonlinear Prob- lems Stochastic MINLPs involving both continuous-valued and discrete-valued variables in dynamic environments have been widely encountered in the wireless communication systems, 21 which have the following general forms: maximize {xi}T 1 ,{zi}T 1 T � t=1 ctft(xt, zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' yt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18a) subject to g(xt) ≤ bt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18b) h(zt) ≤ dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18c) xt ∈ Zn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18d) zt ∈ Rn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ∀t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18e) yt ∼ P(y | yt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' xt−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' bmzt−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (18f) where ft(·) denotes the utility function of wireless net- works,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' gt(·) and ht(·) can be the performance/resource con- straint functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' such as the QoS constraints and the max- imum/average power constraints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' xt and zt denote discrete- valued and continuous-valued network resources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' yt denotes the dynamic states of the environment with Marko- vian properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Obviously, it is computationally expensive and analytically intractable to solve such an NP-hard stochastic non-convex problem with COAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' On the other hand, the value- based DRL algorithms can only deal with the discrete action space to evaluate the state-action values for each possible action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Hence, bypassing the evaluation of the state-action values and directly choosing the action under the currently learned policy, the policy-based methods can be effectively employed to solve the challenging stochastic MINLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To model the stochastic MINLP as an MDP, the state can be defined as the features related to the decision-making, the discrete-valued and continuous-valued variables yet to be optimized can be considered as actions of MDP, which are sampled from the learned policy mapping function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The reward can be defined as the objective function or the metric strongly correlated with the optimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then, the policy- based methods can be employed to learn the optimal policy function directly in the continuous action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Note that the discrete-valued variables in the stochastic MINLPs can be obtained by rounding the optimized continuous-valued variables in the testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We introduce several applications of the policy-based DRL in solving the stochastic MINLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Mobile Edge Network Optimization: Applications mi- grated from cloud to edge have been a prevalent trend in the era of 5G and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By enabling a large number of access points and integrating the widely distributed comput- ing, caching, and communication resources, the mobile edge network (MEN) can significantly reduce the communication latency and improve the network performance by jointly op- timizing the heterogeneous resources at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DRL-based methods [120]–[127] have shown promising performance in the multi-resource joint optimization in MEN attributed to their self-adaptation to dynamic environment, the general- ization power for high-dimensional problem and the real- time inference in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [120] first formulated a joint optimization problem for task offloading, bandwidth allocation, and energy sensing in IoT networks, and then proposed a DDPG-based joint design scheme to minimize the transmission delay and energy consumption, which can well handle the time-varying channel and dynamic bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Further, the authors in [121] developed the Wolpertinger DDPG to eliminate the possible performance degradation of DDPG induced by rounding discrete variables in a similar context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [123] proposed an LSTM-assisted DRL algo- rithm to allocate resources across slices under varying service demands, where the LSTM mechanism was adopted for higher tracking accuracy of user mobility and the system utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [124] further considered the joint optimization of channel allocation and continuous energy harvesting time while taking energy consumption and queue length into account, where the proposed DRL algorithm can achieve higher throughput with stringent performance constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Recently, the block- chain application, as a computational intensive scenario, has been widely combined with the MEN to achieve higher mining efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [127] developed an asynchronous DRL-based algorithm to adaptively allocate the channel resources and establish the pricing policy by maximizing the rational profit among all miners while taking the wireless fading channel into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the advantageous actor-critic algorithm (A3C) was adopted to avoid the overestimation or underestimation of the chosen action, which can achieve better performance than that of the DDPG as numerically verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Space-Air-Ground-Integrated Network: Space-air- ground integrated (SAGI) network provides ubiquitous communication and computing services from cloud to edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the physical distance among layers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', space, air and ground), it is essential to develop a joint optimization framework to coordinate the resources across different layers and timescales to satisfy stringent QoS constraints, where various DRL-based approaches have been proposed [128]– [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [128] developed an actor-critic- based DRL algorithm to jointly optimize the task offloading and computational resource allocation by minimizing the cross-layer queuing delay, where a queuing-aware agent was developed to balance the instantaneous queuing boosting and long-term latency constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Further, they proposed a block- chain and semi-distributed learning-based DRL algorithm by minimizing latency while guaranteeing long-term security requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [131] took the multi-dimensional resource heterogeneity and network dynamics into account and proposed a soft actor-critic-based DRL algorithm by minimizing the energy consumption and the queuing latency of the offloading tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 3: Distributed Constraint Optimization Prob- lems Distributed constraint optimization problems in multi-agent systems (MASs) involving multiple nodes are challenging but very common in wireless systems, which have the following general forms: maximize {xi,t}N 1 T � t=1 ctft({xi,t}N 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' {yi,t}N 1 ), (19a) subject to gi,t(xi,t) ≤ bi,t, ∀i, (19b) {yi,t+1}N 1 ∼ P({yi,t+1}N 1 |{xi,t}N 1 , {yi,t}N 1 ), (19c) 22 where {xi,t}N 1 and {yi,t}N 1 denote the set of variables and system parameters of the wireless systems, respectively, ft(·) denotes the objective function of the MAS, gi,t(·) denotes the performance constraint function of each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Note that the evolution of system states {yi,t+1}N 1 can be modeled as a Markov process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the agent can be the BS or edge device in wireless networks, the system parameters can be wireless fading channels or edge resource status, while the set of variables can be the corresponding resource allocation policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' It is worth noting that the system parameters of MAS at the current time slot depend on the system parameters and action variables of MAS at the last time slot, therefore the interactions among agents determine the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Due to the coupling among the agents and the non-convexity of the objective function, it is challenging to achieve the desired performance within acceptable computational delay using COAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Fortunately, built on the MAS, the multi-agent deep RL (MADRL) shows unmatched performance in deal- ing with the multi-agent co-learning problems, which allows multiple agents to learn multiple individual policies and one global policy collaboratively based on the interactions among agents and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' MADRL enables each agent to develop its own decision- making policy which can be executed decentralized, thereby improving the scalability of network significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, the agents in MAS can build either competitive relationships or cooperative relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the objective of MADRL algorithms can be divided into three categories: maximizing the global reward by coordinating all agents, maximizing the reward of each agent by constructing an equilibrium among agents, or constructing a competitive equilibrium among dif- ferent groups of cooperative agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In wireless networks, it is more general that the MAS operates in cooperative mode, which is the focus of the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Note that the learned policy in MADRL can be unstable due to the fact that the state of each agent depends not only on its own states and actions but also on the actions and states of other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such mutual coupling depends on the communication range of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In some scenarios, the agent can only communicate with the surrounding agents, thereby leading to a partially observable MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, besides the consistent trial-and- error explorations as in a single-agent system, an efficient communication mechanism among agents is also important to achieve the long-term optimal policy in dynamic MAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we illustrate the application examples of MADRL for cooperative MAS design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome the issue of huge CSI overhead for centralized RRM design in large-scale wireless networks, Yasir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [133] firstly adopted the MADRL technique to dynamically allocate transmit power at each BS through mutual coordination for sum-rate maximization of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By modeling each BS as an intelligent agent, each agent determines its transmit power individually according to its local and neigh- boring channel information, achieving the same performance as that of COA with full CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Further, the authors in [134] ex- tended it to the continuous-valued power control, where three different RL algorithms were proposed to feature a promising performance of MADRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the above methods require additional CSI exchange between the neighboring BSs, which can downgrade the spectrum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address this, DEC- MAPC proposed in [26] achieved fully decentralized power control only using local CSI while maximizing the sum- rate of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, to achieve fully distributed implementation, DEC-MAPC was proposed to decompose the global state-action value into a monotonic increasing non- linear function of all local state-action values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, each BS only needs to determine its transmit power by maximizing the local state-action value, then the network utility can be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address the continuous power control while cooperation, an actor-critic framework with a double critic network was adopted in DEC-MAPC, leading to more accurate estimation of state-action values and higher network utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages The advantages of DRL methods are summarized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) High Adaptability: The training data for policy updates are collected from historical interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the training of DRL policy can keep track of real-time wireless dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Compared with generic NNs, the training data in DRL methods are label-free, making it unrestricted to the traditional algorithms and allowing more degrees of freedom to improve the learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Additionally, the DRL is model-free, thus enabling the ex- ploration of unknown and complicated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Suitable for Long-Term Optimization: The DRL-based methods can learn a long-term optimal policy that takes the potential reward in the future and long-term system constraints into account rather than just considering instantaneous system reward and one-shot constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the DRL methods also have some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' First, it is hard to train a global optimal policy because the action space is generally too large to be exhaustively traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, to obtain a good policy, numerous training experiences shall be stored, which however is challenging due to the limited storage capacity of local devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Second, while the DRL methods can be deployed with a relatively simple network structure, there are lots of hyperparameters involved in the training and execution, whose values are chosen man- ually by costly trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The fine-tuning of hyperparameters is labor-intensive and time-consuming, and improperly chosen values can significantly degrade the performance of DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Third, DRL is developed based on the MDP modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For some complicated applications, the definition of MDP can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, it can be hard to acquire some state information efficiently in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' It also can be quite difficult to define a highly featured state space and choose a good reward function in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' END-TO-END LEARNING FOR SEMANTIC OPTIMIZATION The joint optimization of physical layer transmitter and receiver in wireless communication systems is extremely chal- lenging due to infinitely large searching space for modular functions, the complex interactions among modules and the 23 highly non-convexity of optimized global performance in terms of communication rate, transmission reliability, resource consumption, transmission delays, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', in conventional block- based communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DL-enabled end-to-end learn- ing has been studied to merge the transmission blocks and jointly design the transmitter and receiver in a data-driven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To boost the system capacity and improve transmis- sion efficiency and reliability, semantic communication has been envisioned as a new transmission paradigm by delivering the semantic meaning rather than bit stream of transmitted messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this section, we identify the motivation of se- mantic communication and review the classic framework of a semantic system, followed by the DL-enabled semantic system design and the overviews of its applications for different types of transmission tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivation and Challenges In conventional wireless communication systems, message compression and message error correction are achieved by source coding and channel coding, respectively, aiming for high transmission efficiency and reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In view of this, conventional communication networks have been designed to optimize data-oriented performance metrics such as communi- cation data rate, spectrum/energy efficiency, symbol or bit level accuracy and latency, while ignoring the semantic meaning be- hind the transmitted messages [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, the bit-error rate (BER) or symbol-error rate (SER) is usually taken as per- formance metric in communication systems to measure the bit or symbol level accuracy and effectiveness of transmit symbols [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the communication system capacity approaching Shannon limit and the booming development of ML, it is an increasing belief in the community that classical Shannon’s information theory needs to be upgraded for the next evolution of wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A variety of services emerged in 6G wireless systems are service/content-centric, which means they are more concerned about the semantic- related information instead of physical data symbols, which sparks a paradigm shift from the symbol transmission to the semantic meaning transmission in communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By delivering the semantic meaning of the message relevant to the transmission task directly rather than its exact copy, semantic communication is expected to break through the classic design paradigms of Shannon which is targeting at the accurate transference of source signals to the destination re- ceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, the semantic communication can increase the system capacity by identifying and extracting the seman- tic meaning and eliminating the irrelevant information from transmitted messages to realize compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The semantic communication can guarantee the reliability of transmission through exact semantic meaning recovery/interpretation at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, to reap the benefit of semantic communica- tion, semantic-aware optimization for wireless techniques and network structures should be activated to accommodate to the new requirements in semantic communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' There are several challenges for semantic-aware optimiza- tion, which are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) New Metric Design: To enable semantic-aware opti- mization, it is indispensable to design metrics for both exact semantic meaning extraction and accurate semantic meaning transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The first metric measures the meaning behind the transmitted symbols mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The second metric characterizes the semantic errors between recovered and trans- mitted semantic meaning to guarantee successful semantic meaning inference at receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Joint Design of Transmitter and Receiver: Instead of the separate design, the coordinated design of transmitter (source) and receiver (destination) can achieve high system capacity and reliable transmission simultaneously by exploiting seman- tic side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such joint design is expected to compress the transmitted signals maximally while reserving the semantic meaning at transmitter and recover the semantic meaning at receiver to combat the channel fading and semantic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Mathematical Theories: It is still an ongoing research direction to develop efficient and elegant mathematical theories to evaluate the overall performance of a semantic communi- cation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Motivated by recent ML tools, DL-enabled semantic com- munication system has received considerable attention, where the transmitter and receiver implemented by DNNs can be jointly learned targeting at good overall performance [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we will introduce the architecture of a classic semantic communication system, after which we will review the existing techniques to address the challenges for semantic- aware optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Architectures of Semantic Communication Systems Semantic communication system usually contains three components including semantic transmitter and receiver, knowledge base, as well as semantic noise and error [27], [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Semantic Transmitter and Receiver: The desired seman- tic transmitter and receiver are expected to be agents with in- telligence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' humans and smart devices), aiming to perform the functions of semantic communication terminals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', exe- cuting highly intelligent compression/extraction/interpretation algorithms, sensing the environment to obtain high-level data and updating the knowledge bases, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic encoders are typically deployed in the semantic transmitter, which are able to extract the meaning of the source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' text, speech and image messages) and encode these features into symbols (bits) for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The receiver with semantic decoder should be able to recover the compressed features sent by semantic transmitter as well as perform various intelligent tasks based on the inferred semantic information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', automatic speech recognition when transmitting speech signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Knowledge Base: The semantic transmitter and receiver contain certain knowledge bases (KBs) to capture the meaning of the knowledge entities and their complex relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The KBs at transmitter and receiver are expected to be constantly updated by self-learning and both contain the knowledge ele- ments involved in the current communication, which constitute the core of a semantic communication system [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The KBs are the knowledge models that the transmitter and receiver 24 Semantic Encoder Source Message Semantic Information Corrupted Information Noise Wireless Channel Semantic Receiver Semantic Transmitter Semantic Decoder Target Message Transmitter KB Receiver KB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' observed previously and can be shared through communi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' With the transmitter KB, the semantic transmitter extracts the features of the transmitting messages and then the semantic receiver can interpret and infer the meanings of them based on the receiver KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on different types of source messages such as text, image or audio, the KBs could be different for various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition, the KBs at the semantic transmitter and receiver may also be different due to the different abilities for understanding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the transmitter is Chinese language system while the receiver only uses English).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Semantic Noise and Error: Semantic noise that inter- feres with the interpretation of the semantic information dur- ing encoding, data transportation, and decoding processes is introduced as one of the semantic communication components [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic communication system contains two kinds of noises, namely channel noise and semantic noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition to physical channel noise such as additive white Gaussian noise (AWGN), fading channels, and multi-path effect, which are introduced by channel impairments and can cause the signal attenuation and distortion, the semantic noise is defined as a type of disturbance in message interpretation processes due to the ambiguity in words, sentences or symbols used in the message transmission [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic noise can lead to semantic errors in the receiver and misunderstanding of the received message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The semantic noise may occur when KBs between the semantic transmitter and receiver are mismatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' These two kinds of noises will eventually lead to semantic understanding errors at the receiver and it is hard to distinguish which factor causes the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Semantic Meaning Extraction and Interpretation In wireless networks, the core issue of semantic communi- cation is how to extract the semantic information and then perfectly recover it after data transportation, which can be expressed mathematically under information-theoretic view as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For the source signal x and its corresponding received signal y, which belong to a pair of random variables (X, Y ), the probability model involved in semantic communication is represented as a Markov chain Y ↔ X ↔ Z ↔ ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The random variable Z and ˆZ represent the semantic information after semantic encoder and the received semantic information at semantic decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Assuming a wireless fading channel from the semantic encoder to the semantic decoder, then we have ˆZ = HZ + W , where the random variables H and W represent the channel model and the channel noise model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The semantic encoder Cθ(·) can be rep- resented by parameter θ, while the semantic decoder Cφ(·) is parameterized by φ at the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the communica- tion probabilistic model satisfies p(y|x) = pφ(y|ˆz) · pθ(ˆz|x), where pθ(ˆz|x) = pc(ˆz|z) · pθ(z|x) denotes the transmitter and channel probabilistic encoder and pθ, pc, pφ denote the transition probabilities of the semantic encoder, the wireless channel, and the semantic decoder, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The Markov chain of semantic communication thus reduces to Y ↔ X ↔ ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Since the goal of semantic communication is to maximize expected faithfulness in representing observed messages (that is to say to minimize the semantic errors) and minimize the amount of data to be transmitted [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the general form of optimization problem of semantic communication can be expressed as minimize P ˆ Z|X � f(Y , X, ˆZ), g(X, ˆZ) � (20a) subject to (X, Y ) ∈ K, (20b) where P ˆ Z|X denotes a statistical mapping of source infor- mation to received semantic information, function f(·, ·, ·) measures the semantic error, function g(·, ·) characterizes the number of symbols to be transmitted in semantic communica- tion, and K denotes the background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The key of semantic communication is to define the semantic-aware optimization metrics g to quantify the se- mantic information and f to measure the semantic error in (20a) and jointly design the semantic encoder and decoder to obtain the optimal mapping P ˆ Z|X in a task-oriented sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Inspired by the powerful representation ability of DNN and its successful employment in natural language processing (NLP), DL-based end-to-end semantic system has gain much traction, in which the semantic encoder and decoder are implemented by DNNs to represent and interpret the semantic meaning, and are jointly trained in an end-to-end manner to achieve the global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we will detail an information-theoretic framework for semantic communication system design by theoretically characterizing the trade-off between compression of semantic feature extraction and dis- tortion of semantic meaning transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' An Information Bottleneck Optimal Semantic System: The IB framework was proposed in [145], [146] as a principle approach to characterize the trade-off between information compression and target signal reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, IB can be used to provide theoretical guidance for the semantic system design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' From the perspective of reliable transmission, as long as the encoding information entropy remains unchanged, that is, when I( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y ) = I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y ), the semantic communication can recover the target information Y completely and losslessly through the semantic decoder theoretically, where the mutual information I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y ) = � y∈Y � x∈X p(x, y) log( p(x,y) p(x)p(y)) 25 obtained from the joint probability distribution p(x, y) and the marginal probability distribution p(x), p(y) is a measure of the mutual dependence between two random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the loss of information is inevitable in the practice due to the signal compression and system noise, therefore it is natural to maximize I( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y ) while restricting the information flow from source signal X to compressed feature ˆZ in semantic commu- nication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the IB-based optimization problem can be formulated as: maximize P ˆ Z|X I( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y ) (21a) subject to I( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' X) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' (21b) Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' given samples of P ˆ Z|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the objective function of the IB optimization problem is to maximize the mutual information between received semantic information and target signal to minimize the information loss of semantic interpre- tation for reliable transmission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' while the constraint keeps the mutual information between the source signal and the seman- tic information within a certain range to guarantee a target compression ratio of semantic extraction for improved system efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By solving (21), we can learn the parameterized optimal semantic encoder Cθ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To solve the above IB optimization problem, a Lagrangian operator β ≥ 0 can be introduced to maximize its Lagrangian dual equation LIB(θ) = I( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Y )−βI( ˆZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The Lagrangian operator β controls the trade-off between the compression ratio of the received semantic information ˆZ with respect to the source information X and the amount of semantic infor- mation transferred from the transmitter to the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When β = 0, the objective LIB aims at minimal distortion (maximal semantic information transfer), whereas for β → ∞, data rate is minimized (compression ratio is maximized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome the intractability of mutual information in IB optimization, the authors of [147] constructed a lower bound for LIB(θ) using some variational distribution qψ(ˆz), which can be easily opti- mized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, exploiting elementary properties of mutual information, entropy and Kullback–Leibler divergence (KLD), the LIB(θ) is lower bounded by Epθ(y,ˆz)[log pφ(y|ˆz)] − βEp(x)[DKL(pθ(ˆz|x)||qψ(ˆz))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Using the re-parametrization trick with conditions that hold for variational auto-encoder (VAE) [148], this lower bound enables the optimization of the parameters of semantic encoder θ, decoder φ and the variational parameters ψ via gradient-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By parameterizing the semantic encoder and decoder as NNs, the IB optimization problem can be effectively solved by ML techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Applications of Semantic Communications In Table VI, we summarize the different DL-enabled se- mantic systems based on the different transmission tasks (text [34], [149], [150], image [151], [152], speech signals [153], [154] and general signals [155], [156]), from the perspective of performance metrics, semantic quantity module, semantic error module, loss function, KBs as well as the adaptation to dynamic environment for task-specific applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Text Signals: A DL-based semantic communication sys- tem, namely DeepSC, was proposed in [34] for text trans- mission, which was the first work clarifying the concept of semantic information and semantic error at the sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A new metric, namely sentence similarity, was proposed to reflect the learning performance of semantic system for text transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To jointly train the deep encoder and decoder, cross-entropy and lower bound of mutual information (LBMI) constitute the loss function, where the first term measures the semantic errors between transmitted message and recovered message and the second term measures the system capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By minimizing the loss function, the DeepSC can be trained to maximize the system capacity while minimizing the semantic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A transformer-based DNN in DeepSC further makes it applicable to varying communication conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Considering the capacity-limited devices in a more practical scenario, au- thors in [149] proposed a distributed semantic communication system for IoT networks, called L-DeepSC, where bilingual evaluation understudy (BLEU) score and cross-entropy are adopted as performance metric and loss function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To reduce the model sharing cost on IoT devices, the semantic models are compressed through network sparsification and quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A refined CSI estimation scheme based on deep denoising network was proposed to eliminate the impact of fading channels on the semantic model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Image Signals: When aiming at wireless image trans- mission, a joint source-channel coding (JSCC) was proposed by [151], which can be regarded as an early semantic com- munication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In JSCC, two CNNs were considered as autoencoder for image feature extraction at transmitter and autodecoder for image reconstruction at receiver, respectively, where a non-trainable layer in the middle represents the noisy communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Peak signal-to-noise ratio (PSNR) metric was used to measure the learning performance of semantic system for image transmission and the MSE loss function was used to minimize the average distortion between the original input images and its reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To exploit the feedback channel information, Kurka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [152] further extended deep JSCC to DeepJSCC-f by deploying layered autoencoders, where the transmission of each image signal is divided into multiple layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Speech Signals: Other series of works focused on learn- ing semantic information directly from the raw speech signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [153] firstly proposed DL-enabled semantic com- munication system called DeepSC-S for speech signals, where attention-based SE-ResNets semantic encoder and decoder were proposed to identify the essential speech information and recover signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The MSE was used as loss function for training DeepSC-S, and the signal-to-distortion ration (SDR) and the perceptual evaluation of speech distortion (PESQ) score were adopted to measure the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, a novel semantic-oriented speech to text transmission system was considered by [154], where an attention-based network is utilized to identify the semantic representation of the input speech signals and a semantic decoder implemented using MLPs is employed to transform the received speech features to the text form for speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 26 TABLE VI DL-ENABLED SEMANTIC SYSTEMS FOR DIFFERENT TRANSMISSION SIGNALS Transmission Signals Refs Performance Metrics Semantic Quantity Module (SQM) Semantic Error Module (SEM) Loss Function KBs Methods for Dynamic Environment Text [34] Sentence similarity Transformer Transformer Cross-entropy - λ LBMI Datasets of words Transfer learning [149] BLEU score MLPs MLPs Cross-entropy Datasets of words \\ [150] Edge-based similarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' word2vec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' hybrid-based similarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' METEOR and BLEU score Part-of-speech strategy Context-based strategy Log-softmax Datasets of words and speech Context-based dynamic programming algorithm Image [151],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [152] PSNR metric Encoder CNN Decoder CNN MSE Datasets of images \\ Speech [153] SDR and PESQ score SE-ResNet module Multiple SE-ResNet modules MSE Datasets of speeches \\ [154] WER and semantic similarity score Attention-based NNs MLPs Cross-entropy Datasets of words and speeches \\ General [155] Recall@1 for image retrieval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' BLEU score for machine translation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' answer accuracy for VQA Transformer Transformer Cross-entropy and MSE Different dataset \\ [156] PSNR and text accuracy Dob Dpr ESD Library data at receiver Data adaptation 4) General Signals: In addition to the semantic communi- cation systems for specific signals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' general signal transmission was considered in [155],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A transformer-based semantic communication system was proposed in [155] for transmitting multimodal data considering three different tasks, including image retrieval, machine translation and visual question an- swering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the training algorithms, such as the loss functions, for different tasks were designed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address this, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [156] firstly designed a semantic- distortion-based universal loss function adapted to general sig- nals, which consists a hyper-parameter-based linear combina- tion of distortion measure functions for observable information Dob and for pragmatic output Dpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' These functions can be designed to be the KLD, cross entropy, MSE, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', to adapt to the different transmission tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 5) Adaptation to Dynamic Environment: When the com- munication environment is dynamic or the transmission task is changed, it will pose great challenges for the design of DL- enabled semantic communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' First, it requires different KBs to extract and interpret the transmitted messages as the varying of communication environment or transmission tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, if the transmitted message is unseen at the transmitter and receiver, the KBs need to be updated and expanded based on the empirical semantic information, leading to formidable computational costs for the training of semantic encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The KBs matching the dynamic transmission conditions can be learned from the perceived environments/empirical information and can be shared be- tween transmitter and receiver via communication to minimize the semantic inference errors, which however are complex and long-term processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Existing semantic communication designs [149], [151], [152] assumed the shared and fixed KBs, leading to limited scalability and poor generalizability in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Second, for the newly updated KBs and the dynamic changing environment, the semantic coding strategies should quickly adapt to the new environment and KBs with minimal training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Transfer learning [34], [153] was adopted to accelerate the DL model training in dynamic environment by synergizing the past learning experiences to assist the new problem solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the re-training of NNs still requires extra communication and computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A receiver-leading dynamic semantic communication system with non-shareable KBs at receiver was proposed in [156], where an individual data adaptation network was configured at the semantic transmitter to tackle the dynamic data transmission environment leaving the semantic encoder adaptive to dynamic environment without retraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages Semantic communications are expected to improve the communication efficiency and reliability, enhance the quality of experience for human-oriented services as well as support a more robust and upgrade/evolution-friendly communication systems [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, both theoretical and practical imple- mentation of semantic communication are in an early stage, which will spark an explosion of research interests in both academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' FEDERATED LEARNING FOR DISTRIBUTED OPTIMIZATION When optimizing a large-scale model with training data scattered across massive number of edge devices, distributed ML has emerged as a key enabling technology to reduce the communication cost and preserve data privacy in resource limited wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' FL [157], [158] has been proposed as a prominent distributed ML scheme to effectively solve the model optimization problem in ML over large-scale wireless networks, where each edge device participates in the train- ing process by exchanging model parameters with data kept locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this section, we firstly review the FL framework, followed by its applications in wireless communication sys- tems based on different network structures and the summary of its pros and cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Federated Learning Framework Considering more practical scenarios in large-scale wireless networks, where most of training data accounting for a global 27 model learning are generated locally at end devices, in the aforementioned centralized learning framework, edge devices are required to send these data to a central server (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' a BS), triggering high communication costs and data privacy issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To mitigate these problems, FL, a distributed framework to train a global statistical model, was proposed [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Under the coordination of a dedicated central server, FL allows multiple devices to participate in the global model training through local model update and model parameters exchange without sharing their raw data [160], thereby preserving the data privacy and saving the communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The unique challenges of FL compared with centralized learning frame- works include system heterogeneity with various end-device features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', transmission environment, communication re- sources, computational powers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' ), data heterogeneity with non-identical local data distributions and unbalanced local datasets, and dynamic wireless environment with uncertain wireless channels and access links [8], [161]–[163], which have stimulated a growing research interest in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The goal of FL is to minimize a global loss or empirical risk function LFL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', minθ � i∈S wiLi(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Di), where θ represents the model weights, Li denotes the local loss function of device i over local dataset Di, S is the set of participating end-devices and wi denotes the weight for each local loss function with wi ≥ 0 and � wi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In a typical cross-device FL training process, as illustrated in Figure 8, a central server orchestrates and repeats the following steps (referred as federated averaging algorithm) until the convergence of global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Device Selection: The central server selects a subset of devices meeting certain eligibility requirements to participate in each training round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Typical device selection schemes in wireless networks include random scheduling, round robin, proportional fairness as well as incentive mechanism based on auction game [164], [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Global Model Broadcast: The central server broadcasts the current model set to the selected devices that participate in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Local Model Training: Based on the received global model, each selected device takes a batch of samples from its local dataset and utilizes a local model update algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', stochastic gradient descent algorithm) to obtain the updated local model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 4) Model Aggregation: All the local model updates are aggregated at the central server by computing the model aggregation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' weighted average function) to obtain the updated global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To implement federated learning in wireless networks with limited resources and unreliable communication links, the efficiency of information transmission becomes the core issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In wireless FL, the local model shall be transmitted from end devices to central server, followed by the calculation of the aggregation function at the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Given the struc- ture of model aggregation function, the wireless transmission of local model can be divided into orthogonal transmission and non-orthogonal transmission [166]–[169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The practical orthogonal frequency division multiple access (OFDMA) and FDMA techniques are adopted to support interference-free uplink local model transmission and downlink global model transmission in [166], [167] through orthogonalized frequency allocation, which could increase the communication latency instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In order to improve the transmission efficiency, non- orthogonal transmission schemes leveraging the principle of over-the-air computation (AirComp) have been studied in [168]–[171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Unlike the orthogonal transmission which re- quires the decoding of each local model separately,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' AirComp allows the central server to receive a desired aggregation function of local models via their concurrent transmissions on same resource block [168],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [169] by exploiting the waveform superposition nature of wireless multiple access channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' therefore the communication latency and the consumed com- munication resources will not increase with the number of devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' facilitating its usage for large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Training data is essential for ML model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The cen- tralized ML algorithms assume the training data is independent and identical distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=') distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In wireless FL, as the training datasets are generated distributed by end devices, the heterogeneous nature of large-scale wireless networks leads to non-independent and non-identical distribution (non- i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=') datasets at different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The dynamic network envi- ronments, such as the mobility of devices and the randomness of link connections, further make the sensory data not only heterogeneous but also non-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the non- i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and non-stationary feature of on-device data becomes one of the key bottlenecks to accomplish efficient and accurate distributed ML tasks in hyper-scale wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following subsections, we overview the existing wireless FL frameworks to resolve the issue of network/data heterogeneity and instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 1: Cross-Device Federated Learning Cross-device FL involves enormous number of IoT devices or agents, where the data is generated locally and remains typically decentralized [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In order to deploy DNN models on a larger-scale wireless network, avoid huge communication overhead for training data collection in centralized learning as well as protect data privacy, there are already some studies that leverage FL for wireless communication applications including hybrid beamforming [172] and channel estimation [173], [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, in these wireless applications, the training data across devices are typically non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' due to the heterogeneity of the transmission environment where de- vices are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' data can slow down the convergence and deteriorate the learning accuracy of FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To improve the overall performance of FL in dealing with various wireless communication problems with non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' training data, the deeper and wider CNN models were used in FL-based systems [172], [173] to provide better feature extraction and representation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition to the data heterogene- ity, system parameters, such as SNR, antenna numbers, and channel statistics of participated devices are also dynamically changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As shown in [172], [173], the decentralized FL- based wireless communication systems are more robust to the channel imperfections and corruptions compared with the centralized learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To cope with the varying system condi- tions, a federated dynamic detection network was proposed 28 Central Server Global Model Aggregation Broadcast Global Model Upload Local Models Devices Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Illustration of cross-device federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Broadcast Global Model Cellular Networks Upload Local Models Global Model Aggregation Silo Servers Central Server Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Illustration of cross-silo federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' in [175] to perform the dynamic MIMO detection, where two independent detection networks were built leveraging the algorithm unrolling approach, and a specifically designed route network was built to adaptively select a better detector for every sample under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the multi- network design can induce significant training cost and has the limited adaptability to the changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The dynamic and uncertain wireless environment poses great challenges for efficient wireless resource allocation in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The impact of resource allocation policies on the learning performance of wireless FL, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the test accuracy and training efficiency, is generally implicit and non-analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, in the training of the CNN-based wireless FL system, it is hard to obtain an analytical expression of the FL testing accuracy with respect to the resource allocation parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', device selection, power allocation, computational resource allocation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a result, the conventional convex optimization based resource allocation algorithms are infeasible for such scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Fortunately, the dynamic resource allocation problem in the FL system can be formulated as a stochastic optimization problem by modeling the total training process of FL as an MDP with the resource status at each training round being the state, the resource allocation policies being the action, and FL learning performances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', training latency, learning accuracy, energy consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=') being the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, the model-free DRL can be applied to solve the dynamic resource allocation problems in FL while regarding the FL performance changes as a black box [176]–[179], given the diverse and dynamic wireless conditions for FL participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, in each training round, the resource allocation policies shall be optimized based on the real-time resource states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', CSI, calculation resources and available bandwidth), leading to the change of the FL performance of DNN model, which is considered as the immediate reward of the current policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Then the state evolves based on the action performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Through trial-and-error method, the long- term optimal resource allocation policy can be learned to maximize the total amount of reward received over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' We summarize the representative works of DRL-assisted FL in Table VII, detailing the DRL algorithms, state space, action space and reward function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the authors in [182] proposed an experience-based scheduling framework for client selection in each communication round to cope with the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' data distribution, where DQN was adopted to learn the optimal client selection policy aiming for higher test accuracy and fewer communication rounds under dynamic wireless conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DQN-based algorithm was also adopted in [185] to optimize the user access in open radio access network for long-term throughput maximization and efficient FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Further, the authors in [183] developed a double DQN-based algorithm to optimize the amount of data, energy and computational resources allocated to each end device for FL training by minimizing the energy consumption and system delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The joint optimization of radio resource allocation and device scheduling were also investigated in [186], where an actor-critic based DRL approach was proposed to optimize the transmit power at the BS and the computing frequencies at the local devices when participating in the FL training, such that the FL learning performance and the fairness of users can be maximized while reducing the energy and time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [178], a value-based DRL method across computing, communication and caching was proposed for FL optimization in a 5G ultra-dense edge computing networks, where DQN was adopted to solve the complex optimization objectives integrating the QoS metric and communication delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To reduce the back-haul traffic congestion in the large- scale IoT network, the authors in [179] adopted a policy-based algorithm to allocate the back-haul data flow by maximizing the network utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [187], the authors exploited the DRL technique to optimally allocate the available energy and data units at each device and design the block generation rate at miner in a blockchain-assisted FL system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To summarize, by transforming the implicit optimization target related to the FL learning accuracy and efficiency into a numerical reward, such DRL-based algorithms show better fitness in wireless FL for dynamic resource allocation compared with the COAs under dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Application 2: Cross-Silo Federated Learning In contrast with cross-device FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' where a large number of IoT devices participate in FL to complete same learning task,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the clients in cross-silo setting are silo servers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' including 12:2829 TABLE VII DRL-ENABLED FEDERATED LEARNING FRAMEWORKS Algorithm Type DRL Algorithms Refs State Space Action Space Reward Function DQN [178] Task queue state and the available resource state Application partitioning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' subcarrier allocation strategy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and service caching placement Negative summation of normalized execution time Value-Based [180] Selected action in previous time slot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' local information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' interferers’ information and interfered neighbors’ information Transmit power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' beamformer Achievable rate minus the sum of the achievable rate losses of the interfered links [181] Number of available channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' number of power level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' length of the binary representation of the feedback signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and indicator of no transmission Channel selection policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power allocation policy Network utility DDQN [177] Channel conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' resource allocation actions Channel selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' transmit power Sum-rate of all cells minus QoS based penalty [182] Compressed model weights Device scheduling Exponential function of achieved test accuracy [183] Available CPU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' energy unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and wireless bandwidth Device scheduling Resource utilization of MEC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Policy-Based A2C [179] Statistics of data flow Available assignment options Mixed function of delay and PLR DDPG [184] The selected beamformer indexes, the achievable rate and signal power at all users, interferer links, and interfered links, respectively Codeword in the beamformer codebook Achievable rate minus the sum of the achievable rate losses of the interfered links organizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' medical or financial), geo-distributed data- centers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', each of which has an identity or name that allows the system to access it specifically [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In cross-silo setting, a number of organizations can share incentive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the payoff-sharing scheme in financial FL system [188]) rather than data directly to train an ML model due to confidentiality or law issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In cross-device FL, the training data are usually partitioned by examples, while in cross-silo FL, in addition to be partitioned by examples, the training data can be partitioned by features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', different organizations keep the data of differ- ent features corresponding to same batch of customers [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The feature partitioned FL requires multi-clients to train the model collaboratively by exchanging intermediate parameters rather than model parameters to deal with the missing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Lots of works have been done to address the security and privacy challenges [189] and communication bottlenecks [190] in feature-partitioned FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Another source of heterogeneity in cross-silo FL is the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and non-stationary siloed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Typically in large-scale wireless networks, the clients in cross- silo setting can be clusters of cellular or D2D networks that contain non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and non-stationary heterogeneous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To overcome the model divergence and staleness in the training stage and poor accuracy in the inference stage induced by non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' and non-stationary training data of different clients in cross-silo FL, an adaptive federated multi-task learning (FMTL) framework was proposed by [191], which preserves multiple models at the server and the clients through adaptive model updating and cluster splitting to deal with non- stationary environments and multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Specifically, in model update scheme, model dichotomy [192] was adopted to find the geometric centers of local models in two virtual sub- clusters, whose model update directions were compounded to determine the update direction of global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The recursive cluster splitting, through decreasing the sub-clusters’ distances of updating directions, was also proposed to avoid the poor performance induced by the mutation of data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, a binary tree-based low-complexity model se- lection scheme was proposed to choose the best model for fitting the current data in both training and testing stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Such adaptive FMTL has been shown to accelerate the model training convergence and reduce the computation complexity while ensuring model accuracy when it is applied to solve the GNN-based power control problem in cross-silo FL system consisting of D2D networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Advantages and Disadvantages FL can not only embed the training capabilities of DNN for hyper-scale wireless networks, but also build a unified multi- source data application system with privacy preserving among multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, FL can realize data sharing and inte- gration across silo servers for supporting high-precision model construction [159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, FL still faces many challenges in terms of privacy protection, theoretical analysis and wireless deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the privacy preserving techniques usually sacrifice the learning performance and induce addi- tional computational cost, which are undesirable for efficient FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The convergence analysis of FL considering the trade- off between learning performance and resource consumption is difficult due to the highly non-convexity and intractability of optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, the deployment of FL in large-scale wireless networks should jointly consider the issues of dynamic fading channel, communication overhead, low power constraint of end devices, as well as the availability and the willingness of participants, which can be challenging for efficient algorithm design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DISCUSSIONS AND FUTURE RESEARCH DIRECTIONS In light of the appealing benefit of ML for large-scale optimization in 6G wireless networks, significant efforts are still needed to upgrade the existing ML techniques or develop new ones considering the constraints of practical wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To further pave the path for its more comprehensive applications in future wireless communication systems, in this section, we summarize the existing DNN design principles to accommodate to different kinds of op- timization problems in wireless networks, after which, we summarize the existing theoretical tools to characterize the performance of MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Subsequently, we discuss the software 30 platforms and implementation issues for MOAs in 6G wireless networks and some potential research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Neural Network Design for Wireless Communications The design of NNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' loss function design, network architecture design and training algorithm design) for solving complex optimization problems in various wireless commu- nication applications requires careful consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we summarize the design principles of DNNs from different aspects when applied to solve large-scale optimiza- tion problems in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Loss Functions: The design of loss function is generally dependent on the communication problem being solved and the availability of training labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' When training labels are available, the DNN can be effectively trained in a supervised manner by constructing the loss functions using the training labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, regression-based losses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', MSE or weighted MSE) are usually optimized for estimation problems, such as channel estimation [29] and MIMO detection [19], and for resource allocation problems, such as power allocation [82] and beamforming design [194], when the labels of targeting signals are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Cross-entropy losses are adopted for clas- sification problems, such as codebook-based precoder design [195], user scheduling [73], and sub-channel selection [196], when the class labels are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In some specific applica- tions, information theory-based losses can help to fulfill the goals of optimization task, which can be effectively calculated using the empirical data (including training labels), such as the mutual information-based loss in semantic communication [9], min-max generative adversarial net (GAN) loss in channel estimation [197], information-bottleneck-based loss in edge inference system [198], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Alternatively, when training labels are unavailable, the average performance measurements are adopted to formulate the loss function for model training in an unsupervised manner [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, in resource management problem, the average performance measurement E[f(p(h), h)], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', (weighted) sum-rate [84], [200], energy efficiency [201], communication delay [202], secrecy capacity [203], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', are utilized to guide the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Network Architectures: For network architecture de- sign, MLP is usually adopted as a benchmark algorithm for comparison [19], [33], while for the problems with special structures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', data structure, algorithm structure and problem structure), specialized NN architectures are preferred to better serve the underlying purpose of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For data with graph structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', network data [200]), GNN is more suitable for exploiting the inherent graph structure of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For data with spatio-temporal correlations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' traffic prediction [204]), CNN or RNN is specialized in capturing the correlations within data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For data with low dimensional structure, the generative model can be utilized to capture the fundamental sparse structure within data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the underlying probability distribution of spatial channel can be learned by generative network [205]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For algorithms with iterative nature, NN can be designed to imitate the forward operation of iterative algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', deep unrolling NN inherits the structure of original iterative algorithm [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For problems with the distributed data, FL framework can be exploited to meet the distributed requirement [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For problems with complex dynamic environment, DRL constitutes a viable technology to address the stochastic optimization problem through agents and environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To cope with the more stringent requirements in 6G system and support diverse applications for future wireless networks, hybrid DL-based optimizations have attracted increasing atten- tion to fully integrate the intrinsic diverse features of different tasks into the design of customized NNs and make the most of the advantages of various DL techniques to improve the algo- rithm efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The core of hybrid DL is to match different features of the problem with appropriate learning technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, DRL-enabled FL system has been discussed in Section VII to solve dynamic resource allocation problems in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, the algorithm unrolling approaches can be easily combined with not only conventional optimization [52], [54] but also some DL techniques such as GNN [97] to speed up the iterative algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' GNN integrated with DRL has been discussed in [206] for algorithmic and methodologi- cal improvements, where DRL and GNN complement each other for better utility or application-specific enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, the versatility of DRL and the flexible encoding capability of GNNs can be combined to address challenging optimization problems in different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, contrastive learning, a self-supervised learning approach, has been leveraged in GNN to address the challenge of data heterogeneity in graphs [207], [208], where the node features are learned in an unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Training Algorithms: When the network structure is fixed with reasonable loss function, Adam [209], the most popular back propagation algorithm, is usually applied for stable training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To further depict the recurrence relation of gradients between two adjacent layers, the generalized chain rule was proposed in [21] to perform back propagation in unrolling-based DNN algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To obtain better learning performance for algorithm unrolling methods, the layer-wise training approach is widely adopted [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' End-to-end training of DNNs based on a large number of channel samples can bypass the explicit channel modeling procedure and poten- tially provide system-level performance gains compared to the COAs when solving the optimization problems in wireless communication systems [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, to accommodate the NN to the dynamic changing environment in wireless communication systems, several techniques can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, transfer learning has been adopted in [15] to tackle the task mismatch issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the network setting in training is different from that in testing) in LBB algorithm via self-imitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Similarly, transfer learning in semantic commu- nication in [34], [153] enables the trained NN to adapt to the dynamic communication environment quickly with reduced number of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Meta learning is another technique to improve the generalizability of DNN in dynamic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, model-agnostic meta-learning is used in [210] to learn a good model initialization by alternating inner-task and across-task updates, such that the learned model parameters can adapt to a new environment with a small number of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 31 Even though transfer learning and meta learning can signif- icantly speed up the learning process of NN in new environ- ment, they still require batch-sized training data to be available before the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In many scenarios of wireless networks, the training data arrives sequentially in a stream whose inher- ent features can be drifted due to the dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this case, a promising solution is to learn the models on the fly, which can improve the generalizability and scalability of model sufficiently and save the memory of system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Online deep learning [211], to learn the DNNs from the sequentially received data in an online manner, has been investigated in the various contexts of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, in [212], an online DNN framework was proposed to solve the general optimization problems in wireless communication, where the self-defined layers rather than convolutional or full-connected layers were adopted to estimate optimization variables for each data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The CNN-based [213] and GNN-based [214] online learning algorithms were further proposed, where the online module is retrained based on the observed testing samples to overcome the mismatches between training and testing data induced by the dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides the network generalization issues, the complex and unpredictable environment may also lead to implicit system performance functions in resource allocation problem, which hinders the gradient calculation in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To tackle this, a model-free approximation approach was proposed in [215], in which the gradients are approximated by their zeroth-ordered updates through sampling the model functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 4) Optimization Constraints: Optimizations in wireless communications usually need to deal with various complex constraints to meet specific requirements, which brings ad- ditional difficulties to the design of ML algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Normal- ization layer as the output layer of DNN provides a simple and effective way to deal with modulus constraint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' power constraint [84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The proper activation functions can help to restrain the network output within a feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For exam- ple, sigmoid can keep the output between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For discrete constraints, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', quantization, various smooth functions can be constructed to approximate the discontinuous function, or we can directly set the gradient to be 1 at the back propagation to avoid the gradient vanishing and explosion [216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For other more complex constraints, primal-dual learning [85], [217] plays an important role, where DNNs are designed to solve the corresponding unconstrained Lagrangian dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The authors in [90] showed that the duality gap of radio resource management optimization problem in wireless networks is negligible if the parameterized learning through NN is near universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Theoretical Tools ML technology, represented by DNN, has made great achievements in the fields of computer vision, natural language processing and communications in recent years with the great improvement of computing power and the great enrichment of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, due to the “black box” nature of DNN, the lack of interpretability and theoretical guarantee of the DL-based framework is a critical issue that needs to be addressed for applications requiring highly transparent and reliable technolo- gies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' wireless communications, healthcare and automatic driving).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' DNN is a model function characterizing complex relationships among data, and its “black box” nature is mainly manifested in the fact that there is a huge unknown gap between the design of the model and its final performance on specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In other words, it is impossible to accurately predict and control DNN performance when designing the model, to clearly understand the reasons for its good or bad performance, and to systematically improve its performance, but only to rely on some lucky model design and training tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' From ML perspective, this is due to the fact that the ML task (including training data) and DNN theory aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', loss quantities and the generalization performance of the model) have not been thoroughly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Especially when dealing with high-dimensional real data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' images with the millions of dimensions), many of the existing statistical quantities and information-theoretic concepts such as entropy, mutual information, maximum likelihood and KLD suffer from the curse of dimensionality for computation, the ill- posedness for degenerate distributions, as well as the lack of guarantees for finite samples [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To avoid these issues, the principled formulations are replaced with approximate bounds, simplified assumptions, heuristics and special tricks in practice, resulting in a serious performance gap between theory and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In wireless communications, the specialized MOAs poten- tially enable rigorous analytical results within certain per- formance limits [219] due to their task-specific features and the model-inspired property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the following, we summarize the existing research results and progresses on the theoretical aspects of the MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Algorithm Unrolling: Inherited from the traditional it- erative algorithm, the behavior of each layer of unrolled NN is interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The unrolled iterative hard threshold (IHT), used to solve ℓ0 norm constrained sparse recovery problem, has been theoretically analyzed in [220], which provided the optimality condition for the exact sparse recovery of the unrolled NN and proved the linear convergence rate of the unrolled IHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The theoretical studies for the unrolled ISTA can be found in [221], [222], where the linear convergence rate of unrolled ISTA has been established and the structure of optimal network parameters for unrolled ISTA has been analyzed as well to guide the network structure design and the training parameters downsizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The extension of unrolled ISTA to solve the group-sparse matrix estimation problem has been studied in [14] and applied to JADCE problem in wireless networks, which established the linear convergence rate of unrolled ISTA with group sparsity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' While some progresses have been achieved to establish the performance guarantees of algorithm unrolling, the underlying mechanism and the impact of learned parameters on the convergence and learning accuracy are still to be further discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Learning to Branch-and-Bound: The complexity of LBB for solving MINLPs is analyzed in [15] following the common assumptions and analysis of imitation learning [223], which shows the expected number of nodes explored and the number of relaxed problems solved are O(L2) and O(L) with L 32 integer variables under proper parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, the computational complexity of LBB is much lower than that of the traditional BB algorithm especially for large-scale network with large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the theoretical analysis of learning performance is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Graph Neural Network for Structured Optimization: The graph optimization problem for large-scale wireless networks exhibits the property of permutation invariance or permutation equivariance depending on the output features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The classic GNN frameworks enjoy the same properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', permutation invariance or equivariance, which guarantees advantageous performance of GNN when applied to solve the graph- structured optimization problems [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [33] firstly provided the generalization analysis of GNNs to theoretically verify the advantages of GNNs over MLPs in solving wireless communication problems in terms of the gen- eralization error and the required number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Based on the probably approximately correct (PAC) learning framework, they showed that the GNNs’ generalization error and required number of training samples are O(n) and O(n2) lower than those of MLPs, where n is the number of nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, the GNNs can be theoretically proved to solve the graph optimization problem with near-optimal performance and much fewer training samples than generic NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 4) Deep Reinforcement Learning for Stochastic Optimiza- tion: The existing theoretical framework of DRL is established based on the classic control theory and the theoretical results of conventional RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides the optimization performance in terms of the expected accumulated reward, another main concern of RL community is the convergence performance in terms of the sample efficiency over collected experience data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [224] proposed a pessimism-based approach that guarantees the convergence with O(d) sample complexity while only requiring Bellman closedness as standard in the exploratory setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' [225] proposed a novel theoretical analysis framework for DRL according to the primal-dual formulation of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' By relaxing the stringent requirement on the all-policy concentrability and Bellman- completeness, the framework proposed therein enables poly- nomial sample complexity under single-policy concentrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 5) End-to-End Learning for Semantic Optimization: The theory of semantic communication has caught extensive atten- tions in the past several years with some preliminary research results [36], [226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In [9], the IB theory was used to design the semantic communication systems by taking the meaning of semantic information and the compression ratio into consider- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the IB formulation is task and label dependent, that is, the measurement quantity changes as tasks and labels change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides, the IB provides an information-theoretic guidance for semantic information extraction and transfer, where the implementation of each functionality relies on the DNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Accordingly, a theoretical understanding of DNN can facilitate the theoretical analysis of semantic communication system, which hinges on the development of ML theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 6) Federated Leaning for Distributed Optimization: The theoretical development of FL depends on the research progress of the underlying DL theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The existing theoret- ical research of FL focused on the convergence performance analysis with simple ML models, such as convex loss functions in [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The generic performance analysis for DL model in FL is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Furthermore, the theoretical analysis of FL shall be developed in view of various challenging practical issues including expensive communication, system and data heterogeneity as well as privacy and security concerns [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 7) End-to-End Learning for General Non-convex Optimiza- tion: The DL theories also enable the theoretical analysis of deep generative networks [205], ReLU-based DNNs [227], continual learning [25], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', for end-to-end learning frame- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The error bound of generative model for CS was ana- lyzed in [228] for ReLU-based generative NN and L-Lipschitz generative models, which can guide the high dimensional channel estimation applications in wireless communications [205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The ReLU-based DNN was proved mathematically equivalent to a piecewise linear function under some mild conditions, which can be applied to theoretically prove that the end-to-end DNNs can be utilized as a universal approximator of the MMSE channel estimator to supply theoretical support for DNN-based channel estimation algorithm design [227].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Moreover, the convergence analysis of continual learning framework [25] and model-free online DNNs [229] were further established for end-to-end wireless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As the development of theoretical understandings on various learning techniques, more solid and in-depth description of theoretical analysis of MOA designs can be obtained to guide their usage in practical 6G wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Implementation Issues and Software Platforms 1) Implementation Issues: In academic, most existing learning to optimize methods are trained and tested on an offline simulator with designed dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' the DeepMIMO [230] and Raymobtime [231] for collecting realistic training data) or generated dataset from simulator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', the data generated from BB algorithm for training LBB models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The software platforms for offline training simulator of ML-based model shall be discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For distributed ML to be implemented on massive low-power end devices [8], FL, decentralized learning, model-split learning, distributed RL as well as trustworthy learning are considered as promising edge learning algorithms which are amenable to distributed implementations in large-scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The edge inference implementation issues can be categorized as horizontal edge inference and vertical edge inference based on different collab- orative computing mechanisms, which are also well discussed in literatures [232]–[234] to realize real-time inference in practical distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The standardizations of AI/ML for communication project have just been established and discussed in December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The first technical standard for AI/ML was approved in the 3rd generation partnership project (3GPP) Release 18 [235] to investigate the implementation-related issues of AI/ML in physical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In order to ensure that the AI/ML model can be stably applied to the communication system, the general AI/ML architecture, collaboration between user and network, life cycle management of AI/ML models, model 33 activation/deactivation, model monitoring, and model switch- ing/updating shall be further discussed and designed in 3GPP [236]–[238], which can evaluate whether to update the AI model or go back to the traditional algorithm according to the monitoring results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Software Platforms: For offline training of ML models, there are a rapidly growing body of software platforms for simulations and productization of ML algorithms and models, which can greatly simplify the construction of the compli- cated NNs, including the forward operation, gradient back propagation as well as the parallel computing, and provide potentially feasible platforms for the implementation of MOAs in practical wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' MATLAB Neu- ral Network Toolbox, TensorFlow [239] and PyTorch [240] have provided excellent open-source software frameworks, which are compatible with common operating systems and can be installed in various communication devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=', cloud server, BS, AP or a terminal with certain computational power, for real-time/offline data collection, model training, updating and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, Pytorch can be used to build a DNN easily in network environment and the library is well optimized for graphic processing units (GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Tensor- Flow is more suitable for building advanced and large-scale NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As a production-oriented DL framework, Caffe2 [241] has been developed in Facebook to train NNs on multiple GPUs in distributed setting for supporting mobile operating systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' iOS and Android).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition, there are many other software platforms such as Blocks [242], CNTK [243] and Lasagne [244] that can also support mobile systems for commercial-grade distributed DL implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' From the perspective of promoting scientific research, Open-L2O [44], a research-oriented open software package, has recently been established to support both model-free and model-based “learning to optimize” approaches, which facilitates a fair performance comparison of different algorithms in a simulated environment and a fully automatic algorithm design of various kinds of optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The advances in computing capacities and data storage techniques further fertilize the development of more sophisticated and advanced MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For example, the GPU can be utilized to execute the DL algorithms much faster than traditional processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides general-purpose GPU, customizable field-programmable gate array (FPGA) and dedicated application specific integrated circuit (ASIC) also encourage the research progress of DL in big data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Challenges and Future Research Directions Even though great progresses have been made in the field of MOA designs, a large amount of data are still required to train the DNNs to achieve near-optimal performance, which leads to several challenges for the practical deployment of MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The acquisition of training data in practice can be difficult due to the hard-measurable environment, high storage cost and dynamic nature of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To address this, the task- oriented MOAs as introduced in this paper can significantly improve the sample efficiency and the generalizability of NNs by incorporating the prior knowledge and task-specific features into the DL design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' To avoid the transmission cost of data acquisition, local dataset can be exploited to train the DL models locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, how to address the heterogeneity of the distributed computing nodes and guarantee a satisfactory overall performance is another issue to be carefully looked at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides the quantify of training data, the quality of training data can also greatly affect the learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Robust MOA design, which is robust to data errors, measurement noises, hardware faults and mismatched training/testing con- ditions, is critical to generate reliable and trustworthy results in the practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition to the challenges related to the data acquisition, we suggest following possible directions for future research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 1) Theoretical Analysis of MOAs: A huge amount of pa- rameters need to be optimized when using ML-based al- gorithm to solve large-scale wireless optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Hence, the research progress of formal and rigorous ML theories shall help us to understand the optimal parameters to be learned, which can significantly reduce the training time of ML-based algorithm and guide the design of MOAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' For instance, the good model parameters were analyzed in [14] and can be obtained by solving a simple convex opti- mization problem rather than through computational intensive back propagation algorithm, which significantly accelerates the training process of large-scale DL models for JADCE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' However, the theoretical understandings of MOAs for wireless communication applications are still in the initial stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 2) Ultra-Lite Neural Network Design: Most of the existing MOAs are highly demanding on computational power and storage space, which renders their deployment on small size and low computational power end devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' As motivated by the edge computing, the DL should be implemented in a distributed manner based on the local datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' The straight- forward network sparsification or pruning can alleviate the storage and computational burden, while it can deteriorate the learning performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Therefore, a light-weight and low-complexity MOA achieving on-par performance with the traditional algorithm is attracting increasing attention in edge computing systems, which allows on-device model train- ing and computing with high accuracy, small model size and low computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 3) Advanced Methodologies and Extended Applications of MOAs: A trend of MOA is to exploit more sophisticated un- derlying features of specific optimization problems and design more advanced and task-specific MOAs to embed expert prior knowledge into DL techniques to speed up the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In particular, the model-based approaches can be utilized to in- spire/assist the design of MOAs and provide theoretical insight for the designed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In addition to the development of the methodologies, another trend is to explore new applications in wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Besides the optimization problems men- tioned before, learning to optimize techniques are expected to solve other problems involved in wireless communication for future research directions, including multi-object optimization problems [245], bi-level optimization problems [246], conic programming [247], maximum-likelihood estimation problems [248], etc, in various emerging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' 34 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' CONCLUSION Integrating high-performance intelligent algorithm into the wireless networks has been an inevitable trend and disruptive shift for supporting highly transparent, reliable and large-scale 6G communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In this paper, we investigated some of the most groundbreaking ML technologies applied for solving challenging large-scale optimization problems in 6G wireless networks, including algorithm unrolling, learning to branch-and-bound, graph neural network for structured optimization, deep reinforcement learning for stochastic opti- mization, end-to-end learning for semantic-aware optimization as well as the federated wireless learning for distributed optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In each section, the general algorithm was first introduced, followed by its case studies of the formulated optimization problems arising from wireless applications as well as the summary of its advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' In the last part, the neural network design for wireless communi- cations, theoretical tools, implementation issues together with the future research directions were also discussed to implement ML algorithms in wireless communications from theory to practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE1T4oBgHgl3EQftgVE/content/2301.03377v1.pdf'} +page_content=' Note that the research contributions discussed are representative but not complete, we hope that this article will serve as a valuable reference and guideline for ML-based optimization algorithm design in wireless networks across algorithmic regulation, theoretical understanding, systematic design and the practical deployment to support 6G intelligent communications.' metadata={'source': 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b/NdFKT4oBgHgl3EQfey4y/content/tmp_files/2301.11826v1.pdf.txt @@ -0,0 +1,717 @@ +DEEP CLUSTERING SURVIVAL MACHINES WITH INTERPRETABLE EXPERT +DISTRIBUTIONS +Bojian Hou⋆, Hongming Li⋆, Zhicheng Jiao†, Zhen Zhou⋆, Hao Zheng⋆, Yong Fan⋆ +⋆ Department of Radiology, Perelman School of Medicine, University of Pennsylvania, USA +† Department of Diagnostic Imaging, Warren Albert Medical School, Brown University, USA +ABSTRACT +Conventional survival analysis methods are typically ineffec- +tive to characterize heterogeneity in the population while such +information can be used to assist predictive modeling. In this +study, we propose a hybrid survival analysis method, referred +to as deep clustering survival machines, that combines the dis- +criminative and generative mechanisms. Similar to the mix- +ture models, we assume that the timing information of sur- +vival data is generatively described by a mixture of certain +numbers of parametric distributions, i.e., expert distributions. +We learn weights of the expert distributions for individual in- +stances according to their features discriminatively such that +each instance’s survival information can be characterized by +a weighted combination of the learned constant expert distri- +butions. This method also facilitates interpretable subgroup- +ing/clustering of all instances according to their associated +expert distributions. Extensive experiments on both real and +synthetic datasets have demonstrated that the method is capa- +ble of obtaining promising clustering results and competitive +time-to-event predicting performance. +Index Terms— Survival analysis, clustering, time-to- +event prediction +1. INTRODUCTION +In survival analysis, it is desired to know individual subjects’ +probability of an event of interest to occur such as the oc- +currence of a disease or even death beyond a certain time t +according to their data /features X [1]. As a result, the proba- +bility can be modeled as a survival function S(·|X) = P(T > +t|X). This task is also referred to as time-to-event prediction, +and one of its main challenges is censoring when the event +outcomes of some instances are unobservable after a certain +time or some instances do not experience any event during +follow-up. +Many methods have been proposed for time-to-event +prediction in survival analysis. The most conventional and +prevalent method is a semi-parametric method called the +Cox PH model [2]. It assumes that the hazard rate for ev- +ery instance is constant over time known as the proportional +hazard (PH) assumption. Some nonparametric methods such +as Kaplan-Meier [3], Nelson-Aalen [4], and Life-Table [5] +are also widely used in survival analysis. Nevertheless, they +suffer from the curse of dimensionality. +Survival analysis +also attracts the attention of the machine learning community +and many machine learning methods [6, 7, 8, 9, 10, 11] have +been developed to reveal the relationship between the features +and the survival information. Particularly, a fully paramet- +ric method, referred to as deep survival machines (DSM) +[12], has demonstrated competitive predicting performance +compared with state-of-the-art methods. Nevertheless, DSM +learns different base distributions for different instances, +which makes its inner mechanism hard to interpret [13, 14]. +In addition to the time-to-event prediction task, the task +of clustering cohorts is also crucial in survival analysis. With +the clustering results, clinicians can provide customized treat- +ments [15] for groups with different risks. The methods men- +tioned above usually cluster the cohorts in a post-hoc way, +i.e., they will artificially stratify the cohorts according to the +predicted risks. This usually leads to even groups, thus lack- +ing interpretability. Lately, two studies consider both clus- +tering and time-to-event prediction simultaneously and they +can cluster data in an uneven manner. Particularly, survival +clustering analysis (SCA) [16] assumes that the latent space +is a mixture of distributions and uses the truncated Dirich- +let process to realize the automatic identification of the clus- +ter numbers. However, SCA cannot control the number of +clusters and thus cannot validate its advantages compared to +those post-hoc methods. Variational deep survival clustering +(VaDeSC) [17], as a fully generative method, uses Gaussian +mixture distribution to model the features in a latent space +and uses the Weibull distribution to model the survival timing +information. This work builds a good bridge between the fea- +tures and survival information by jointly optimizing both like- +lihoods. However, there is a trade-off between the discrimi- +native and generative learning paradigms. A fully generative +framework may not be a good fit for all kinds of data since it +is hard to let both the features and survival information obey +the prior assumption of distributions at the same time. +In this study, we propose a hybrid method to leverage both +the discriminative and generative strategies. Specifically, we +assume that there are certain numbers of expert distributions +in a latent space and each expert distribution can be modeled +arXiv:2301.11826v1 [cs.LG] 27 Jan 2023 + +by parameterized Weibull distributions in a generative way. +The survival function for each instance is a weighted combi- +nation of all the expert distributions and the weight for each +instance is learned by a multi-layer perceptron (MLP) directly +from the features in a discriminative manner. Consequently, +we can naturally cluster all the instances according to their +weights allocated to different expert distributions. In sum- +mary, our contributions are threefold:ZhZhen +• We propose a hybrid survival analysis method that in- +tegrates the advantage of discriminative and generative +ideas, and can perform both clustering and time-to-event +prediction simultaneously. +• We conduct extensive experiments on several real-world +datasets and abundant synthetic datasets, and the results +show promising clustering results as well as competitive +time-to-event prediction performance. +• Our method is interpretable in that the expert distributions +are constant for all the instances. Different weight shows +different attention to the expert distributions and thus we +can easily tell which subgroup the instance belongs to. +Fig. 1. The model structure of the proposed DCSM. Part 1 +learns each instance’s survival function by a weighted combi- +nation of the expert distributions. Part 2 clusters instances by +the learned weights allocated to each expert distribution. +2. METHODS +The data we tackled are right-censored, i.e., our data D is +a set of tuples {xi, ti, δi}N +i=1 where xi is the feature vector +associated with the ith instance, ti is the last-followed time, +δi is the event indicator, and N is the number of instances. +When δi = 1 (it means the ith instance is uncensored), ti will +be the time when the event happens whereas when δi = 0 (it +means the ith instance is censored), ti will be the time when +the instance quits the study or the study ends. Denote the +DU as the uncensored subset where the corresponding event +indicator δ = 1 and DC as the censored subset where δ = 0. +In Part 1 of Fig. 1, the deep clustering survival machines +are designed to learn a conditional distribution P(T|X = x) +by optimizing the maximum likelihood estimation (MLE) of +the time T. Similar to the mixture model learning paradigm, +the conditional distribution P(T|X += +x) is character- +ized by learning a mixture over K well-defined paramet- +ric distributions, referred to as expert distributions. In or- +der to use gradient-based methods to optimize MLE, we +choose the Weibull distributions as the expert distribu- +tions that are flexible to fit various distributions and have +closed-form solutions for the PDF and CDF: PDF(t) = +µ +σ +� t +σ +�µ−1 e−( t +σ) +µ +, CDF(t) = e−( t +σ) +µ +, where µ and σ are +the shape and scale parameters separately. +Part 1 in Fig. 1 indicates that we first need to learn an +encoder for the input features x to obtain a compact and in- +formative representation ˜x. Here we use a multi-layer percep- +tron (MLP) φθ(·) parameterized by θ as the backbone model. +This representation will be multiplied by a parameter w with +softmax to obtain the mixture weight αk with respect to each +(kth) expert distribution that is parameterized by µk and σk. +The final survival distribution for the time T conditioned on +each instance is a weighted combination over all K constant +expert distributions. Eventually, we have a set of parame- +ters {θ, w, {µk, σk}K +k=1} to learn during the training process. +Noting that µk and σk are the same for different input in- +stances so that we can cluster each instance/subject according +to its weight αk that is allocated to each expert distribution, as +illustrated in Part 2 of Fig. 1. Specifically, we assign an sub- +group/cluster indicator i to each instance when the instance’s +corresponding αi is the largest among all K weights. +According to the framework of MLE, our goal is to max- +imize the likelihood with respect to the timing information T +conditioned on x. Given that the likelihood functions are dif- +ferent for uncensored and censored data, we calculate them +separately. +For the uncensored data, the log-likelihood of +T is computed as follows where α is the hidden variable +and ELBO is the lower bound of the likelihood derived by +Jensen’s Inequality: +ln P(DU|θ) = ln +� +Π|DU| +i=1 P(T = ti|X = xi, θ) +� += +�|DU| +i=1 ln +��K +k=1 P(T = ti|α, µk, σk)P(α|X = xi, w) +� += +�|DU| +i=1 ln +� +Eα∼(·|xi,w)[P(T = ti|α, µk, σk)] +� +≥ +�|DU| +i=1 +� +Eα∼(·|xi,w)[ln P(T = ti|α, µk, σk)] +� += +�|DU| +i=1 (softmaxK(ln PDF(ti|µki, σki))) = ELBOU(θ). +Similarly, the log-likelihood of T for the censored data is: +ln P(DC|θ) = ln +� +Π|DC| +i=1 P(T > ti|X = xi, θ) +� +≥ +�|DC| +i=1 +� +Eα∼(·|xi,w)[ln P(T > ti|α, µk, σk)] +� += +�|DC| +i=1 (softmaxK(ln CDF(ti|µki, σki))) = ELBOC(θ). + +:Part 1: Distribution Learning +MLP Encoder +Mixture +P(T = tX) +softmax(wx) += +μk k +Expert Distribution +:Part 2: Clustering/Stratification +Age +Age +Age +Treatment +Treatment +Treatment +BMI +BMI +BMI +Group 1 +Group 2 +Group kTable 1. C Index and LogRank results compared to Cox PH, Deep Cox, DSM, SCA, and VaDeSC. The best ones are bold. +Metric +Dataset +SUPPORT +PBC +FRAMINGHAM +FLCHAIN +C Index +Cox PH +0.8401±0.0070 +0.8476±0.0126 +0.7580±0.0063 +0.7984±0.0046 +Deep Cox +0.8053±0.0058 +0.8474±0.0181 +0.7612±0.0057 +0.7893±0.0063 +DSM +0.8300±0.0045 +0.8363±0.0133 +0.7593±0.0050 +0.8009±0.0036 +SCA +0.8203±0.0121 +0.8251±0.0258 +0.5311±0.1235 +0.7467±0.0091 +VaDeSC +0.8419±0.0041 +0.8278±0.0085 +0.5802±0.0406 +0.7886±0.0100 +DCSM (Ours) +0.8305±0.0028 +0.8359±0.0109 +0.7530±0.0053 +0.7916±0.0074 +LogRank +Cox PH +500.3282±60.4977 +198.2686±17.3940 +576.1450±22.9621 +399.0243±25.7657 +Deep Cox +326.1931±54.7026 +203.3091±22.8343 +593.7317±14.4697 +403.4643±35.8034 +DSM +563.4841±0.0045 +196.0912±0.0133 +587.5718±0.0050 +406.4549±0.0036 +SCA +212.5712±26.2629 +260.5682±67.4875 +278.3525±51.1866 +536.1056±109.1680 +VaDeSC +196.8495±19.6887 +118.9605±77.4716 +348.5500±697.1000 +95.5291±108.9488 +DCSM (Ours) +1067.6184±271.6551 +302.5395±30.1043 +751.9770±48.9725 +571.0441±99.0101 +In addition, to stabilize the performance, we incorporate prior +knowledge for µk and σk. Specifically, we minimize the prior +loss Lprior to make them as close as to the prior µ and σ that +are determined by the MLE result with single distribution: +Lprior = +�K +k=1 ∥µk − µ∥2 +2 + ∥σk − σ∥2 +2. +(1) +Our final objective Lall is the sum of the negative of the log- +likelihood of both the uncensored and censored data as well +as the prior loss where λ is a trade-off hyperparameter: +Lall = Lprior − ELBOU(θ) − λ · ELBOC(θ). +(2) +3. EXPERIMENTS +We conducted extensive experiments to validate the effective- +ness of the proposed method in terms of both time-to-event +prediction and clustering. +Table 2. Statistics of datasets used in the experiments. The +time range tmax in PBC is noted in years while others are +noted in days. “FRAM” refers to “FRAMINGHAM”. +Dataset +SUPPORT +PBC +FRAM +FLCHAIN +Events (%) +68.11 +37.28 +30.33 +30.07 +N +9105 +1945 +11627 +6524 +d (categorical) +44 (26) +25 (17) 18 (10) +8 (2) +tmax +2029 +14.31 +8766 +5167 +3.1. Datasets +We conducted experiments on four real-world datasets (as +shown in Table 2) and 36 synthetic datasets with different +numbers of instances, ranging in 200, 500, 1000, 3000, 5000, +and 10000, and different numbers of features ranging in 10, +20, 50, 200, 500, and 1000. For all the synthetic datasets, +the percentage of censoring was set to 30%. The simulation +process followed VaDeSC [17] except that we changed the +distribution of the features from Gaussian to Uniform to val- +idate the limitation of the fully generative method, which is +discussed in section 3.4. +3.2. Baseline Methods, Metrics and Settings +We compared our method to five methods. +Two of them +are the state-of-the-art methods SCA [16] and VaDeSC [17] +which can perform both time-to-event prediction and clus- +tering. The other three methods are Cox PH [2], Deep Cox +[7], and DSM [12], which only provide the time-to-event +prediction function. We used their predicted risks to cluster +data evenly. +We used two metrics to evaluate the performance of all +the methods. +Specifically, “concordance index” (C Index) +was used to evaluate the time-to-event prediction perfor- +mance. For the clustering task, we leveraged LogRank test to +evaluate the performance. +We conducted five-fold cross validation to estimate the C +Index and LogRank measures and obtained their average val- +ues along with the standard deviation. The parameters were +chosen by grid search. Specifically, the trade-off parameter +λ was chosen from [0.5, 0.75, 1], and the learning parame- +ter step size was chosen from [1e-3, 1e-4]. The layer setting +of the multiple perceptron was chosen from [[50], [50, 50]] +where “50” is the number of neurons in each layer. +3.3. Quantitative Results on Real Data +Table 1 shows the C Index values on real data, including the +average results of five independent runs and their standard de- +viations. These results indicated that our method achieved a +competitive performance compared to other baselines. Al- +though our model’s performance was not the best on some +datasets, the difference with the best performance was not sig- +nificant at a 95% confidence interval. +Table 1 also summarizes the results of the LogRank tests. +LogRank statistic evaluates how well the clustering results are + +(a) C Index on synthetic data +(b) KM plot of Cox PH +(c) KM plot of Deep Cox +(d) KM plot of DSM +(e) KM plot of SCA +(f) KM plot of VaDeSC +(g) KM plot of DCSM (ours) (h) Expert distribution of DCSM +Fig. 2. (a) The C Index comparison among the 36 synthetic datasets. A radar plot is used to illustrate the performance +comparison. A bigger area means better performance. We fill the area of our method and we can see that on most synthetic +datasets (30 among 36), the baseline methods’ curves fall inside our method. (b-g) The Kaplan-Meier plots of all the methods +on data PBC. The cross mark means censoring. The learned expert distributions are shown in (h). The shape of the two expert +distributions resembles our Kaplan-Meier curves, facilitating effective data stratification. +regarding the survival information and with a larger value in- +dicating a better performance. The results demonstrated that +our method outperformed all the baselines. This could be +more useful than the time-to-event prediction because such +information can facilitate personalized treatment planning. +3.4. Quantitative Results on Synthetic Data +Fig. 2(a) shows the comparison of C Index values on syn- +thetic data. We used radar plot to highlight the performance +difference. The bigger the area surrounded by the curves, +the better the performance. Fig. 2(a) demonstrated that our +method generated the biggest area surrounded by the curve, +indicating that our method outperformed all the baselines on +30 among 36 datasets. Our method learned the survival in- +formation generatively by assuming the survival information +follows the Weibull distribution. As Weibull distribution is +rather flexible and can simulate many different distributions +from reality, therefore our method can fit well to the survival +information and obtain the best performance in most cases. +VaDeSC as a fully parametric method also assumes the +survival information obeys the Weibull distribution, but it as- +sumes the features follow the Gaussian distribution whereas +we generate the features using Uniform distribution. In this +way, VaDeSC cannot model the feature distribution well and +thus has inferior performance. Our method learns the features +in a discriminative way. Thus our method can learn a likely +pattern no matter what the real distribution of the features is. +3.5. Qualitative Results on Real Data +Kaplan-Meier (KM) curves according to the clustering re- +sults of all the methods are shown in Fig. 2(b-g). Due to the +page limit, we only show the results on the PBC dataset. The +LogRank of one trial of these methods was 175.81 (Cox PH), +188.59 (Deep Cox), 153.85 (DSM), 162.08 (SCA), 87.62 +(VaDeSC), and 357.72 (DCSM, ours). It is worth noting that +SCA and VaDeSC in Fig. 2(e, f) can automatically determine +the numbers of instances in different groups. VaDeSC had +more unbalanced results, which results in a low LogRank. +Our method obtained the best performance. Fig. 2(h) shows +that the shapes of the two expert distributions resemble the +KM curves, facilitating effective data stratification. +4. CONCLUSION +We propose a deep hybrid method that integrates the discrim- +inative and generative strategies into one framework. Assum- +ing the survival function for each instance is a weighted com- +bination of constant expert distributions, our method is ca- +pable of learning the weight for each expert distribution dis- +criminatively and the distribution of the survival information +generatively. Extensive experimental results along with the +quantitative and qualitative analyses have demonstrated the +advantages of our method. The constant expert distributions +also enhance the interpretability of data stratification. + +5000x20 +3000x20 +1000x20 +10000x20 +500x20 +Cox PH +200x50 +200x20 +Deep Cox +500x50 +10000x10 +DSM +SCA +1000x50 +5000x10 +VaDeSC +DCSM (ours) +3000x50 +3000x10 +5000x50 +1000x10 +10000x50 +500x10 +200x200 +200x10 +0 +0.1 +0.2 +0.3 +0.4 +500x200 +10000x1000 +1000x200 +5000x1000 +3000x200 +3000x1000 +5000x200 +1000x1000 +10000x200 +500x1000 +200x500 +200x1000 +500x500 +10000x500 +1000x500 +3000x500 +5000x500LogRank:175.81 +1.0 +Cluster0,#292 +Cluster1,#292 +Survival Probability +0.8 +0.6 +0.4 +S +0.2 +0 +2 +4 +6 +8 +10 +12 +14 +TimeLogRank:188.59 +1.0 +Cluster0,#292 +Cluster1,#292 +Survival Probability +0.8 +0.6 +0.4 +S +0.2 +0 +2 +4 +. +8 +10 +12 +14 +TimeLogRank:153.85 +1.0 +Cluster0,#292 +Cluster1,#292 +Survival Probability +0.8 +0.6 +0.4 +S +0.2 +0 +2 +4 +6 +8 +10 +12 +14 +TimeLogRank:162.08 +1.0 +Survival Probability +0.8 +Cluster0,#43 +Cluster1,#2 +0.4 +Cluster2,#15 +Cluster3,#5 +S +0.2 +Cluster4.#428 +Cluster5,#72 +Cluster6,#19 +0.0 +0 +2 +6 +10 +12 +14 +TimeLogRank:87.62 +1.0 +Cluster0,#440 +Cluster1,#144 +0.6 +Survival +0.4 +0.2 +0 +2 +4 +6 +8 +10 +12 +14 +TimeLogRank:357.72 +1.0 +Cluster0#189 +Cluster1,#395 +Probability +0.8 +0.6 +Survival +0.4 +S +0.2 +2 +4 +6 +8 +10 +12 +14 +TimeWeibullCDEData:PBC +1.0 +Expert Distribution o +Expert Distribution 1 +0.8 +0.6 +0.4 +0.2 +0.0 +2 +4 +6 +8 +10 +12 +145. COMPLIANCE WITH ETHICAL STANDARDS +Our method complies with ethical standards. All the datasets +we studied are public benchmark datasets. +6. 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IEEE, 2021, pp. 497– +498. +[11] Zhicheng Jiao, Hongming Li, Ying Xiao, Jay Dorsey, +Charles B Simone, Steven Feigenberg, Gary Kao, and +Yong Fan, “Integration of deep learning radiomics and +counts of circulating tumor cells improves prediction +of outcomes of early stage nsclc patients treated with +stereotactic body radiation therapy,” International Jour- +nal of Radiation Oncology* Biology* Physics, vol. 112, +no. 4, pp. 1045–1054, 2022. +[12] Chirag Nagpal, Xinyu Li, and Artur Dubrawski, “Deep +survival machines: Fully parametric survival regression +and representation learning for censored data with com- +peting risks,” IEEE Journal of Biomedical and Health +Informatics, vol. 25, no. 8, pp. 3163–3175, 2021. +[13] Bo-Jian Hou and Zhi-Hua Zhou, +“Learning with +interpretable structure from rnn,” +arXiv preprint +arXiv:1810.10708, 2018. +[14] Bo-Jian Hou and Zhi-Hua Zhou, “Learning with inter- +pretable structure from gated rnn,” IEEE transactions +on neural networks and learning systems, vol. 31, no. 7, +pp. 2267–2279, 2020. +[15] Bojian Hou, Hao Zhang, Gur Ladizhinsky, Stephen +Yang, Volodymyr Kuleshov, Fei Wang, and Qian +Yang, “Clinical evidence engine: proof-of-concept for +a clinical-domain-agnostic decision support infrastruc- +ture,” arXiv preprint arXiv:2111.00621, 2021. +[16] Paidamoyo Chapfuwa, Chunyuan Li, Nikhil Mehta, +Lawrence Carin, and Ricardo Henao, “Survival clus- +ter analysis,” in Proceedings of the ACM Conference on +Health, Inference, and Learning, 2020, pp. 60–68. +[17] Laura Manduchi, Riˇcards Marcinkeviˇcs, Michela C +Massi, Thomas Weikert, Alexander Sauter, Verena +Gotta, Timothy M¨uller, Flavio Vasella, Marian C Nei- +dert, Marc Pfister, et al., +“A deep variational ap- +proach to clustering survival data,” +arXiv preprint +arXiv:2106.05763, 2021. + diff --git a/NdFKT4oBgHgl3EQfey4y/content/tmp_files/load_file.txt b/NdFKT4oBgHgl3EQfey4y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ac5abf878e8d1bfa643584ec39231a8e795d01d --- /dev/null +++ b/NdFKT4oBgHgl3EQfey4y/content/tmp_files/load_file.txt @@ -0,0 +1,379 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf,len=378 +page_content='DEEP CLUSTERING SURVIVAL MACHINES WITH INTERPRETABLE EXPERT DISTRIBUTIONS Bojian Hou⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Hongming Li⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Zhicheng Jiao†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Zhen Zhou⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Hao Zheng⋆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Yong Fan⋆ ⋆ Department of Radiology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Perelman School of Medicine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' USA † Department of Diagnostic Imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Warren Albert Medical School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Brown University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' USA ABSTRACT Conventional survival analysis methods are typically ineffec- tive to characterize heterogeneity in the population while such information can be used to assist predictive modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In this study, we propose a hybrid survival analysis method, referred to as deep clustering survival machines, that combines the dis- criminative and generative mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Similar to the mix- ture models, we assume that the timing information of sur- vival data is generatively described by a mixture of certain numbers of parametric distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=', expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We learn weights of the expert distributions for individual in- stances according to their features discriminatively such that each instance’s survival information can be characterized by a weighted combination of the learned constant expert distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This method also facilitates interpretable subgroup- ing/clustering of all instances according to their associated expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Extensive experiments on both real and synthetic datasets have demonstrated that the method is capa- ble of obtaining promising clustering results and competitive time-to-event predicting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Index Terms— Survival analysis, clustering, time-to- event prediction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' INTRODUCTION In survival analysis, it is desired to know individual subjects’ probability of an event of interest to occur such as the oc- currence of a disease or even death beyond a certain time t according to their data /features X [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' As a result, the proba- bility can be modeled as a survival function S(·|X) = P(T > t|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This task is also referred to as time-to-event prediction, and one of its main challenges is censoring when the event outcomes of some instances are unobservable after a certain time or some instances do not experience any event during follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Many methods have been proposed for time-to-event prediction in survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The most conventional and prevalent method is a semi-parametric method called the Cox PH model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' It assumes that the hazard rate for ev- ery instance is constant over time known as the proportional hazard (PH) assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Some nonparametric methods such as Kaplan-Meier [3], Nelson-Aalen [4], and Life-Table [5] are also widely used in survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Nevertheless, they suffer from the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Survival analysis also attracts the attention of the machine learning community and many machine learning methods [6, 7, 8, 9, 10, 11] have been developed to reveal the relationship between the features and the survival information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Particularly, a fully paramet- ric method, referred to as deep survival machines (DSM) [12], has demonstrated competitive predicting performance compared with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Nevertheless, DSM learns different base distributions for different instances, which makes its inner mechanism hard to interpret [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In addition to the time-to-event prediction task, the task of clustering cohorts is also crucial in survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' With the clustering results, clinicians can provide customized treat- ments [15] for groups with different risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The methods men- tioned above usually cluster the cohorts in a post-hoc way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=', they will artificially stratify the cohorts according to the predicted risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This usually leads to even groups, thus lack- ing interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Lately, two studies consider both clus- tering and time-to-event prediction simultaneously and they can cluster data in an uneven manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Particularly, survival clustering analysis (SCA) [16] assumes that the latent space is a mixture of distributions and uses the truncated Dirich- let process to realize the automatic identification of the clus- ter numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' However, SCA cannot control the number of clusters and thus cannot validate its advantages compared to those post-hoc methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Variational deep survival clustering (VaDeSC) [17], as a fully generative method, uses Gaussian mixture distribution to model the features in a latent space and uses the Weibull distribution to model the survival timing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This work builds a good bridge between the fea- tures and survival information by jointly optimizing both like- lihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' However, there is a trade-off between the discrimi- native and generative learning paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' A fully generative framework may not be a good fit for all kinds of data since it is hard to let both the features and survival information obey the prior assumption of distributions at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In this study, we propose a hybrid method to leverage both the discriminative and generative strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Specifically, we assume that there are certain numbers of expert distributions in a latent space and each expert distribution can be modeled arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='11826v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='LG] 27 Jan 2023 by parameterized Weibull distributions in a generative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The survival function for each instance is a weighted combi- nation of all the expert distributions and the weight for each instance is learned by a multi-layer perceptron (MLP) directly from the features in a discriminative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Consequently, we can naturally cluster all the instances according to their weights allocated to different expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In sum- mary, our contributions are threefold:ZhZhen We propose a hybrid survival analysis method that in- tegrates the advantage of discriminative and generative ideas, and can perform both clustering and time-to-event prediction simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We conduct extensive experiments on several real-world datasets and abundant synthetic datasets, and the results show promising clustering results as well as competitive time-to-event prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Our method is interpretable in that the expert distributions are constant for all the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Different weight shows different attention to the expert distributions and thus we can easily tell which subgroup the instance belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The model structure of the proposed DCSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Part 1 learns each instance’s survival function by a weighted combi- nation of the expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Part 2 clusters instances by the learned weights allocated to each expert distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' METHODS The data we tackled are right-censored, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=', our data D is a set of tuples {xi, ti, δi}N i=1 where xi is the feature vector associated with the ith instance, ti is the last-followed time, δi is the event indicator, and N is the number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' When δi = 1 (it means the ith instance is uncensored), ti will be the time when the event happens whereas when δi = 0 (it means the ith instance is censored), ti will be the time when the instance quits the study or the study ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Denote the DU as the uncensored subset where the corresponding event indicator δ = 1 and DC as the censored subset where δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In Part 1 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 1, the deep clustering survival machines are designed to learn a conditional distribution P(T|X = x) by optimizing the maximum likelihood estimation (MLE) of the time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Similar to the mixture model learning paradigm, the conditional distribution P(T|X = x) is character- ized by learning a mixture over K well-defined paramet- ric distributions, referred to as expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In or- der to use gradient-based methods to optimize MLE, we choose the Weibull distributions as the expert distribu- tions that are flexible to fit various distributions and have closed-form solutions for the PDF and CDF: PDF(t) = µ σ � t σ �µ−1 e−( t σ) µ , CDF(t) = e−( t σ) µ , where µ and σ are the shape and scale parameters separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Part 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 1 indicates that we first need to learn an encoder for the input features x to obtain a compact and in- formative representation ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Here we use a multi-layer percep- tron (MLP) φθ(·) parameterized by θ as the backbone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This representation will be multiplied by a parameter w with softmax to obtain the mixture weight αk with respect to each (kth) expert distribution that is parameterized by µk and σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The final survival distribution for the time T conditioned on each instance is a weighted combination over all K constant expert distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Eventually, we have a set of parame- ters {θ, w, {µk, σk}K k=1} to learn during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Noting that µk and σk are the same for different input in- stances so that we can cluster each instance/subject according to its weight αk that is allocated to each expert distribution, as illustrated in Part 2 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Specifically, we assign an sub- group/cluster indicator i to each instance when the instance’s corresponding αi is the largest among all K weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' According to the framework of MLE, our goal is to max- imize the likelihood with respect to the timing information T conditioned on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Given that the likelihood functions are dif- ferent for uncensored and censored data, we calculate them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' For the uncensored data, the log-likelihood of T is computed as follows where α is the hidden variable and ELBO is the lower bound of the likelihood derived by Jensen’s Inequality: ln P(DU|θ) = ln � Π|DU| i=1 P(T = ti|X = xi, θ) � = �|DU| i=1 ln ��K k=1 P(T = ti|α, µk, σk)P(α|X = xi, w) � = �|DU| i=1 ln � Eα∼(·|xi,w)[P(T = ti|α, µk, σk)] � ≥ �|DU| i=1 � Eα∼(·|xi,w)[ln P(T = ti|α, µk, σk)] � = �|DU| i=1 (softmaxK(ln PDF(ti|µki, σki))) = ELBOU(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Similarly, the log-likelihood of T for the censored data is: ln P(DC|θ) = ln � Π|DC| i=1 P(T > ti|X = xi, θ) � ≥ �|DC| i=1 � Eα∼(·|xi,w)[ln P(T > ti|α, µk, σk)] � = �|DC| i=1 (softmaxK(ln CDF(ti|µki, σki))) = ELBOC(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' :Part 1: Distribution Learning MLP Encoder Mixture P(T = tX) softmax(wx) = μk k Expert Distribution :Part 2: Clustering/Stratification Age Age Age Treatment Treatment Treatment BMI BMI BMI Group 1 Group 2 Group kTable 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' C Index and LogRank results compared to Cox PH, Deep Cox, DSM, SCA, and VaDeSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The best ones are bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Metric Dataset SUPPORT PBC FRAMINGHAM FLCHAIN C Index Cox PH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8401±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8476±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7580±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7984±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0046 Deep Cox 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8053±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8474±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7612±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7893±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0063 DSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8300±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8363±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7593±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8009±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0036 SCA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8203±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8251±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5311±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7467±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0091 VaDeSC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8419±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8278±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5802±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7886±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0100 DCSM (Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8305±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8359±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7530±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7916±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0074 LogRank Cox PH 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3282±60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4977 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2686±17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3940 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1450±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='9621 399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0243±25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7657 Deep Cox 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1931±54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7026 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3091±22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8343 593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='7317±14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4697 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4643±35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8034 DSM 563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4841±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0045 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0912±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0133 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5718±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0050 406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4549±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0036 SCA 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5712±26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2629 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5682±67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4875 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3525±51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1866 536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1056±109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1680 VaDeSC 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8495±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6887 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='9605±77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4716 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5500±697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1000 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5291±108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='9488 DCSM (Ours) 1067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6184±271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6551 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5395±30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1043 751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='9770±48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='9725 571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0441±99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0101 In addition, to stabilize the performance, we incorporate prior knowledge for µk and σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Specifically, we minimize the prior loss Lprior to make them as close as to the prior µ and σ that are determined by the MLE result with single distribution: Lprior = �K k=1 ∥µk − µ∥2 2 + ∥σk − σ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' (1) Our final objective Lall is the sum of the negative of the log- likelihood of both the uncensored and censored data as well as the prior loss where λ is a trade-off hyperparameter: Lall = Lprior − ELBOU(θ) − λ · ELBOC(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' EXPERIMENTS We conducted extensive experiments to validate the effective- ness of the proposed method in terms of both time-to-event prediction and clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Statistics of datasets used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The time range tmax in PBC is noted in years while others are noted in days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' “FRAM” refers to “FRAMINGHAM”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Dataset SUPPORT PBC FRAM FLCHAIN Events (%) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='11 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='28 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='33 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='07 N 9105 1945 11627 6524 d (categorical) 44 (26) 25 (17) 18 (10) 8 (2) tmax 2029 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='31 8766 5167 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Datasets We conducted experiments on four real-world datasets (as shown in Table 2) and 36 synthetic datasets with different numbers of instances, ranging in 200, 500, 1000, 3000, 5000, and 10000, and different numbers of features ranging in 10, 20, 50, 200, 500, and 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' For all the synthetic datasets, the percentage of censoring was set to 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The simulation process followed VaDeSC [17] except that we changed the distribution of the features from Gaussian to Uniform to val- idate the limitation of the fully generative method, which is discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Baseline Methods, Metrics and Settings We compared our method to five methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Two of them are the state-of-the-art methods SCA [16] and VaDeSC [17] which can perform both time-to-event prediction and clus- tering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The other three methods are Cox PH [2], Deep Cox [7], and DSM [12], which only provide the time-to-event prediction function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We used their predicted risks to cluster data evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We used two metrics to evaluate the performance of all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Specifically, “concordance index” (C Index) was used to evaluate the time-to-event prediction perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' For the clustering task, we leveraged LogRank test to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We conducted five-fold cross validation to estimate the C Index and LogRank measures and obtained their average val- ues along with the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The parameters were chosen by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Specifically, the trade-off parameter λ was chosen from [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='75, 1], and the learning parame- ter step size was chosen from [1e-3, 1e-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The layer setting of the multiple perceptron was chosen from [[50], [50, 50]] where “50” is the number of neurons in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Quantitative Results on Real Data Table 1 shows the C Index values on real data, including the average results of five independent runs and their standard de- viations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' These results indicated that our method achieved a competitive performance compared to other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Al- though our model’s performance was not the best on some datasets, the difference with the best performance was not sig- nificant at a 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Table 1 also summarizes the results of the LogRank tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' LogRank statistic evaluates how well the clustering results are (a) C Index on synthetic data (b) KM plot of Cox PH (c) KM plot of Deep Cox (d) KM plot of DSM (e) KM plot of SCA (f) KM plot of VaDeSC (g) KM plot of DCSM (ours) (h) Expert distribution of DCSM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' (a) The C Index comparison among the 36 synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' A radar plot is used to illustrate the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' A bigger area means better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We fill the area of our method and we can see that on most synthetic datasets (30 among 36), the baseline methods’ curves fall inside our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' (b-g) The Kaplan-Meier plots of all the methods on data PBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The cross mark means censoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The learned expert distributions are shown in (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The shape of the two expert distributions resembles our Kaplan-Meier curves, facilitating effective data stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' regarding the survival information and with a larger value in- dicating a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The results demonstrated that our method outperformed all the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' This could be more useful than the time-to-event prediction because such information can facilitate personalized treatment planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Quantitative Results on Synthetic Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2(a) shows the comparison of C Index values on syn- thetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' We used radar plot to highlight the performance difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The bigger the area surrounded by the curves, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2(a) demonstrated that our method generated the biggest area surrounded by the curve, indicating that our method outperformed all the baselines on 30 among 36 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Our method learned the survival in- formation generatively by assuming the survival information follows the Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' As Weibull distribution is rather flexible and can simulate many different distributions from reality, therefore our method can fit well to the survival information and obtain the best performance in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' VaDeSC as a fully parametric method also assumes the survival information obeys the Weibull distribution, but it as- sumes the features follow the Gaussian distribution whereas we generate the features using Uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' In this way, VaDeSC cannot model the feature distribution well and thus has inferior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Our method learns the features in a discriminative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Thus our method can learn a likely pattern no matter what the real distribution of the features is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Qualitative Results on Real Data Kaplan-Meier (KM) curves according to the clustering re- sults of all the methods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2(b-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Due to the page limit, we only show the results on the PBC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The LogRank of one trial of these methods was 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='81 (Cox PH), 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='59 (Deep Cox), 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='85 (DSM), 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='08 (SCA), 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='62 (VaDeSC), and 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='72 (DCSM, ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' It is worth noting that SCA and VaDeSC in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2(e, f) can automatically determine the numbers of instances in different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' VaDeSC had more unbalanced results, which results in a low LogRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Our method obtained the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 2(h) shows that the shapes of the two expert distributions resemble the KM curves, facilitating effective data stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' CONCLUSION We propose a deep hybrid method that integrates the discrim- inative and generative strategies into one framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Assum- ing the survival function for each instance is a weighted com- bination of constant expert distributions, our method is ca- pable of learning the weight for each expert distribution dis- criminatively and the distribution of the survival information generatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' Extensive experimental results along with the quantitative and qualitative analyses have demonstrated the advantages of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' The constant expert distributions also enhance the interpretability of data stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 5000x20 3000x20 1000x20 10000x20 500x20 Cox PH 200x50 200x20 Deep Cox 500x50 10000x10 DSM SCA 1000x50 5000x10 VaDeSC DCSM (ours) 3000x50 3000x10 5000x50 1000x10 10000x50 500x10 200x200 200x10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4 500x200 10000x1000 1000x200 5000x1000 3000x200 3000x1000 5000x200 1000x1000 10000x200 500x1000 200x500 200x1000 500x500 10000x500 1000x500 3000x500 5000x500LogRank:175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0 Cluster0,#292 Cluster1,#292 Survival Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2 0 2 4 6 8 10 12 14 TimeLogRank:188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0 Cluster0,#292 Cluster1,#292 Survival Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2 0 2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 8 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0 Expert Distribution o Expert Distribution 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content='0 2 4 6 8 10 12 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFKT4oBgHgl3EQfey4y/content/2301.11826v1.pdf'} +page_content=' 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gate mechanosensory +processing in C. elegans +Sandeep Kumar1, Anuj K Sharma 2, Andrew M Leifer1,2* +1 Princeton Neuroscience Institute , Princeton University, Princeton, NJ, United States +of America +2 Department of Physics, Princeton University, Princeton, NJ, United States of America +* leifer@princeton.edu +Abstract +Animals must integrate sensory cues with their current behavioral context to generate a +suitable response, but how this integration occurs is poorly understood. Here we report +that Caenorhabditis elegans uses inhibitory signals from turning-associated neurons to +rapidly modulate mechanosensory processing depending on the animal’s behavioral +context. Using high-throughput optogenetic perturbations triggered on behavior, we +show that turning associated neurons SAA, RIV and/or SMB suppress +mechanosensory-evoked reversals during turns. We find that activation of the +gentle-touch mechanosensory neurons or of any of the interneurons AIZ, RIM, AIB and +AVE during a turn is less likely to evoke a reversal than activation during forward +movement. Adding inhibition of SAA, RIV and SMB during a turn restores the +likelihood with which mechanosensory activation evokes reversals. Seperately, activation +of premotor interneuron AVA evokes reversals regardless of whether the animal is +turning or moving forward. We therefore propose that inhibitory signals from SAA, RIV +and/or SMB gate mechanosensory signals upstream of neuron AVA, and identify +putative synapses and receptors where this gating occurs. We conclude that C. elegans +relies on inhibitory feedback from the motor circuit to modulate its response to sensory +stimuli on fast timescales. This need for motor signals in sensory processing may +explain the ubiquity of motor-related neural activity patterns across the brain, including +in sensory processing areas. +Introduction +A critical role of the nervous system is to detect sensory information and select a +suitable motor response, taking into consideration the animal’s environment and current +behavior. How the brain integrates sensory stimuli with broader context is an active +area of research. For example, primates integrate a primary visual cue with a contextual +visual cue to flexibly alter their neural computations [1,2]. In Drosophila, dopaminergic +signals reflect mating drive, a long-lived internal state, that in turn gates the animal’s +courtship response to auditory and visual cues [3]. In C. elegans long-lived internal +states lasting many minutes such as hunger [4], quiescence [5–9] and arousal [10] are all +thought to alter the animal’s response to stimuli via various synaptic or +neuromodulatory mechanisms. In those investigations, sensory signals are combined +with one another or are integrated with long-lived internal state. Less is known about +how sensory processing is modulated by short-timescale behavior. Short +seconds-timescale modulation of sensory processing is of particular interest because 1) it +January 10, 2023 +1/19 +arXiv:2301.02709v1 [q-bio.NC] 6 Jan 2023 + +allows the animal to respond to urgent signals, such as threats and 2) because the +timescale suggests a circuit level mechanism, instead of other longer timescale +mechanisms, such as neuromodulation or changes in gene expression. Here we +investigate short-timescale behavioral modulation of the C. elegans gentle-touch +response. +We study the nematode C. elegans because its compact brain is well suited for +investigations spanning sensory input to motor output [11,12]. The C. elegans +gentle-touch circuitry allows the animal to avoid predation and is one of the most +well-studied circuits of the worm [13–15]. We discovered that animals traveling forward +are much more likely to respond to a mechanosensory stimulus by backing up (reversal), +than animals that receive the same stimulus while they are in the middle of a turn. In +other words the worm’s response to mechanosensory stimuli is gated by the animal’s +short-timescale behavioral context [16,17]. Suppressing mechanosensory-evoked +reversals during turns may be part of a prey avoidance strategy. Turns are an important +part of the C. elegans escape response, and by preventing turns from being interrupted +prematurely, the animal may be ensuring that the escape response continues to +completion [16,18,19]. +The neural mechanism underlying this rapid modulation of sensorimotor processing +has not previously been described. Because turns are short-lived, lasting less than 2 +seconds, we suspect gating is mediated by fast neural dynamics at the circuit level. +In mouse, fly and C. elegans, regions across the brain exhibit activity patterns +related to the animal’s locomotory state and body pose [20–23]. A leading hypothesis is +that these motor signals may be important to modulate sensory representations +including vision [24], thermosensation [25], and for corollary discharge [26]. In this +study, we sought to investigate how locomotory signals interact with mechanosensory +signals on short timescales to modulate mechanosensory processing. +We used a high-throughput closed-loop optogenetic approach [17] to interrogate the +mechanosensorimotor circuitry in C. elegans and measured the animal’s behavior in +response to over 39,000 stimulus events. From these measurements, we identified a +putative circuit by which inhibitory signals from turning-associated neurons disrupt +mechanosensory processing and modulates the likelihood of a reversal depending on the +animal’s behavior. +Results +Turns on their own decrease the likelihood of +mechanosensory-evoked reversals +Previously we reported that optogenetic activation of gentle-touch mechanosensory +neurons delivered during forward locomotion was more likely to evoke a transition to +backward locomotion, called a “reversal,” than activation delivered during the onset of a +turn, (Fig. 1a,b) [16]. From those measurements we had concluded that either turning +itself or possibly some other behavior related to turning modulates +mechanosensory-evoked reversals [17]. +In this work we first sought to distinguish whether turns themselves modulated the +reversals or whether it was another ancillary behavior related to turns. Turns in our +recordings most often occurred immediately after backward locomotion– part of a fixed +action pattern called the “escape response” that consists of backward locomotion, a +turn and then finally forward locomotion [18]. By contrast, about 44% of the turns we +observed were preceded by only forward locomotion, what we call “isolated” turns. We +sought to test whether isolated turns also exhibited a reduction in mechanosensory +evoked responses. By re-analyzing our prior measurements [17], we found that isolated +January 10, 2023 +2/19 + +d. +a. +c. +Stim during +Forward +Stim during +Turn +Reversal +Stim +Forward +Turn Onset +Stim +No +Reversal +Time +Camera +IR LED Ring +Projector +b. +Fwd +Turn +Fwd +Turn +0 +0.1 +0.2 +0.3 +Probability of reversal +Stim +No Stim +Iso. +Turn +n.s. +Fwd +Esc. Iso. +Turn +Fwd +Esc. +*** +n.s. +*** +n.s. +n.s. +0 +0.1 +0.2 +0.3 +Probability of reversal +*** +n.s. +Fig 1. Turns decrease the likelihood of mechanosensory-evoked reversals. a) +Closed-loop optogenetic stimulation is delivered to animals as they crawl based on their +current behavior. b) Optogenetic stimulation is delivered to gentle-touch +mechanosensory neurons in worms that are either moving forward (top row) or turning +(bottom row). c) The probability of a reversal is shown in response to stimulation during +forward movement or turn. Responses are also shown for a low-light no-stimulation +control. The number of stimulation events, from left to right: 6,002, 1,114, 5,996, and +1,050. Reanalysis of recordings from [17]. d.) The probability of reversal in response to +stimulation during turning is shown broken down further by turn subtype: escape-like +turns and isolated turns. N =6,002, 602, 512, 5,996, 599 and 451 stim events, from left +to right. Error bars are 95 percent confidence intervals calculated via bootstrap. *** +indicates p < 0.001. ‘n.s.’ indicates p > 0.05 via two-proportion Z-test. +January 10, 2023 +3/19 + +turns also reduced the likelihood of a reversal response (Fig. 1c,d). This finding suggests +that turns alone are sufficient to modulate the likelihood of a mechanosensory-evoked +reversal response. +We therefore focused on the turn regardless of what behavior preceded it, and sought +to identify the circuit level mechanisms with which the turn interacts with the +mechanosensorimotor response pathway. From here onwards, we consider both isolated- +and escape-like turns together. +Turns decrease the likelihood of interneuron-evoked reversals, +except for those evoked by AVA +Mechanosensory signals from the anterior gentle-touch mechanosensory neurons AVM +and ALM are thought to evoke a reversal response by traveling downstream through a +network of interneurons that are associated with backward locomotion [13,14,19,28–31]. +These include neurons AVA [32–34], AIZ [35], RIM [33,36], AIB [33] and +AVE [37](Fig. 2a). Like the anterior mechanosensory neurons, these interneurons are +known to induce reversals upon stimulation [33,35]. To better understand where this +network interacts with turning, we sought to investigate whether these interneurons’ +ability to evoke reversals also depends on turning. We used a collection of transgenic +strains with cell-specific or near-cell-specific promoters that drive expression of the +optogenetic proteins Chrimson or ChR2 in each of these interneurons (Table 1). We +then used a high-throughput closed-loop optogenetic delivery system of our own design +to stimulate the interneuron with 3 s illumination when the worm was either crawling +forward or beginning to turn [17]. In this way we measured the animal’s response to +many thousands of optogenetic stimulation events. +As expected, optogenetic activation of any of these interneurons during forward +locomotion evoked reversals (Fig. 2b) compared to the baseline probability of a +spontaneous reversal (Supplementary Fig. S1). Activating any of the interneurons we +tested, except for AVA, showed a statistically significant decrease in the probability of +evoking reversals when activated during turns, compared to during forward locomotion, +Fig. 2b. In other words, activation of these interneurons showed a turning-dependent +response, similar to the mechanosensory neurons. By contrast, there was no significant +difference in AVA’s ability to evoke reversals when stimulated during turning compared +to during forward locomotion. +From these measurements within the mechanosensorimotor pathway, we conclude +that neurons AIZ, RIM, AIB and AVE lie either at or upstream of the junction in which +turning signals arrive to modulate the reversal response. AVA, in contrast, lies in the +pathway downstream of the arrival of turning related signals. We therefore sought to +investigate neural sources of this turning related signal. +We note that in the measurements of Fig. 2, we have emphasized how the +probability of evoking a reversal changes between forward and turning. We do not +concern ourselves with neuron-to-neuron variability in the probability of evoking +reversals because that likely reflects strain-specific differences in gene expression or in +the efficiencies of ChR2 compared to Chrimson. +Turning associated neurons RIV, SMB and SAA regulate +reversals +Turning in the worm occurs either when the animal is moving forward, is paused or is +transitioning from backward to forward locomotion, but not during sustained backward +locomotion [38]. Neurons RIV, SMB and SAA are among those neurons associated with +turning. RIV, SMB and SAAD have increased calcium activity during turns [19,39], +January 10, 2023 +4/19 + +FwdTurn +FwdTurn +FwdTurn +FwdTurn +FwdTurn +0 +0.5 +1 +Probability of reversal +AIZ +AVA +AVE +AIB +RIM +Context +*** +*** +*** +*** +n.s. +a. +b. +Touch +neurons +Reversal +associated +interneurons +Turning +neurons +ALM +AVA +AVE +AIB +AVM +AIZ +RIM +SMB +RIV +AVD +SAA +Fig 2. Turns decrease the likelihood of interneuron evoked reversals, +except for AVA. a) Diagram showing chemical (arrows) and electrical (resistor +symbol) synapses among the anterior mechanosensory neurons, downstream +interneurons, and turning associated neurons. Adapted from nemanode.org [27] b) +Probability of a reversal response is shown for optogenetic stimulation of reversal +associated interneurons delivered either during forward movement or during the onset of +turn. Strains are listed in Table 1. 3s illumination of 80 µW/mm2 red light (AVE or +AVA) or >300 µW/mm2 blue light (neuron AIZ, RIM, or AIB). Error bars are 95 +percent confidence intervals calculated via bootstrap. *** indicates p < 0.001 via +two-proportion Z-test. p value for AVA stimulation group is 0.1. N =2,612, 601, 883, +107, 880, 511, 1,007, 342, 409, 191 stimulus events (from left-to-right). +January 10, 2023 +5/19 + +0 +2 +4 +Reversal duration +post stimulus onset (s) +Inhibit RIV, +SMB, SAA +o ++ +*** +gtACR2 in +RIV, SMB, SAA +Fig 3. RIV, SMB and SAA neurons influence reversal duration. Time spent +going backwards in a 10 s window coinciding with optogenetic inhibition. Worms +expressing inhibitory opsin gtACR2 in neurons RIV, SMB, and SAA under lim-4 +promoter were inhibited by blue light of either 180 µW/mm2 (‘+’) or 2µW/mm2 (‘o’) +for 10 s during backward locomotion. Worms spent more time reversing when these +neurons were inhibited than in the control. *** indicates p < 0.001. N = 612 and 695 +stimulus events for ‘o’ and ‘+’ conditions, respectively. +and ablation of RIV, SMB or SAA show defects in turning or head bending +amplitude [28,39]. Wang and colleagues observed that inhibiting RIV, SMB and SAA +when the animal is backing up prolongs the reversal. They therefore proposed that +activity from turning-related neurons may inhibit reversals [19]. We independently +confirm that inhibiting RIV, SMB and SAA increases reversal duration, Fig. 3 and +Supplementary Fig. S2. We therefore sought to investigate whether these turning +neurons also inhibit reversals during turns, and whether they may explain why +mechanosensory stimulation is less likely to evoke reversals during turning. +Inhibiting RIV, SMB and SAA abolishes the turning dependent +modulation of mechanosensory processing +We reasoned that if the turning neurons RIV, SMB and SAA inhibit reversals, then +releasing this inhibition during the onset of turning should allow mechanosensory +stimuli delivered during a turn to evoke reversals as effectively as if they were delivered +during forward locomotion. We designed an experiment to simultaneously inhibit these +turning neurons while stimulating the touch neurons during the onset of a turn. We +expressed a blue-light inhibitory opsin, gtACR2, in the turning associated neurons RIV, +SMB and SAA and a red-light activating opsin Chrimson in the gentle touch neurons. +When we activated the touch neurons using red light, this strain behaved similarly +to our other strains: stimuli delivered during turns were less likely to evoke a reversal +than those delivered during forward locomotion, Fig. 4. But when we also inhibited the +turning associated neurons with blue light while stimulating the touch neurons during +the onset of a turn, the likelihood of evoking reversals was significantly higher and, +crucially, not significantly different than for stimuli delivered during forward locomotion. +In other words, inhibiting these turning associated neurons during turns abolished the +turning-dependence of the mechanosensory response. This is consistent with a model in +which signals from RIV, SMB and/or SAA disrupt mechanosensory processing during +turning. By inhibiting those neurons after the onset of a turn, we prevent this +disruption, presumably by inhibiting an inhibitory signal. +January 10, 2023 +6/19 + +Chrimson in +touch neurons +gtACR2 in +RIV, SMB, SAA +Fig 4. Optogenetic inhibition of neurons RIV, SAA and SMB during turns +restore mechanosensory-evoked reversal response. Probability of reversals when +touch neurons are activated or when touch neurons are activated and RIV, SMB and +SAA are inhibited simultaneously; during either forward movement or turn onset. +Touch neurons express Chrimson and are activated with red light. RIV, SMB and SAA +express gtACR2 and are inhibited with blue light. Strains are listed in Table 1. *** +indicates p < 0.001, two-sample Z-test for proportions. N =5,381, 1,525 and 1,115 +stimulation events from left to right. Additional controls are shown in Fig S3. +We performed additional experiments to rule out alternative explanations +(Supplementary Fig S3 and Supplementary text). For example, we find that blue light +illumination without an inhibitory opsin in the turning-associated neurons is insufficient +to restore mechanosensory evoked reversal responses during turns (Supplementary Fig +S3b). Taken together we conclude that inhibition of the turning neurons during turns +disinhibits mechanosensory evoked response. +Inhibitory signals from turning neurons gate mechanosensory +processing +Taken together, our measurements supports a model in which the turning neurons RIV, +SMB and/or SAA gate mechanosensory information and prevent it from propagating +further downstream to evoke a reversal, Fig. 5a. In this model, mechanosensory signals +from the gentle-touch mechanosensory neurons ALM and AVM propagate in a +feedforward manner to reversal-associated interneurons RIM, AIZ, AIB and AVE. If the +animal is moving forward, the mechanosensory signals continue to propagate to AVA +and evoke reversals. But if the animal is turning, inhibitory signals originating from +RIV/SMB/SAA suppress or disrupt mechanosensory-related signals within the +interneurons and prevent propagation to AVA. This model is consistent with our +measurements and leads us to conclude that turning-related inhibitory signals gate +mechanosensory processing. +January 10, 2023 +7/19 + +Mechanosensory +neurons +AIZ, RIM, AIB, AVE +AVA +Reverse +RIV, SMB, SAA +Turn +Releases +ACh +AIB +SMB +RIV +SAA +RIM +lgc-47 +acc-1 +lgc-47 +Inhibitory Ach Receptors +a. +b. +Fig 5. Putative circuit mechanism. a) In response to gentle touch, +mechanosensory neurons propagate signals to promoter neuron AVA and evoke a +reversal. But during turning, neurons RIV, SMB, and SAA send inhibitory signals that +disrupt sensory signals before they reach AVA thus gating the likelihood of a reversal. +b) Wiring and gene expression is consistent with the following inhibitory path (yellow +arrows): SAA releases acetylcholine via synapses onto RIM and AIB, each of which +expresses inhibitory acetylcholine receptors. +Discussion and Conclusions +Here we show that putative inhibitory signals from turning associated neurons +RIV/SMB/SAA modulate mechanosensory evoked reversals downstream of the gentle +touch neurons and upstream of neuron AVA. But within those constraints, where +exactly might those signals combine? Neuron wiring and gene expression data suggests +that one location may be across the inhibitory synapses from SAA to AIB and RIM, +Fig. 5b. SAA is cholinergic and makes synapses to AIB and RIM [27,40], which both +express inhibitory acetylcholine receptors [41,42]: AIB expresses the inhibitory +acetylcholine receptors lgc-47, and acc-1; while RIM expresses inhibitory (e.g. lgc-47) +and excitatory (e.g. acr-3) acetylcholine receptors. We predict that RIM may spatially +localize its lgc-47 receptors to its synapse with SAA such that SAA’s input is net +inhibitory. Because AIB and RIM both synapse onto AVA, the inhibition of AIB and +RIM is well positioned to interrupt the propagation of mechanosensory signals to AVA. +Gene expression and wiring therefore suggest a plausible path by which inhibition from +the turning circuitry interrupts mechanosensory signals from reaching AVA. +Wang and colleagues had predicted that turning circuitry may inhibit reversal +circuitry through a yet-to-be-identified pathway [19]. Our findings suggest that SAA to +AIB and RIM may be that pathway. More broadly our findings reinforce a longstanding +hypothesis that different motor programs in the worm inhibit one another, such as +forward and reverse locomotion [43]. +In our model, AVA performs a role similar to that of a “decision neuron” with +respect to reversals [44]. This is consistent with our previous observation that AVA’s +calcium activity more closely reflects the animal’s decision to reverse, and is less +reflective of the strength of the stimulus (e.g. AVA’s activity does not reflect how many +touch neurons are activated) [32]. The simple model we describe assumes feed-forward +January 10, 2023 +8/19 + +propagation of signals from ALM and AVM to AVA and omits recurrent connections +among the neurons in between. Future investigations are needed to explore additional +contributions from recurrence. +More broadly, we show that motor related signals are directly influencing neural +activity in areas that contain a mix of sensory and motor information. This is +reminiscent of saccadic suppression in vision [45–47] and corollary discharge [25,26] in +which motor related activity modulates or impinges upon sensory representations. Our +findings add to a growing body of evidence suggesting that behavior information is +necessary for sensory processing, and this may explain why behavior-related neural +activity patterns are seen across the brain in mice, fly and worms, including in +nominally sensory areas [20–23]. +Because turning events are infrequent, spontaneous and brief, they are rare +compared to the time the animal spends moving forward or backwards. But obtaining +sufficient statistical power to probe sensory processing during turns required hundreds +of observations per condition. In total we measured over 40,000 behavior responses to +stimulation, including more than 16,000 during turns. This was only made feasible by +leveraging computer-vision and targeted illumination to track many worms in parallel +and to automatically deliver stimuli triggered upon the animal’s turns. Such closed-loop +automated experimental paradigms will be important for future investigations into +other rare and brief spontaneous behaviors. +Materials and methods +Strains +Strains used in this work are listed in Table 1. In each strain light-gated ion channels +have been expressed to either excite or inhibit specific neurons. We expressed excitatory +opsin Chrimson in the six gentle touch neurons using the mec-4 promoter. Promoters +ser-2, tdc-1, npr-9, opt-3, rig-3 are used to express excitatory opsin in neurons AIZ, +RIM, AIB, AVE, and AVA respectively. To express gtACR2 in RIV, SMB, and SAA, we +used the lim-4 promoter and performed integration using a mini-SOG approach. We +injected into CZ20310 worms, followed by a blue light treatment (450nm, M450LP1, +Thorlabs) for 30 minutes as described in [48]. Before conducting experiments, we +outcrossed integrated worms with the wild type N2 strains for at least six generations to +generate AML496. AML496 worms were then crossed into AML67 worms to create +AML499 strain. +Nematode handling +All worm strains were maintained at 20 C, on regular NGM media plates seeded with E. +coli (OP-50) as food source. Experiments were performed on young adult animals. To +obtain young adults, worms were bleached three days prior to the experiments. +Bleached eggs were washed and centrifuged in M9 (0.8rcf for two mins) three times. +Bleached eggs were suspended in M9 and stored in a shaker overnight. The following +morning hatched L1 larvae were centrifuged and transferred to freshly seeded plates +consisting of 1 ml of 0.5 mM all-trans-retinal mixed with OP50 and stored in the dark +at 20 C until young adulthood. +For experiments, young adult worms were washed in M9 and transferred to an empty +agarose plate for experiments. Excess M9 solution was absorbed with a kim wipe as +described in [16,17]. +January 10, 2023 +9/19 + +Table 1. +Strains used. +Strain name +Target neu- +ron expres- +sion +additional +expression +Genotype +Figure +Ref +AML67 +ALML, +ALMR, +AVM, +PLML, +PLMR, +PVM +wtfIs46[mec- +4P::Chrimson::SL2::mCherry::unc-54 +40ng/ul] +Fig 1c,d and Fig S3b +[16] +TQ3301 +AIZ +xuIs198[Pser-2(2)::frt::ChR2::YFP,Podr- +2(2b)::flp, Punc-122::YFP]; lite-1(xu7) +Fig 2b and Fig S1 +[35] +QW910 +RIM +zfIs9[Ptdc-1::ChR2::GFP, lin-15+]; lite- +1(ce314) +Fig 2b and Fig S1 +[36] +QW1097 +AIB +zfIs112[Pnpr-9::ChR2::GFP, lin15+]; lite- +1(ce314) +Fig 2b and Fig S1 +[36] +Not provided +AVE +opt-3::Chrimson +Fig 2b and Fig S1 +[37] +AML17 +AVA +I1, I4, M4, +and NSM +wtfIs2[rig-3::Chrimson::SL2::mCherry] +Fig 2b and Fig S1 +[32] +AML496 +RIV, SMB, +SAA +wtfIs465 +[lim- +4P::gtACR2::SL2::eGFP::unc-54 80ng/ul ++ unc-122::RFP 50ng/ul] +Fig 3 and Fig S3c +This +work +AML499 +RIV, SMB, +SAA; +ALML, +ALMR, +AVM, +PLML, +PLMR, +PVM +wtfIs46[mec- +4P::Chrimson::SL2::mCherry::unc- +54 +40ng/ul]; +wtfIs465 +[lim- +4P::gtACR2::SL2::eGFP::unc-54 80ng/ul ++ unc-122::RFP 50ng/ul] +Fig 4, Fig S2 and Fig +S3a +This +work +January 10, 2023 +10/19 + +Target neuron(s) +Perturbation +Target +Behavior +Stim +Triggered +on +Stim +Duration +(s) +ISI +(s) +Illumination +Intensity +(uW/mm2) +Illumination +Color +Strain +# Plates +Total stim +events +Figures +Ref. +Forward +- +30 +29 +11,998 +Turn +Turns +>30 +47 +2,164 +Forward +- +30 +16 +5,258 +Turn +Turns +>30 +27 +1,184 +Forward +- +30 +12 +1,766 +Turn +Turns +>30 +19 +238 +Forward +- +30 +4 +1,747 +Turn +Turns +>30 +24 +1,038 +Forward +- +30 +8 +2,413 +Turn +Turns +>30 +16 +832 +Forward +- +30 +8 +1,035 +Turn +Turns +>30 +20 +411 +RIV, SMB, SAA +Inhibit gtACR2 +Reversal +Reversals +10 +>30 +2, 180 +Blue +AML496 +14 +1,307 +Figure 3 +ALML, ALMR, AVM, +PLML, PLMR, PVM, +RIV, SMB, SAA +Inhibit gtACR2 +Reversal +Reversals +10 +>30 +2, 180 +Blue +AML499 +12 +2,532 +Supplementary +Figure S2 +Forward +- +30 +8 +5,381 +Turn +Turns +>30 +16 +1,525 +Excite +Chrimson and +Inhibit gtACR2 +Turn +Turns +>30 Red=60, Blue=180 +Red + Blue +17 +1,115 +Inhibit gtACR2 +Turn +Turns +>30 +180 +Blue +15 +954 +Control +Turn +Turns +>30 +Red=0.5, Blue=2 +Red + Blue +8 +1,961 +Forward +- +30 +6 +3,722 +Turn +Turns +>30 +12 +903 +Turn +Turns +>30 Red=60, Blue=180 +Red + Blue +15 +794 +Turn +Turns +>30 +180 +Blue +16 +579 +Control +Turn +Turns +>30 +Red=0.5, Blue=2 +Red + Blue +15 +772 +RIV, SMB, SAA +Inhibit gtACR2 +Turn +Turns +3 +>30 +2, 180 +Blue +AML496 +16 +2,074 +Supplementary +Figure S3c +This work +TOTAL: +400 +53,703 +AVE +Excite +Chrimson +Excite +Chrimson +ALML, ALMR, AVM, +PLML, PLMR, PVM +AVA +Excite +Chrimson +Excite +Chrimson +ALML, ALMR, AVM, +PLML, PLMR, PVM, +RIV, SMB, SAA +ALML, ALMR, AVM, +PLML, PLMR, PVM +Excite +Chrimson +Excite ChR2 +AIZ +3 +RIM +Excite ChR2 +AIB +Excite ChR2 +2, 300 +3 +3 +0.5, 80 +2, 340 +3 +2, 300 +Blue +QW1097 +Figure 2; +Supplementary +Figure S1 +Figure 2; +Supplementary +Figure S1 +Figure 2; +Supplementary +Figure S1 +Blue +Figure 1c, 1d +AML67 +TQ3301 +Blue +QW910 +Red +Figure 2; +Supplementary +Figure S1 +Figure 2; +Supplementary +Figure S1 +60 +Red +3 +Figure 4; +Supplementary +Figure S3a +AML499 +3 +0.5, 80 +Red +AVE +3 +0.5, 80 +Red +AML17 +3 +60 +Red +AML67 +Supplementary +Figure S3b +This work +This work +This work +This work +[17] +This work +This work +This work +This work +Table 2. Summary of optogenetic measurements performed during behavior. +Behavior Analysis +Computer-vision based behavior analysis was used to identify when the animal is +moving forward, when it is undergoing a reversal or when it is turning. Analysis was +performed as reported previously using two different sets of algorithms for real time vs +post-processing [17]. All figures in this work reflect behavior classifications from the +off-line retrospective analysis. +Briefly, animals are segmented and a centerline is detected. Additional logic is used +to find centerlines even when the animal touches itself [16]. The animal’s center of mass +velocity is also computed. Behavior classification is first performed by classifying pose +dynamics in a behavior map [16,49] and then refined by inspecting the animal’s ellipse +ratio and center of mass velocity to catch any omitted turns, or instances when the +behavior mapper fails to classify. Compared to our previous recent work [17] we +changed two parameters to be more conservative in classifying animals as turning or +reversing. Specifically, to be classified as turning we now require that the binary image +of the animal have an ellipse ratio of 3.1, compared to 3.6 previously. Similarly, to be +classified as a reversal, the animal must now achieve a center of mass velocity of -0.11 +January 10, 2023 +11/19 + +mm/s, instead of -0.1 mm/s. +For experiments probing reversal duration, we report the time the animal spent going +backwards in a 10 s window, coinciding with optogenetic inhibition. So for example, if +after stimulus onset the animal continued moved backwards for 3 s, then paused for 1 s, +and moved backwards for 2 s more, we report a “reversal duration” of 5 s. +Optogenetic activation and inhibition +In this work we seek to deliver optogenetic illumination specifically when the animal is +either moving forward, or turning, or reversing. We conduct different sets of +experiments for each of these three conditions, using different sets of animals for each +experiment. In all cases we use a projector-based illumination system that tracks many +individuals on a plate full of animals, segments them in real time, and addresses each +animal individually to shine light on them, as described previously [17]. All experiments +are performed on plates containing approximately 30 to 40 animals. +To measure the animal’s response to optogenetic perturbations during forward +locomotion, we optogenetically illuminated all tracked animals on the plate every 30 s, +in open loop. In post-processing we then only considered those animals that were +moving forward at the time of illumination. +To measure the animal’s response to optogenetic activation or inhibition delivered +during the onset of turns, our system waited until it detected that an animal was +beginning to turn, and then delivered a stimulus automatically. In post-processing we +retrospectively evaluated whether the turn was valid at time of stimulus onset, and only +included those stimuli events that met our more stringent criteria, as described in [17]. +To measure the animal’s response to optogenetic inhibition during reversals, our +system waited until it detected that an animal had been reversing for 1 second, and +then delivered the illumination. As before, we retrospectively confirmed that the animal +was reversing before including it for further analysis. +Illumination color, intensity and duration are listed in Table 2. As a control, we +compare optogenetic stimulation to illumination with very low intensity light that does +not discernably alter behavior [17]. +Statistical Analysis +To reject the null hypothesis that two empirically observed probabilities are the same, +we use a two-proportion Z-test [50]. Error bars report 95% confidence interval +calculated via a bootstrap procedure. +Data availability +Computer-readable files showing processed tracked behavior trajectories and stimulus +events for all experiments are publicly posted at +https://doi.org/10.6084/m9.figshare.21699668. +Code availability +All analysis code used in this manuscript are available at +https://github.com/leiferlab/analysis-code-kumar-2022.git. +Strains and plasmid availability +All genetic strains and plasmids generated as part of this manuscript are being made +available through Caenorhabditis Genetics Center (CGC) and Addgene respectively. +January 10, 2023 +12/19 + +Acknowledgments +We thank Zhaoyu Li (Queensland Brain Institute), Shawn Xu (University of Michigan) +and Mark Alkema (University of Massachusetts Worcester) for the strains. This work +used computing resources from the Princeton Institute for Computational Science and +Engineering. Research reported in this work was supported by the Simons Foundation +under award SCGB #543003 to AML; and by the National Science Foundation, through +an NSF CAREER Award to AML (IOS-1845137) and through the Center for the +Physics of Biological Function (PHY-1734030). Strains from this work are being +distributed by the CGC, which is funded by the NIH Office of Research Infrastructure +Programs (P40 OD010440). The content is solely the responsibility of the authors and +does not represent the official views of any funding agency. +References +1. Mante V, Sussillo D, Shenoy KV, Newsome WT. Context-dependent +computation by recurrent dynamics in prefrontal cortex. Nature. +2013;503(7474):78–84. doi:10.1038/nature12742. +2. Remington ED, Narain D, Hosseini EA, Jazayeri M. Flexible Sensorimotor +Computations through Rapid Reconfiguration of Cortical Dynamics. Neuron. +2018;98(5):1005–1019.e5. doi:10.1016/j.neuron.2018.05.020. +3. Zhang SX, Rogulja D, Crickmore MA. 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The gray bars represent the trials when the stimulus was delivered during +the forward state while the purple bar represents the trials during which stimulus was +delivered on turning onset. For all these measurements control stimulus was delivered. +Three seconds of only 0.5 µW/mm2 of red stimulus (neuron AVE and AVA) or 2 +µW/mm2 of blue stimulus (neuron AIZ, RIM, and AIB). Error bars are 95 percent +confidence intervals calculated via 10,000 bootstraps. Two sample Z-test was used to +calculate significance. p value for AIZ, RIM, AIB, AVE, and AVA stimulation group is +0.596, 0.936, 0.045, 0.565, 0.262 respectively. The number of stimulus events for each +condition (from left-most bar to right-most bar) are: 2,646, 583, 883, 131, 867, 527, +1,406, 490, 626, 220. +0 +2 +4 +6 +Reversal duration +post stimulus onset (s) +Inhibit RIV, +SMB, SAA +o ++ +*** +Chrimson in +touch neurons +gtACR2 in +RIV, SMB, SAA +Supplementary Fig S2. Inhibition of RIV, SMB and SAA prolong +reversals, in a second transgenic background. Same experiment as in Fig. 3, but +in a transgenic background that also expresses Chrimson in the mechanosensory +neurons. Results are consistent with Fig. 3. Worm spent more time reversing when the +RIV, SMB, and SAA neurons were inhibited compared to when a control stimulus +intensity was used. *** indicates p < 0.001. The number of stimulus events for mock +and experimental conditions are 1,168, 1,364 respectively. +January 10, 2023 +17/19 + +a. +b. +c. +Chrimson in +touch neurons +Chrimson in +touch neurons +gtACR2 in +RIV, SMB, SAA +gtACR2 in +RIV, SMB, SAA +Supplementary Fig S3. Additional control experiments show that +blue-light alone cannot restore mechanosensory-evoked reversal response. +a) Probability of reversals when either touch neurons are activated, or RIV, SMB and +SAA are inhibited, or both simultaneously; during either forward movement or turn +onset. First three bars are same as in Figure S3. Touch neurons express Chrimson and +are activated with red light. RIV, SMB and SAA expressing gtACR2 are inhibited with +blue light. Strains are listed in Table 1. *** indicates p < 0.001, two-sample z-test for +proportions. N =5,381, 1,525, 1,115, 954 and 1,961 stim events, from left to right. b) +Same experiments were repeated in a strain that expressed Chrimson in the +gentle-touch mechanosensory neurons, but no inhibitory opsins. N =3,722, 903, 794, +579 and 772 stim events. c) Same experiments are shown for animals that only express +inhibitory opsin gtACR2 in RIV, SMB and SAA, but no Chrimson. N =1,041 and +1,033 stim events. +January 10, 2023 +18/19 + +Additional control experiments show that blue-light alone cannot restore +mechanosensory evoked reversals +We sought to rule out alternative explanations for why shining blue light during turns +may cause an increase in reversals. It is known that the nominally red-light sensitive +Chrimson can also be mildly activated by blue light [51]. We therefore we tested +whether the increase in responsiveness to the mechanosensory stimulus was the result of +blue-light activation of Chrimson expressed in the touch neurons. +Consistent with mild blue-light activation of Chrimson, shining only blue light on +animals expressing Chrimson in the touch neurons (Supplementary Fig. S3a,b far right +bar) but not on animals expressing only gtACR2 in the turning neurons (Supplementary +Fig. S3c far right bar) caused a significant increase in the probability of reversals +compared to no stimulation (second to right bar). However, this mild blue-light +activation of Chrimson is insufficient to explain the increase in reversal probability we +observed when inhibiting turning neurons via gtACR2. Even when shining both blue +and red light on animals that lack the inhibitory opsin gtACR2, but do contain +Chrimson in their touch neurons, we still observed a large and significant reduction in +the likelihood of reversing in response to stimuli delivered during turns compared to +delivered during forward locomotion (Supplementary Fig. S3b middle bar, compared to +far left bar). This suggests that it is the inhibition of neurons RIV, SMB and SAA that +abolishes the turning-dependence of mechanosensory processing and not mild blue-light +activation of the touch neurons. Further consistent with this view, adding blue light to +red light in those animals that lack the inhibitory opsin does not significantly increase +the probability of reversing (Supplementary Fig. S3b second compared to third bar). +A simple and fully consistent explanation is that our red light illumination strongly +activates the touch neurons and that any additional blue light contributes only very +modest additional activation to the touch neurons, and not enough to explain the +increase we see when inhibiting RIV/SMB/SAA. Moreover, this modest additional +blue-light activation of the touch neurons is only significant in control experiments +without any red light. Taken together we conclude that inhibition of the turning +neurons, and not mild blue-light activation of Chrimson, is responsible for abolishing +the turning dependence of the mechanosensory response. +January 10, 2023 +19/19 + diff --git a/TdE0T4oBgHgl3EQf2QI-/content/tmp_files/load_file.txt b/TdE0T4oBgHgl3EQf2QI-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f54b07c03986c15d9896c5878dbdff87777e87c --- /dev/null +++ b/TdE0T4oBgHgl3EQf2QI-/content/tmp_files/load_file.txt @@ -0,0 +1,909 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf,len=908 +page_content='Inhibitory motor signals gate mechanosensory processing in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans Sandeep Kumar1, Anuj K Sharma 2, Andrew M Leifer1,2* 1 Princeton Neuroscience Institute , Princeton University, Princeton, NJ, United States of America 2 Department of Physics, Princeton University, Princeton, NJ, United States of America leifer@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='edu Abstract Animals must integrate sensory cues with their current behavioral context to generate a suitable response, but how this integration occurs is poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Here we report that Caenorhabditis elegans uses inhibitory signals from turning-associated neurons to rapidly modulate mechanosensory processing depending on the animal’s behavioral context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Using high-throughput optogenetic perturbations triggered on behavior, we show that turning associated neurons SAA, RIV and/or SMB suppress mechanosensory-evoked reversals during turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We find that activation of the gentle-touch mechanosensory neurons or of any of the interneurons AIZ, RIM, AIB and AVE during a turn is less likely to evoke a reversal than activation during forward movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Adding inhibition of SAA, RIV and SMB during a turn restores the likelihood with which mechanosensory activation evokes reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Seperately, activation of premotor interneuron AVA evokes reversals regardless of whether the animal is turning or moving forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We therefore propose that inhibitory signals from SAA, RIV and/or SMB gate mechanosensory signals upstream of neuron AVA, and identify putative synapses and receptors where this gating occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We conclude that C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans relies on inhibitory feedback from the motor circuit to modulate its response to sensory stimuli on fast timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This need for motor signals in sensory processing may explain the ubiquity of motor-related neural activity patterns across the brain, including in sensory processing areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Introduction A critical role of the nervous system is to detect sensory information and select a suitable motor response, taking into consideration the animal’s environment and current behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' How the brain integrates sensory stimuli with broader context is an active area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' For example, primates integrate a primary visual cue with a contextual visual cue to flexibly alter their neural computations [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In Drosophila, dopaminergic signals reflect mating drive, a long-lived internal state, that in turn gates the animal’s courtship response to auditory and visual cues [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans long-lived internal states lasting many minutes such as hunger [4], quiescence [5–9] and arousal [10] are all thought to alter the animal’s response to stimuli via various synaptic or neuromodulatory mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In those investigations, sensory signals are combined with one another or are integrated with long-lived internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Less is known about how sensory processing is modulated by short-timescale behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Short seconds-timescale modulation of sensory processing is of particular interest because 1) it January 10, 2023 1/19 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='02709v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='NC] 6 Jan 2023 allows the animal to respond to urgent signals, such as threats and 2) because the timescale suggests a circuit level mechanism, instead of other longer timescale mechanisms, such as neuromodulation or changes in gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Here we investigate short-timescale behavioral modulation of the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans gentle-touch response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We study the nematode C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans because its compact brain is well suited for investigations spanning sensory input to motor output [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans gentle-touch circuitry allows the animal to avoid predation and is one of the most well-studied circuits of the worm [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We discovered that animals traveling forward are much more likely to respond to a mechanosensory stimulus by backing up (reversal), than animals that receive the same stimulus while they are in the middle of a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In other words the worm’s response to mechanosensory stimuli is gated by the animal’s short-timescale behavioral context [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Suppressing mechanosensory-evoked reversals during turns may be part of a prey avoidance strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turns are an important part of the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans escape response, and by preventing turns from being interrupted prematurely, the animal may be ensuring that the escape response continues to completion [16,18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The neural mechanism underlying this rapid modulation of sensorimotor processing has not previously been described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Because turns are short-lived, lasting less than 2 seconds, we suspect gating is mediated by fast neural dynamics at the circuit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In mouse, fly and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans, regions across the brain exhibit activity patterns related to the animal’s locomotory state and body pose [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' A leading hypothesis is that these motor signals may be important to modulate sensory representations including vision [24], thermosensation [25], and for corollary discharge [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In this study, we sought to investigate how locomotory signals interact with mechanosensory signals on short timescales to modulate mechanosensory processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We used a high-throughput closed-loop optogenetic approach [17] to interrogate the mechanosensorimotor circuitry in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' elegans and measured the animal’s behavior in response to over 39,000 stimulus events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' From these measurements, we identified a putative circuit by which inhibitory signals from turning-associated neurons disrupt mechanosensory processing and modulates the likelihood of a reversal depending on the animal’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Results Turns on their own decrease the likelihood of mechanosensory-evoked reversals Previously we reported that optogenetic activation of gentle-touch mechanosensory neurons delivered during forward locomotion was more likely to evoke a transition to backward locomotion, called a “reversal,” than activation delivered during the onset of a turn, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 1a,b) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' From those measurements we had concluded that either turning itself or possibly some other behavior related to turning modulates mechanosensory-evoked reversals [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In this work we first sought to distinguish whether turns themselves modulated the reversals or whether it was another ancillary behavior related to turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turns in our recordings most often occurred immediately after backward locomotion– part of a fixed action pattern called the “escape response” that consists of backward locomotion, a turn and then finally forward locomotion [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' By contrast, about 44% of the turns we observed were preceded by only forward locomotion, what we call “isolated” turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We sought to test whether isolated turns also exhibited a reduction in mechanosensory evoked responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' By re-analyzing our prior measurements [17], we found that isolated January 10, 2023 2/19 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Stim during Forward Stim during Turn Reversal Stim Forward Turn Onset Stim No Reversal Time Camera IR LED Ring Projector b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Fwd Turn Fwd Turn 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='3 Probability of reversal Stim No Stim Iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Fwd Esc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turn Fwd Esc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='3 Probability of reversal *** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turns decrease the likelihood of mechanosensory-evoked reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a) Closed-loop optogenetic stimulation is delivered to animals as they crawl based on their current behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b) Optogenetic stimulation is delivered to gentle-touch mechanosensory neurons in worms that are either moving forward (top row) or turning (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' c) The probability of a reversal is shown in response to stimulation during forward movement or turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Responses are also shown for a low-light no-stimulation control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The number of stimulation events, from left to right: 6,002, 1,114, 5,996, and 1,050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Reanalysis of recordings from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=') The probability of reversal in response to stimulation during turning is shown broken down further by turn subtype: escape-like turns and isolated turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =6,002, 602, 512, 5,996, 599 and 451 stim events, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Error bars are 95 percent confidence intervals calculated via bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' ‘n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.’ indicates p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='05 via two-proportion Z-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 3/19 turns also reduced the likelihood of a reversal response (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 1c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This finding suggests that turns alone are sufficient to modulate the likelihood of a mechanosensory-evoked reversal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We therefore focused on the turn regardless of what behavior preceded it, and sought to identify the circuit level mechanisms with which the turn interacts with the mechanosensorimotor response pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' From here onwards, we consider both isolated- and escape-like turns together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turns decrease the likelihood of interneuron-evoked reversals, except for those evoked by AVA Mechanosensory signals from the anterior gentle-touch mechanosensory neurons AVM and ALM are thought to evoke a reversal response by traveling downstream through a network of interneurons that are associated with backward locomotion [13,14,19,28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' These include neurons AVA [32–34], AIZ [35], RIM [33,36], AIB [33] and AVE [37](Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Like the anterior mechanosensory neurons, these interneurons are known to induce reversals upon stimulation [33,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To better understand where this network interacts with turning, we sought to investigate whether these interneurons’ ability to evoke reversals also depends on turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We used a collection of transgenic strains with cell-specific or near-cell-specific promoters that drive expression of the optogenetic proteins Chrimson or ChR2 in each of these interneurons (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We then used a high-throughput closed-loop optogenetic delivery system of our own design to stimulate the interneuron with 3 s illumination when the worm was either crawling forward or beginning to turn [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In this way we measured the animal’s response to many thousands of optogenetic stimulation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' As expected, optogenetic activation of any of these interneurons during forward locomotion evoked reversals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2b) compared to the baseline probability of a spontaneous reversal (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Activating any of the interneurons we tested, except for AVA, showed a statistically significant decrease in the probability of evoking reversals when activated during turns, compared to during forward locomotion, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In other words, activation of these interneurons showed a turning-dependent response, similar to the mechanosensory neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' By contrast, there was no significant difference in AVA’s ability to evoke reversals when stimulated during turning compared to during forward locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' From these measurements within the mechanosensorimotor pathway, we conclude that neurons AIZ, RIM, AIB and AVE lie either at or upstream of the junction in which turning signals arrive to modulate the reversal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' AVA, in contrast, lies in the pathway downstream of the arrival of turning related signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We therefore sought to investigate neural sources of this turning related signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We note that in the measurements of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2, we have emphasized how the probability of evoking a reversal changes between forward and turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We do not concern ourselves with neuron-to-neuron variability in the probability of evoking reversals because that likely reflects strain-specific differences in gene expression or in the efficiencies of ChR2 compared to Chrimson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turning associated neurons RIV, SMB and SAA regulate reversals Turning in the worm occurs either when the animal is moving forward, is paused or is transitioning from backward to forward locomotion, but not during sustained backward locomotion [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Neurons RIV, SMB and SAA are among those neurons associated with turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' RIV, SMB and SAAD have increased calcium activity during turns [19,39], January 10, 2023 4/19 FwdTurn FwdTurn FwdTurn FwdTurn FwdTurn 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5 1 Probability of reversal AIZ AVA AVE AIB RIM Context *** *** *** *** n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Touch neurons Reversal associated interneurons Turning neurons ALM AVA AVE AIB AVM AIZ RIM SMB RIV AVD SAA Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Turns decrease the likelihood of interneuron evoked reversals, except for AVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a) Diagram showing chemical (arrows) and electrical (resistor symbol) synapses among the anterior mechanosensory neurons, downstream interneurons, and turning associated neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Adapted from nemanode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='org [27] b) Probability of a reversal response is shown for optogenetic stimulation of reversal associated interneurons delivered either during forward movement or during the onset of turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 3s illumination of 80 µW/mm2 red light (AVE or AVA) or >300 µW/mm2 blue light (neuron AIZ, RIM, or AIB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Error bars are 95 percent confidence intervals calculated via bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001 via two-proportion Z-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' p value for AVA stimulation group is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =2,612, 601, 883, 107, 880, 511, 1,007, 342, 409, 191 stimulus events (from left-to-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 5/19 0 2 4 Reversal duration post stimulus onset (s) Inhibit RIV, SMB, SAA o + *** gtACR2 in RIV, SMB, SAA Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' RIV, SMB and SAA neurons influence reversal duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Time spent going backwards in a 10 s window coinciding with optogenetic inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Worms expressing inhibitory opsin gtACR2 in neurons RIV, SMB, and SAA under lim-4 promoter were inhibited by blue light of either 180 µW/mm2 (‘+’) or 2µW/mm2 (‘o’) for 10 s during backward locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Worms spent more time reversing when these neurons were inhibited than in the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N = 612 and 695 stimulus events for ‘o’ and ‘+’ conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' and ablation of RIV, SMB or SAA show defects in turning or head bending amplitude [28,39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Wang and colleagues observed that inhibiting RIV, SMB and SAA when the animal is backing up prolongs the reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' They therefore proposed that activity from turning-related neurons may inhibit reversals [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We independently confirm that inhibiting RIV, SMB and SAA increases reversal duration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 3 and Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We therefore sought to investigate whether these turning neurons also inhibit reversals during turns, and whether they may explain why mechanosensory stimulation is less likely to evoke reversals during turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Inhibiting RIV, SMB and SAA abolishes the turning dependent modulation of mechanosensory processing We reasoned that if the turning neurons RIV, SMB and SAA inhibit reversals, then releasing this inhibition during the onset of turning should allow mechanosensory stimuli delivered during a turn to evoke reversals as effectively as if they were delivered during forward locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We designed an experiment to simultaneously inhibit these turning neurons while stimulating the touch neurons during the onset of a turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We expressed a blue-light inhibitory opsin, gtACR2, in the turning associated neurons RIV, SMB and SAA and a red-light activating opsin Chrimson in the gentle touch neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' When we activated the touch neurons using red light, this strain behaved similarly to our other strains: stimuli delivered during turns were less likely to evoke a reversal than those delivered during forward locomotion, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' But when we also inhibited the turning associated neurons with blue light while stimulating the touch neurons during the onset of a turn, the likelihood of evoking reversals was significantly higher and, crucially, not significantly different than for stimuli delivered during forward locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In other words, inhibiting these turning associated neurons during turns abolished the turning-dependence of the mechanosensory response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This is consistent with a model in which signals from RIV, SMB and/or SAA disrupt mechanosensory processing during turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' By inhibiting those neurons after the onset of a turn, we prevent this disruption, presumably by inhibiting an inhibitory signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 6/19 Chrimson in touch neurons gtACR2 in RIV, SMB, SAA Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Optogenetic inhibition of neurons RIV, SAA and SMB during turns restore mechanosensory-evoked reversal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Probability of reversals when touch neurons are activated or when touch neurons are activated and RIV, SMB and SAA are inhibited simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' during either forward movement or turn onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Touch neurons express Chrimson and are activated with red light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' RIV, SMB and SAA express gtACR2 and are inhibited with blue light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001, two-sample Z-test for proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =5,381, 1,525 and 1,115 stimulation events from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Additional controls are shown in Fig S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We performed additional experiments to rule out alternative explanations (Supplementary Fig S3 and Supplementary text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' For example, we find that blue light illumination without an inhibitory opsin in the turning-associated neurons is insufficient to restore mechanosensory evoked reversal responses during turns (Supplementary Fig S3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Taken together we conclude that inhibition of the turning neurons during turns disinhibits mechanosensory evoked response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Inhibitory signals from turning neurons gate mechanosensory processing Taken together, our measurements supports a model in which the turning neurons RIV, SMB and/or SAA gate mechanosensory information and prevent it from propagating further downstream to evoke a reversal, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In this model, mechanosensory signals from the gentle-touch mechanosensory neurons ALM and AVM propagate in a feedforward manner to reversal-associated interneurons RIM, AIZ, AIB and AVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' If the animal is moving forward, the mechanosensory signals continue to propagate to AVA and evoke reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' But if the animal is turning, inhibitory signals originating from RIV/SMB/SAA suppress or disrupt mechanosensory-related signals within the interneurons and prevent propagation to AVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This model is consistent with our measurements and leads us to conclude that turning-related inhibitory signals gate mechanosensory processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 7/19 Mechanosensory neurons AIZ, RIM, AIB, AVE AVA Reverse RIV, SMB, SAA Turn Releases ACh AIB SMB RIV SAA RIM lgc-47 acc-1 lgc-47 Inhibitory Ach Receptors a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Putative circuit mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a) In response to gentle touch, mechanosensory neurons propagate signals to promoter neuron AVA and evoke a reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' But during turning, neurons RIV, SMB, and SAA send inhibitory signals that disrupt sensory signals before they reach AVA thus gating the likelihood of a reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b) Wiring and gene expression is consistent with the following inhibitory path (yellow arrows): SAA releases acetylcholine via synapses onto RIM and AIB, each of which expresses inhibitory acetylcholine receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Discussion and Conclusions Here we show that putative inhibitory signals from turning associated neurons RIV/SMB/SAA modulate mechanosensory evoked reversals downstream of the gentle touch neurons and upstream of neuron AVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' But within those constraints, where exactly might those signals combine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Neuron wiring and gene expression data suggests that one location may be across the inhibitory synapses from SAA to AIB and RIM, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' SAA is cholinergic and makes synapses to AIB and RIM [27,40], which both express inhibitory acetylcholine receptors [41,42]: AIB expresses the inhibitory acetylcholine receptors lgc-47, and acc-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' while RIM expresses inhibitory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' lgc-47) and excitatory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' acr-3) acetylcholine receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We predict that RIM may spatially localize its lgc-47 receptors to its synapse with SAA such that SAA’s input is net inhibitory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Because AIB and RIM both synapse onto AVA, the inhibition of AIB and RIM is well positioned to interrupt the propagation of mechanosensory signals to AVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Gene expression and wiring therefore suggest a plausible path by which inhibition from the turning circuitry interrupts mechanosensory signals from reaching AVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Wang and colleagues had predicted that turning circuitry may inhibit reversal circuitry through a yet-to-be-identified pathway [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Our findings suggest that SAA to AIB and RIM may be that pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' More broadly our findings reinforce a longstanding hypothesis that different motor programs in the worm inhibit one another, such as forward and reverse locomotion [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In our model, AVA performs a role similar to that of a “decision neuron” with respect to reversals [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This is consistent with our previous observation that AVA’s calcium activity more closely reflects the animal’s decision to reverse, and is less reflective of the strength of the stimulus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' AVA’s activity does not reflect how many touch neurons are activated) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The simple model we describe assumes feed-forward January 10, 2023 8/19 propagation of signals from ALM and AVM to AVA and omits recurrent connections among the neurons in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Future investigations are needed to explore additional contributions from recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' More broadly, we show that motor related signals are directly influencing neural activity in areas that contain a mix of sensory and motor information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This is reminiscent of saccadic suppression in vision [45–47] and corollary discharge [25,26] in which motor related activity modulates or impinges upon sensory representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Our findings add to a growing body of evidence suggesting that behavior information is necessary for sensory processing, and this may explain why behavior-related neural activity patterns are seen across the brain in mice, fly and worms, including in nominally sensory areas [20–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Because turning events are infrequent, spontaneous and brief, they are rare compared to the time the animal spends moving forward or backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' But obtaining sufficient statistical power to probe sensory processing during turns required hundreds of observations per condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In total we measured over 40,000 behavior responses to stimulation, including more than 16,000 during turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This was only made feasible by leveraging computer-vision and targeted illumination to track many worms in parallel and to automatically deliver stimuli triggered upon the animal’s turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Such closed-loop automated experimental paradigms will be important for future investigations into other rare and brief spontaneous behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Materials and methods Strains Strains used in this work are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In each strain light-gated ion channels have been expressed to either excite or inhibit specific neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We expressed excitatory opsin Chrimson in the six gentle touch neurons using the mec-4 promoter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Promoters ser-2, tdc-1, npr-9, opt-3, rig-3 are used to express excitatory opsin in neurons AIZ, RIM, AIB, AVE, and AVA respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To express gtACR2 in RIV, SMB, and SAA, we used the lim-4 promoter and performed integration using a mini-SOG approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We injected into CZ20310 worms, followed by a blue light treatment (450nm, M450LP1, Thorlabs) for 30 minutes as described in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Before conducting experiments, we outcrossed integrated worms with the wild type N2 strains for at least six generations to generate AML496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' AML496 worms were then crossed into AML67 worms to create AML499 strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Nematode handling All worm strains were maintained at 20 C, on regular NGM media plates seeded with E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' coli (OP-50) as food source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Experiments were performed on young adult animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To obtain young adults, worms were bleached three days prior to the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Bleached eggs were washed and centrifuged in M9 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='8rcf for two mins) three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Bleached eggs were suspended in M9 and stored in a shaker overnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The following morning hatched L1 larvae were centrifuged and transferred to freshly seeded plates consisting of 1 ml of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5 mM all-trans-retinal mixed with OP50 and stored in the dark at 20 C until young adulthood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' For experiments, young adult worms were washed in M9 and transferred to an empty agarose plate for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Excess M9 solution was absorbed with a kim wipe as described in [16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 9/19 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strain name Target neu- ron expres- sion additional expression Genotype Figure Ref AML67 ALML, ALMR, AVM, PLML, PLMR, PVM wtfIs46[mec- 4P::Chrimson::SL2::mCherry::unc-54 40ng/ul] Fig 1c,d and Fig S3b [16] TQ3301 AIZ xuIs198[Pser-2(2)::frt::ChR2::YFP,Podr- 2(2b)::flp, Punc-122::YFP];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' lite-1(xu7) Fig 2b and Fig S1 [35] QW910 RIM zfIs9[Ptdc-1::ChR2::GFP, lin-15+];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' lite- 1(ce314) Fig 2b and Fig S1 [36] QW1097 AIB zfIs112[Pnpr-9::ChR2::GFP, lin15+];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' lite- 1(ce314) Fig 2b and Fig S1 [36] Not provided AVE opt-3::Chrimson Fig 2b and Fig S1 [37] AML17 AVA I1, I4, M4, and NSM wtfIs2[rig-3::Chrimson::SL2::mCherry] Fig 2b and Fig S1 [32] AML496 RIV, SMB, SAA wtfIs465 [lim- 4P::gtACR2::SL2::eGFP::unc-54 80ng/ul + unc-122::RFP 50ng/ul] Fig 3 and Fig S3c This work AML499 RIV, SMB, SAA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' ALML, ALMR, AVM, PLML, PLMR, PVM wtfIs46[mec- 4P::Chrimson::SL2::mCherry::unc- 54 40ng/ul];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' wtfIs465 [lim- 4P::gtACR2::SL2::eGFP::unc-54 80ng/ul + unc-122::RFP 50ng/ul] Fig 4, Fig S2 and Fig S3a This work January 10, 2023 10/19 Target neuron(s) Perturbation Target Behavior Stim Triggered on Stim Duration (s) ISI (s) Illumination Intensity (uW/mm2) Illumination Color Strain # Plates Total stim events Figures Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Forward 30 29 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='998 Turn Turns >30 47 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='164 Forward 30 16 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='258 Turn Turns >30 27 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='184 Forward 30 12 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='766 Turn Turns >30 19 238 Forward 30 4 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='747 Turn Turns >30 24 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='038 Forward 30 8 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='413 Turn Turns >30 16 832 Forward 30 8 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='035 Turn Turns >30 20 411 RIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' SMB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' SAA Inhibit gtACR2 Reversal Reversals 10 >30 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 180 Blue AML496 14 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='307 Figure 3 ALML,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' ALMR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' AVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' PLML,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' PLMR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' PVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' RIV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' SMB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' SAA Inhibit gtACR2 Reversal Reversals 10 >30 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 180 Blue AML499 12 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='532 Supplementary Figure S2 Forward 30 8 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='381 Turn Turns >30 16 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='525 Excite Chrimson and Inhibit gtACR2 Turn Turns >30 Red=60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Blue=180 Red + Blue 17 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='115 Inhibit gtACR2 Turn Turns >30 180 Blue 15 954 Control Turn Turns >30 Red=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5, Blue=2 Red + Blue 8 1,961 Forward 30 6 3,722 Turn Turns >30 12 903 Turn Turns >30 Red=60, Blue=180 Red + Blue 15 794 Turn Turns >30 180 Blue 16 579 Control Turn Turns >30 Red=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5, Blue=2 Red + Blue 15 772 RIV, SMB, SAA Inhibit gtACR2 Turn Turns 3 >30 2, 180 Blue AML496 16 2,074 Supplementary Figure S3c This work TOTAL: 400 53,703 AVE Excite Chrimson Excite Chrimson ALML, ALMR, AVM, PLML, PLMR, PVM AVA Excite Chrimson Excite Chrimson ALML, ALMR, AVM, PLML, PLMR, PVM, RIV, SMB, SAA ALML, ALMR, AVM, PLML, PLMR, PVM Excite Chrimson Excite ChR2 AIZ 3 RIM Excite ChR2 AIB Excite ChR2 2, 300 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5, 80 2, 340 3 2, 300 Blue QW1097 Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S1 Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S1 Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S1 Blue Figure 1c, 1d AML67 TQ3301 Blue QW910 Red Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S1 Figure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S1 60 Red 3 Figure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Figure S3a AML499 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5, 80 Red AVE 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5, 80 Red AML17 3 60 Red AML67 Supplementary Figure S3b This work This work This work This work [17] This work This work This work This work Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Summary of optogenetic measurements performed during behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Behavior Analysis Computer-vision based behavior analysis was used to identify when the animal is moving forward, when it is undergoing a reversal or when it is turning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Analysis was performed as reported previously using two different sets of algorithms for real time vs post-processing [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' All figures in this work reflect behavior classifications from the off-line retrospective analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Briefly, animals are segmented and a centerline is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Additional logic is used to find centerlines even when the animal touches itself [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The animal’s center of mass velocity is also computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Behavior classification is first performed by classifying pose dynamics in a behavior map [16,49] and then refined by inspecting the animal’s ellipse ratio and center of mass velocity to catch any omitted turns, or instances when the behavior mapper fails to classify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Compared to our previous recent work [17] we changed two parameters to be more conservative in classifying animals as turning or reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Specifically, to be classified as turning we now require that the binary image of the animal have an ellipse ratio of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1, compared to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='6 previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Similarly, to be classified as a reversal, the animal must now achieve a center of mass velocity of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='11 January 10, 2023 11/19 mm/s, instead of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1 mm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' For experiments probing reversal duration, we report the time the animal spent going backwards in a 10 s window, coinciding with optogenetic inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' So for example, if after stimulus onset the animal continued moved backwards for 3 s, then paused for 1 s, and moved backwards for 2 s more, we report a “reversal duration” of 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Optogenetic activation and inhibition In this work we seek to deliver optogenetic illumination specifically when the animal is either moving forward, or turning, or reversing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We conduct different sets of experiments for each of these three conditions, using different sets of animals for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In all cases we use a projector-based illumination system that tracks many individuals on a plate full of animals, segments them in real time, and addresses each animal individually to shine light on them, as described previously [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' All experiments are performed on plates containing approximately 30 to 40 animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To measure the animal’s response to optogenetic perturbations during forward locomotion, we optogenetically illuminated all tracked animals on the plate every 30 s, in open loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In post-processing we then only considered those animals that were moving forward at the time of illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To measure the animal’s response to optogenetic activation or inhibition delivered during the onset of turns, our system waited until it detected that an animal was beginning to turn, and then delivered a stimulus automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' In post-processing we retrospectively evaluated whether the turn was valid at time of stimulus onset, and only included those stimuli events that met our more stringent criteria, as described in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' To measure the animal’s response to optogenetic inhibition during reversals, our system waited until it detected that an animal had been reversing for 1 second, and then delivered the illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' As before, we retrospectively confirmed that the animal was reversing before including it for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Illumination color, intensity and duration are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' As a control, we compare optogenetic stimulation to illumination with very low intensity light that does not discernably alter behavior [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Statistical Analysis To reject the null hypothesis that two empirically observed probabilities are the same, we use a two-proportion Z-test [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Error bars report 95% confidence interval calculated via a bootstrap procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Data availability Computer-readable files showing processed tracked behavior trajectories and stimulus events for all experiments are publicly posted at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='21699668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Code availability All analysis code used in this manuscript are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='com/leiferlab/analysis-code-kumar-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains and plasmid availability All genetic strains and plasmids generated as part of this manuscript are being made available through Caenorhabditis Genetics Center (CGC) and Addgene respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 12/19 Acknowledgments We thank Zhaoyu Li (Queensland Brain Institute), Shawn Xu (University of Michigan) and Mark Alkema (University of Massachusetts Worcester) for the strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This work used computing resources from the Princeton Institute for Computational Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Research reported in this work was supported by the Simons Foundation under award SCGB #543003 to AML;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' and by the National Science Foundation, through an NSF CAREER Award to AML (IOS-1845137) and through the Center for the Physics of Biological Function (PHY-1734030).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains from this work are 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Visual and motor signatures of locomotion dynamically shape a population code for feature detection in Drosophila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' eLife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='11:e82587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='7554/eLife.' metadata={'source': 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Shaevitz JW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Mapping the stereotyped behaviour of freely moving fruit flies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Journal of the Royal Society, Interface / the Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='11(99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='1098/rsif.' metadata={'source': 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proportion of defectives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Available from: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='itl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='gov/div898/handbook/prc/section3/prc33.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Supplementary Fig S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Probability of reversal upon low-light (control) illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a) Five key interneurons were probed using whole body optogenetic stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The gray bars represent the trials when the stimulus was delivered during the forward state while the purple bar represents the trials during which stimulus was delivered on turning onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' For all these measurements control stimulus was delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Three seconds of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='5 µW/mm2 of red stimulus (neuron AVE and AVA) or 2 µW/mm2 of blue stimulus (neuron AIZ, RIM, and AIB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Error bars are 95 percent confidence intervals calculated via 10,000 bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Two sample Z-test was used to calculate significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' p value for AIZ, RIM, AIB, AVE, and AVA stimulation group is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='596, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='936, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='565, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='262 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The number of stimulus events for each condition (from left-most bar to right-most bar) are: 2,646, 583, 883, 131, 867, 527, 1,406, 490, 626, 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 0 2 4 6 Reversal duration post stimulus onset (s) Inhibit RIV, SMB, SAA o + *** Chrimson in touch neurons gtACR2 in RIV, SMB, SAA Supplementary Fig S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Inhibition of RIV, SMB and SAA prolong reversals, in a second transgenic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Same experiment as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 3, but in a transgenic background that also expresses Chrimson in the mechanosensory neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Results are consistent with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Worm spent more time reversing when the RIV, SMB, and SAA neurons were inhibited compared to when a control stimulus intensity was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' The number of stimulus events for mock and experimental conditions are 1,168, 1,364 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 17/19 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Chrimson in touch neurons Chrimson in touch neurons gtACR2 in RIV, SMB, SAA gtACR2 in RIV, SMB, SAA Supplementary Fig S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Additional control experiments show that blue-light alone cannot restore mechanosensory-evoked reversal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' a) Probability of reversals when either touch neurons are activated, or RIV, SMB and SAA are inhibited, or both simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' during either forward movement or turn onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' First three bars are same as in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Touch neurons express Chrimson and are activated with red light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' RIV, SMB and SAA expressing gtACR2 are inhibited with blue light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Strains are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' *** indicates p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content='001, two-sample z-test for proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =5,381, 1,525, 1,115, 954 and 1,961 stim events, from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' b) Same experiments were repeated in a strain that expressed Chrimson in the gentle-touch mechanosensory neurons, but no inhibitory opsins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =3,722, 903, 794, 579 and 772 stim events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' c) Same experiments are shown for animals that only express inhibitory opsin gtACR2 in RIV, SMB and SAA, but no Chrimson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' N =1,041 and 1,033 stim events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 18/19 Additional control experiments show that blue-light alone cannot restore mechanosensory evoked reversals We sought to rule out alternative explanations for why shining blue light during turns may cause an increase in reversals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' It is known that the nominally red-light sensitive Chrimson can also be mildly activated by blue light [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' We therefore we tested whether the increase in responsiveness to the mechanosensory stimulus was the result of blue-light activation of Chrimson expressed in the touch neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Consistent with mild blue-light activation of Chrimson, shining only blue light on animals expressing Chrimson in the touch neurons (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S3a,b far right bar) but not on animals expressing only gtACR2 in the turning neurons (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S3c far right bar) caused a significant increase in the probability of reversals compared to no stimulation (second to right bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' However, this mild blue-light activation of Chrimson is insufficient to explain the increase in reversal probability we observed when inhibiting turning neurons via gtACR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Even when shining both blue and red light on animals that lack the inhibitory opsin gtACR2, but do contain Chrimson in their touch neurons, we still observed a large and significant reduction in the likelihood of reversing in response to stimuli delivered during turns compared to delivered during forward locomotion (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S3b middle bar, compared to far left bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' This suggests that it is the inhibition of neurons RIV, SMB and SAA that abolishes the turning-dependence of mechanosensory processing and not mild blue-light activation of the touch neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Further consistent with this view, adding blue light to red light in those animals that lack the inhibitory opsin does not significantly increase the probability of reversing (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' S3b second compared to third bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' A simple and fully consistent explanation is that our red light illumination strongly activates the touch neurons and that any additional blue light contributes only very modest additional activation to the touch neurons, and not enough to explain the increase we see when inhibiting RIV/SMB/SAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Moreover, this modest additional blue-light activation of the touch neurons is only significant in control experiments without any red light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' Taken together we conclude that inhibition of the turning neurons, and not mild blue-light activation of Chrimson, is responsible for abolishing the turning dependence of the mechanosensory response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} +page_content=' January 10, 2023 19/19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQf2QI-/content/2301.02709v1.pdf'} diff --git a/U9FKT4oBgHgl3EQfmC51/content/tmp_files/2301.11856v1.pdf.txt b/U9FKT4oBgHgl3EQfmC51/content/tmp_files/2301.11856v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b13c7a7e60abbe26b40d2ec5a157ce1b487c27e --- /dev/null +++ b/U9FKT4oBgHgl3EQfmC51/content/tmp_files/2301.11856v1.pdf.txt @@ -0,0 +1,1257 @@ +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +Hui Wen Goh 1 Jonas Mueller 1 +Abstract +In real-world data labeling applications, annota- +tors often provide imperfect labels. It is thus com- +mon to employ multiple annotators to label data +with some overlap between their examples. We +study active learning in such settings, aiming to +train an accurate classifier by collecting a dataset +with the fewest total annotations. Here we pro- +pose ActiveLab, a practical method to decide what +to label next that works with any classifier model +and can be used in pool-based batch active learn- +ing with one or multiple annotators. ActiveLab +automatically estimates when it is more informa- +tive to re-label examples vs. labeling entirely new +ones. This is a key aspect of producing high qual- +ity labels and trained models within a limited an- +notation budget. In experiments on image and +tabular data, ActiveLab reliably trains more accu- +rate classifiers with far fewer annotations than a +wide variety of popular active learning methods. +1. Introduction +Model-agnostic active learning methods use outputs from +some arbitrary type of trained prediction model in order to +identify the most informative data to label, so that a more +accurate version of the same model can be trained. Such +general approaches are popular because they can be directly +applied to many data modalities (image, text, etc.) as long as +a reasonable model can be trained. Focusing on highly prac- +tical settings, we consider model-agnostic pool-based active +learning with multiple data annotators that label a batch of +many examples in between model (re)training runs. This +setting is easy to setup and allows us to address common +issues in real-world active learning such as: labelers who +are imperfect, or expensive model (re)training that cannot +be executed every time a new example is labeled. Working +with annotators that may provide incorrect labels, it is useful +to sometimes ask new annotators to provide extra labels for +examples previously labeled by others. This allows us to +verify the current consensus label or estimate a better one. +*Equal +contribution +1Cleanlab. +Correspondence +to: +Hui +Wen +Goh +, +Jonas +Mueller +. +Here we introduce ActiveLab1, a straightforward active +learning algorithm that estimates when such re-labeling will +be more effective than labeling an entirely new example. A +very general approach, ActiveLab can be used: with any +type of classifier model (or ensemble of multiple models) +and data modality, for active learning with multiple anno- +tators where the set of annotators changes over time, for +traditional active learning where each example is labeled +at most once (Appendix D), and for active label cleaning +where all data is already labeled by at least one annotator +and the goal is to establish the highest quality consensus +labels within a limited annotation budget. ActiveLab is +easy-to-implement and computationally efficient. +2. Methods +This paper focuses on classification tasks with K classes, for +which some (arbitrary) classifier model M can be trained. +For our ith example with feature values Xi, this model +predicts a class probability vector �pM(Yi | Xi) estimating +the likelihood that X belongs to each class k ∈ [K] := +{1, 2, . . . , K}. +In the pool-based batch active learning settings we consider, +each round involves the steps described below. In the be- +ginning, we start with a training set D of examples that +have at least one (noisy) annotation, where some of these +examples may have been labeled by multiple annotators. +We also have a pool of unlabeled examples U that have +zero annotations. Our proposed active learning method may +choose to collect new labels for examples in either D or U. +Based on classifier predictions �p and the currently-observed +annotations D, ActiveLab estimates an acquisition score si +for each example. Examples with the lowest si values are +those for which collecting an additional label is expected +to be most informative when subsequently training M. To +avoid overfit/biased results, classifier predictions �p should +be out-of-sample, coming from a copy of the model M that +has never been trained with the example it is asked to predict +the class of. +We can obtain out-of-sample predictions for every xi ∈ D +by fitting our model via k-fold cross-validation in Step 3. +1Our +code: +https://github.com/cleanlab/ +multiannotator-benchmarks/tree/main/active_ +learning_benchmarks +arXiv:2301.11856v1 [cs.LG] 27 Jan 2023 + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +Active learning with multiple annotators +Input: D: labeled examples with at least one annotation +Input: U: unlabeled pool of examples (not yet annotated) +1: for r = 1, 2, . . . {rounds of active learning} do +2: +Estimate consensus labels �Yi for annotated examples +xi ∈ D (some of which have multiple annotations) +3: +Train classifier model Mr with these labels: (xi, �Yi) +4: +Obtain (out-of-sample) predicted class probabilities +for all examples: �p = Mr(x) for x ∈ D ∪ U +5: +Use active learning method to score all examples: +si = A(�pi; D) for all xi ∈ D ∪ U +6: +Assemble batch B of the B best-scoring examples, +collect one additional label Yij for each xi ∈ B, and +add new {Yij} to the training data (updating D, U) +7: end for +For examples currently in the unlabeled pool x ∈ U, Step 6 +can collect their first label, and there may be already-labeled +examples x ∈ D in the selected batch B for which we collect +yet another label. There are many ways to operationalize +the collection of labels in Step 6 of active learning. The +examples to acquire an extra label for could be divided +amongst a limited pool of annotators (some of which labeled +other examples in previous active learning rounds), or these +examples could be given to new annotators to label. +While one can envision alternate methods that suggest which +annotator should label which example (Huang et al., 2017), +we find such a tightly-controlled setting too rigid for many +applications. Step 6 is intentionally flexible. We also do not +consider methods that can ask more than one annotator to +review the same example within a round as such methods +can be brittle (Baldridge & Osborne, 2004). +Notation. +In the remaining notation, all definitions of ob- +jects are given with respect to the current round. Here we +omit subscripts r and how objects change between rounds. +In the current round, the set of annotated examples D con- +tains n examples labeled by m annotators in total. Yij ∈ [K] +denotes the class annotator Aj chose for example xi ∈ D, +with Yij = ∅ if annotator Aj did not label example i. Yi is +the set of collected labels for example xi, with |Yi| = 0 if +xi ∈ U. Ij is the subset of examples labeled by annotator +Aj, and Ji is the subset of annotators that labeled xi. +2.1. ActiveLab +ActiveLab extends the CROWDLAB estimator of Goh et al. +(2022). Some equations in this paper overlap with CROWD- +LAB, but we present them for completeness. Not every +CROWDLAB equation is motivated here, curious readers +can refer to detailed explanations by Goh et al. (2022). +Unlike ActiveLab, which is intended for guiding collection +of additional labels, CROWDLAB is intended for analyzing +a static dataset labeled by multiple annotators. Empirically +it performs poorly when used for active learning. While both +approaches estimate consensus labels in a similar fashion, +they score examples differently. CROWDLAB estimates +the likelihood that each current consensus label is correct +or not, whereas ActiveLab estimates the utility of collecting +another label to further improve the consensus and model +trained therewith. CROWDLAB assigns very low scores +to examples annotated by many labelers that heavily dis- +agree, but even though their consensus label is unreliable, +ActiveLab recognizes there is less utility in collecting one +more label for such fundamentally difficult examples (vs. +examples that currently have fewer annotations). Unlike +CROWDLAB, ActiveLab also scores examples which cur- +rently have not been labeled yet. It must trade-off the po- +tential information gain from collecting the 1st label for an +example from U vs. the jth label for an example already +labeled j − 1 times. Both methods can utilize any type of +classifier model M trained in any fashion. +We first describe how ActiveLab computes the score si for +examples that have at least one annotation. Both CROWD- +LAB and ActiveLab are straightforward weighted ensem- +bles which linearly combine multiple predictors to form a +single estimate of class probabilities. In prediction compe- +titions, such ensembles are often more accurate and better +calibrated. One of these predictors is the (out-of-sample +predictions from a) trained classifier M, abbreviated as +�pM,i,k := �pM(Yi = k | X = xi). The other predictors are +the annotators who previously labeled xi. From the label Yij +chosen by annotator Aj, we form an annotator-estimated +class probability vector �pAj,i,k ≈ p(Yi = k | Yij) that is +directly comparable to the classifier predicted class probabil- +ities (details further below). ActiveLab and CROWDLAB +take a weighted average of this collection of probabilistic +predictions to form a single vector of ensemble predicted +class probabilities for each xi. +CROWDLAB subsequently selects the most likely class +under this ensemble estimate as the consensus label �Yi rep- +resenting our best guess of the true label Yi. In Step 2 of +each active learning round, we use CROWDLAB to estimate +a single consensus label �Yi that aggregates the available an- +notations Yi for each example xi ∈ D. Subsequently in +Step 5, ActiveLab scores xi ∈ D via the likelihood that +class �Yi is correct under its ensemble estimate, expressed as: +If xi ∈ D : +(1) +si = +wM · �pM,i,�Yi + w ¯ +A · 1 +K + � +j∈Ji wj · �pAj,i,�Yi +wM + w ¯ +A + � +j∈Ji wj +If xi ∈ U : si = wM · maxk �pM,i,k + w ¯ +A · 1 +K +wM + w ¯ +A +(2) +The above estimates depend on wM, wj which determine +how much to weigh the model M and each annotator Aj. + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +We estimate their relative trustworthiness (based on the +observed annotations {Yij}) in order to select these weights, +via the same procedure as CROWDLAB (details further +below). Intuitively our estimate should down-weigh untrust- +worthy annotators or a poorly trained classifier, see Goh +et al. (2022) for further discussion on this estimate’s robust- +ness against bad annotators/models. Unlike CROWDLAB, +equation (1) also contains a uniform 1/K predictor that re- +ceives weight w ¯ +A := 1 +m +�m +j=1 wj, representing the weight +assigned to our average annotator (across all examples). +Here is a fundamental difference between ActiveLab and +CROWDLAB: under the former, the estimated likelihood +that �Yi is the correct class for xi ∈ D is much lower (closer +to uniform) for examples with few annotations. This regu- +larization has smaller effect on examples with many annota- +tions. Thus amongst the x ∈ D, ActiveLab naturally favors +acquiring labels for examples that currently have fewer an- +notations. ActiveLab also favors examples where annotators +disagree with the consensus (note �pAj,i,�Yi is much smaller +if Yij ̸= �Yi) or the classifier predicts the consensus to be +unlikely. These are the xi ∈ D whose current consensus +label may be wrong, warranting re-labeling to determine +whether a better label can be established. +Scoring examples from the unlabeled pool. +Before +delving into the details of wM, wj, and �pAj, we describe +how ActiveLab scores xi ∈ U. This is detailed in equa- +tion (2). Since we have no annotations for xi ∈ U, Ac- +tiveLab scores such examples only using the probabilistic +predictions from our classifier �pM. Many traditional active +learning methods also operate this way (Munro, 2021). As +seen in (2), the score si for xi ∈ U is similarly computed +as for xi ∈ D, except for modifications required to handle +missing information. Since Ji = ∅ in this case, we simply +drop the annotator-predictors �pAj from the weighted ensem- +ble in order to obtain its estimate for unlabeled examples. +And we simply take �Yi = arg maxk �pM,i,k, the class pre- +dicted by our classifier, since CROWDLAB cannot estimate +a consensus label for xi ∈ U. Amongst the unlabeled ex- +amples, ActiveLab thus favors acquiring labels for those xi +for which the classifier is least confident, as in traditional +uncertainty sampling (Munro, 2021). +To label or re-label? +Since they are computed in a similar +fashion, the si are directly comparable between xi ∈ D vs. +U. ActiveLab thus naturally suggests when it is better to +re-label an example from D vs. labeling a new example +from U. Cases when this might be true for some example +xi ∈ D include settings where: its annotations disagree +(indicating that some annotators are noisy), or the model +has atypically low confidence in its prediction (indicating xi +may be an outlier or high-influence datapoint whose label we +should really get right), or the model confidently disagrees +with the annotations. This last case is especially pertinent +for examples xi that only have a single annotation, where +we may prefer to trust a confident prediction from a well- +trained classifier over the given label which may be wrong +(Northcutt et al., 2021a; Kuan & Mueller, 2022). Fixing +labels for existing training data can improve a classifier more +than noisily labeling additional data (Northcutt et al., 2021b; +Iraola & Yepes, 2021). Section 5.1 empirically explores this. +Mathematically, it is evident that ActiveLab will always +prefer to label new examples from U if every annotation and +the classifier (confidently) agree for all xi ∈ D. +Example. +Consider xi with a single annotation Yij and a +different xℓ ∈ U, such that our classifier is equally confident +in its predictions for both. In this case, deciding whether to +re-label xi vs. labeling xℓ specifically depends on: whether +Yij matches the classifier’s predicted class arg maxk �pM,i,k, +and how much ActiveLab weights this annotator (wj) vs. +the average annotator (w ¯ +A) and the classifier (wM). If +the classifier’s prediction matches Yij, then ActiveLab will +prefer to label xℓ. If the classifier disagrees with the annota- +tion, then ActiveLab will prefer to re-label xi whenever the +CROWDLAB consensus label �Yi ̸= Yij. This occurs if: +wM +� +�pM,i,k∗ − �pM,i,Yij +� +> wj +� +�pAj,i,Yij − �pAj,i,k∗� +where k∗ := arg maxk �pM,i,k ̸= Yij in this example. The +inequality is satisfied if: wM ≫ wj (i.e. ActiveLab esti- +mates the classifier is more trustworthy than annotator Aj) +and �pM,i,k∗ − �pM,i,Yij ≫ �pAj,i,Yij − �pAj,i,k∗ (i.e. the clas- +sifier predicts Yij is not the correct label confidently relative +to the estimated accuracy of the data annotators). +Details for estimating weights and annotator likelihood. +ActiveLab estimates wM, wj, and �pAj in the same fashion +as CROWDLAB. We present the mathematical details here +but refer readers to the explanations/motivations articulated +by Goh et al. (2022). In equation (1), �pAj ∈ Rk is an +“annotator likelihood” vector containing the probabilities +that xi belongs to each class given that annotator Aj chose +the label Yij. It is very simply defined: +�pAj,i,k ≈ p(Yi = k | Yij) := +� +P +when Yij = k +1−P +K−1 +when Yij ̸= k +P ≥ 0 is a global scalar parameter shared across all anno- +tators. It is estimated by computing the average annotator +agreement across all examples that have more than one an- +notation. P estimates the probability that a typical annotator +would select the consensus label for an arbitrary example +they are given (Goh et al., 2022). +The weights wM, wj in equation (1) estimate the trustwor- +thiness of our classifier model and each annotator. The +model weight is defined in terms of the normalized accuracy + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +of the classifier’s predictions with respect to the consensus +label, over the subset of examples with more than one anno- +tation. The weight wj for annotator Aj is defined in terms of +how much labels chosen by Aj agree with other annotators +when they labeled the same examples as Aj. More formally: +wj := 1 − +1 − gj +1 − AMLC +wM := +� +1 − 1 − AM +1 − AMLC +� +· +� +1 +n +� +i∈D +|Ji| +Above gj is the agreement between Aj and other annotators: +gj := +� +i∈Ij +� +ℓ∈Ji,ℓ̸=j 1(Yij = Yiℓ) +� +i∈Ij(|Ji| − 1) +AM represents the empirical accuracy of the classifier +model’s predictions with respect to the consensus labels: +AM := +1 +|I+| +� +i∈I+ +1 +� +�Yi = arg max +k +�pM,i,k +� +(3) +AMLC is a normalization factor, the baseline accuracy (with +respect to consensus labels) achieved by predicting the +overall most labeled class YMLC (amongst all annotations +for the dataset) always for every example. +AMLC := +1 +|I+| +� +i∈I+ +1(YMLC = �Yi) +(4) +Note that to avoid bias (Goh et al., 2022), the accuracy +estimates which determine P, wj, and wM are always +computed over the subset of labeled examples that received +more than one annotation: I+ := {i ∈ D : |Ji| > 1}. +2.2. Calibration of Classifier Predictions +While cross-validation enables us to produce out-of-sample +predictions for each xi ∈ D, some types of models tend +to nonetheless output overconfident predictions (Guo et al., +2017). Our active learning methods rely on the classifier +to determine what data to label next and subsequently re- +train another version of this same classifier. In this self- +reinforcing process, overconfident predictions may be ex- +tremely detrimental. +To mitigate overconfidence (or underconfidence), we cali- +brate the classifier’s predicted class probabilities in Step 4 +of each active learning round, before we compute ActiveLab +scores via equation (1). We perform this calibration against +the empirical distribution of the annotators’ labels Yi for +each example in D. Calibration is done by temperature scal- +ing (Guo et al., 2017) the classifier’s predicted probabilities +�pM(Yi | Xi) to minimize their (soft) cross entropy against +the empirical distribution �pemp of classes in Yi. That is, we +choose the temperature T to maximize: +� +i∈D +K +� +k=1 +�pemp(Yi = k | {Yij}j∈Ji) · log �p(T ) +M,i,k +where �p(T ) +M,i,k = σ +��pM,i,k +T +� +for softmax σ(zk) = +ezk +� +k ezk +After identifying the best value of T, we calibrate the pre- +dictions for all examples in both D and U and compute +ActiveLab scores using �p(T ) +M,i,k in place of �pM,i,k. In our ex- +periments, this calibration step improved a variety of active +learning methods, allowing them to more robustly improve +the accuracy of various types of models. +2.3. ActiveLab (Ensemble) +Ensemble methods aggregate outputs from multiple (inde- +pendently trained) models into a single set of predictions +that can be more accurate than any of the constituent models +(Dietterich, 2000). Model ensembles are also popular in +active learning; disagreeing predictions between models in- +dicate areas of high epistemic uncertainty where annotating +more data can greatly improve at least one of the constituent +models (Seung et al., 1992). Here we present an straightfor- +ward extension of ActiveLab to ensemble settings. +Assuming there are L trained models in an ensemble, let +�pMℓ(Yi | Xi) denote the class probabilities for xi predicted +by model Mℓ for ℓ = 1, 2, ..., L. Here we can apply Active- +Lab similarly as in the single-model case, but now allowing +each model to have its own weight wM1, wM2, ..., wML +used for averaging estimates. We use the following Active- +Lab scores in ensemble settings: +If xi ∈ D : +(5) +si = +w ¯ +A · 1 +K + +L +� +ℓ=1 +wMℓ · �pMℓ,i,�Yi + � +j∈Ji +wj · �pAj,i,�Yi +w ¯ +A + +L +� +ℓ=1 +wMℓ + � +j∈Ji +wj +If xi ∈ U : si = +w ¯ +A · 1 +K + �L +ℓ=1 wMℓ · �pMℓ,i,�Yi +w ¯ +A + �L +ℓ=1 wMℓ +(6) +Above the annotator weights wj, w ¯ +A and likelihoods �pAj +have the same definitions as in ActiveLab with a single +model. Here consensus labels �Yi are estimated from an +similar ensemble extension of CROWDLAB, in which we +propose to set w ¯ +A = 0 in equation (5) and identify which +class �Yi ∈ [K] maximizes the expression. Equation (6) +shows we handle examples from the unlabeled pool in the +same fashion as in the single-model case. For each xi ∈ U, +we obtain a predicted class �Yi ∈ [K] from the ensemble +classifier and treat �Yi as a proxy for its consensus label. + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +The weights wMℓ for each model are computed the same +way as in ActiveLab with a single model. Each model’s +prediction accuracy with respect to consensus labels is again +used to infer how trustworthy each model is relative to the +annotators, with AMℓ and AMLC defined as in (3) and (4). +wMℓ := +� +1 − 1 − AMℓ +1 − AMLC +� +· +� +1 +n +� +i +|Ji| +To predict with our ensemble classifier after training, we +can also take a weighted average of each model’s predicted +class probabilities using the same weights wMℓ. +3. Related Work +The most popular active learning methods are those like +ActiveLab that can used with any classifier model for any +data modality (Munro, 2021). While there has been exten- +sive research on active learning (Zhan et al., 2022) and +analyzing crowdsourced labels (Paun et al., 2018), few +model/modality-agnostic active learning methods have been +developed for settings with multiple annotators and data +re-labeling (Lin et al., 2014). Many of the active learning +methods proposed for such settings are specific to certain +types of models or data types (Rodrigues et al., 2014; Zhao +et al., 2011; Yan et al., 2011; Yang et al., 2018; Huang et al., +2017; Gilyazev & Turdakov, 2018; Iraola & Yepes, 2021). +Other approaches like impact sampling (Lin et al., 2016) are +too computationally expensive to run on problems like the +image classification task in Section 4. +3.1. Baseline Methods +Our subsequent experiments benchmark ActiveLab against +the following commonly used model/modality-agnostic +methods for active learning and data re-labeling. Each +method is applied in the same manner as ActiveLab to itera- +tively label a dataset, except which xi are labeled is chosen +via different si, and consensus labels for all xi ∈ D are +computed via majority-vote as used by Zheng et al. (2010). +Random. This method selects which examples to annotate +entirely at random. It uses score: si = x where x ∈ [0, 1] is +sampled uniformly at random and independently of i. +Good Random. This is a better variant of random selection +that accounts for the number of annotations xi already has: +si = x + |Yi| where x ∈ [0, 1] is sampled uniformly at +random. This pseudo-random selection prioritizes examples +with the fewest number of labels collected thus far, a simpler +variant of the approach of Chen et al. (2022). The unlabeled +pool is labeled first prior to any re-labeling. +Entropy (Cohn et al., 1996). This method scores examples +via the entropy of the model-predicted probabilities. +si = +K +� +k=1 +�pM,i,k · log �pM,i,k +(7) +Uncertainty (Cohn et al., 1996). Measures how confident +the model is in its predicted class: si = maxk �pM,i,k. +Active Label Cleaning (Bernhardt et al., 2022). This ap- +proach was recently proposed for efficiently re-labeling an +already-labeled dataset with multiple annotators. To select +which data to collect an extra annotation for, Bernhardt et al. +(2022) introduce a score that is a difference of two terms. +The first term is the cross-entropy between the M-predicted +class probabilities and the empirical distribution of the anno- +tators’ labels for a particular example, and the second term +is the entropy of the M-predicted class probabilities. +si = +K +� +k=1 +�pM,i,k · log �pM,i,k +(8) +− +K +� +k=1 +�pemp(Yi = k | {Yij}j∈Ji) · log �pM,i,k +Disagreement (Ensemble) (Seung et al., 1992). Like Ac- +tiveLab (Ensemble), disagreement also employs an ensem- +ble of multiple classifier models. This method measures the +level of disagreement between different individual models’ +predictions. We employ a standard measure of disagreement +for predicted class probabilities, where the score is defined +as the total (soft) cross entropy between each model’s pre- +dicted probabilities and the average estimate over all the +models (McCallum et al., 1998). +si = − 1 +L +L +� +ℓ=1 +K +� +k=1 +�pMℓ,i,k · �p ¯ +M,i,k +(9) +where �p ¯ +M,i,k = 1 +L +L +� +ℓ=1 +�pMℓ,i,k +To produce predictions from our ensemble classifier after +running this method, we simply average the predictions +from the individual models. +4. Experiments +In our experiments, each dataset is partitioned into train, +test, and unlabeled pools. We have high-quality (i.e. ground +truth) labels for the test set, which facilitates accurate evalu- +ation of trained classifiers. No such ground-truth labels are +available for the training set. Instead, all examples in the +training set have been labeled by one or more (potentially +noisy) annotators, and we consider this to be the dataset +D for training an initial classifier, collected prior to active + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.90 +0.92 +0.94 +0.96 +0.98 +Model Accuracy +Model Accuracy of Single-Model Methods +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.92 +0.93 +0.94 +0.95 +0.96 +0.97 +0.98 +Model Accuracy +Model Accuracy of Ensemble Methods +Random +Good Random +Entropy +Uncertainty +ActiveLab +Disagrement (ensemble) +ActiveLab (ensemble) +Figure 1. Evaluating active learning methods on the Wall Robot dataset to train an: ExtraTrees classifier (left) or ensemble of 3 models +(right). Curves show test accuracy after each active learning iteration, averaged over 5 runs with the standard deviation in results shaded. +learning. At the outset, no labels are available for exam- +ples in the unlabeled pool. The train/test/unlabeled pools +and the initial training annotations are identical across all +runs/methods evaluated for the dataset. After training the +model in Step 3 of each round of active learning, we evalu- +ate its test accuracy against ground truth labels (only used +for evaluation purposes). To acquire labels in Step 6, our +experiments use a single new annotator to label the entire +selected batch of data from a round of active learning. +4.1. Datasets and Models +We evaluate active learning methods on datasets of different +modalities and sizes, training various classification models +for these datasets to ensure our methods are model agnostic. +Wall Robot Navigation (Freire et al., 2009). This is a tabu- +lar dataset with 4 classes corresponding to directions a robot +should navigate which are to be predicted from its sensor +measurements. The initial train set for this dataset contains +500 examples, the unlabeled pool contains 1500 examples, +and the test set used to measure the model accuracy contains +1000 examples. In each round of active learning between +model training runs, we collect additional labels for the 100 +examples with the lowest active learning scores from a sin- +gle new annotator. We simulate imperfect annotators for +this dataset. Some of these 100 examples may already have +been previously labeled by other annotators and some may +not have been labeled at all yet. +We consider 3 types of classifier models: Extremely Ran- +domized Trees (Extra Trees) (Geurts et al., 2006), which +was the most accurate model from the sklearn package +on this dataset, fully-connected neural networks (MLP), +K-Nearest Neighbors, and an ensemble composed of all 3. +CIFAR-10H (Peterson et al., 2019). This image classifica- +tion dataset offers many annotated labels for each image +in the CIFAR-10 test set, provided by different human an- +notators. Our experiment uses a subset of 1000 images as +the initial training set, 4000 images in the unlabeled pool, +and 5000 images in the test set. Our high-quality test set +labels to measure model accuracy are those from the orig- +inal CIFAR-10 dataset (Krizhevsky & Hinton, 2009), as +Northcutt et al. (2021a) found the CIFAR-10 labels contain +few errors. In each round of active learning, we collect +additional labels from one new human annotator for the 500 +images with the lowest scores si. +We use an Imagenet-pretrained ResNet-18 classifier for +single-model active learning. For ensemble-model active +learning, our ensemble consists of three classifiers: ResNet- +18, ResNet-34 and ResNet-50 (He et al., 2016). +Wall Robot Complete. Similar to the Wall Robot Navi- +gation tabular dataset, a key difference is that Wall Robot +Complete has 2000 labeled examples in the initial training +set, 1000 examples in the test set, and there is no unlabeled +pool. As for Wall Robot Navigation, we collect additional +labels for the 100 examples with the lowest active learning +scores in each active learning round. Since all the exam- +ples already start out with some labels, this is a re-labeling +(i.e. label cleaning) task, where we aim to obtain accurate +consensus labels by having multiple annotators review the +examples where this is necessary (Bernhardt et al., 2022). + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Iteration +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Model Accuracy +Model Accuracy of Single-Model Methods +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Iteration +0.775 +0.800 +0.825 +0.850 +0.875 +0.900 +0.925 +Model Accuracy +Model Accuracy of Ensemble Methods +Random +Good Random +Entropy +Uncertainty +ActiveLab +Disagrement (ensemble) +ActiveLab (ensemble) +Figure 2. Evaluating active learning methods on CIFAR-10H to train a: ResNet-18 classifier (left) or ensemble of ResNet-18/34/50 models. +Curves show the test accuracy after each iteration of active learning, averaged over 5 runs with the standard deviation in results shaded. +5. Results +Our main evaluation criterion is the test accuracy of classi- +fier trained in each round of active learning. Each experi- +ment (sequential active learning run) is repeated 5 times and +we report the average model accuracy across the trials. +Figures 1, 2 and S1 illustrate that ActiveLab significantly +outperforms the other active learning methods in both the +single-model and ensemble setting. These findings demon- +strate that ActiveLab effectively selects examples to label +and re-label in data of various modalities modeled with dif- +ferent types of classifiers. Unsurprisingly, active learning +with ensemble models can produce higher accuracy than +achieved with single models. Although note that single +model accuracy when collecting data with ActiveLab can +attain comparable performance to the ensemble models, es- +pecially for strong single models like in Figure 1 +Figure 3 shows that ActiveLab is also the best method for +active label cleaning (re-labeling an already labeled dataset). +It even outperforms the method Bernhardt et al. (2022) de- +signed specifically for this setting. Unlike Bernhardt et al. +(2022), ActiveLab estimates account for the number of an- +notations each example has and the quality of the annotators +behind them. Existing active learning methods do not appear +well-suited for such label cleaning tasks. +5.1. Labeling New Examples vs Re-labeling +Traditional active learning only considers collecting at most +one label per example and focuses entirely on the unlabeled +pool rather than considering the option to re-label. If we +have a huge unlabeled pool and a limited labeling budget, +is there any utility in re-labeling? Our previous results +clearly demonstrate the value of smart re-labeling when the +size of U and labeling budgets are suitably matched. But +with near-perfect annotators and an near-infinite unlabeled +pool, re-labeling might not seem like a good idea (Lin et al., +2014). Thus we empirically investigate the question: At +what degree of annotation-noise is there value in re-labeling +when the size of U greatly exceeds our labeling budget? +We consider two settings: one where we only label new +examples in each active learning round (single label case), +and another where we can re-label examples if ActiveLab +chooses to do so (multiannotator label case). We run these +approaches on a few variants of the Wall Robot Navigation +dataset where we simulate annotators with different label +noise rates. A higher noise rate annotator produces labels +which are often wrong, while an annotator with noise rate 0 +always selects labels that are correct. Similar to our previous +Wall Robot benchmark, we conduct this experiment with +an initial train set of 500 labeled examples, an unlabeled +pool of 1500 examples, and test set of 1000 well-labeled +examples. We label batches of 100 examples in each ac- +tive learning round. Both single label and multiannotator +label experiments start with the same labeled subset D (and +always have the same annotator noise rates). In the single +label experiment, active learning is done using the tradi- +tional entropy score only considering examples in U. In the +multiannotator label experiment, active learning is done via +ActiveLab, which often selects a mixture of examples from +D and U to collect an additional label for. +Figure 4 reveals that across all annotator noise levels, the + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.90 +0.92 +0.94 +0.96 +0.98 +Model Accuracy +Model Accuracy of Single-Model Methods +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.950 +0.955 +0.960 +0.965 +0.970 +0.975 +0.980 +Model Accuracy +Model Accuracy of Ensemble Methods +Random +Good Random +Entropy +Uncertainty +ActiveLab +Active Label Cleaning +Disagrement (ensemble) +ActiveLab (ensemble) +Figure 3. Evaluating active learning methods on the Wall Robot Complete dataset to train an: ExtraTrees classifier (left) or ensemble of 3 +models (right). Curves show test accuracy after each iteration of re-labeling, averaged over 5 runs with the standard deviation shaded. +model accuracy for the mulitannotator labels case is equal +or better than for single labels. As expected, the difference +in model accuracy between single labels and mulitannotator +labels is larger when annotators are more noisy. This sug- +gests it is rarely a bad idea to allow re-labeling if you have a +method to do it adaptively like ActiveLab. It appears vital to +re-label in settings with over 20% label noise. Our findings +run contrary to the study of Lin et al. (2014), who acknowl- +edged they were missing an effective active learning method +with re-labeling at the time of their study. +6. Discussion +Relying on model predictions to infer what data is most +informative, model-agnostic active learning methods will +improve automatically as supervised learning architectures +and training procedures continue to advance. More sophisti- +cated active learning methods designed for specific models +or training procedures will not enjoy these benefits and +may become irrelevant if incompatible with tomorrow’s +state-of-the-art models. Unlike traditional model-agnostic +active learning that solely relies on model predictions to +determine which examples to label next, ActiveLab consid- +ers re-labeling examples xi ∈ D and estimates the value +of this based on additional information like the: number of +available annotations for xi, disagreement amongst these an- +notations, and relative trustworthiness of the trained model +vs. the annotators. Re-labeling facilitates more robust model +training when data annotators are imperfect. Future work +might seek to achieve further robustness by filtering bad +annotators (e.g. based on their weights wj) and data/labels +from the training set D of each active learning round. +0 +1 +2 +3 +4 +5 +6 +7 +8 +Iteration +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Model Accuracy +Multiannotator labels +Single labels +Noise rate = 0.4 +Noise rate = 0.3 +Noise rate = 0.2 +Noise rate = 0.1 +Noise rate = 0 +Figure 4. Comparing active learning methods that exclusively label +new examples (single labels) vs. can also re-label examples instead +(multiannotator labels), when annotators have different noise rates. +Shown is the test accuracy of an ExtraTrees classifier trained on a +certain number of total labels (corresponding to each iteration of +active learning) for the Wall Robot Dataset. Curves are the average +over 5 runs, and the standard deviation in results is shaded. + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +References +Baldridge, J. and Osborne, M. 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To simulate +human annotators that make imperfect decisions (i.e. occasional labeling errors), we take the original set of labels from +the Wall Robot dataset as ground truth labels. For each annotator, we add some random noise to their labels (noise rate += 0.15 for Wall Robot Navigation and noise rate = 0.2 for Wall Robot Complete), representing mislabeled examples. The +randomly selected noisy annotations have an incorrect class (flipped probabilistically) that does not match the ground truth +label. Using this method, we obtained 30 sets of labels, representing 30 annotators. +To setup the initial labeled and unlabeled pools D and U, we completely dropped all the annotator labels for examples that +begin unlabeled, while dropping a random fraction of the annotator labels for the examples in D that are labeled from the +start, ensuring we keep at least one annotation for these examples. When collecting additional labels in each round of active +learning, we simulate another annotator in the same fashion who labels the entire batch. +We also considered a second version of this benchmark with more heterogeneous annotators (including some very inaccurate +outliers), and the results of the evaluation remained mostly the same as those presented here. +B. Experiment Details +In each round of active learning, we fit all models to D using 5-fold cross-validation. Additional details not mentioned +here can be found in the code2 for reproducing our experiments, as can raw the results of all active learning methods on all +datasets. +For the experiments on the tabular Wall Robot dataset, we fit our classifier models using the sklearn package (Pedregosa +et al., 2011). The models used were the: ExtraTreesClassifier with default hyperparameters, MLPClassifier with default +hyperparameters except the max iteration set to 500 (to ensure convergence), and KNeighborsClassifier with default +hyperparameters. +The image classifier models for our experiments on CIFAR-10H were fit using the AutoGluon AutoML package (Erickson +et al., 2020) in order to avoid having to manually tune models and their optimization. We used various ResNet models +initialized with default Imagenet-pretrained weights and then fine-tuned them on our dataset D (in a cross-valdated manner). +C. Additional Results for Wall Robot +In addition to the Extra Trees model reported in Section 4.1, we repeat our single-model active learning experiments on +the Wall Robot dataset using a Multilayer Perceptron (feedforward neural network) classifier. These additional results +demonstrate that ActiveLab reliably produces larger improvement in model accuracy than other active learning methods, +regardless which type of classifier model is being trained. +2https://github.com/cleanlab/multiannotator-benchmarks/tree/main/active_learning_ +benchmarks +1 + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.89 +0.90 +0.91 +0.92 +0.93 +0.94 +0.95 +0.96 +Model Accuracy +Model Accuracy of Single-Model Methods +Random +Good Random +Entropy +Uncertainty +ActiveLab +Figure S1. Evaluating active learning methods on the Wall Robot dataset to train a MLP classifier. Curves show the test accuracy after +each active learning iteration, averaged over 5 runs with the standard deviation in results shaded. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Iteration +0.945 +0.950 +0.955 +0.960 +0.965 +0.970 +Model Accuracy +Model Accuracy of Single-Model Methods +Random +Good Random +Entropy +Uncertainty +ActiveLab +Active Label Cleaning +Figure S2. Evaluating active learning methods on the Wall Robot Complete dataset to train a MLP classifier. Curves show the test accuracy +after each iteration of re-labeling, averaged over 5 runs with the standard deviation shaded. + +ActiveLab: Active Learning with Re-Labeling by Multiple Annotators +D. Active Learning in Single-Label Settings +While ActiveLab is designed for scenarios where multiple annotators can label the same example, the method can also be +applied for traditional active learning settings where we collect at most one label for each example. In this singly-labeled +setting, we only score the xi ∈ U, as is common practice in pool-based active learning. +In such settings, we do not have data from multiple annotators to estimate the relative trustworthiness of the annotators and +our model. Thus ActiveLab weights are undefined, but they are also not needed since we are only scoring unlabeled data +without annotations in this setting. As a result, the natural ActiveLab score in such settings is: +si = maxk �pM,i,k + 1 +K +2 +for xi ∈ U +(10) +This is equivalent to simply relying on the confidence of the classifier, and thus equivalent to selecting examples via the +aforementioned Uncertainty baseline method, a classic technique for active learning (Cohn et al., 1996; Munro, 2021). +In this singly-labeled setting, we provide a benchmark of this active learning method against two alternatives: randomly +selecting examples to label, or using the entropy score. We use the same version of the Wall Robot Navigation dataset as our +other benchmarks (with a noisy annotator). The initial train set contains 500 examples, and there are 1500 examples in the +unlabeled pool. Each round, we select the 100 unlabeled examples with the lowest score to label and add them to the labeled +subset. After training, model accuracy is similarly measured on a held-out test set of 1000 examples. +Figure S3 shows that ActiveLab exhibits comparable performance to the entropy baseline method in the singly-labeled +setting. Both methods significantly outperform random selection of examples, even in this setting with noisy labels. +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Iteration +0.60 +0.62 +0.64 +0.66 +0.68 +0.70 +Model Accuracy +Random +Entropy +ActiveLab +Figure S3. Evaluating active learning methods in the traditional singly-labeled setting on the Wall Robot dataset to train an ExtraTrees +classifier. Curves show the test accuracy after each active learning iteration, averaged over 5 runs with the standard deviation shaded. + diff --git a/U9FKT4oBgHgl3EQfmC51/content/tmp_files/load_file.txt b/U9FKT4oBgHgl3EQfmC51/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6fa9dce395a6ce6717e07993379fe0dac0c44196 --- /dev/null +++ b/U9FKT4oBgHgl3EQfmC51/content/tmp_files/load_file.txt @@ -0,0 +1,698 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf,len=697 +page_content='ActiveLab: Active Learning with Re-Labeling by Multiple Annotators Hui Wen Goh 1 Jonas Mueller 1 Abstract In real-world data labeling applications, annota- tors often provide imperfect labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' It is thus com- mon to employ multiple annotators to label data with some overlap between their examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' We study active learning in such settings, aiming to train an accurate classifier by collecting a dataset with the fewest total annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Here we pro- pose ActiveLab, a practical method to decide what to label next that works with any classifier model and can be used in pool-based batch active learn- ing with one or multiple annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' ActiveLab automatically estimates when it is more informa- tive to re-label examples vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' labeling entirely new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' This is a key aspect of producing high qual- ity labels and trained models within a limited an- notation budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' In experiments on image and tabular data, ActiveLab reliably trains more accu- rate classifiers with far fewer annotations than a wide variety of popular active learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Introduction Model-agnostic active learning methods use outputs from some arbitrary type of trained prediction model in order to identify the most informative data to label, so that a more accurate version of the same model can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Such general approaches are popular because they can be directly applied to many data modalities (image, text, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=') as long as a reasonable model can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Focusing on highly prac- tical settings, we consider model-agnostic pool-based active learning with multiple data annotators that label a batch of many examples in between model (re)training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' This setting is easy to setup and allows us to address common issues in real-world active learning such as: labelers who are imperfect, or expensive model (re)training that cannot be executed every time a new example is labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Working with annotators that may provide incorrect labels, it is useful to sometimes ask new annotators to provide extra labels for examples previously labeled by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' This allows us to verify the current consensus label or estimate a better one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Equal contribution 1Cleanlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FKT4oBgHgl3EQfmC51/content/2301.11856v1.pdf'} +page_content=' Correspondence to: Hui Wen Goh 0, unless P = NP [17]. +We view an independent set as a collection of tokens placed on the vertices of a graph such +that no two tokens are adjacent. This gives rise to two natural adjacency relations between +∗This work was supported by ANR project GrR (ANR-18-CE40-0032) +1 +arXiv:2301.02020v1 [math.CO] 5 Jan 2023 + +independent sets, also called reconfiguration steps. +In the Token Jumping (TJ) problem, +introduced by Kamiński et al. [8], a single reconfiguration step consists of first removing a token +on some vertex u and then immediately adding it back on any other vertex v, as long as no +two tokens become adjacent. +The token is said to jump from vertex u to vertex v. +In the +Token Sliding (TS) problem, introduced by Hearn and Demaine [7], two independent sets +are adjacent if one can be obtained from the other by a token jump from vertex u to vertex v +with the additional requirement of uv being an edge of the graph. The token is then said to +slide from vertex u to vertex v along the edge uv. Note that, in both the TJ and TS problems, +the size of independent sets is fixed. Generally speaking, in the Token Jumping and Token +Sliding problems, we are given a graph G and two independent sets Is and It of G. The goal +is to determine whether there exists a sequence of reconfiguration steps (called a reconfiguration +sequence) that transforms Is into It (where the reconfiguration step depends on the problem). +We can reformulate the problem with the configuration graph. Given a graph G we can +define the configuration graph Rk(G) as the graph whose vertices correspond to independent +sets of size k and where we put an edge between I and J if one can transform I into J in one +step (under the token jumping variant). There exists a reconfiguration sequence from I to J if +and only if I and J belong to the same connected component of Rk(G). +Both problems have been extensively studied, albeit under different names. +They are +PSPACE-complete, even restricted to bounded bandwidth (and hence pathwidth) graphs [16] +and planar graphs [7]. Their complexity is also known (respectively PSPACE and NP) on bi- +partite graphs [10] and several polynomial algorithms exist in simpler classes such as trees [4] +and interval graphs [2]. +All along the paper we mainly focus on the Token Jumping model but all our lower bounds +also hold for the Token Sliding version. +Diameter of the configuration graph and the (6, 3)-problem. +In many cases, the di- +ameter of the configuration graph, even if connected, is not polynomial (and that is one of the +reasons why most of the reconfiguration problems do not belong to NP). An important line of +research has focused on finding conditions that ensure that the configuration. But the asymp- +totic behavior of maximum possible length of a shortest reconfiguration sequence has not been +really studied. The problem of determining "which graphs on n vertices have the largest amount +of independent sets?" has received considerable attention. On the contrary, the question "which +graphs on n vertices have a configuration graph of independent sets has the largest diameter?" +has not, as far as we know, received any attention. +The 3n-vertex graph with the largest number of maximum independent sets is a disjoint +collection of triangles which admits 3n independent sets. In that case, one can easily remark +that we can easily transform any maximum independent set into any other in O(n) steps by +replacing a vertex of a triangle by another (which can be done without conflict since the triangles +are independent). +So a graph whose configuration graph of independent sets has maximum +diameter must have a completely different behavior. In this paper, we consider the following +questions: what is the largest possible diameter of (a connected component of) the configuration +graph amongst all the graphs of size n? What if we fix the size k of the independent set we want +to consider? +Let k, n be two integers. Let us denote by D(n, k) the maximum diameter, amongst all the +graph G on n vertices, of a connected component of Rk(G). The goal of this paper mainly focuses +on finding lower and upper bounds on D(n, k). There is a natural upper bound for D(n, k) which +is the maximum number +�n +k +� +of subsets of vertices size k. We will prove in Section 2 that this +bound cannot be reached and that the D(n, k) is actually at most O(nk−1). More precisely, we +will prove that D(n, k) ⩽ +� n +k−1 +� +. +2 + +We can easily prove that the order of magnitude of this bound is tight since, for k = 2, the +following holds as we will prove in Section 3: +Theorem 1. D(n, 2) = n − 2, and the complement of the n-vertex path is the unique tight +example. +One can naturally wonder if this bound is still tight for larger values of k. The answer is +negative since we can prove that this upper bound can actually be very slightly improved for +every k ⩾ 3. Namely, we will prove that: +Theorem 2. For k ⩾ 3, we have D(n, k) = o(nk−1). +The proof of Theorem 2 is inspired from the upper bound proof of the (6, 3)-problem and is +based on an application of the hypergraph removal lemma. A hypergraph H is (s, t)-free if no set +of s vertices of H contains at least t hyperedges. The (6, 3)-problem (or Ruzsa–Szemerédi prob- +lem) asks for the maximum number of hyperedges in a (6, 3)-free n-vertex 3-uniform hypergraph. +The so-called (6, 3)-theorem of Ruzsa-Szemerédi [13] ensures this value is o(n2). +This gain (o(nk−1) versus O(nk−1)) might appear marginal but we can prove that, again, it +cannot be widely improved. Namely we prove that the following holds: +Theorem 3. +D(n, 3) = Ω(n2/eO(√log n)). +The value n/eO(√log n) corresponds to the largest known asymptotic size for a subset of [1, n] +without arithmetic progressions of length 3 [1]. Any improvement of this bound would also imply +an improvement of the bound of Theorem 3. Note that the best bound for the (6, 3)-problem +also has this order of magnitude [13]. +The 3-reconfiguration problem is actually very close to the (6, 3)-problem. Indeed, if we +consider a shortest path in the 3-configuration graph of G and only consider even (resp. odd) +vertices of that path, then we have a set of hyperedges of size 3. And one can easily check that +this set of hyperedges satisfy the (6, 3)-property. So our result implies in particular that, given +a set of size n, we can find two sets X1, X2 of n2/eO(√log n) 3-hyperedges such that both of them +are (6, 3)-free but whose union is "path-like", meaning that for every hyperedge (but at most +two which are the endpoints of the path) there are two others hyperedges that intersect it on +two vertices. +The idea of the proof of Theorem 3 consists in starting from a clique. We will then remove +edges to create almost linearly many paths in the configuration graph of linear length. The +involved part of the proof consists in showing that these paths remain independent of each other +(i.e. there is no edge between them in the configuration graph) using a set S of integers with no +arithmetic progression of size 3. We finally use a last trick to glue these paths together in order +to obtain the claimed diameter. Note that the classical construction giving n/eO(√log n) hyper- +edges [13] for the (6, 3)-problem cannot be easily used in our construction since the construction +is tripartite and then hard to reconnect into a configuration graph. +We were not able to prove that the lower bounds and the upper bounds almost match for +larger values of k. In particular, it is open to determine if the 4-configuration graph can have +super-quadratic diameter (while the upper bound is o(n3)). We conjecture that the following +holds: +Conjecture 4. +D(n, 4) = n3−o(1). +3 + +A first step to prove super-quadratic diameter is to ensure that there exists a graph with a +lot of copies of K4 such that no two of them intersect on a triangle. This was recently shown to +be true for any value of k. Namely, Gower and Janzer proved in [5] that, for every k and every +n, there exists an n-vertex graph with nk−1−o(1) copies of Kk such that every Kk−1 is contained +in at most one Kk. This result might suggest that D(n, k) = nk−1−o(1). +Our construction for k = 3 has to be drastically modified in order to work. Indeed, our +construction is heavily based on the fact that we can find a graph with an almost linear number +of linear paths in its 3-configuration graph. To get a super-quadratic bound, we need to either +increase the number of paths or their lengths. We failed trying both options. +However, in general, we were able to show that the following holds: +Theorem 5. For every integer k we have +D(n, k) = +n2⌊k/3⌋ +eOk(√log n) +For k = 4, 5, we can also ensure that the lower bound is quadratic. Actually, what we prove +is slightly stronger but can be asymptotically summarized with Theorem 5. +The idea of the proof of Theorem 5 consists in successively adding a graph (inspired by) the +construction of Theorem 3 and connecting it in a clever way to the previous graph to increase +the diameter quadratically while increasing the size of the independent set by 3. Note that a +super-quadratic lower bound for k = 4 might lead to an improvement of this general lower bound +as long as there is a clever gluing. +Observe that the asymptotic estimate in Theorem 5 depends on k, and hence may not hold +when k is not constant, for example when k is linear in n. Constructing graphs that maximize +the diameter of a connected component in their k-configuration graphs (regardless of the value +of k) is a question raised during the Core Challenge 2022 [14] for graphs on 10, 50 and 100 +vertices. Our team proposed a generic construction that obtained the best results. Rewritten +in the current formalism, our statement from [14] becomes: +Lemma 6. For every integer n, there exists a graph G on 10n vertices such that its R3n(G) is +a path of length Θ(4n). In particular D(n, 3n +10 ) = Ω(2n/5). +Note that we also give a construction showing that D(n, 2n +5 ) = Ω(2n/5) (with a slightly worse +constant than in Lemma 6). Roughly speaking, these graphs are constructed by adding edges +between complements of paths on 10 and 5 vertices respectively, in a similar fashion to the proof +of the upcoming Lemma 18. In particular, those two constructions can be combined and yield +the following. +Theorem 7. For every n and every k such that 3n/10 ⩽ k ⩽ 2n/5, D(n, k) = Ω(2n/5). +We believe it is quite surprising that this lower bound holds for such a range of values of k, +and thus raise the following question. +Question 8. What is the asymptotic behavior of maxk D(n, k)? +2 +Generic upper bounds +We start this section with a preliminary upper bound on D(n, k). +Lemma 9. D(n, k) ⩽ +� n +k−1 +� +. +4 + +Proof. Consider a shortest path P in the k-configuration graph of an n-vertex graph G. With +each edge of P, we associate the k−1 vertices of the intersection of the independent corresponding +to its endpoints. This defines a mapping from E(P) to sets of k − 1 vertices of G. Since there +are nk−1 such sets, we simply have to show that this mapping is injective. Assume that two +distinct edges are mapped to the same set X of k − 1 vertices. Then X belongs to at least +three distinct independent sets that are vertices of P. These three independent sets are pairwise +adjacent, which is impossible since P is a shortest path. +We will see that this bound is sharp for k = 2. However, when k increases this bound can +be slightly improved, as summarized in Theorem 2 that we recall below. +Theorem 2. For k ⩾ 3, we have D(n, k) = o(nk−1). +Proof. Consider a graph G on n vertices whose configuration graph has maximum diameter d. +Let P = Z1, Z2, . . . , Zd be a shortest path of length d in Rk(G). Let us partition the nodes in +P into two sets P1 and P2 where P1 (resp. P2) is the set of odd (resp. even) nodes of P. Note +that if we consider two subsets of Pi for i ⩽ 2 then their intersection has size at most k − 2 +(otherwise P would not be induced). +For every i ⩽ 2, let Hi be the (k − 1)-uniform hypergraph whose vertices are the same as +for G and whose hyperedges are the independent sets of size k − 1 contained in some set of +Pi. Moreover, denote by K the (k − 1)-uniform hyperclique on k vertices. Observe that by +construction, each Z ∈ Pi creates (exactly) one copy of K in Hi. Also note that every subset +of size k − 1 of such a Z belongs to exactly one independent set of Pi since otherwise the two +independent sets would be adjacent, contradicting the minimality of P. We now distinguish two +cases: +Case 1. Hi contains more than nk−1 copies of K. +Since by Lemma 9, at most nk−1 are created by some Z ∈ Pi, there exists a copy of K in Hi +such that V (K) /∈ Pi. Consider now three hyperedges e1, e2, e3 in K that pairwise intersect on +k − 2 vertices (note that this is possible since k ⩾ 3). +By construction, each of these hyperedges are contained in some element of Pi, so there exist +x1, x2, x3 ∈ V (G) such that ej ∪{xj} ∈ Pi for j = 1, 2, 3. In particular, each ej is an independent +set of G, therefore e1 ∪ e2 ∪ e3 is also an independent set of G of size k. Therefore all the +ej ∪ {xj}’s are at distance at most 2 from each other in Rk(G) since all of them are adjacent to +e1 ∪ e2 ∪ e3. This is a contradiction since P is a shortest path and P1 (resp. P2) only contains +even (resp. odd) vertices of P and then two of the three independent sets ej ∪ {xj} ∈ Pi for +j ⩽ 3 should be at distance at least 4. +Case 2. Hi contains at most nk−1 = o(nk) copies of K. +By the hypergraph removal lemma [12, 6], there exists a set S of hyperedges of H such that +|S| = o(nk−1) and H − S contains no copy of K. Recall that each hyperedge of S is contained +in exactly one element of P1, and each element of Pi creates a copy of K in H, therefore we get +|Pi| ⩽ |S| = o(nk−1). +To conclude, observe that d ⩽ 2|Pi| = o(nk−1) for every i ⩽ 2. +3 +Lower bounds +3.1 +Independent sets of size 2 +In this section, our main goal is to prove Theorem 1 that we recall below. +Theorem 1. D(n, 2) = n − 2, and the complement of the n-vertex path is the unique tight +example. +5 + +We thus consider the independent sets of size 2 of a graph G. Note that these sets are +exactly the non-edges of G, i.e. the edges of G = (V, P2(V )\E). Therefore, we get the following +observation. +Observation 10. The configuration graph R2(G) is the line graph of G. +Note that for every graph G, any induced path on p vertices in L(G) corresponds to a path +on p edges in G. In particular, we derive two consequences. +Observation 11. Let A, B be two independent sets of G of size 2 and a ∈ A, b ∈ B. There is a +TJ-transformation from A to B if and only if a, b are in the same connected component of G. +We can also obtain the following which ensures that the bound of Lemma 9 is tight: +Lemma 12. For every n-vertex graph G, +diam(R2(G)) = diam(L(G)) ⩽ diam(G) − 1 ⩽ n − 2. +Note that the last bound is tight only when G is a path, which concludes the proof of +Theorem 1. +Since the diameter is linear, one might wonder if we can determine in linear time if there +exists such a transformation (and find it). Note that we cannot just compute the line graph of +the complement and run a BFS on it. Indeed, even if a BFS can be computed in linear time +with respect to the number of edges of its input, this number may be quadratic with respect to +the number of edges of the original graph. However, by complementing only the graph induced +by vertices of large degree, we obtain the following. +Theorem 13. Let A, B be two independent sets of G of size 2. We can decide if there exists a +TJ-transformation from A to B in time O(|V (G)| + |E(G)|). +Proof. Let G be an n-vertex m-edge graph, and s, t two vertices of G. We start by precomputing +the degrees of the vertices of G in O(n + m) time. Let B be the set of vertices of degree at +least n−1 +2 , and S = V (G) \ B. Observe that by the pigeonhole principle, any two vertices in S +must have a common non-neighbor, hence are connected in G. Let us denote by H the graph +obtained by identifying all the vertices of S into a single vertex x and where we put an edge +between x and y /∈ S if y is adjacent to a vertex of S. It is easy to check that there is a path +between s and t in G if and only if there is such a path in H (up to replacing s or t by x if they +lie in S). Note moreover that one can easily compute the graph H in O(n + m) time. +One can notice that the graph H might be sparse and then its complement can have size +Ω(|V (H)|2). However, observe that +(|V (H)| − 1) × n − 1 +2 +⩽ +� +v∈B +degG(v) ⩽ 2m, +hence |V (H)| = O( m +n ). In particular, one can compute H and use a BFS in H in time O(|V (H)|2) = +O( m2 +n2 ) = O(m). +Note that the algorithm we provide can easily be adapted to return a (possibly non-optimal) +transformation when it exists. +3.2 +Almost quadratic construction for independent sets of size 3 +The rest of this section is devoted to prove the following result: +6 + +Theorem 3. +D(n, 3) = Ω(n2/eO(√log n)). +The proof is based on two steps. First, we prove that there exists a graph whose configuration +graph is the disjoint union of n/eO(√log n) paths of linear length. We then prove that, starting +from a graph whose configuration graph is disconnected, we can (up to adding few vertices), +obtain a graph whose configuration graph is connected and whose diameter is at least the sum +of the diameter of the connected components of the initial configuration graph. While the first +step is specific to k = 3 and is based on the existence of almost linear subsets of integers without +arithmetic sequences of length 3, the gluing process is general and holds for any possible value +of k. Let us first prove the gluing lemma. +Lemma 14. Let k ⩾ 3. Let G be a graph on n vertices whose k-configuration graph contains r +connected components C1, . . . , Cr of diameter respectively d1, . . . , dr. Then there exists a graph +on at most n + (3k − 2) · (r − 1) vertices whose configuration graph has diameter at least (4k − +4)(r − 1) + � +i⩽r di. +x1 +x2 +x3 +x4 +x5 +x6 +x7 +b1 +b2 +b3 +Bi +a3 +a2 +a1 +Ai+1 +Figure 1: Construction for Lemma 14 with k = 3, where only non-edges are drawn. +Proof. For every i ⩽ r, let us denote by Ai and Bi two independent sets at distance di in the +component Ci of the k-configuration graph. For every i, we denote by ai +j (resp. bi +j) the j-th vertex +of Ai (resp. Bi). We moreover assume that ai +k and bi +1 are respectively the first and last vertices +modified in a shortest sequence Pi from Ai to Bi. Note that the sets (A1, . . . , Ar, B1, . . . , Br) +might intersect. +For every i ⩽ r−1, we create 3k−2 new vertices xi +1, . . . , xi +3k−2 in order to connect Bi to Ai+1 +in the configuration graph of the new graph (see Figure 1 for an illustration of the construction). +We first add all the edges between the new vertices and V (G) and, for every i ̸= j, the vertices +xi +a and xj +b are adjacent regardless of a, b. Moreover, for every p < q ⩽ 3k − 2, xi +p is adjacent to +xi +q if and only if q − p > k − 1. We finally remove the following edges: for every j we remove +the edges between xi +j (resp. x3k−2−j) and bi +j′ (resp. ai+1 +k−j′) with j′ > j. Let us denote by H the +resulting graph. +Let Xi be the (ordered) set of vertices {bi +1, . . . , bi +k, xi +1, . . . , xi +3k−2, ai+1 +1 +, . . . , ai+1 +k +}. Let us first +prove the following simple claim on the structure of independent sets: +Claim 15. Let i ⩽ r − 2. Every k-independent set S containing one vertex in {xi +1, . . . , xi +3k−2}: +• consists of k consecutive vertices of Xi and, +• has degree 2 in Rk(H) if it contains a vertex in {xi +2, . . . , xi +3k−3}. +Proof. Let us first prove that if an independent set S contains a vertex in {xi +1, . . . , xi +3k−2} then +it contains consecutive vertices of Xi. +7 + +If S ⊆ {xi +1, . . . , xi +3k−2}, then let us denote by xi +a and xi +b the first and last vertices in S. By +construction, since S is an independent set, we must have b − a ⩽ k − 1. And then S contains +only non-neighbors of xi +a and xi +b, hence b − a = k − 1 and S = {xi +a, . . . , xi +b}. So from now on we +can assume that S contains a vertex of V (G). +Since xi +k−1, . . . , xi +2k−2 are complete to G, the set S cannot contain one of these vertices. +And since {xi +1, . . . , xi +k−2} is complete to {xi +2k−1, . . . , xi +3k−2}, we can assume by symmetry that S +contains vertices in {xi +1, . . . , xi +k−2} but not in {xi +2k−1, . . . , xi +3k−2}. Let us denote by a ⩽ k −1 the +largest index such that xi +a ∈ S. By construction, xi +a is non-adjacent to the k−1 vertices before it +in the sequence and complete to all the other vertices of H\Xi. So S = {bi +a+1, . . . , bi +k, xi +1, . . . , xi +a}. +For the second item, observe that indeed, each set of k consecutive vertices in Xi is indepen- +dent, and is connected to the independent sets corresponding to the k vertices just before and +after them in the ordering. Moreover, each independent set S intersecting {xi +2, . . . , xi +3k−3} con- +tains at least two vertices in {xi +1, . . . , xi +3k−2}. Therefore S is only adjacent to independent sets +containing at least one vertex in {xi +1, . . . , xi +3k−2}. By the first item, they consist of k consecutive +vertices of Xi, hence S has exactly two neighbors in Rk(H). +So the k-configuration graph of H restricted to Xi induces a path Pi from Bi to Ai+1 +of length 4k − 2. By concatenating these paths with shortest reconfiguration sequences from +the Ai to Bi for every i ⩽ r, we get a reconfiguration sequence P from A1 to Br of length +(4k − 2)(r − 1) + � +i⩽r di. +To complete the proof we have to prove that we can shorten the sequence P by exactly +2(r − 1) steps (and that no shorter transformation exists). Indeed, for every i ⩽ r − 1, consider +a reconfiguration sequence from Ai to Bi of minimum size where ai +1 is the vertex deleted from +Ai at the beginning of the sequence and bi +1 is the last vertex to be moved on Bi at the end +of the sequence (this reconfiguration sequence exists by assumption). If we denote by B′ +i the +independent set before Bi, B′ +i contains Bi \ {bi +1}. Then B′ +i is adjacent to (Bi ∪ xi +1) \ bi +1 in the +configuration graph and then we can remove Bi in the reconfiguration sequence from A1 to Br +and still have a reconfiguration sequence. Similarly, we can find a shortcut of the sequence on +Ai for every 2 ⩽ i ⩽ r. So we can shorten P by 2(r − 1) steps. +We claim that this transformation has shortest length. Let us briefly argue why it is true. +Consider an independent set of H containing exactly one vertex in {xi +j | i ⩽ r −1, j ⩽ 3k −2}. It +should contain the vertex xi +1 or xi +3k−2 for some i by Claim 15 and k−1 vertices of an independent +set in Ci or in Ci+1. Thus it can only be adjacent to an independent set of the component of Ci +or Ci+1 or an independent set containing two vertices of Xi by Claim 15. +Using Claim 15 again, it means that from an independent set only containing xi +1 (resp. +xi +3k−3) we can only reach an independent set of Ci+1 (resp. Ci) containing Bi−1 \ bi−1 +1 +(resp. +Ai+1 \ ai+1 +k +). +Before proving that it is possible to obtain an almost linear number of components of almost +linear size, we need some definitions and results of group theory. +Let S be a set of integers. We say that S is 3-AP-free if it does not contain an arithmetic +progression of length 3, i.e. there does not exist s1 < s2 < s3 in S such that s2 − s1 = s3 − s2. +Determining the size of the largest possible 3-AP-free subset of [1, n] is a heavily studied problem +whose exact answer is not known. It was shown that there does not exist any 3-AP-free set of +positive density in N [9]. However Behnrend proved in [1] that there exist 3-AP-free subsets of +{1, . . . , n} of size n/eO(√log n). Note that if S is 3-AP-free, then the set 4S + 1 also is 3-AP-free. +So, there exists a 3-AP-free sequence of size n/eO(√log n) only containing integers whose value is +1 modulo 4. Such a set will be called an odd 3-AP-free sequence in the rest of the paper. +Now let p be a prime number. It is well known that, for every α ∈ {1, . . . , p−1}, the sequence +of the kα modulo p is a periodic sequence of period p. That is {0, α, . . . , (p−1)α} = {0, . . . , p−1} +8 + +modulo p. +We claim that the following holds: +Lemma 16. Let n be a prime number. Let S be an odd 3-AP-free sequence where the maximum +integer is at most n/8. Then, there exists a graph on n − 1 vertices whose configuration graph +is the disjoint union of |S| paths of length n − 3. +Proof. Let G be the graph obtained from a clique on n vertices by removing the edges (i, i + s) +and (i, i + 2s) for s ∈ S and i ⩽ n, where all integers are understood modulo n. +We claim that R3(G) consists of |S| induced cycles of length n. First observe that for each +s ∈ S we have n independent sets, namely {ks, (k + 1)s, (k + 2)s} (0 ⩽ k < n), which induce a +cycle. In particular, R3(G) contains |S| cycles of length n. +Let us now show that this union of cycles is induced. To this end, let I = {a, b, c} and I′ = +{a′, b′, c′} be two independent sets such that c−b = b−a = s ∈ S and c′−b′ = b′−a′ = s′ ∈ S with +s ̸= s′. If I ∩I′ contains two elements x and y, then observe that x−y ∈ {±s, ±2s}∩{±s′, ±2s′}, +which is impossible since s and s′ are distinct integers below n/8 and are both 1 mod 4. Therefore +I and I′ are not adjacent. +Finally, we prove that R3(G) has no other vertex. Let {a, b, c} be an independent set of G. +If b−a ∈ S and c−b ∈ S, then c−a is even and lies in 2S, hence we can write b−a = s, c−b = s′ +and c − a = 2s′′ = s′ − s. Since S is 3-AP-free, we have s = s′ = s′′ and then b − a = c − b (and +then {a, b, c} is one of the sets described above). Otherwise, b − a or c − b does not belong to S, +thus c − a is equal to 0 or 3 modulo 4 and then c − a /∈ S ∪ 2S, which is a contradiction. +So R3(G) is the disjoint union of cycles with no edges between them. Now if we remove the +vertex 0 from G, each of these cycles now becomes a path (we remove the three independent +sets containing it for every S), which completes the proof. +Let us combine Lemma 14 and 16 to prove Theorem 3. Let S be an odd 3-AP-free sequence +of size n/eO(√log n). By Lemma 14, there exists a graph G on n−1 vertices whose 3-configuration +graph admits |S| connected components of diameter n − 3. By applying Lemma 16 to G, we +obtain a new graph on n − 1 + 7 · (|S| − 1) ⩽ 8n vertices and whose 3-configuration graph has +diameter at least 8 · (|S| − 1) + |S| · (n − 3) = n2/eO(√log n), which completes the proof. +We end this section with the following question which would extend Theorem 13 to indepen- +dent sets of size 3. +Question 17. Can we compute the diameter or the existence of a transformation between two +independent sets of size 3 in (sub)quadratic time? +3.3 +General lower bound +The goal of this section is to generalize the construction of Section 3.2 to larger values of k. +Unfortunately, we were not able to obtain a lower bound that almost fits the upper bound for +larger values of k, but we still obtain the following. +Theorem 5. For every integer k we have +D(n, k) = +n2⌊k/3⌋ +eOk(√log n) +Let us first give the flavour of the proof with an intermediate construction. +Lemma 18. Let G be a graph with independence number k such that Rk(G) has diameter d. +Then for every integer n, there exists a graph H on |V (G)| + 6n + 2 vertices such that Rk+2(H) +has diameter at least 2dn. +9 + +B +A +x1 +x2 +x3 +x6k +x6k+2 +. . . +G +Figure 2: Construction of the proof of Lemma 18. For readability, we have only represented the +non-edges incident with the vertices of X. +Proof. Let A and B be two independent sets of size k at distance d in Rk(G). +Let us create 6n + 2 vertices X = {x1, x2, . . . , x6n+2}, inducing the complement of a path. +For every 1 ⩽ ℓ ⩽ n, link the vertices x6ℓ−3 and x6ℓ to respectively V (G)\B and V (G)\A. Let us +denote by H the resulting graph (see Figure 2 for an illustration). Let C, D be the independent +sets A ∪ {x1, x2} and A ∪ {x6n+1, x6n+2}. Since k is the independence number of G and X is +the complement of a path, the maximum independent sets of H have size k + 2 and all of them +contain k vertices in V (G) and two vertices in X. +In particular, in every transformation from C to D, two tokens are present at each time in +X, and the movements of these tokens correspond to a path in R2(X). Therefore, in order to +slide the tokens from {x1, x2} to {x6n+1, x6n+2}, each transformation must hit in order some +independent sets S1, . . . , S6n+1 such that Si ∩ X = {xi, xi+1}. Amongst all such independent +sets, we select Si as the first such independent set. Note that for every i = 3 mod 6 (resp. 0 +mod 6), Si = B ∪{xi, xi+1} (resp. Si = A∪{xi, xi+1}). So, for every i = 0 or 3 mod 6, in order +to transform Si into Si+3 we need at least d + 3 steps (since we need to transform A into B plus +three token slides on X). +Therefore, the length of the reconfiguration sequence from C to D is at least 2n(d+3), which +completes the proof. +Since the graph H constructed in Lemma 18 has maximum independent sets of size k + 2, +we can iterate the process starting from the complement of a path and prove that the following +holds: +Corollary 19. For any k, D(n, k) = Ωk(n⌊k/2⌋). +In the proof of Lemma 18, we can see the vertices of indices 0 modulo 3 as toll booths which +enforces us to perform a lot of modifications in G in order to pass though these vertices. To +improve the bound of Corollary 19, we will generalize Lemma 18. Indeed instead of gluing the +complement of a path to G and increasing the size of the independent sets by 2, we will copy a +(slightly modified) copy of the graph of Theorem 3 and increase the size of the independent sets +by 3. Note that we cannot automatically, when we have a graph with a large reconfiguration +diameter, glue it with another graph and get a large reconfiguration diameter since we need +to be careful on where we put the toll booths. (That is why the graph of Theorem 3 has to +be slightly modified to work in our setting.) The main ingredient for Theorem 5 is thus the +following analogue of Lemma 18. +Lemma 20. Let n, k be two integers. Let G be a graph with maximum independent sets of size +k such that the k-configuration graph of G has diameter d. Then there exists a graph on at most +|V (G)| + 3kn vertices whose (k + 3)-reconfiguration diameter is at least +dn2 +eO(√log n) . +Proof. The construction is inspired from Section 3.2. Let S′ be a largest 3-AP-free sequence in +[1, n/64] and S be the set 8S′ + 1. Recall that |S| = +n +eO(√log n) . Denote by H the graph obtained +10 + +applying Lemma 16 to S. Recall that vertices of H can be labeled from 1 to n − 1 in such a way +that: +1. the independent sets of size 3 have consecutive values modulo 8, and +2. if two such sets are adjacent in R3(H), then their symmetric difference contains two +elements whose difference is ±3 mod 8. +3. each vertex of H appears in at least one independent set in each connected component of +R3(H). +Let A and B two independent sets of G at distance d from each other in Rk(G). Denote by +G′ the graph obtained by taking a copy of G and H and adding all the edges: +• between V (G) \ A and vertices of H that are 0 mod 8. +• between V (G) \ B and vertices of H that are 4 mod 8. +Note that since G has no independent set of size more than k, then any independent set of G′ +of size k+3 decomposes as k vertices of G and 3 vertices of H. In particular, any reconfiguration +sequence from I to J in G′ yields a reconfiguration sequence from I∩V (H) to J∩V (H) in H (and +the same holds for G). Therefore Rk+3(G′) contains at least as many connected components as +R3(H). +Let X1, . . . , Xr be a connected component of R3(H) which induces a path. By construction, +we may assume that X1 contains vertices that are 1,2, and 3 mod 8, and by (1) and (2), each +Xi contains vertices that are i, i + 1 and i + 2 mod 8. Up to reducing r by at most 7, we may +even assume that Xr also contains vertices that are 1, 2 and 3 mod 8. Thus X1 ∪ A and Xr ∪ A +are independent sets of G′. +Observe that if i = 1 mod 4, there is no edge between Xi and V (G), hence there is a +reconfiguration sequence of length d between Xi∪A and Xi∪B. Therefore, one can reach Xr ∪A +from X1∪A going through the following steps: X1∪B, X2∪B, X3∪B, X4∪B, X5∪B, X5∪A, . . .. +Therefore, X1 ∪ A and Xr ∪ A are in the same connected component of Rk+3(G′). Let us +now compute (a lower bound on) their distance. Consider a shortest reconfiguration sequence +between X1 ∪ A and Xr ∪ A. Recall that this yields a (non-necessarily shortest) reconfiguration +sequence from X1 to Xr in H. By (3), for every vertex of H which is 0 mod 8, we may choose +a set Xi that contains it, and denote by Xi1, . . . , Xin/8 the subsequence they form (observe that +this is well-defined since no Xi can contain two vertices that are 0 mod 8 by (1)). +Now by (1) and (2), note that in the reconfiguration sequence between every Xij and Xij+1, +there must exist some independent set Xi′ +j containing a vertex that is 4 mod 8. By construction, +the only independent set of G′ containing each Xij (resp. Xi′ +j) is Xij ∪ A (resp. Xi′ +j ∪ B). +Therefore the distance between Xij ∪ A and Xi′ +j ∪ B is at least d, and so is the distance between +Xi′ +j ∪ B and Xij+1 ∪ A. +Therefore, the distance between X1 ∪ A and Xr ∪ A is at least 2d × ( n +8 − 1) = dn +4 − 2d. Since +Rk+3(H) contains at least |S| components of diameter at least dn +4 − 2d, applying Lemma 14 to +G′ yields a graph on |V (G′)|+(3k −2)(|S|−1) = |V (G)|+n−1+(3k −2)(|S|−1) vertices with +diameter at least (4k − 4)(|S| − 1) + |S|dn/4 − 2d|S|, which concludes since |S| = n/eO(√log n). +As an immediate corollary, we obtain Theorem 5. +11 + +4 +Conclusion +Let us start the conclusion with this simple remark: +Lemma 21. Let G be a graph. There exists a super-graph G′ of G with the same number of +vertices such that R3(G′) is a path and the largest diameters of the connected components of +R3(G) and R3(G′) are the same. +Proof. We start by adding arbitrarily edges to G while the largest diameter of a component in +R3(G) stays unchanged. To conclude, we show that R3(G) is a path. Let P be a shortest path +of maximal length in R3(G). +Assume that there is a node Z = {u, v, w} of R3(G) that is not in P. For each x ̸= y ∈ Z, +adding the edge xy to G decreases the diameter of the component of P in R3(G). Hence there +must be an independent set of P that contains both x and y. +Therefore, we can assume that P contains three independent sets {u, v, a}, {u, w, b} and +{v, w, c}, which are all pairwise distinct since Z /∈ P. Since P is a shortest path and these sets +are all neighbors of Z, they must be consecutive in P. Therefore, we have a = b = c. But +then, {u, v, a}, {v, w, a} and {u, w, a} induces a triangle in R3(G), a contradiction since P is +induced. +Note that all the graphs obtained from our constructions also satisfy that their configuration +graphs are paths. We conjecture that the following is true in general: +Conjecture 22. For every k ⩾ 2 and every n, there exists a graph G on n vertices maximizing +D(n, k) and such that the Rk(G) is a path. +Note that this result holds for k = 2 (since complement of paths are tight) and for k = 3 as +proven in Lemma 21. +Another interesting question is the following. We have remarked that both lower and upper +bounds of the (6, 3)-problem correspond to the bounds obtained for the largest possible diameter +of R3(G). +Are the two problems equivalent (up to a multiplicative constant)? +While the +existence of a better diameter for some graph G would immediately imply a better bound for +the (6, 3)-problem (by simply considering even vertices of a shortest path), the converse is not +immediate. +Acknowledgments. +This work started thanks to the CoRe programming challenge [14] whose +topic in 2022 consisted in finding the graphs on 10, 50 and 100 vertices with the largest possible +independent set reconfiguration diameter. +References +[1] F. A. Behrend. 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Theory of Computing, 3(1):103–128, 2007. +13 + diff --git a/UtA0T4oBgHgl3EQfEf_h/content/tmp_files/load_file.txt b/UtA0T4oBgHgl3EQfEf_h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58d3353aa86dac248af521d75fe185a2cc514901 --- /dev/null +++ b/UtA0T4oBgHgl3EQfEf_h/content/tmp_files/load_file.txt @@ -0,0 +1,623 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf,len=622 +page_content='Extremal Independent Set Reconfiguration∗ Nicolas Bousquet1, Bastien Durain1,2, Théo Pierron1, and Stéphan Thomassé2 1Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France 2ENS Lyon, Département informatique, Lyon, France Abstract The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Extremal problems on independent sets are widely studied: for example, it is well known that an n-vertex graph has at most 3n/3 maximum independent sets (and this is tight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size k among all n-vertex graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We give a tight bound for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We generalize our results for larger values of k by proving an n2⌊k/3⌋ lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 1 Introduction Many questions can be formalized as follows: given the description of a system state and the description of a state we would “prefer” the system to be in, is it possible to transform the system from its current state into the more desired one without “breaking” the system in the process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' And if yes, how many steps are needed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Such problems naturally arise in the fields of mathematical puzzles, operational research, computational geometry, bioinformatics, and quantum computing for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' These questions received a substantial amount of attention under the so-called combinatorial reconfiguration framework in the last few years from both structural and algorithmic point of views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We refer the reader to the surveys [15, 11, 3] for more background on combinatorial reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Given a reconfiguration problem, one can naturally define the (re)configuration graph where the vertices correspond to solutions and there is an edge between two vertices if one can transform one into the other in one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Structural properties of configuration graphs have been studied under various names in different fields, for instance by looking for its connectivity (irreducibility of Markov chains) or hamiltonian paths (Gray codes for hypercubes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Independent set reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Given a simple undirected graph G, a set of vertices S ⊆ V (G) is an independent set if the vertices of S are pairwise non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Finding an independent set of maximum cardinality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=', the Independent Set problem, is a fundamental problem in algorithmic graph theory and is known to be not only NP-hard, but also W[1]-hard and not approximable within O(n1−ε), for any ε > 0, unless P = NP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We view an independent set as a collection of tokens placed on the vertices of a graph such that no two tokens are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This gives rise to two natural adjacency relations between ∗This work was supported by ANR project GrR (ANR-18-CE40-0032) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='02020v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='CO] 5 Jan 2023 independent sets, also called reconfiguration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In the Token Jumping (TJ) problem, introduced by Kamiński et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' [8], a single reconfiguration step consists of first removing a token on some vertex u and then immediately adding it back on any other vertex v, as long as no two tokens become adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The token is said to jump from vertex u to vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In the Token Sliding (TS) problem, introduced by Hearn and Demaine [7], two independent sets are adjacent if one can be obtained from the other by a token jump from vertex u to vertex v with the additional requirement of uv being an edge of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The token is then said to slide from vertex u to vertex v along the edge uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that, in both the TJ and TS problems, the size of independent sets is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Generally speaking, in the Token Jumping and Token Sliding problems, we are given a graph G and two independent sets Is and It of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The goal is to determine whether there exists a sequence of reconfiguration steps (called a reconfiguration sequence) that transforms Is into It (where the reconfiguration step depends on the problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We can reformulate the problem with the configuration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Given a graph G we can define the configuration graph Rk(G) as the graph whose vertices correspond to independent sets of size k and where we put an edge between I and J if one can transform I into J in one step (under the token jumping variant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' There exists a reconfiguration sequence from I to J if and only if I and J belong to the same connected component of Rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Both problems have been extensively studied, albeit under different names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' They are PSPACE-complete, even restricted to bounded bandwidth (and hence pathwidth) graphs [16] and planar graphs [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Their complexity is also known (respectively PSPACE and NP) on bi- partite graphs [10] and several polynomial algorithms exist in simpler classes such as trees [4] and interval graphs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' All along the paper we mainly focus on the Token Jumping model but all our lower bounds also hold for the Token Sliding version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Diameter of the configuration graph and the (6, 3)-problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In many cases, the di- ameter of the configuration graph, even if connected, is not polynomial (and that is one of the reasons why most of the reconfiguration problems do not belong to NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' An important line of research has focused on finding conditions that ensure that the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' But the asymp- totic behavior of maximum possible length of a shortest reconfiguration sequence has not been really studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The problem of determining "which graphs on n vertices have the largest amount of independent sets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='" has received considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' On the contrary, the question "which graphs on n vertices have a configuration graph of independent sets has the largest diameter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='" has not, as far as we know, received any attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The 3n-vertex graph with the largest number of maximum independent sets is a disjoint collection of triangles which admits 3n independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In that case, one can easily remark that we can easily transform any maximum independent set into any other in O(n) steps by replacing a vertex of a triangle by another (which can be done without conflict since the triangles are independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So a graph whose configuration graph of independent sets has maximum diameter must have a completely different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In this paper, we consider the following questions: what is the largest possible diameter of (a connected component of) the configuration graph amongst all the graphs of size n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' What if we fix the size k of the independent set we want to consider?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let k, n be two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us denote by D(n, k) the maximum diameter, amongst all the graph G on n vertices, of a connected component of Rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The goal of this paper mainly focuses on finding lower and upper bounds on D(n, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' There is a natural upper bound for D(n, k) which is the maximum number �n k � of subsets of vertices size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We will prove in Section 2 that this bound cannot be reached and that the D(n, k) is actually at most O(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' More precisely, we will prove that D(n, k) ⩽ � n k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 2 We can easily prove that the order of magnitude of this bound is tight since, for k = 2, the following holds as we will prove in Section 3: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, 2) = n − 2, and the complement of the n-vertex path is the unique tight example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' One can naturally wonder if this bound is still tight for larger values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The answer is negative since we can prove that this upper bound can actually be very slightly improved for every k ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Namely, we will prove that: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For k ⩾ 3, we have D(n, k) = o(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The proof of Theorem 2 is inspired from the upper bound proof of the (6, 3)-problem and is based on an application of the hypergraph removal lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' A hypergraph H is (s, t)-free if no set of s vertices of H contains at least t hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The (6, 3)-problem (or Ruzsa–Szemerédi prob- lem) asks for the maximum number of hyperedges in a (6, 3)-free n-vertex 3-uniform hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The so-called (6, 3)-theorem of Ruzsa-Szemerédi [13] ensures this value is o(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This gain (o(nk−1) versus O(nk−1)) might appear marginal but we can prove that, again, it cannot be widely improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Namely we prove that the following holds: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, 3) = Ω(n2/eO(√log n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The value n/eO(√log n) corresponds to the largest known asymptotic size for a subset of [1, n] without arithmetic progressions of length 3 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Any improvement of this bound would also imply an improvement of the bound of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that the best bound for the (6, 3)-problem also has this order of magnitude [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The 3-reconfiguration problem is actually very close to the (6, 3)-problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Indeed, if we consider a shortest path in the 3-configuration graph of G and only consider even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' odd) vertices of that path, then we have a set of hyperedges of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' And one can easily check that this set of hyperedges satisfy the (6, 3)-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So our result implies in particular that, given a set of size n, we can find two sets X1, X2 of n2/eO(√log n) 3-hyperedges such that both of them are (6, 3)-free but whose union is "path-like", meaning that for every hyperedge (but at most two which are the endpoints of the path) there are two others hyperedges that intersect it on two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The idea of the proof of Theorem 3 consists in starting from a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We will then remove edges to create almost linearly many paths in the configuration graph of linear length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The involved part of the proof consists in showing that these paths remain independent of each other (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' there is no edge between them in the configuration graph) using a set S of integers with no arithmetic progression of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We finally use a last trick to glue these paths together in order to obtain the claimed diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that the classical construction giving n/eO(√log n) hyper- edges [13] for the (6, 3)-problem cannot be easily used in our construction since the construction is tripartite and then hard to reconnect into a configuration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We were not able to prove that the lower bounds and the upper bounds almost match for larger values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, it is open to determine if the 4-configuration graph can have super-quadratic diameter (while the upper bound is o(n3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We conjecture that the following holds: Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, 4) = n3−o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 3 A first step to prove super-quadratic diameter is to ensure that there exists a graph with a lot of copies of K4 such that no two of them intersect on a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This was recently shown to be true for any value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Namely, Gower and Janzer proved in [5] that, for every k and every n, there exists an n-vertex graph with nk−1−o(1) copies of Kk such that every Kk−1 is contained in at most one Kk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This result might suggest that D(n, k) = nk−1−o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Our construction for k = 3 has to be drastically modified in order to work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Indeed, our construction is heavily based on the fact that we can find a graph with an almost linear number of linear paths in its 3-configuration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To get a super-quadratic bound, we need to either increase the number of paths or their lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We failed trying both options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' However, in general, we were able to show that the following holds: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every integer k we have D(n, k) = n2⌊k/3⌋ eOk(√log n) For k = 4, 5, we can also ensure that the lower bound is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Actually, what we prove is slightly stronger but can be asymptotically summarized with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The idea of the proof of Theorem 5 consists in successively adding a graph (inspired by) the construction of Theorem 3 and connecting it in a clever way to the previous graph to increase the diameter quadratically while increasing the size of the independent set by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that a super-quadratic lower bound for k = 4 might lead to an improvement of this general lower bound as long as there is a clever gluing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observe that the asymptotic estimate in Theorem 5 depends on k, and hence may not hold when k is not constant, for example when k is linear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Constructing graphs that maximize the diameter of a connected component in their k-configuration graphs (regardless of the value of k) is a question raised during the Core Challenge 2022 [14] for graphs on 10, 50 and 100 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Our team proposed a generic construction that obtained the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Rewritten in the current formalism, our statement from [14] becomes: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every integer n, there exists a graph G on 10n vertices such that its R3n(G) is a path of length Θ(4n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular D(n, 3n 10 ) = Ω(2n/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that we also give a construction showing that D(n, 2n 5 ) = Ω(2n/5) (with a slightly worse constant than in Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Roughly speaking, these graphs are constructed by adding edges between complements of paths on 10 and 5 vertices respectively, in a similar fashion to the proof of the upcoming Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, those two constructions can be combined and yield the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every n and every k such that 3n/10 ⩽ k ⩽ 2n/5, D(n, k) = Ω(2n/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We believe it is quite surprising that this lower bound holds for such a range of values of k, and thus raise the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Question 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' What is the asymptotic behavior of maxk D(n, k)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 2 Generic upper bounds We start this section with a preliminary upper bound on D(n, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, k) ⩽ � n k−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Consider a shortest path P in the k-configuration graph of an n-vertex graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' With each edge of P, we associate the k−1 vertices of the intersection of the independent corresponding to its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This defines a mapping from E(P) to sets of k − 1 vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since there are nk−1 such sets, we simply have to show that this mapping is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Assume that two distinct edges are mapped to the same set X of k − 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then X belongs to at least three distinct independent sets that are vertices of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' These three independent sets are pairwise adjacent, which is impossible since P is a shortest path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We will see that this bound is sharp for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' However, when k increases this bound can be slightly improved, as summarized in Theorem 2 that we recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For k ⩾ 3, we have D(n, k) = o(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Consider a graph G on n vertices whose configuration graph has maximum diameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let P = Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Zd be a shortest path of length d in Rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us partition the nodes in P into two sets P1 and P2 where P1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' P2) is the set of odd (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' even) nodes of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that if we consider two subsets of Pi for i ⩽ 2 then their intersection has size at most k − 2 (otherwise P would not be induced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every i ⩽ 2, let Hi be the (k − 1)-uniform hypergraph whose vertices are the same as for G and whose hyperedges are the independent sets of size k − 1 contained in some set of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Moreover, denote by K the (k − 1)-uniform hyperclique on k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observe that by construction, each Z ∈ Pi creates (exactly) one copy of K in Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Also note that every subset of size k − 1 of such a Z belongs to exactly one independent set of Pi since otherwise the two independent sets would be adjacent, contradicting the minimality of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We now distinguish two cases: Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Hi contains more than nk−1 copies of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since by Lemma 9, at most nk−1 are created by some Z ∈ Pi, there exists a copy of K in Hi such that V (K) /∈ Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Consider now three hyperedges e1, e2, e3 in K that pairwise intersect on k − 2 vertices (note that this is possible since k ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By construction, each of these hyperedges are contained in some element of Pi, so there exist x1, x2, x3 ∈ V (G) such that ej ∪{xj} ∈ Pi for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, each ej is an independent set of G, therefore e1 ∪ e2 ∪ e3 is also an independent set of G of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore all the ej ∪ {xj}’s are at distance at most 2 from each other in Rk(G) since all of them are adjacent to e1 ∪ e2 ∪ e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This is a contradiction since P is a shortest path and P1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' P2) only contains even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' odd) vertices of P and then two of the three independent sets ej ∪ {xj} ∈ Pi for j ⩽ 3 should be at distance at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Hi contains at most nk−1 = o(nk) copies of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By the hypergraph removal lemma [12, 6], there exists a set S of hyperedges of H such that |S| = o(nk−1) and H − S contains no copy of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Recall that each hyperedge of S is contained in exactly one element of P1, and each element of Pi creates a copy of K in H, therefore we get |Pi| ⩽ |S| = o(nk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To conclude, observe that d ⩽ 2|Pi| = o(nk−1) for every i ⩽ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 3 Lower bounds 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='1 Independent sets of size 2 In this section, our main goal is to prove Theorem 1 that we recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, 2) = n − 2, and the complement of the n-vertex path is the unique tight example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 5 We thus consider the independent sets of size 2 of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that these sets are exactly the non-edges of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' the edges of G = (V, P2(V )\\E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, we get the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The configuration graph R2(G) is the line graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that for every graph G, any induced path on p vertices in L(G) corresponds to a path on p edges in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, we derive two consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let A, B be two independent sets of G of size 2 and a ∈ A, b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' There is a TJ-transformation from A to B if and only if a, b are in the same connected component of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We can also obtain the following which ensures that the bound of Lemma 9 is tight: Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every n-vertex graph G, diam(R2(G)) = diam(L(G)) ⩽ diam(G) − 1 ⩽ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that the last bound is tight only when G is a path, which concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since the diameter is linear, one might wonder if we can determine in linear time if there exists such a transformation (and find it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that we cannot just compute the line graph of the complement and run a BFS on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Indeed, even if a BFS can be computed in linear time with respect to the number of edges of its input, this number may be quadratic with respect to the number of edges of the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' However, by complementing only the graph induced by vertices of large degree, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let A, B be two independent sets of G of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We can decide if there exists a TJ-transformation from A to B in time O(|V (G)| + |E(G)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be an n-vertex m-edge graph, and s, t two vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We start by precomputing the degrees of the vertices of G in O(n + m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let B be the set of vertices of degree at least n−1 2 , and S = V (G) \\ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observe that by the pigeonhole principle, any two vertices in S must have a common non-neighbor, hence are connected in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us denote by H the graph obtained by identifying all the vertices of S into a single vertex x and where we put an edge between x and y /∈ S if y is adjacent to a vertex of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' It is easy to check that there is a path between s and t in G if and only if there is such a path in H (up to replacing s or t by x if they lie in S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note moreover that one can easily compute the graph H in O(n + m) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' One can notice that the graph H might be sparse and then its complement can have size Ω(|V (H)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' However, observe that (|V (H)| − 1) × n − 1 2 ⩽ � v∈B degG(v) ⩽ 2m, hence |V (H)| = O( m n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, one can compute H and use a BFS in H in time O(|V (H)|2) = O( m2 n2 ) = O(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that the algorithm we provide can easily be adapted to return a (possibly non-optimal) transformation when it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='2 Almost quadratic construction for independent sets of size 3 The rest of this section is devoted to prove the following result: 6 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' D(n, 3) = Ω(n2/eO(√log n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The proof is based on two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' First, we prove that there exists a graph whose configuration graph is the disjoint union of n/eO(√log n) paths of linear length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We then prove that, starting from a graph whose configuration graph is disconnected, we can (up to adding few vertices), obtain a graph whose configuration graph is connected and whose diameter is at least the sum of the diameter of the connected components of the initial configuration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' While the first step is specific to k = 3 and is based on the existence of almost linear subsets of integers without arithmetic sequences of length 3, the gluing process is general and holds for any possible value of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us first prove the gluing lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let k ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be a graph on n vertices whose k-configuration graph contains r connected components C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Cr of diameter respectively d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then there exists a graph on at most n + (3k − 2) · (r − 1) vertices whose configuration graph has diameter at least (4k − 4)(r − 1) + � i⩽r di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' x1 x2 x3 x4 x5 x6 x7 b1 b2 b3 Bi a3 a2 a1 Ai+1 Figure 1: Construction for Lemma 14 with k = 3, where only non-edges are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every i ⩽ r, let us denote by Ai and Bi two independent sets at distance di in the component Ci of the k-configuration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every i, we denote by ai j (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' bi j) the j-th vertex of Ai (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We moreover assume that ai k and bi 1 are respectively the first and last vertices modified in a shortest sequence Pi from Ai to Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that the sets (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Ar, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Br) might intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every i ⩽ r−1, we create 3k−2 new vertices xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2 in order to connect Bi to Ai+1 in the configuration graph of the new graph (see Figure 1 for an illustration of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We first add all the edges between the new vertices and V (G) and, for every i ̸= j, the vertices xi a and xj b are adjacent regardless of a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Moreover, for every p < q ⩽ 3k − 2, xi p is adjacent to xi q if and only if q − p > k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We finally remove the following edges: for every j we remove the edges between xi j (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' x3k−2−j) and bi j′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' ai+1 k−j′) with j′ > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us denote by H the resulting graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let Xi be the (ordered) set of vertices {bi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , bi k, xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2, ai+1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , ai+1 k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us first prove the following simple claim on the structure of independent sets: Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let i ⩽ r − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Every k-independent set S containing one vertex in {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}: consists of k consecutive vertices of Xi and, has degree 2 in Rk(H) if it contains a vertex in {xi 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us first prove that if an independent set S contains a vertex in {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2} then it contains consecutive vertices of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 7 If S ⊆ {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}, then let us denote by xi a and xi b the first and last vertices in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By construction, since S is an independent set, we must have b − a ⩽ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' And then S contains only non-neighbors of xi a and xi b, hence b − a = k − 1 and S = {xi a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So from now on we can assume that S contains a vertex of V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since xi k−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 2k−2 are complete to G, the set S cannot contain one of these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' And since {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi k−2} is complete to {xi 2k−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}, we can assume by symmetry that S contains vertices in {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi k−2} but not in {xi 2k−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us denote by a ⩽ k −1 the largest index such that xi a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By construction, xi a is non-adjacent to the k−1 vertices before it in the sequence and complete to all the other vertices of H\\Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So S = {bi a+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , bi k, xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For the second item, observe that indeed, each set of k consecutive vertices in Xi is indepen- dent, and is connected to the independent sets corresponding to the k vertices just before and after them in the ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Moreover, each independent set S intersecting {xi 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−3} con- tains at least two vertices in {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore S is only adjacent to independent sets containing at least one vertex in {xi 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , xi 3k−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By the first item, they consist of k consecutive vertices of Xi, hence S has exactly two neighbors in Rk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So the k-configuration graph of H restricted to Xi induces a path Pi from Bi to Ai+1 of length 4k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By concatenating these paths with shortest reconfiguration sequences from the Ai to Bi for every i ⩽ r, we get a reconfiguration sequence P from A1 to Br of length (4k − 2)(r − 1) + � i⩽r di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To complete the proof we have to prove that we can shorten the sequence P by exactly 2(r − 1) steps (and that no shorter transformation exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Indeed, for every i ⩽ r − 1, consider a reconfiguration sequence from Ai to Bi of minimum size where ai 1 is the vertex deleted from Ai at the beginning of the sequence and bi 1 is the last vertex to be moved on Bi at the end of the sequence (this reconfiguration sequence exists by assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' If we denote by B′ i the independent set before Bi, B′ i contains Bi \\ {bi 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then B′ i is adjacent to (Bi ∪ xi 1) \\ bi 1 in the configuration graph and then we can remove Bi in the reconfiguration sequence from A1 to Br and still have a reconfiguration sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Similarly, we can find a shortcut of the sequence on Ai for every 2 ⩽ i ⩽ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So we can shorten P by 2(r − 1) steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We claim that this transformation has shortest length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us briefly argue why it is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Consider an independent set of H containing exactly one vertex in {xi j | i ⩽ r −1, j ⩽ 3k −2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' It should contain the vertex xi 1 or xi 3k−2 for some i by Claim 15 and k−1 vertices of an independent set in Ci or in Ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Thus it can only be adjacent to an independent set of the component of Ci or Ci+1 or an independent set containing two vertices of Xi by Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Using Claim 15 again, it means that from an independent set only containing xi 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' xi 3k−3) we can only reach an independent set of Ci+1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Ci) containing Bi−1 \\ bi−1 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Ai+1 \\ ai+1 k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Before proving that it is possible to obtain an almost linear number of components of almost linear size, we need some definitions and results of group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let S be a set of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We say that S is 3-AP-free if it does not contain an arithmetic progression of length 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' there does not exist s1 < s2 < s3 in S such that s2 − s1 = s3 − s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Determining the size of the largest possible 3-AP-free subset of [1, n] is a heavily studied problem whose exact answer is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' It was shown that there does not exist any 3-AP-free set of positive density in N [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' However Behnrend proved in [1] that there exist 3-AP-free subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , n} of size n/eO(√log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that if S is 3-AP-free, then the set 4S + 1 also is 3-AP-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So, there exists a 3-AP-free sequence of size n/eO(√log n) only containing integers whose value is 1 modulo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Such a set will be called an odd 3-AP-free sequence in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Now let p be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' It is well known that, for every α ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , p−1}, the sequence of the kα modulo p is a periodic sequence of period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' That is {0, α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , (p−1)α} = {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , p−1} 8 modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We claim that the following holds: Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let n be a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let S be an odd 3-AP-free sequence where the maximum integer is at most n/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then, there exists a graph on n − 1 vertices whose configuration graph is the disjoint union of |S| paths of length n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be the graph obtained from a clique on n vertices by removing the edges (i, i + s) and (i, i + 2s) for s ∈ S and i ⩽ n, where all integers are understood modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We claim that R3(G) consists of |S| induced cycles of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' First observe that for each s ∈ S we have n independent sets, namely {ks, (k + 1)s, (k + 2)s} (0 ⩽ k < n), which induce a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, R3(G) contains |S| cycles of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us now show that this union of cycles is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To this end, let I = {a, b, c} and I′ = {a′, b′, c′} be two independent sets such that c−b = b−a = s ∈ S and c′−b′ = b′−a′ = s′ ∈ S with s ̸= s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' If I ∩I′ contains two elements x and y, then observe that x−y ∈ {±s, ±2s}∩{±s′, ±2s′}, which is impossible since s and s′ are distinct integers below n/8 and are both 1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore I and I′ are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Finally, we prove that R3(G) has no other vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let {a, b, c} be an independent set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' If b−a ∈ S and c−b ∈ S, then c−a is even and lies in 2S, hence we can write b−a = s, c−b = s′ and c − a = 2s′′ = s′ − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since S is 3-AP-free, we have s = s′ = s′′ and then b − a = c − b (and then {a, b, c} is one of the sets described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Otherwise, b − a or c − b does not belong to S, thus c − a is equal to 0 or 3 modulo 4 and then c − a /∈ S ∪ 2S, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So R3(G) is the disjoint union of cycles with no edges between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Now if we remove the vertex 0 from G, each of these cycles now becomes a path (we remove the three independent sets containing it for every S), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us combine Lemma 14 and 16 to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let S be an odd 3-AP-free sequence of size n/eO(√log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By Lemma 14, there exists a graph G on n−1 vertices whose 3-configuration graph admits |S| connected components of diameter n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By applying Lemma 16 to G, we obtain a new graph on n − 1 + 7 · (|S| − 1) ⩽ 8n vertices and whose 3-configuration graph has diameter at least 8 · (|S| − 1) + |S| · (n − 3) = n2/eO(√log n), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We end this section with the following question which would extend Theorem 13 to indepen- dent sets of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Question 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Can we compute the diameter or the existence of a transformation between two independent sets of size 3 in (sub)quadratic time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='3 General lower bound The goal of this section is to generalize the construction of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='2 to larger values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Unfortunately, we were not able to obtain a lower bound that almost fits the upper bound for larger values of k, but we still obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every integer k we have D(n, k) = n2⌊k/3⌋ eOk(√log n) Let us first give the flavour of the proof with an intermediate construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be a graph with independence number k such that Rk(G) has diameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then for every integer n, there exists a graph H on |V (G)| + 6n + 2 vertices such that Rk+2(H) has diameter at least 2dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 9 B A x1 x2 x3 x6k x6k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' G Figure 2: Construction of the proof of Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For readability, we have only represented the non-edges incident with the vertices of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let A and B be two independent sets of size k at distance d in Rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us create 6n + 2 vertices X = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , x6n+2}, inducing the complement of a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every 1 ⩽ ℓ ⩽ n, link the vertices x6ℓ−3 and x6ℓ to respectively V (G)\\B and V (G)\\A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us denote by H the resulting graph (see Figure 2 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let C, D be the independent sets A ∪ {x1, x2} and A ∪ {x6n+1, x6n+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since k is the independence number of G and X is the complement of a path, the maximum independent sets of H have size k + 2 and all of them contain k vertices in V (G) and two vertices in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, in every transformation from C to D, two tokens are present at each time in X, and the movements of these tokens correspond to a path in R2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, in order to slide the tokens from {x1, x2} to {x6n+1, x6n+2}, each transformation must hit in order some independent sets S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , S6n+1 such that Si ∩ X = {xi, xi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Amongst all such independent sets, we select Si as the first such independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that for every i = 3 mod 6 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 0 mod 6), Si = B ∪{xi, xi+1} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Si = A∪{xi, xi+1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' So, for every i = 0 or 3 mod 6, in order to transform Si into Si+3 we need at least d + 3 steps (since we need to transform A into B plus three token slides on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, the length of the reconfiguration sequence from C to D is at least 2n(d+3), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since the graph H constructed in Lemma 18 has maximum independent sets of size k + 2, we can iterate the process starting from the complement of a path and prove that the following holds: Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For any k, D(n, k) = Ωk(n⌊k/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In the proof of Lemma 18, we can see the vertices of indices 0 modulo 3 as toll booths which enforces us to perform a lot of modifications in G in order to pass though these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To improve the bound of Corollary 19, we will generalize Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Indeed instead of gluing the complement of a path to G and increasing the size of the independent sets by 2, we will copy a (slightly modified) copy of the graph of Theorem 3 and increase the size of the independent sets by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that we cannot automatically, when we have a graph with a large reconfiguration diameter, glue it with another graph and get a large reconfiguration diameter since we need to be careful on where we put the toll booths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' (That is why the graph of Theorem 3 has to be slightly modified to work in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=') The main ingredient for Theorem 5 is thus the following analogue of Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let n, k be two integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be a graph with maximum independent sets of size k such that the k-configuration graph of G has diameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Then there exists a graph on at most |V (G)| + 3kn vertices whose (k + 3)-reconfiguration diameter is at least dn2 eO(√log n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' The construction is inspired from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let S′ be a largest 3-AP-free sequence in [1, n/64] and S be the set 8S′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Recall that |S| = n eO(√log n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Denote by H the graph obtained 10 applying Lemma 16 to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Recall that vertices of H can be labeled from 1 to n − 1 in such a way that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' the independent sets of size 3 have consecutive values modulo 8, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' if two such sets are adjacent in R3(H), then their symmetric difference contains two elements whose difference is ±3 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' each vertex of H appears in at least one independent set in each connected component of R3(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let A and B two independent sets of G at distance d from each other in Rk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Denote by G′ the graph obtained by taking a copy of G and H and adding all the edges: between V (G) \\ A and vertices of H that are 0 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' between V (G) \\ B and vertices of H that are 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that since G has no independent set of size more than k, then any independent set of G′ of size k+3 decomposes as k vertices of G and 3 vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' In particular, any reconfiguration sequence from I to J in G′ yields a reconfiguration sequence from I∩V (H) to J∩V (H) in H (and the same holds for G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore Rk+3(G′) contains at least as many connected components as R3(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Xr be a connected component of R3(H) which induces a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By construction, we may assume that X1 contains vertices that are 1,2, and 3 mod 8, and by (1) and (2), each Xi contains vertices that are i, i + 1 and i + 2 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Up to reducing r by at most 7, we may even assume that Xr also contains vertices that are 1, 2 and 3 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Thus X1 ∪ A and Xr ∪ A are independent sets of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Observe that if i = 1 mod 4, there is no edge between Xi and V (G), hence there is a reconfiguration sequence of length d between Xi∪A and Xi∪B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, one can reach Xr ∪A from X1∪A going through the following steps: X1∪B, X2∪B, X3∪B, X4∪B, X5∪B, X5∪A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content='. Therefore, X1 ∪ A and Xr ∪ A are in the same connected component of Rk+3(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let us now compute (a lower bound on) their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Consider a shortest reconfiguration sequence between X1 ∪ A and Xr ∪ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Recall that this yields a (non-necessarily shortest) reconfiguration sequence from X1 to Xr in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By (3), for every vertex of H which is 0 mod 8, we may choose a set Xi that contains it, and denote by Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' , Xin/8 the subsequence they form (observe that this is well-defined since no Xi can contain two vertices that are 0 mod 8 by (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Now by (1) and (2), note that in the reconfiguration sequence between every Xij and Xij+1, there must exist some independent set Xi′ j containing a vertex that is 4 mod 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' By construction, the only independent set of G′ containing each Xij (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Xi′ j) is Xij ∪ A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Xi′ j ∪ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore the distance between Xij ∪ A and Xi′ j ∪ B is at least d, and so is the distance between Xi′ j ∪ B and Xij+1 ∪ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, the distance between X1 ∪ A and Xr ∪ A is at least 2d × ( n 8 − 1) = dn 4 − 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since Rk+3(H) contains at least |S| components of diameter at least dn 4 − 2d, applying Lemma 14 to G′ yields a graph on |V (G′)|+(3k −2)(|S|−1) = |V (G)|+n−1+(3k −2)(|S|−1) vertices with diameter at least (4k − 4)(|S| − 1) + |S|dn/4 − 2d|S|, which concludes since |S| = n/eO(√log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' As an immediate corollary, we obtain Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' 11 4 Conclusion Let us start the conclusion with this simple remark: Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' There exists a super-graph G′ of G with the same number of vertices such that R3(G′) is a path and the largest diameters of the connected components of R3(G) and R3(G′) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We start by adding arbitrarily edges to G while the largest diameter of a component in R3(G) stays unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' To conclude, we show that R3(G) is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Let P be a shortest path of maximal length in R3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Assume that there is a node Z = {u, v, w} of R3(G) that is not in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For each x ̸= y ∈ Z, adding the edge xy to G decreases the diameter of the component of P in R3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Hence there must be an independent set of P that contains both x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, we can assume that P contains three independent sets {u, v, a}, {u, w, b} and {v, w, c}, which are all pairwise distinct since Z /∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Since P is a shortest path and these sets are all neighbors of Z, they must be consecutive in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Therefore, we have a = b = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' But then, {u, v, a}, {v, w, a} and {u, w, a} induces a triangle in R3(G), a contradiction since P is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that all the graphs obtained from our constructions also satisfy that their configuration graphs are paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We conjecture that the following is true in general: Conjecture 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' For every k ⩾ 2 and every n, there exists a graph G on n vertices maximizing D(n, k) and such that the Rk(G) is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Note that this result holds for k = 2 (since complement of paths are tight) and for k = 3 as proven in Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Another interesting question is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' We have remarked that both lower and upper bounds of the (6, 3)-problem correspond to the bounds obtained for the largest possible diameter of R3(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Are the two problems equivalent (up to a multiplicative constant)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' While the existence of a better diameter for some graph G would immediately imply a better bound for the (6, 3)-problem (by simply considering even vertices of a shortest path), the converse is not immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtA0T4oBgHgl3EQfEf_h/content/2301.02020v1.pdf'} +page_content=' This work started thanks to the CoRe programming challenge [14] whose topic in 2022 consisted in finding the graphs on 10, 50 and 100 vertices with the largest possible independent set reconfiguration diameter.' metadata={'source': 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0000000000000000000000000000000000000000..2bb279e80c66427610c5e8b40e65c83815177e4c --- /dev/null +++ b/V9AyT4oBgHgl3EQfhvhg/content/tmp_files/2301.00382v1.pdf.txt @@ -0,0 +1,1031 @@ +Semi-analytical technique for the design of +disordered coatings with tailored optical +properties +BHRIGU RISHI MISHRA,1 NITHIN JO VARGHESE,1 AND KARTHIK +SASIHITHLU1,* +1Department of Energy Science and Engineering, Indian Institute of Technology Bombay +*ksasihithlu@iitb.ac.in +Abstract: Disordered media coatings are finding increasing use in applications such as day-time +radiative cooling paints and solar thermal absorber plate coatings which require tailored optical +properties over a broad spectrum ranging from visible to far-IR wavelengths. Both monodisperse +and polydisperse configurations with thickness of coatings up to 500 𝜇𝑚 are currently being +explored for use in these applications. In such cases it becomes increasingly important to explore +utility of analytical and semi-analytical methods for design of such coatings to help reduce +the computational cost and time for design. While well-known analytical methods such as +Kubelka-Munk and four-flux theory have previously been used for analysis of disordered coatings, +analysis of their utility has so far in literature been restricted to either solar spectrum or IR but not +simultaneously over the combined spectrum as required for the above applications. In this work, +we have analysed the applicability of these two analytical methods for such coatings over the +entire wavelength range from visible to IR, and based on observed deviation from exact numerical +simulation we propose a semi-analytical technique to aid in the design of these coatings with +significant computational cost savings. +© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement +1. +Introduction +Disordered coatings, which consist of dielectric/metal nanoparticles dispersed randomly in +a matrix, find their application in several fields such as solar thermal absorber coatings [1], +solar reflecting coatings [2], color paints [3], translucent paints [4], tissue optics [5], daytime +passive radiative cooling coatings [6–8], and many more. The main advantages that such +disordered media offer that make them an attractive proposition for use in these applications +are their cost-effective means of fabrication, and tunability of the desired optical properties of +the coating - since the spectral position of Mie (plasmon) resonance of the embedded dielectric +(metal) particles strongly depend on the size of the particles. The main challenging task in the +design of such disordered media is the modelling of its optical properties. Techniques based on +homogenization of the composite structures that predict effective permittivity and permeability +of the disordered media, such as Maxwell-Garnett theory [9] and Bruggeman’s model [10], are +valid only when the particle sizes are much smaller than the incident wavelength [11]. Doyle et +al. [12] showed that the use of Mie coefficients in this effective medium theory provides good +accuracy to the calculation of effective optical properties of metal spheres suspended in a polymer. +However, the theory predicts absorption for a nonabsorbing particle [11] and thus cannot be used +to predict the effective refractive index of disordered media for solar reflecting paint/coatings +where non absorbing particles are utilized. In literature, other analytical techniques developed +for this objective include those which consider diffusion of photons [13], and those which solve +the radiative transfer equation under 𝑁-flux (2 ≤ 𝑁 ≤ 4) approximations [14, 15]. Of these +methods Kubelka Munk (KM) theory [16] (for which 𝑁 = 2) and the four-flux (FF) method [17] +are commonly used. In addition, simulation techniques such as the Monte Carlo method [18], +arXiv:2301.00382v1 [physics.optics] 1 Jan 2023 + +and exact electromagnetic solvers are employed to model the optical/radiative properties of +disordered coatings. However, these simulation techniques do not present a clear picture linking +the microscopic properties of the particles, such as the scattering and absorption coefficients, +to the macroscopic optical properties of the coating. Moreover, exact electromagnetic solvers +which solve Maxwell’s equations numerically to obtain radiative properties of the coating, put a +premium on computational resources and the time for design when the thickness of random media +is in the order of tens/hundreds of microns - as is currently being deployed in these applications. +Particularly when several parameters of the configuration are in play, such as that encountered in +disordered media, analytical techniques such as KM and FF theories provide important means to +arrive at an optimum combination of the parameters with minimal computational resources while +also explicitly linking the properties of the micro constituents to the observed optical properties +of the coating. +KM and FF theories have so far in literature been used in applications where the spectrum of +interest has been limited to either the visible spectrum or IR separately. For example, KM theory +has been used extensively in paints and coatings [1, 3], paper industry [19], tissue optics [5] +among others. Similarly, the FF method has been used extensively by researchers to model, +predict and optimize the optical properties of light scattering coatings [2,20–22]. However, the +applicability of these theories over a broad spectrum covering both visible and IR spectrum +simultaneously has not been a subject of attention. This becomes important when designing +coatings for applications such as day-time passive radiative cooling and solar thermal absorber +plate coatings where tailored optical properties over a broad spectrum covering both the solar +spectrum as well as far-infrared are crucial. For example, coatings for day-time passive radiative +cooling [23] require high reflectivity in solar spectrum (0.3-3 𝜇m wavelength range) and high +emissivity in the infra-red spectrum (5-15 𝜇m wavelength range). It is not obvious that the +analytical techniques retain their accuracy over such a broad spectrum since with increasing +wavelength there is a possibility that the nature of scattering transitions from the independent +scattering regime (where scattering cross-section of particles can be superposed) to dependent +scattering regime (where near-field effects, and interference between far-field scattered radiation +become important). Previously reported relation [24] demarcating the two regimes has been +obtained from experimental observations carried out in the visible spectrum only. Thus there is a +need to explore the applicability of these analytical techniques over a broad spectrum in greater +depth. In regimes where the predictions from these analytical techniques are not satisfactory, +other possible methods of design which combine the accuracy of exact electromagnetic solvers +with the minimal computational requirements of analytical methods are expected to be of pressing +need to researchers interested in designing such coatings. +With this in mind, the manuscript has been arranged as follows. In Section 2 we have compared +the reflectivity and emissivity predictions of KM and FF techniques with results from exact +numerical solvers for different degrees of absorption in the particles (imaginary index of particle += 0.0, 10−2, and 10−1) and in the matrix (imaginary index of matrix = 0.0, 10−4, and 10−2) for +different thickness of the coating (10 𝜇m and 50 𝜇m) in the wavelength range 0.3 − 15 𝜇𝑚. We +show that these techniques are accurate over the entire spectrum when particles are in the limit of +independent scattering and under low absorption conditions but fail when volume fraction of +particles is high such that interaction among particles is no longer negligible or when absorption +in the matrix/particles is high. For such conditions where analytical techniques fail to predict the +optical properties accurately we propose an alternative technique which combines the use of exact +numerical solver and KM theory which we show can predict optical properties with accuracy as +well as with minimal computational requirements. This ‘semi-analytical’ technique is detailed +in Section 3. In the end, as an example to showcase the applicability of this semi-analytical +technique, we predict the properties of a disordered coating suitable for the application of passive +radiative cooling and compare these with experimental measurements previously reported in + +literature. +2. +Analytical techniques - Kubelka Munk (KM) and the Four-Flux (FF) methods +We start with the expressions for reflectivity and transmissivity of the coating as obtained from +KM and FF theories which we use in the work to analyze the optical properties of the disordered +coating. Detailed derivations of these expressions can be found in several references [17,25,26]. +The optical coating considered in this work is a plane-parallel slab of particulate composite on a +substrate as shown in Fig. 1. The composite is considered to be of finite thickness and infinite +extension in the lateral direction. The randomly distributed spherical particles embedded within +the host medium (also called the matrix) act as inhomogeneities to the propagating EM wave, +thereby causing its scattering (and absorption, in case the particle is lossy). The objective is to +predict the optical properties of this coating including the total reflectance, transmittance and +absorption. +Fig. 1. Schematic of coating considered in this work with incident plane wave source. +The expression for the reflectivity (𝑅KM) and transmissivity (𝑇KM) from KM theory is given +by [3,25]: +𝑅KM = (1 − 𝛾)(1 + 𝛾)(exp(𝐴𝐿) − exp(−𝐴𝐿)) +(1 + 𝛾)2 exp(𝐴𝐿) − (1 − 𝛾)2 exp(−𝐴𝐿) +(1) +𝑇KM = +4𝛾 +(1 + 𝛾)2 exp(𝐴𝐿) − (1 − 𝛾)2 exp(−𝐴𝐿) +(2) +where, 𝐿 is the thickness of the layer, the coefficients 𝛾 and 𝐴 are given by [3,27]: +𝛾 = +√︁ +𝑘/(𝑘 + 𝑠′); 𝐴 = +√︁ +2𝑘(2𝑘 + 𝑠′) +(3) +with +𝑠′ = 3𝑠(1 − 𝑔) − 𝑘 +4 +; +(4) +and the factors 𝑠 and 𝑘 got using Mie theory [1] +𝑠 = 3 𝑓 𝑄sca +4𝑟 +; +𝑘 = 3 𝑓 𝑄abs +4𝑟 +, +(5) +where 𝑓 is the volume fraction, 𝑟 is the radius of the sphere, 𝑄sca (𝑄abs) is the Mie scattering +(absorption) efficiency of a single particle embedded in a host medium of index 𝑛ℎ, and 𝑔 is the +asymmetry parameter. Expressions for 𝑄sca, 𝑄abs, and 𝑔 in terms of standard Mie coefficients can + +Source +Matrix +Spherical particles +Substratebe found in Ref. [28]. It should be pointed out that the relations between the coating properties +𝛾 and 𝐴, and the particle properties 𝑠 and 𝑘 given in Eq. 3 are not unique - several other +relations [4,5,29–33] have been proposed over the years. The expressions in Eq. 3 and Eq. 4, +taken from Ref. [3,27], is representative and have been chosen for demonstrative reasons. As we +will see in Sec. 3 the semi-analytical method being proposed in this work does not depend on +such relations and hence do not affect the central results of this work. +In the limit of low absorption, 𝑘𝐿 → 0, the reflectivity in Eq. (1) can be shown to reduce +to [25]: +𝑅KM = +𝑠′𝐿 +𝑠′𝐿 + 1. +(6) +It must be noted that Eq. (1) and Eq. (2) do not take into account surface reflection of incident +radiation at interface (1). Modified reflectance 𝑅0 and transmittance 𝑇0 which take into account +surface reflection correction are calculated using [34]: +𝑅0 = 𝑅c + (1 − 𝑅c)(1 − 𝑅i)𝑅KM +1 − 𝑅i𝑅KM +; +𝑇0 = (1 − 𝑅c)𝑇KM +1 − 𝑅i𝑅KM +(7) +where, 𝑅c is the specular reflectance of incident light got from Fresnel reflection which for normal +incidence from a medium of index 𝑛surr reads: +𝑅c = +�𝑛 − 1 +𝑛 + 1 +�2 +(8) +with 𝑛 = 𝑛h/𝑛surr and 𝑅i is the diffuse reflectance of internal radiation at interface (1), marked in +Fig. 1, which is calculated using: +𝑅𝑖 = 2 +∫ +𝜋/2 +0 +𝜌(𝜃) sin𝜃 cos𝜃 d𝜃. +(9) +where, from Fresnel’s coefficients: +𝜌(𝜃) = 1 +2 +������ +�√ +𝑛2 − sin𝜃 − cos𝜃 +√ +𝑛2 − sin𝜃 + cos𝜃 +�2 ++ +� +𝑛2cos𝜃 − +√ +𝑛2 − sin𝜃 +𝑛2cos𝜃 + +√ +𝑛2 − sin𝜃 +�2������ +. +(10) +The expression for 𝑅i from Eq. 9 can be used even the limit of low diffuse scattering since the +contribution from the product 𝑅𝑖𝑅KM will be negligible in this regime. +Many configurations developed for radiative cooling application [7,35,36] and solar absorber +plates [37,38] involve use of a substrate. In the presence of a substrate, the net reflectance and +transmittance from Eq. 7 will have to be further modified as [39]: +𝑅 = 𝑅0 + +𝑇2 +0 𝑅g +1 − 𝑅0𝑅g +; 𝑇 = (1 − 𝑅g)𝑇0 +1 − 𝑅0𝑅g +(11) +Here 𝑅g is the diffuse reflectance at interface (2) obtained from Eq. 9 with 𝑛 = 𝑛h/𝑛g. The +substrate index 𝑛𝑔 is taken to be 1.5 in this work. +The derivation of reflection and transmission coefficients from KM theory assumes that +the incident light is diffuse. When the incident radiation is collimated, alternate methods +such as the four-flux theory, which take into account the propagation of both collimated and +diffuse radiation across the interfaces in two directions, are expected to be more accurate. +This careful consideration of both collimated and diffuse components leads to expressions for +the optical properties being far more complicated than in KM theory. The net reflection and +transmission coefficients when incident radiation is fully collimated can be expressed in terms of + +a summation over collimated-collimated reflectivity (𝑅cc), collimated-diffuse reflectivity (𝑅cd), +collimated-collimated transmissivity (𝑇cc), and collimated-diffuse transmissivity (𝑇cd) as: +𝑅 = 𝑅cc + 𝑅cd + 𝑅dd; 𝑇 = 𝑇cc + 𝑇cd + 𝑇dd +(12) +Expressions for 𝑅cc, 𝑅cd, 𝑇cc and 𝑇cd are quite elaborate and have been included in the supple- +mentary document (Section S1) for reference. +(a) +(b) +(c) +(d) +Fig. 2. Reflectivity and transmissivity spectrum for 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.05, +and (a) 𝑛p = 2.5, 𝐿 = 10 𝜇m; (b) 𝑛p = 2.5, 𝐿 = 50 𝜇m. Reflectivity and absorptivity +spectrum for 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.05, and (c) 𝑛p = 2.5 + 𝑖0.1, 𝐿 = 10 𝜇m; (d) +𝑛p = 2.5 + 𝑖0.1, 𝐿 = 50 𝜇m. Here, FF stands for four flux, KM for Kubelka Munk, and +LM for Lumerical. +In Sections 2.1, 2.2, and 2.3, we use the expressions for 𝑅 and 𝑇 given in Eqs. 11 for KM +theory and Eqs. 12 for FF to predict the optical properties of disordered coatings and compare +these with the results obtained from Lumerical FDTD solver [40]. We analyze situations where +both the particles and host medium are absorbing as well as non-absorbing, and also consider the +effect of different thickness of the coating. The degree of absorption in particles considered in +this work are relevant for dielectric inclusions typically included in coatings for use in radiative +cooling and solar thermal applications. In addition, to facilitate the parametric study we assume +non-dispersive form of refractive index for both the particles as well as the host matrix. We first +confine our analysis to the independent scattering regime in Sections 2.1 and 2.2, and extend + +1.0 +Reflectivity/Transmissivity +、二 +0.8 +0.6 +D +0.4 +0.2 +0.0 +10 +15 +Wavelength (um)1.0 +Reflectivity/Transmissivity +L= 50 μm +2.5 +0.8 +0.6 +RkM + KM +0.4 +LM +0.2 +0.0 +10 +15 +Wavelength (μum)1.0 +L= 50 μm +Reflectivity/ Absorptivity +np = 2.5+i0.1 +0.8 +0.6 +M +A +KM +R. +0.4 +0.2 +0.0 +10 +15 +Wavelength (μum)1.0 +Reflectivity/ Absorptivity +0.8 +L= 50 μm +np = 2.5+i0.1 +FF +0.6 +0.4 +KM +0.2 +0.0 +10 +15 +Wavelength (um)the analysis to dependent scattering regime later in Sec. 2.3. The FDTD simulations were set +up in ANSYS Lumerical. Periodic boundary conditions were applied in the lateral 𝑥 and 𝑦 +directions, and coating is illuminated with a plane wave source from 𝑧 direction. A mesh size of +30 nm was used which we find is sufficient for convergence (mesh convergence study is shown in +supplementary Fig. S3). +2.1. +Comparison of predictions from KM, FF theories and FDTD solver in the indepen- +dent scattering regime for monodisperse inclusions with and without absorption in +particles and in host medium +Figure 2a and 2b show the comparison between KM, FF, and FDTD results for the case when +particles are non-absorbing and Fig. 2c and 2d show the corresponding comparison when +particles are absorbing with imaginary index of particles 𝑛′′ +𝑝 = 10−1. We compare the predictions +for different coating thicknesses 10 𝜇𝑚 and 50 𝜇𝑚 keeping the other parameters 𝑟 = 0.25 𝜇𝑚, +𝑓 = 0.05, 𝑛ℎ = 1.5 non varying. It is observed that particularly for smaller thickness of coating +and in the absence of absorption the predictions from FF method deviates significantly from +the FDTD simulations as compared to KM method in both the visible as well as IR spectrum. +However, for larger thicknesses of the coating and in the presence of absorption in particles, +FF is relatively more accurate than the KM method across the spectrum, more so for higher +wavelengths. +In the presence of absorbing host media, the expressions for 𝛾 and 𝐴 in Eqs. 1 and 2 needs +to be modified to account for absorption in the matrix [20, 22] as: 𝛾 = +√︁ +𝑘′/(𝑘′ + 𝑠′), and +𝐴 = +√︁ +2𝑘′(2𝑘′ + 𝑠′) where, 𝑘′ = 𝑘 + (1 − 𝑓 )𝛼. Here, 𝛼 = 4𝜋𝑛′′ +h /𝜆, with 𝑛′′ +h being the imaginary +part of refractive index of the matrix, and 𝜆 the wavelength in vacuum. In addition, expressions +for 𝑄sca and 𝑄abs in Eq. 5 needs to be modified as shown by Mischenko et al. [41]. Figure 3a +and 3b show the comparison between KM, FF, and FDTD results for the case when host medium +is weakly absorbing with 𝑛′′ +ℎ = 10−4 and Fig. 3c and 3d show the corresponding comparison +when it is more strongly absorbing with 𝑛′′ +ℎ = 10−2. In the presence of weakly absorbing matrix +and for smaller thickness of the coating FF is again observed to deviate significantly from the +FDTD simulations. As absorption increases we observe significant deviation from FDTD results +in both FF and KM theories particularly for the higher wavelengths. +2.2. +Comparison of predictions from KM, FF theories and FDTD solver in the indepen- +dent scattering regime for polydisperse inclusions with and without absorption in +particles +In this section we explore the predictive capability of KM and FF theories for polydisperse +medium which consists of randomly positioned particles with different sized radius. The study +is motivated from the observation that synthesis of nanoparticles via various methods such as +sol-gel [42], microemulsion [43], hydrothermal [44], results in a polydisperse size distribution of +particles. Moreover, some recent studies [45,46] have also deliberately adopted coatings with +different size distribution of particles to make use of the property of size-dependent scattering of +particles to obtain wavelength-selective coatings. Such a particulate medium can be analyzed by +considering the particles to be distributed about a mean radius 𝑟 with standard deviation 𝜎, with +the expressions for 𝑠 and 𝑘 to be used in Eqs. 5 got by summing over the respective coefficients +for individual particle volume fractions 𝑓𝑖 [22] as: +𝑠 = +𝑁 +∑︁ +𝑖=1 +𝑠𝑖; +𝑘 = +𝑁 +∑︁ +𝑖=1 +𝑘𝑖 +(13) +where 𝑠𝑖 and 𝑘𝑖 are the Mie scattering and absorption coefficients respectively of the particle +with fill fraction 𝑓𝑖. Equation (13) can also be used to calculate 𝑠 and 𝑘 when there are two or + +(a) +(b) +(c) +(d) +Fig. 3. Reflectivity and absorptivity spectra for 𝑛p = 2.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.05, and +(a) 𝑛h = 1.5+𝑖10−4, 𝐿 = 10 𝜇m; (b) 𝑛h = 1.5+𝑖10−4, 𝐿 = 50 𝜇m; (c) 𝑛h = 1.5+𝑖10−2, +𝐿 = 10 𝜇m; (d) 𝑛h = 1.5 + 𝑖10−2, 𝐿 = 50 𝜇m. Here, FF stands for four flux, KM for +Kubelka Munk, and LM for Lumerical. +more type of particles present in the matrix (with different refractive index). For demonstration +we consider a Gaussian distribution of spherical particles about mean radius 𝑟 = 0.25 𝜇𝑚 with +standard deviation 𝜎 = 0.016 𝜇𝑚 with and without absorption. The particle size distribution +curve has been shown in Fig. S2. Figure 4a and 4b show the comparison between KM, FF, +and FDTD results for the case when particles are non-absorbing and Fig. 4c and 4d show the +corresponding comparison when particles are absorbing with 𝑛′′ +𝑝 = 10−2. Other parameter values +are retained as for the case of monodisperse particulate coating. The observations follow the trend +seen for the case of monodisperse coating with significant deviations observed in the predictions +of FF method for lower values of thicknesses of coatings and when particles are nonabsorbing. +For larger thicknesses of the coating and in the presence of absorption, both FF and KM are +observed to predict the optical properties with reasonable accuracy across the spectrum. +2.3. +Comparison of predictions from KM, FF and FDTD solvers in the dependent +scattering regime +So far we have analyzed for the situations where the fill fraction 𝑓 of particles in the composite is +small enough so that the particles can be assumed to independently scatter from one another. + +1.0 +L=10 μm +Reflectivity/ Absorptivity +0.8 +nh = 1.5+il0-4 +M +0.6 +0.4 +N +0.2 +0.0 +10 +15 +Wavelength (μum)1.0 +Reflectivity/Absorptivity +L = 50 μm +0.8 +nh = 1.5+il0-4 +A +0.6 +0.4 +0.2 +0.0 +10 +15 +Wavelength (um)1.0 +L= 10 μm +Reflectivity/ Absorptivity +0.8 +nh = 1.5+il0-2 +0.6 +R +KM +0.4 +0.2 +0.0 +10 +15 +Wavelength (μum)1.0 +Reflectivity/ Absorptivity +0.8 +L= 50 μm +nh = 1.5+i10-2 +0.6 +R +KM +KM +0.4 +A +0.2 +0.0 +10 +15 +Wavelength (um)(a) +(b) +(c) +(d) +Fig. 4. Reflectivity and transmissivity for 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝜎 = 0.016, 𝑓 = 0.05 +and (a) 𝑛p = 2.5, 𝐿 = 10 𝜇m; (b) 𝑛p = 2.5, 𝐿 = 50 𝜇m. Reflectivity and absorptivity +for 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝜎 = 0.016, 𝑓 = 0.05 and (c) 𝑛p = 2.5 + 0.01𝑖, 𝐿 = 10 𝜇m; +(d) 𝑛p = 2.5 + 0.01𝑖, 𝐿 = 50 𝜇m. Here, FF stands for four flux, KM for Kubelka Munk, +and LM for Lumerical. +However, as the fill fraction of particles increases, there will be a transition to dependent- +scattering regime where both the near-field interaction between the particles as well as far-field +interference between the scattered field of individual particles have a significant impact on the +overall properties of the coating. Hotel [24] empirically determined this transition to occur when +𝑓 > 0.27 and 𝑑/𝜆 > 0.3 where 𝑑 is the mean inter-particle spacing and 𝜆 is the wavelength. +Several coatings reported in literature [7,46–49] have fill fractions in the range 0.1-0.6 where +such effects cannot be neglected. We thus explore here the predictive capability of FF and KM +theories for such coatings by considering a monodisperse distribution of particles with increased +fill fraction 𝑓 = 0.3 while retaining other parameter values to be same as that included for Fig. +2b. This comparison is shown in Fig. 5, where we observe that the predictions from both FF and +KM theories deviate significantly from FDTD simulations across the spectra, and thus cannot be +relied on for predicting optical properties of such coatings. +A comparison between the weighted average of the optical properties across the spectra as +predicted by KM and FF theories for the different cases considered so far has been tabulated in +Table 1. The weighted averages are calculated as: 𝑅solar = +∫ +𝐼AM1.5(𝜆)𝑅(𝜆)𝑑𝜆/ +∫ +𝐼AM1.5(𝜆) 𝑑𝜆, + +1.0 +Reflectivity/Transmissivity +L= 10 μm +0.8 +=2.5 +0.6 +R +KM +KM +0.4 +0.2 +0.0 +10 +15 +Wavelength (um1.0 +Reflectivity/Transmissivity +L= 50 μm +2.5 +0.8 +FF +0.6 +R +KM +0.4 +LM +0.2 +0.0 +10 +15 +Wavelength (um)1.0 +Reflectivity/Absorptivity +L= 10 μm +0.8 +np = 2.5+i0.01 +0.6 +V +0.4 +N +0.2 +0.0 +10 +15 +Wavelength (μum)1.0 +L = 50 μum +Reflectivity/ Absorptivity +np = 2.5+i0.01 +0.8 +R +0.6 +0.4 +0.2 +0.0 +10 +15 +Wavelength (um)Fig. 5. Reflectivity and transmissivity for 𝑛p = 2.5, 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.3, +and 𝐿 = 50 𝜇m. Here, FF stands for four flux, KM for Kubelka Munk, and LM for +Lumerical. +where 𝐼AM1.5(𝜆) is the spectral solar irradiance [50] and 𝜖IR = +∫ +𝐼BB(𝜆)𝜖(𝜆)𝑑𝜆/ +∫ +𝐼BB(𝜆) 𝑑𝜆, +where 𝐼BB(𝜆) is the black body irradiance. For the relevant applications in consideration for this +study i.e. coatings suitable for radiative cooling application and for use in solar thermal absorber +plates, the reflection over the solar spectrum i.e. over wavelength range 0.3 − 3 𝜇𝑚 and emissivity +over the infra-red spectrum i.e. over wavelength range 5 − 15 𝜇𝑚 is of primary importance, and +the weighted average over this spectral range is reported in Table 1 along with the deviation from +FDTD simulations expressed in % error in brackets. +3. +Semi-analytical method +The comparison with FDTD simulations shown in Section 2 demonstrate the failure of KM +and FF analytical methods in configurations where dependent scattering is not negligible and +when matrix/particles are absorbing. This failure can be attributed to the actual scattering +and absorption coefficients of these coatings diverging from the values calculated using Mie +scattering coefficients of the individual particles. At present no single analytical technique +exists that can correctly predict the optical properties of particulate media in the presence of +dependent scattering effects as well as correctly account for the absorption in matrix/particles. +One can then resort to using exact numerical solvers to accurately estimate the optical properties +of the coating in such cases. However, as Fig. 6 shows, the computational time required to +simulate such structures increases exponentially with thickness of the coating. For coatings +of thickness in the range 100-500 microns which are currently being adopted in literature for +the radiative cooling application [6,8,36,46,49] the design time is clearly prohibitive. In such +cases it becomes imperative to develop alternate techniques which can combine the accuracy +power of exact FDTD solvers with the simplicity and minimal computational requirements of the +analytical techniques. Particularly when multiple parameters are involved in design - such as that +observed for disordered media - such a method will prove to be useful in reducing the design +time to find the optimum combination of parameters necessary to obtain the required optical +properties of the coating. In order to obtain a better estimate for the absorption and scattering +coefficients of such media where dependent scattering effects are non-negligible, researchers +have previously [27,51–53] relied on experimental measurements of the optical properties of a +fabricated coating and then using the KM theory results from Section 2 to extract the required +coefficients. Instead of relying on experimental measurements which is not always feasible +especially at the initial state of design, we modify this technique and instead propose the following + +1.0 +Reflectivity/Transmissivity +0.8 +FF +0.6 +TKM +0.4 +0.2 +0.0 +10 +15 +Wavelength (um)Table 1. Weighted-average reflectivity in solar spectrum 𝑅solar,KM (𝑅solar,FF) and +emissivity in IR spectrum 𝜖IR,KM (𝜖IR,FF) calculated using KM (FF) theory. Values in +brackets denote deviation of prediction from FDTD results. +Sr. no. +Fig. no. +𝑅solar,KM +𝜖IR,KM +𝑅solar,FF +𝜖IR,FF +1 +2a +0.391 (21.4%) +- +0.509 (58.1%) +- +2 +2b +0.745 (0.67%) +- +0.747 (0.4%) +- +3 +2c +0.076 (13.4%) +0.08 (95%) +0.081 (20.9%) +0.043 (4.87%) +4 +2d +0.078 (18.2%) +0.331 (70.6%) +0.079 (19.7%) +0.197 (1.55%) +5 +3a +0.376 (18.6%) +- +0.483 (52.4%) +- +6 +3b +0.619 (8.22%) +0.012 (100%) +0.612 (6.99%) +0.005 (16.67%) +7 +3c +0.112 (30.2%) +0.205 (83.0%) +0.110 (27.9%) +0.086 (23.2%) +8 +3d +0.112 (36.6%) +0.661 (51.3%) +0.108 (31.7%) +0.357 (18.3%) +9 +4a +0.392 (19.1%) +- +0.510 (55.0%) +- +10 +4b +0.746 (4.63%) +- +0.755 (5.89%) +- +11 +4c +0.260 (25.0%) +0.008 (60%) +0.279 (34.1%) +0.004 (20%) +12 +4d +0.304 (16.5%) +0.041 (78.3%) +0.268 (2.68%) +0.023 (0.01%) +13 +5 +0.944 (8.13%) +- +0.935 (6.98%) +- +two-step semi-analytical method to estimate the optical properties of random media of thickness +𝐿 when usage of exact numerical solvers to simulate the properties of such a thick coating is +prohibitive. +• Step 1: Use a numerical solver to obtain the optical properties 𝑅 and 𝑇 of a similar coating +but with much smaller thickness 𝑡𝑠 ≪ 𝐿 and extract the 𝛾 and 𝐴 parameters by inverting +Eq. 1 and 2. Care must be taken at this step to ensure that the configuration set up in +the solver considers incident light to be in the same medium as the index of matrix i.e., +𝑛surr = 𝑛h in order to ensure that reflection from surfaces and substrates are not included +in this step. In case the host matrix is absorbing then only the real part is considered i.e., +𝑛surr = Re(𝑛h). Care must also be taken to ensure that when scattering efficiency of the +particles is high, the value of 𝑡𝑠 should be chosen such that 𝑡𝑠 ≫ 𝑙𝑠 where 𝑙𝑠 ≈ 1/(𝑁𝜎𝑠) +is the scattering mean-free path with 𝑁 being the particle number density and 𝜎𝑠 the +scattering cross section. At the other limit when scattering efficiency is low the optical +properties of the coating are primarily determined from surface reflection and transmission +which are accounted for in step 2. Thus the choice of 𝑡𝑠 is determined from the scattering +mean-free path calculated in the high-scattering regime. +• Step 2: From the 𝛾 and 𝐴 parameters extracted from step 1 use the analytical expressions +from KM theory i.e. Eqs. 1, 2, 7 and 11 to predict the optical properties of the coating of +the required thickness 𝐿 ≫ 𝑡𝑠. Specular reflection at the surfaces as well as at the substrate +are accounted for here. + +Fig. 6. Comparison of computational time as a function of thickness of the disordered +media coating. Simulations are carried out in ANSYS Lumerical using an eight-core +Intel Xeon workstation for the configuration: 𝑛𝑝 = 2.5, 𝑛ℎ = 1.5, 𝑟 = 0.25 𝜇m, +𝑓 = 0.05, with mesh size 30 nm. Auto shutoff level (simulation termination criteria) is +set at 10−3. +A more elaborate procedure, along with details of a supporting convergence test which may need +to be incorporated in some cases to arrive at the value of thickness 𝑡𝑠 is included in Section S4 of +supplementary. +(a) +(b) +Fig. 7. (a) Reflectivity and transmissivity for 𝑛p = 2.5, 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, +𝑓 = 0.3, and 𝐿 = 50 𝜇m; (b) Reflectivity and absorptivity for 𝑛p = 2.5 + 0.1𝑖, 𝑛h = 1.5, +𝑟 = 0.25 𝜇m, 𝑓 = 0.3, and 𝐿 = 50 𝜇m. Here, SM stands for semi-analytical method +and LM for Lumerical. +We now apply this technique for the cases considered in Section 2 where the predictions from +analytical methods deviated significantly from those of FDTD solver, such as for the dependent +scattering regime, as well as when the absorption in the particles/host matrix is significant. Fig. +7 (Fig. 8) shows the comparison between the predictions from the semi-analytical technique +and from FDTD simulations when absorption in particles (host matrix) is varied. In both these +cases the semi-analytical technique uses the results of exact FDTD simulations of a 10 𝜇m thick +coating to predict the optical properties of a larger 50 𝜇m thick coating. A volume fill fraction + +100 +Computational time (Hrs) +90 +- Computational time (Hrs) +80 +-Exponential fit +70 +60 +50 +40 +30 +20 +10 +0 +0 +10 +20 +3040 +50 +60 +70 +80 +90 100110 +Thickness (μm)1.0 +Reflectivity/Transmissivity +0.8 +0.6 +M +R +0.4 +0.2 +0.0 +10 +15 +Wavelength (um1.0 +Reflectivity/ Absorptivity +np = 2.5+i0.1 +0.8 +0.6 +R +0.4 +0.2 +w +0.0 +10 +15 +Wavelength (um)𝑓 = 0.3 is maintained in both these cases where dependent scattering effects are known to be +dominant, while keeping other parameter values same as that analysed for the monodisperse case +of Sec. 2.1. For these cases we observe a close match in the predictions of the semi-analytical +(a) +(b) +Fig. 8. Reflectivity and absorptivity for 𝑛p = 2.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.3, 𝐿 = 50 𝜇m, +and (a) 𝑛h = 1.5 + 𝑖10−2; (b) 𝑛h = 1.5 + 𝑖10−1. Here, SM stands for semi-analytical +method and LM for Lumerical. +method with the FDTD results over the entire spectrum, with only a slight deviation observed for +the higher wavelengths when absorption is high. The weighted-average reflectivity of the coating +for solar spectrum, and emissivity over the infra-red spectrum for the cases considered in Figs. 7 +and 8 are listed in Table 2 along with the deviation from FDTD simulations expressed in % error +in brackets. Particularly illustrative of the effectiveness of the semi-analytical technique is the +reduction in error (1.03 % in Sr. no. 1) as compared to those obtained from analytical techniques +and reported in Table 1 ( 8.13 % using KM theory and 6.98 % using FF theory in Sr. no. 13) for +the configuration: 𝑛p = 2.5, 𝑛h = 1.5, 𝑟 = 0.25 𝜇m, 𝑓 = 0.3, and 𝐿 = 50 𝜇m where dependent +scattering is expected to be dominant. +Table 2. Weighted-average reflectivity in solar spectrum 𝑅solar,SM, and emissivity in +IR spectrum 𝜖IR,SM calculated using the semi-analytical technique. Values in brackets +denote deviation of prediction from FDTD results. +Sr. no. +Fig. no. +𝑅solar,SM +𝜖IR,SM +1 +7a +0.864 (1.03%) +- +2 +7b +0.059 (0.01%) +0.710 (8.73%) +3 +8a +0.178 (2.73%) +0.391 (6.25%) +4 +8b +0.062 (19.2%) +0.935 (5.17%) +4. +Comparison with experimental data +We now apply the semi-analytical technique described in Section 3 to predict the optical properties +of fabricated coatings reported in literature which have been designed for radiative cooling +application. We choose two such disordered coatings where dependent scattering is expected to + +1.0 +Reflectivity/ Absorptivity +0.8 +nh = 1.5+il0-2 +0.6 +0.4 +0.2 +0.0 +10 +15 +Wavelength (um1.0 +Reflectivity/ Absorptivity +nh = 1.5+i10-1 +0.8 +R +0.6 +M +0.4 +0.2 +0.0 +10 +15 +Wavelength (um)be dominant so that analytical techniques are not applicable, and the thickness of the coating +prohibits the use of exact electromagnetic solvers to predict the optical properties to good +accuracy. +In Ref. [48], a hierarchically porous polymer (P(VdF-HFP)) coating of thickness 300 𝜇m +containing air voids with sizes ranging from 0.05-5 𝜇m in P(VdF-HFP) matrix has been fabricated, +and experimentally characterized to have solar reflectivity value of 0.96 and emissivity in the +8-13 𝜇m wavelength range to be 0.97. In order to apply semi-analytical technique to predict the +properties of this coating, we set up a simulation in FDTD solver with a smaller coating thickness +𝑡𝑠 = 50 𝜇m (determined using the convergence test explained in Section S4 of supplementary). +This thickness is chosen to ensure sufficient number of larger sized air voids (𝑟 ≈ 2.5 𝜇m) in this +P(VdF-HFP) matrix. The size distribution of nano-micro air voids used in the simulation is given +in supplementary (Fig. S4). Refractive index data of P(VDF-HFP) is extracted from Ref. [48]. +The reflectivity data in the wavelength range 0.3 − 16 𝜇 m, predicted using the semi-analytical +method for 𝐿 = 300 𝜇m thickness, is compared with that reported in Ref. [48] in Fig. 9a. While +an appreciable match is noticed in the predicted values across the spectrum, small deviation +observed in the reflectivity values can be attributed to our inability to incorporate exact size +distribution of both micro and nano voids as present in the fabricated structure, in ANSYS +Lumerical. +In Ref. [46] an ultrawhite BaSO4 film of thickness 400 𝜇m has been developed with 60 % +volume fraction of BaSO4 nanoparticles, and has been characterized to have reflectivity of 0.976 +in the solar spectrum and emissivity of 0.96 in 8-13 𝜇m wavelength range. In order to apply +the semi-analytical technique to predict the properties of this coating, we set up a simulation +in FDTD solver with structure thickness 𝑡𝑠 = 15 𝜇m and BaSO4 spherical particles randomly +distributed with volume fraction 60 %. The particles are taken to be of uniform size distribution +with diameters spread over the range 398 ± 130 nm to match that reported in Ref. [46]. Matrix is +considered to be air for BaSO4 film. Refractive index data of BaSO4 is extracted from Ref. [54]. +The emissivity data in the wavelength range 0.3 − 16 𝜇m, predicted using the semi-analytical +method for 𝐿 = 400 𝜇m thickness, is compared with that reported in Ref. [46] in Fig. 9b. While +we again observe an appreciable match across the spectrum, some deviation observed particularly +around wavelength of 2 𝜇m is suspected to be due to difference in the refractive index of the +fabricated film and that calculated from first-principles in Ref. [54]. +(a) +(b) +Fig. 9. (a) Reflectivity of hierarchically porous P(VDF-HFP) coating calculated using +semi-analytical technique is compared with experimental result given by Mandal et +al. [48]; (b) Absorptivity/emissivity of BaSO4 film calculated using semi-analytical +technique is compared with experimental result given by Li et al. [46]. + +1.0 +SM +0.8 +Ref.[48] +Reflectivity +0.6 +0.4 +0.2 +0.0 +10 +57 +16 +1 +Wavelength (μum)1.0 +Absorptivity/Emissivity +0.8 +SM +Ref. [46] +0.6 +0.4 +0.2 +0.0 +1 +38 +10 +16 +Wavelength (um)5. +Conclusion +In this study we have analyzed the applicability of well-known analytical techniques of KM +and FF theories to predict optical properties of a disordered metamaterial coating over a broad +spectrum ranging from 300 nm to 15 𝜇m wavelength. Recent advancements in the use of +disordered coatings in applications such as radiative cooling and solar thermal absorber plates +which require tailored optical properties over this wavelength range necessitates such a study. +Based on deviations observed between the predictions of these analytical techniques and exact +FDTD solver in the dependent scattering regime, a two-step semi-analytical technique has been +proposed which can be used to predict optical properties of such coatings with good accuracy +and minimal computational resources. Such a method is expected to be resourceful for designing +coatings with specific optical properties where several parameter combinations need to be +investigated to arrive at an optimal combination. Small deviations observed when absorption in +host matrix is high warrants further research to improve this technique. +Acknowledgments +B.R.M. acknowledges support from Prime Minister’s Research Fellowship (PMRF). K.S. ac- +knowledges support from La Fondation Dassault Systèmes and SERB Grant No. SRG/2020/001 +511. +Disclosures +The authors declare no conflicts of interest. +Supplementary information +See Supplement 1 for supporting content. +References +1. +M. Gunde and Z. Orel, “Absorption and scattering of light by pigment particles in solar-absorbing paints,” Appl. Opt. +39, 622–628 (2000). +2. +T. Nilsson and G. Niklasson, “Radiative cooling during the day: simulations and experiments on pigmented +polyethylene cover foils,” Sol. Energy Mater. & Sol. Cells 37, 93–118 (1995). +3. +M. Quinten, “The color of finely dispersed nanoparticles,” Appl. Phys, B 73, 317–326 (2001). +4. +M. Bandpay, F. Ameri, K. Ansari, and S. Moradian, “Mathematical and empirical evaluation of accuracy of the +kubelka-munk model for color match prediction of opaque and translucent surface coatings,” J. Coat. Technol. Res. +15, 1117–1131 (2018). +5. +A. Roy, R. Ramasubramaniam, and H. 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Ruan, “Atomistic metrics of BaSO4 as an ultra-efficient radiative +cooling material: a first-principles prediction,” arXiv preprint arXiv:2101.05053 (2021). + diff --git a/V9AyT4oBgHgl3EQfhvhg/content/tmp_files/load_file.txt b/V9AyT4oBgHgl3EQfhvhg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..685b39b83ffc0e01c0b367415dff6514be840264 --- /dev/null +++ b/V9AyT4oBgHgl3EQfhvhg/content/tmp_files/load_file.txt @@ -0,0 +1,1042 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf,len=1041 +page_content='Semi-analytical technique for the design of disordered coatings with tailored optical properties BHRIGU RISHI MISHRA,1 NITHIN JO VARGHESE,1 AND KARTHIK SASIHITHLU1,* 1Department of Energy Science and Engineering, Indian Institute of Technology Bombay ksasihithlu@iitb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='in Abstract: Disordered media coatings are finding increasing use in applications such as day-time radiative cooling paints and solar thermal absorber plate coatings which require tailored optical properties over a broad spectrum ranging from visible to far-IR wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Both monodisperse and polydisperse configurations with thickness of coatings up to 500 𝜇𝑚 are currently being explored for use in these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In such cases it becomes increasingly important to explore utility of analytical and semi-analytical methods for design of such coatings to help reduce the computational cost and time for design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' While well-known analytical methods such as Kubelka-Munk and four-flux theory have previously been used for analysis of disordered coatings, analysis of their utility has so far in literature been restricted to either solar spectrum or IR but not simultaneously over the combined spectrum as required for the above applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In this work, we have analysed the applicability of these two analytical methods for such coatings over the entire wavelength range from visible to IR, and based on observed deviation from exact numerical simulation we propose a semi-analytical technique to aid in the design of these coatings with significant computational cost savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' © 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Introduction Disordered coatings, which consist of dielectric/metal nanoparticles dispersed randomly in a matrix, find their application in several fields such as solar thermal absorber coatings [1], solar reflecting coatings [2], color paints [3], translucent paints [4], tissue optics [5], daytime passive radiative cooling coatings [6–8], and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The main advantages that such disordered media offer that make them an attractive proposition for use in these applications are their cost-effective means of fabrication, and tunability of the desired optical properties of the coating - since the spectral position of Mie (plasmon) resonance of the embedded dielectric (metal) particles strongly depend on the size of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The main challenging task in the design of such disordered media is the modelling of its optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Techniques based on homogenization of the composite structures that predict effective permittivity and permeability of the disordered media, such as Maxwell-Garnett theory [9] and Bruggeman’s model [10], are valid only when the particle sizes are much smaller than the incident wavelength [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Doyle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [12] showed that the use of Mie coefficients in this effective medium theory provides good accuracy to the calculation of effective optical properties of metal spheres suspended in a polymer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, the theory predicts absorption for a nonabsorbing particle [11] and thus cannot be used to predict the effective refractive index of disordered media for solar reflecting paint/coatings where non absorbing particles are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In literature, other analytical techniques developed for this objective include those which consider diffusion of photons [13], and those which solve the radiative transfer equation under 𝑁-flux (2 ≤ 𝑁 ≤ 4) approximations [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Of these methods Kubelka Munk (KM) theory [16] (for which 𝑁 = 2) and the four-flux (FF) method [17] are commonly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In addition, simulation techniques such as the Monte Carlo method [18], arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='00382v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='optics] 1 Jan 2023 and exact electromagnetic solvers are employed to model the optical/radiative properties of disordered coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, these simulation techniques do not present a clear picture linking the microscopic properties of the particles, such as the scattering and absorption coefficients, to the macroscopic optical properties of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Moreover, exact electromagnetic solvers which solve Maxwell’s equations numerically to obtain radiative properties of the coating, put a premium on computational resources and the time for design when the thickness of random media is in the order of tens/hundreds of microns - as is currently being deployed in these applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Particularly when several parameters of the configuration are in play, such as that encountered in disordered media, analytical techniques such as KM and FF theories provide important means to arrive at an optimum combination of the parameters with minimal computational resources while also explicitly linking the properties of the micro constituents to the observed optical properties of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' KM and FF theories have so far in literature been used in applications where the spectrum of interest has been limited to either the visible spectrum or IR separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For example, KM theory has been used extensively in paints and coatings [1, 3], paper industry [19], tissue optics [5] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Similarly, the FF method has been used extensively by researchers to model, predict and optimize the optical properties of light scattering coatings [2,20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, the applicability of these theories over a broad spectrum covering both visible and IR spectrum simultaneously has not been a subject of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This becomes important when designing coatings for applications such as day-time passive radiative cooling and solar thermal absorber plate coatings where tailored optical properties over a broad spectrum covering both the solar spectrum as well as far-infrared are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For example, coatings for day-time passive radiative cooling [23] require high reflectivity in solar spectrum (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3-3 𝜇m wavelength range) and high emissivity in the infra-red spectrum (5-15 𝜇m wavelength range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' It is not obvious that the analytical techniques retain their accuracy over such a broad spectrum since with increasing wavelength there is a possibility that the nature of scattering transitions from the independent scattering regime (where scattering cross-section of particles can be superposed) to dependent scattering regime (where near-field effects, and interference between far-field scattered radiation become important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Previously reported relation [24] demarcating the two regimes has been obtained from experimental observations carried out in the visible spectrum only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Thus there is a need to explore the applicability of these analytical techniques over a broad spectrum in greater depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In regimes where the predictions from these analytical techniques are not satisfactory, other possible methods of design which combine the accuracy of exact electromagnetic solvers with the minimal computational requirements of analytical methods are expected to be of pressing need to researchers interested in designing such coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' With this in mind, the manuscript has been arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In Section 2 we have compared the reflectivity and emissivity predictions of KM and FF techniques with results from exact numerical solvers for different degrees of absorption in the particles (imaginary index of particle = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0, 10−2, and 10−1) and in the matrix (imaginary index of matrix = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0, 10−4, and 10−2) for different thickness of the coating (10 𝜇m and 50 𝜇m) in the wavelength range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 − 15 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We show that these techniques are accurate over the entire spectrum when particles are in the limit of independent scattering and under low absorption conditions but fail when volume fraction of particles is high such that interaction among particles is no longer negligible or when absorption in the matrix/particles is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For such conditions where analytical techniques fail to predict the optical properties accurately we propose an alternative technique which combines the use of exact numerical solver and KM theory which we show can predict optical properties with accuracy as well as with minimal computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This ‘semi-analytical’ technique is detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In the end, as an example to showcase the applicability of this semi-analytical technique, we predict the properties of a disordered coating suitable for the application of passive radiative cooling and compare these with experimental measurements previously reported in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Analytical techniques - Kubelka Munk (KM) and the Four-Flux (FF) methods We start with the expressions for reflectivity and transmissivity of the coating as obtained from KM and FF theories which we use in the work to analyze the optical properties of the disordered coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Detailed derivations of these expressions can be found in several references [17,25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The optical coating considered in this work is a plane-parallel slab of particulate composite on a substrate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The composite is considered to be of finite thickness and infinite extension in the lateral direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The randomly distributed spherical particles embedded within the host medium (also called the matrix) act as inhomogeneities to the propagating EM wave, thereby causing its scattering (and absorption, in case the particle is lossy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The objective is to predict the optical properties of this coating including the total reflectance, transmittance and absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Schematic of coating considered in this work with incident plane wave source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The expression for the reflectivity (𝑅KM) and transmissivity (𝑇KM) from KM theory is given by [3,25]: 𝑅KM = (1 − 𝛾)(1 + 𝛾)(exp(𝐴𝐿) − exp(−𝐴𝐿)) (1 + 𝛾)2 exp(𝐴𝐿) − (1 − 𝛾)2 exp(−𝐴𝐿) (1) 𝑇KM = 4𝛾 (1 + 𝛾)2 exp(𝐴𝐿) − (1 − 𝛾)2 exp(−𝐴𝐿) (2) where, 𝐿 is the thickness of the layer, the coefficients 𝛾 and 𝐴 are given by [3,27]: 𝛾 = √︁ 𝑘/(𝑘 + 𝑠′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝐴 = √︁ 2𝑘(2𝑘 + 𝑠′) (3) with 𝑠′ = 3𝑠(1 − 𝑔) − 𝑘 4 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (4) and the factors 𝑠 and 𝑘 got using Mie theory [1] 𝑠 = 3 𝑓 𝑄sca 4𝑟 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑘 = 3 𝑓 𝑄abs 4𝑟 , (5) where 𝑓 is the volume fraction, 𝑟 is the radius of the sphere, 𝑄sca (𝑄abs) is the Mie scattering (absorption) efficiency of a single particle embedded in a host medium of index 𝑛ℎ, and 𝑔 is the asymmetry parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Expressions for 𝑄sca, 𝑄abs, and 𝑔 in terms of standard Mie coefficients can Source Matrix Spherical particles Substratebe found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' It should be pointed out that the relations between the coating properties 𝛾 and 𝐴, and the particle properties 𝑠 and 𝑘 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3 are not unique - several other relations [4,5,29–33] have been proposed over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 4, taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [3,27], is representative and have been chosen for demonstrative reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' As we will see in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3 the semi-analytical method being proposed in this work does not depend on such relations and hence do not affect the central results of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In the limit of low absorption, 𝑘𝐿 → 0, the reflectivity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (1) can be shown to reduce to [25]: 𝑅KM = 𝑠′𝐿 𝑠′𝐿 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (6) It must be noted that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (2) do not take into account surface reflection of incident radiation at interface (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Modified reflectance 𝑅0 and transmittance 𝑇0 which take into account surface reflection correction are calculated using [34]: 𝑅0 = 𝑅c + (1 − 𝑅c)(1 − 𝑅i)𝑅KM 1 − 𝑅i𝑅KM ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑇0 = (1 − 𝑅c)𝑇KM 1 − 𝑅i𝑅KM (7) where, 𝑅c is the specular reflectance of incident light got from Fresnel reflection which for normal incidence from a medium of index 𝑛surr reads: 𝑅c = �𝑛 − 1 𝑛 + 1 �2 (8) with 𝑛 = 𝑛h/𝑛surr and 𝑅i is the diffuse reflectance of internal radiation at interface (1), marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1, which is calculated using: 𝑅𝑖 = 2 ∫ 𝜋/2 0 𝜌(𝜃) sin𝜃 cos𝜃 d𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (9) where, from Fresnel’s coefficients: 𝜌(𝜃) = 1 2 ������ �√ 𝑛2 − sin𝜃 − cos𝜃 √ 𝑛2 − sin𝜃 + cos𝜃 �2 + � 𝑛2cos𝜃 − √ 𝑛2 − sin𝜃 𝑛2cos𝜃 + √ 𝑛2 − sin𝜃 �2������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (10) The expression for 𝑅i from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 9 can be used even the limit of low diffuse scattering since the contribution from the product 𝑅𝑖𝑅KM will be negligible in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Many configurations developed for radiative cooling application [7,35,36] and solar absorber plates [37,38] involve use of a substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In the presence of a substrate, the net reflectance and transmittance from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 7 will have to be further modified as [39]: 𝑅 = 𝑅0 + 𝑇2 0 𝑅g 1 − 𝑅0𝑅g ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑇 = (1 − 𝑅g)𝑇0 1 − 𝑅0𝑅g (11) Here 𝑅g is the diffuse reflectance at interface (2) obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 9 with 𝑛 = 𝑛h/𝑛g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The substrate index 𝑛𝑔 is taken to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The derivation of reflection and transmission coefficients from KM theory assumes that the incident light is diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' When the incident radiation is collimated, alternate methods such as the four-flux theory, which take into account the propagation of both collimated and diffuse radiation across the interfaces in two directions, are expected to be more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This careful consideration of both collimated and diffuse components leads to expressions for the optical properties being far more complicated than in KM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The net reflection and transmission coefficients when incident radiation is fully collimated can be expressed in terms of a summation over collimated-collimated reflectivity (𝑅cc), collimated-diffuse reflectivity (𝑅cd), collimated-collimated transmissivity (𝑇cc), and collimated-diffuse transmissivity (𝑇cd) as: 𝑅 = 𝑅cc + 𝑅cd + 𝑅dd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑇 = 𝑇cc + 𝑇cd + 𝑇dd (12) Expressions for 𝑅cc, 𝑅cd, 𝑇cc and 𝑇cd are quite elaborate and have been included in the supple- mentary document (Section S1) for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and transmissivity spectrum for 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05, and (a) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and absorptivity spectrum for 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05, and (c) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 𝑖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (d) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 𝑖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1, 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, FF stands for four flux, KM for Kubelka Munk, and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, we use the expressions for 𝑅 and 𝑇 given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 11 for KM theory and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 12 for FF to predict the optical properties of disordered coatings and compare these with the results obtained from Lumerical FDTD solver [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We analyze situations where both the particles and host medium are absorbing as well as non-absorbing, and also consider the effect of different thickness of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The degree of absorption in particles considered in this work are relevant for dielectric inclusions typically included in coatings for use in radiative cooling and solar thermal applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In addition, to facilitate the parametric study we assume non-dispersive form of refractive index for both the particles as well as the host matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We first confine our analysis to the independent scattering regime in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2, and extend 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity 、二 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity L= 50 μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 RkM KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 LM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 L= 50 μm Reflectivity/ Absorptivity np = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 M A KM R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/ Absorptivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 L= 50 μm np = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1 FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)the analysis to dependent scattering regime later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The FDTD simulations were set up in ANSYS Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Periodic boundary conditions were applied in the lateral 𝑥 and 𝑦 directions, and coating is illuminated with a plane wave source from 𝑧 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' A mesh size of 30 nm was used which we find is sufficient for convergence (mesh convergence study is shown in supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Comparison of predictions from KM, FF theories and FDTD solver in the indepen- dent scattering regime for monodisperse inclusions with and without absorption in particles and in host medium Figure 2a and 2b show the comparison between KM, FF, and FDTD results for the case when particles are non-absorbing and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2c and 2d show the corresponding comparison when particles are absorbing with imaginary index of particles 𝑛′′ 𝑝 = 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We compare the predictions for different coating thicknesses 10 𝜇𝑚 and 50 𝜇𝑚 keeping the other parameters 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇𝑚, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05, 𝑛ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 non varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' It is observed that particularly for smaller thickness of coating and in the absence of absorption the predictions from FF method deviates significantly from the FDTD simulations as compared to KM method in both the visible as well as IR spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, for larger thicknesses of the coating and in the presence of absorption in particles, FF is relatively more accurate than the KM method across the spectrum, more so for higher wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In the presence of absorbing host media, the expressions for 𝛾 and 𝐴 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1 and 2 needs to be modified to account for absorption in the matrix [20, 22] as: 𝛾 = √︁ 𝑘′/(𝑘′ + 𝑠′), and 𝐴 = √︁ 2𝑘′(2𝑘′ + 𝑠′) where, 𝑘′ = 𝑘 + (1 − 𝑓 )𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, 𝛼 = 4𝜋𝑛′′ h /𝜆, with 𝑛′′ h being the imaginary part of refractive index of the matrix, and 𝜆 the wavelength in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In addition, expressions for 𝑄sca and 𝑄abs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 5 needs to be modified as shown by Mischenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Figure 3a and 3b show the comparison between KM, FF, and FDTD results for the case when host medium is weakly absorbing with 𝑛′′ ℎ = 10−4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3c and 3d show the corresponding comparison when it is more strongly absorbing with 𝑛′′ ℎ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In the presence of weakly absorbing matrix and for smaller thickness of the coating FF is again observed to deviate significantly from the FDTD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' As absorption increases we observe significant deviation from FDTD results in both FF and KM theories particularly for the higher wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Comparison of predictions from KM, FF theories and FDTD solver in the indepen- dent scattering regime for polydisperse inclusions with and without absorption in particles In this section we explore the predictive capability of KM and FF theories for polydisperse medium which consists of randomly positioned particles with different sized radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The study is motivated from the observation that synthesis of nanoparticles via various methods such as sol-gel [42], microemulsion [43], hydrothermal [44], results in a polydisperse size distribution of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Moreover, some recent studies [45,46] have also deliberately adopted coatings with different size distribution of particles to make use of the property of size-dependent scattering of particles to obtain wavelength-selective coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Such a particulate medium can be analyzed by considering the particles to be distributed about a mean radius 𝑟 with standard deviation 𝜎, with the expressions for 𝑠 and 𝑘 to be used in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 5 got by summing over the respective coefficients for individual particle volume fractions 𝑓𝑖 [22] as: 𝑠 = 𝑁 ∑︁ 𝑖=1 𝑠𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑘 = 𝑁 ∑︁ 𝑖=1 𝑘𝑖 (13) where 𝑠𝑖 and 𝑘𝑖 are the Mie scattering and absorption coefficients respectively of the particle with fill fraction 𝑓𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Equation (13) can also be used to calculate 𝑠 and 𝑘 when there are two or (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and absorptivity spectra for 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05, and (a) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+𝑖10−4, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+𝑖10−4, 𝐿 = 50 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (c) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+𝑖10−2, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (d) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 𝑖10−2, 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, FF stands for four flux, KM for Kubelka Munk, and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' more type of particles present in the matrix (with different refractive index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For demonstration we consider a Gaussian distribution of spherical particles about mean radius 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇𝑚 with standard deviation 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='016 𝜇𝑚 with and without absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The particle size distribution curve has been shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Figure 4a and 4b show the comparison between KM, FF, and FDTD results for the case when particles are non-absorbing and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 4c and 4d show the corresponding comparison when particles are absorbing with 𝑛′′ 𝑝 = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Other parameter values are retained as for the case of monodisperse particulate coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The observations follow the trend seen for the case of monodisperse coating with significant deviations observed in the predictions of FF method for lower values of thicknesses of coatings and when particles are nonabsorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For larger thicknesses of the coating and in the presence of absorption, both FF and KM are observed to predict the optical properties with reasonable accuracy across the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Comparison of predictions from KM, FF and FDTD solvers in the dependent scattering regime So far we have analyzed for the situations where the fill fraction 𝑓 of particles in the composite is small enough so that the particles can be assumed to independently scatter from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 L=10 μm Reflectivity/ Absorptivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+il0-4 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Absorptivity L = 50 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+il0-4 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 L= 10 μm Reflectivity/ Absorptivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+il0-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 R KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/ Absorptivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 L= 50 μm nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 R KM KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)(a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and transmissivity for 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='016, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05 and (a) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and absorptivity for 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='016, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05 and (c) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01𝑖, 𝐿 = 10 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (d) 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01𝑖, 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, FF stands for four flux, KM for Kubelka Munk, and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, as the fill fraction of particles increases, there will be a transition to dependent- scattering regime where both the near-field interaction between the particles as well as far-field interference between the scattered field of individual particles have a significant impact on the overall properties of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Hotel [24] empirically determined this transition to occur when 𝑓 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='27 and 𝑑/𝜆 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 where 𝑑 is the mean inter-particle spacing and 𝜆 is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Several coatings reported in literature [7,46–49] have fill fractions in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 where such effects cannot be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We thus explore here the predictive capability of FF and KM theories for such coatings by considering a monodisperse distribution of particles with increased fill fraction 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 while retaining other parameter values to be same as that included for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 5, where we observe that the predictions from both FF and KM theories deviate significantly from FDTD simulations across the spectra, and thus cannot be relied on for predicting optical properties of such coatings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' A comparison between the weighted average of the optical properties across the spectra as predicted by KM and FF theories for the different cases considered so far has been tabulated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The weighted averages are calculated as: 𝑅solar = ∫ 𝐼AM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5(𝜆)𝑅(𝜆)𝑑𝜆/ ∫ 𝐼AM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5(𝜆) 𝑑𝜆, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity L= 10 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 R KM KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity L= 50 μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 R KM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 LM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Absorptivity L= 10 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 np = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 L = 50 μum Reflectivity/ Absorptivity np = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and transmissivity for 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, and 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, FF stands for four flux, KM for Kubelka Munk, and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' where 𝐼AM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5(𝜆) is the spectral solar irradiance [50] and 𝜖IR = ∫ 𝐼BB(𝜆)𝜖(𝜆)𝑑𝜆/ ∫ 𝐼BB(𝜆) 𝑑𝜆, where 𝐼BB(𝜆) is the black body irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For the relevant applications in consideration for this study i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' coatings suitable for radiative cooling application and for use in solar thermal absorber plates, the reflection over the solar spectrum i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' over wavelength range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 − 3 𝜇𝑚 and emissivity over the infra-red spectrum i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' over wavelength range 5 − 15 𝜇𝑚 is of primary importance, and the weighted average over this spectral range is reported in Table 1 along with the deviation from FDTD simulations expressed in % error in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Semi-analytical method The comparison with FDTD simulations shown in Section 2 demonstrate the failure of KM and FF analytical methods in configurations where dependent scattering is not negligible and when matrix/particles are absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This failure can be attributed to the actual scattering and absorption coefficients of these coatings diverging from the values calculated using Mie scattering coefficients of the individual particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' At present no single analytical technique exists that can correctly predict the optical properties of particulate media in the presence of dependent scattering effects as well as correctly account for the absorption in matrix/particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' One can then resort to using exact numerical solvers to accurately estimate the optical properties of the coating in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' However, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 6 shows, the computational time required to simulate such structures increases exponentially with thickness of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For coatings of thickness in the range 100-500 microns which are currently being adopted in literature for the radiative cooling application [6,8,36,46,49] the design time is clearly prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In such cases it becomes imperative to develop alternate techniques which can combine the accuracy power of exact FDTD solvers with the simplicity and minimal computational requirements of the analytical techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Particularly when multiple parameters are involved in design - such as that observed for disordered media - such a method will prove to be useful in reducing the design time to find the optimum combination of parameters necessary to obtain the required optical properties of the coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In order to obtain a better estimate for the absorption and scattering coefficients of such media where dependent scattering effects are non-negligible, researchers have previously [27,51–53] relied on experimental measurements of the optical properties of a fabricated coating and then using the KM theory results from Section 2 to extract the required coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Instead of relying on experimental measurements which is not always feasible especially at the initial state of design, we modify this technique and instead propose the following 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 TKM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Weighted-average reflectivity in solar spectrum 𝑅solar,KM (𝑅solar,FF) and emissivity in IR spectrum 𝜖IR,KM (𝜖IR,FF) calculated using KM (FF) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Values in brackets denote deviation of prediction from FDTD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑅solar,KM 𝜖IR,KM 𝑅solar,FF 𝜖IR,FF 1 2a 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='023 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01%) 13 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='944 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='13%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='935 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='98%) two-step semi-analytical method to estimate the optical properties of random media of thickness 𝐿 when usage of exact numerical solvers to simulate the properties of such a thick coating is prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Step 1: Use a numerical solver to obtain the optical properties 𝑅 and 𝑇 of a similar coating but with much smaller thickness 𝑡𝑠 ≪ 𝐿 and extract the 𝛾 and 𝐴 parameters by inverting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Care must be taken at this step to ensure that the configuration set up in the solver considers incident light to be in the same medium as the index of matrix i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=', 𝑛surr = 𝑛h in order to ensure that reflection from surfaces and substrates are not included in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In case the host matrix is absorbing then only the real part is considered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=', 𝑛surr = Re(𝑛h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Care must also be taken to ensure that when scattering efficiency of the particles is high, the value of 𝑡𝑠 should be chosen such that 𝑡𝑠 ≫ 𝑙𝑠 where 𝑙𝑠 ≈ 1/(𝑁𝜎𝑠) is the scattering mean-free path with 𝑁 being the particle number density and 𝜎𝑠 the scattering cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' At the other limit when scattering efficiency is low the optical properties of the coating are primarily determined from surface reflection and transmission which are accounted for in step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Thus the choice of 𝑡𝑠 is determined from the scattering mean-free path calculated in the high-scattering regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Step 2: From the 𝛾 and 𝐴 parameters extracted from step 1 use the analytical expressions from KM theory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1, 2, 7 and 11 to predict the optical properties of the coating of the required thickness 𝐿 ≫ 𝑡𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Specular reflection at the surfaces as well as at the substrate are accounted for here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Comparison of computational time as a function of thickness of the disordered media coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Simulations are carried out in ANSYS Lumerical using an eight-core Intel Xeon workstation for the configuration: 𝑛𝑝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑛ℎ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05, with mesh size 30 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Auto shutoff level (simulation termination criteria) is set at 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' A more elaborate procedure, along with details of a supporting convergence test which may need to be incorporated in some cases to arrive at the value of thickness 𝑡𝑠 is included in Section S4 of supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (a) Reflectivity and transmissivity for 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, and 𝐿 = 50 𝜇m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) Reflectivity and absorptivity for 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1𝑖, 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, and 𝐿 = 50 𝜇m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, SM stands for semi-analytical method and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We now apply this technique for the cases considered in Section 2 where the predictions from analytical methods deviated significantly from those of FDTD solver, such as for the dependent scattering regime, as well as when the absorption in the particles/host matrix is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 7 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 8) shows the comparison between the predictions from the semi-analytical technique and from FDTD simulations when absorption in particles (host matrix) is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In both these cases the semi-analytical technique uses the results of exact FDTD simulations of a 10 𝜇m thick coating to predict the optical properties of a larger 50 𝜇m thick coating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' A volume fill fraction 100 Computational time (Hrs) 90 Computational time (Hrs) 80 Exponential fit 70 60 50 40 30 20 10 0 0 10 20 3040 50 60 70 80 90 100110 Thickness (μm)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/Transmissivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 M R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/ Absorptivity np = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 is maintained in both these cases where dependent scattering effects are known to be dominant, while keeping other parameter values same as that analysed for the monodisperse case of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' For these cases we observe a close match in the predictions of the semi-analytical (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Reflectivity and absorptivity for 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, 𝐿 = 50 𝜇m, and (a) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 𝑖10−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 + 𝑖10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Here, SM stands for semi-analytical method and LM for Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' method with the FDTD results over the entire spectrum, with only a slight deviation observed for the higher wavelengths when absorption is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The weighted-average reflectivity of the coating for solar spectrum, and emissivity over the infra-red spectrum for the cases considered in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 7 and 8 are listed in Table 2 along with the deviation from FDTD simulations expressed in % error in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Particularly illustrative of the effectiveness of the semi-analytical technique is the reduction in error (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='03 % in Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1) as compared to those obtained from analytical techniques and reported in Table 1 ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='13 % using KM theory and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='98 % using FF theory in Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 13) for the configuration: 𝑛p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑛h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5, 𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25 𝜇m, 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3, and 𝐿 = 50 𝜇m where dependent scattering is expected to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Weighted-average reflectivity in solar spectrum 𝑅solar,SM, and emissivity in IR spectrum 𝜖IR,SM calculated using the semi-analytical technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Values in brackets denote deviation of prediction from FDTD results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 𝑅solar,SM 𝜖IR,SM 1 7a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='864 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='03%) 2 7b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='059 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='01%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='710 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='73%) 3 8a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='178 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='73%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='391 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='25%) 4 8b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='062 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='935 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='17%) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Comparison with experimental data We now apply the semi-analytical technique described in Section 3 to predict the optical properties of fabricated coatings reported in literature which have been designed for radiative cooling application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' We choose two such disordered coatings where dependent scattering is expected to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/ Absorptivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+il0-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Reflectivity/ Absorptivity nh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5+i10-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 15 Wavelength (um)be dominant so that analytical techniques are not applicable, and the thickness of the coating prohibits the use of exact electromagnetic solvers to predict the optical properties to good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [48], a hierarchically porous polymer (P(VdF-HFP)) coating of thickness 300 𝜇m containing air voids with sizes ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='05-5 𝜇m in P(VdF-HFP) matrix has been fabricated, and experimentally characterized to have solar reflectivity value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='96 and emissivity in the 8-13 𝜇m wavelength range to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In order to apply semi-analytical technique to predict the properties of this coating, we set up a simulation in FDTD solver with a smaller coating thickness 𝑡𝑠 = 50 𝜇m (determined using the convergence test explained in Section S4 of supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' This thickness is chosen to ensure sufficient number of larger sized air voids (𝑟 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='5 𝜇m) in this P(VdF-HFP) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The size distribution of nano-micro air voids used in the simulation is given in supplementary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Refractive index data of P(VDF-HFP) is extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The reflectivity data in the wavelength range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 − 16 𝜇 m, predicted using the semi-analytical method for 𝐿 = 300 𝜇m thickness, is compared with that reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [48] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' While an appreciable match is noticed in the predicted values across the spectrum, small deviation observed in the reflectivity values can be attributed to our inability to incorporate exact size distribution of both micro and nano voids as present in the fabricated structure, in ANSYS Lumerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [46] an ultrawhite BaSO4 film of thickness 400 𝜇m has been developed with 60 % volume fraction of BaSO4 nanoparticles, and has been characterized to have reflectivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='976 in the solar spectrum and emissivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='96 in 8-13 𝜇m wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' In order to apply the semi-analytical technique to predict the properties of this coating, we set up a simulation in FDTD solver with structure thickness 𝑡𝑠 = 15 𝜇m and BaSO4 spherical particles randomly distributed with volume fraction 60 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The particles are taken to be of uniform size distribution with diameters spread over the range 398 ± 130 nm to match that reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Matrix is considered to be air for BaSO4 film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Refractive index data of BaSO4 is extracted from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' The emissivity data in the wavelength range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='3 − 16 𝜇m, predicted using the semi-analytical method for 𝐿 = 400 𝜇m thickness, is compared with that reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [46] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' While we again observe an appreciable match across the spectrum, some deviation observed particularly around wavelength of 2 𝜇m is suspected to be due to difference in the refractive index of the fabricated film and that calculated from first-principles in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (a) Reflectivity of hierarchically porous P(VDF-HFP) coating calculated using semi-analytical technique is compared with experimental result given by Mandal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' (b) Absorptivity/emissivity of BaSO4 film calculated using semi-analytical technique is compared with experimental result given by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 SM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [48] Reflectivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 10 57 16 1 Wavelength (μum)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 Absorptivity/Emissivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='8 SM Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' [46] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='0 1 38 10 16 Wavelength (um)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Conclusion In this study we have analyzed the applicability of well-known analytical techniques of KM and FF theories to predict optical properties of a disordered metamaterial coating over a broad spectrum ranging from 300 nm to 15 𝜇m wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Recent advancements in the use of disordered coatings in applications such as radiative cooling and solar thermal absorber plates which require tailored optical properties over this wavelength range necessitates such a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Based on deviations observed between the predictions of these analytical techniques and exact FDTD solver in the dependent scattering regime, a two-step semi-analytical technique has been proposed which can be used to predict optical properties of such coatings with good accuracy and minimal computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Such a method is expected to be resourceful for designing coatings with specific optical properties where several parameter combinations need to be investigated to arrive at an optimal combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Small deviations observed when absorption in host matrix is high warrants further research to improve this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Acknowledgments B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' acknowledges support from Prime Minister’s Research Fellowship (PMRF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' ac- knowledges support from La Fondation Dassault Systèmes and SERB Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' SRG/2020/001 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Disclosures The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfhvhg/content/2301.00382v1.pdf'} +page_content=' Supplementary information See 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100644 index 0000000000000000000000000000000000000000..514c689e2450160a2265b8c88463803e5ab1f10e --- /dev/null +++ b/XtAyT4oBgHgl3EQfvfkj/content/tmp_files/2301.00630v1.pdf.txt @@ -0,0 +1,719 @@ +Prediction of the most strange tri-baryon with lattice QCD constrained potentials +Tian-Wei Wu,1, 2 Si-Qiang Luo,3, 4 Ming-Zhu Liu,5, 6 Li-Sheng Geng,6, 7, 8, 9, 4, ∗ and Xiang Liu3, 10, 4, 11, † +1School of Fundamental Physics and Mathematical Sciences, +Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China +2University of Chinese Academy of Sciences, Beijing, 100049, China +3School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China +4Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China +5School of Space and Environment, Beihang University, Beijing 102206, China +6School of Physics, Beihang University, Beijing 102206, China +7Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China +8Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing, 102206, China +9School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China +10Research Center for Hadron and CSR Physics, Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000, China +11Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China +( Dated: January 3, 2023) +Motivated by the existence of two-body hadronic molecules composed of ΩΩ, ΩcccΩccc and ΩbbbΩbbb pre- +dicted by lattice QCD simulations, we employ the Gaussian expansion method to study whether three-body +systems composed of ΩΩΩ, ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb can bind with the two-body 1S0 interactions pro- +vided by lattice QCD. Our results show that none of the three-body systems binds. On the other hand, we find +that supplemented with a synthetic 5S2 potential, the ΩΩΩ system develops a bound state, for which both the +1S0 and 5S2 interactions play an important role. Our predictions are further corroborated by explicit studies +employing the one-boson exchange potentials constrained by the lattice QCD simulations. Our studies support +the existence of the 3 +2 ++ ΩΩΩ bound state and the non-existence of the 3 +2 ++ ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb +bound states, due to the suppressed 5S2 interactions in heavier systems. +Introduction.—The quark model, as a classification scheme +for light-flavor hadrons, was proposed by Gell-Mann [1] and +Zweig [2] in 1964, which was established when the predicted +Ω baryon with the highest strangeness number was observed +experimentally [3]. +It is often viewed as the first stage in +hadron physics. Since 2003, we have witnessed a new stage +in hadron physics because of the observations of many new +hadronic states such as the charmoniumlike XY Z states and +the pentaquark states [4–12], which have stimulated extensive +studies, both theoretically and experimentally. Although re- +markable progress has been made, a unified understanding of +exotic hadronic states is still missing. At present, it is widely +acknowledged that one should pay more attention to new con- +figurations, exotic quantum numbers, and special systems, to +better understand the nature of exotic hadrnoic matter and the +non-perturbative strong interaction. +In recent years, fully strange and fully heavy di-baryon sys- +tems have attracted considerable attention. With increasing +computational power, lattice QCD has become the primary +force to derive hadron-hadron interactions in a quantitative +way from first principles. In Ref. [13], the authors investigated +the ΩΩ interaction in the 1S0 channel, and concluded that +there exists a weakly bound state regardless of the Coulomb +interaction, which is even shallower than the deuteron. In +Ref. [14], the existence of a 1S0 ΩcccΩccc shallow bound state +is predicted while it disappears once the Coulomb interaction +is taken into account. Very recently, the existence of a deeply +bound 1S0 ΩbbbΩbbb state was also predicted [15]. For the +ΩΩ, ΩcccΩccc, and ΩbbbΩbbb systems, some of us developed +an extended one-boson-exchange (OBE) model to derive their +interactions in Ref. [16], and obtained results consistent with +those of lattice QCD [13–15]. In Ref. [17], the authors found +the existence of fully heavy dibaryon bound states, ΩcccΩccc +and ΩbbbΩbbb, in the constituent quark model, while the corre- +sponding fully heavy hexaquark states are found to be above +the ΩcccΩccc and ΩbbbΩbbb mass thresholds in both the con- +stituent quark models [18–20] and the QCD sum rules [21]. +On the experimental side, studies of fully heavy multi- +quarks have made important breakthroughs. +In 2020, the +LHCb Collaboration reported the observation of the first fully +heavy tetraquark state, X(6900) [22]. It was confirmed by +the CMS Collaboration with a statistical significance of 9.4σ, +and in addition, two new states X(6600) and X(7200) were +observed [23]. The ATLAS Collaboration further confirmed +the discovery of the LHCb Collaboration [24]. Clearly, the +existence of fully heavy multiquark states can be considered +as firmly established. +It is natural to expect the existence of tri-baryon systems +given the (predicted) existence of di-baryon systems. We note +that experimental and theoretical studies of tribaryon systems +other than atomic nuclei and hypernuclei have continued for +many years without conclusive results [25–31]. In this letter, +motivated by the remarkable progress achieved on studies of +the fully heavy multiquark states from both lattice QCD [13– +15] and experiments [22–24], we study the ΩΩΩ system, as +well as the ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb systems. Notice +that this study is different from all the previous works, where +two or three different species of baryons are involved. The +systems we study contain only one species of baryons and +therefore have the highest symmetries. This means that we +only need the interactions of a pair of identical baryons and +the number of allowed configurations is much reduced as well, +arXiv:2301.00630v1 [hep-ph] 2 Jan 2023 + +2 +thus allowing for more robust predictions. +Most strange tribaryon.—We adopt the Gaussian expansion +method (GEM) [32–34] to study the ΩΩΩ system. To solve +the Schr¨odinger equation with GEM, one needs to derive +the two-body interactions and construct the three-body wave +functions. +The ΩΩ interaction has been derived in lattice QCD [13], +where it was shown that the S-wave ΩΩ system can bind +with a binding energy of 1.6(6)+0.7 +−0.6 MeV (without taking +into account the Coulomb interaction). +In addition to lat- +tice QCD, other methods such as the extended one-boson +exchange (OBE) model can also provide the ΩΩ interac- +tion [16]. In this work, we utilize both interactions to study +the ΩΩΩ three-body system. +The ΩΩ lattice QCD potentials for the +1S0 channel +is expressed with three Gaussian functions V +1S0 +L +(r) += +�3 +i=1 aie−bir2 [13]. +Since the Ω baryon is charged, the +Coulomb interaction plays an important role in the ΩΩ and +ΩΩΩ systems. The Coulomb potential between a pair of ΩΩ +is VC(r) = −α/r, where α = 1/137 is the electromagnetic +fine structure constant. +The lattice QCD simulations only provided the 1S0 poten- +tial between the ΩΩ pair. As we see later, the 5S2 potential +plays an important role in the three-body system as well. In +the following, we derive a synthetic “LQCD” 5S2 potential +for the ΩΩ system. That is to say, we obtain the 5S2 poten- +tial from the 1S0 potential. Assuming that there exists a linear +transformation between the 1S0 and 5S2 ΩΩ interactions +V ΩΩ +5S2 (r) = βV ΩΩ +1S0 (ωr + γ) + δ, +(1) +which is a combination of scale and translation transforma- +tions. Because the potential should approach to zero as r ap- +proaches infinity, the vertical translation parameter δ is fixed +at 0. We choose two characteristic points, the minimum and +zero points, to fix the other three parameters, i.e., +V +5S2 +ΩΩ (r′ +0) = βV +1S0 +ΩΩ (r0), +V +5S2 +ΩΩ (r′ +min) = βV +1S0 +ΩΩ (rmin), +r0 = ωr′ +0 + γ, +rmin = ωr′ +min + γ, +(2) +where r0, rmin (r′ +0, r′ +min) are the radial positions where V +1S0 +ΩΩ +(V +5S1 +ΩΩ ) are either vanishing or the most attractive, respectively. +Using the OBE potentials of Ref. [16], the values of β, ω and +γ are determined to be 0.7619, 1.001 and 0.1395 fm, respec- +tively. With these parameters, one can obtain a synthetic lat- +tice QCD 5S2 potential from the genuine lattice QCD 1S0 po- +tential. These potentials and the corresponding OBE poten- +tials of Ref. [16] are shown in Fig. 2. A few remarks are in +order. First, with Eq. (1) and the OBE 1S0 potential, the syn- +thetic 5S2 potential is nearly identical to that of the OBE 5S2 +potential, which shows that the linear transformation can in- +deed faithfully relate the 1S0 and 5S2 potentials. Second, the +synthetic 5S2 potential is more short ranged, more repulsive +but less attractive compared to the lattice QCD 1S0 potential. +Although less attractive, as we show later, without it, the ΩΩΩ +system does not bind. +0.0 +0.5 +1.0 +1.5 +2.0 +-200 +-100 +0 +100 +200 +r [fm] +V [MeV] +OBE 1S0 +OBE 5S2 +LQCD 1S0 +sLQCD 5S2 +FIG. 1. OBE and lattice QCD potentials for the ΩΩ system. The +blue, orange and green solid lines denote the 1S0 OBE, the 5S2 OBE, +and the 1S0 lattice QCD potentials, respectively. The red dashed +lines denote the synthetic 5S2 lattice QCD potential. +The ΩΩΩ three-body wave function can be written as a sum +of three Jacobi channels +ΨJ(⃗r1, ⃗R1) = +� +Ac +αΦc +J(⃗r1, ⃗R1), +(3) +where Ac +α is the expansion coefficients, c = 1 − 3 denote +the three Jacobi channels, α is the set of quantum numbers +characterizing the wave function in each Jacobi channel. The +wave function of each Jacobi channel reads as +Φ1 +J(⃗r1, ⃗R1) = +� +[[χ3χ2]s1χ1]S ⊗ [ψl1(⃗r1)φL1(⃗R1)]Λ +� +J , +Φ2 +J(⃗r2, ⃗R2) = +� +[[χ1χ3]s2χ2]S ⊗ [ψl2(⃗r2)φL2(⃗R2)]Λ +� +J , +Φ3 +J(⃗r3, ⃗R3) = +� +[[χ2χ1]s3χ3]S ⊗ [ψl3(⃗r3)φL3(⃗R3)]Λ +� +J , +where χi is the spin wave function of the ith particle, Hc +s,S = +[[χiχj]sχk]S is the spin wave function of Jacobi channel c, +ψ(ri)φ(Ri) is the spatial wave function, s is the spin of the +sub ΩΩ two-body system, S = 3/2 is the total spin of ΩΩΩ, li +(Li) is the orbit angular momentum corresponding to ri(Ri), +Λ is the total orbit angular momentum built from l and L, and +J is the total angular momentum built from Λ and S. +Fermi-Dirac statistics dictates that only the 1S0 and 5S2 +interactions contribute to the formation of an ΩΩΩ 3 +2 ++ state. +The spin coupling coefficients of different spin configurations +between Jacobi channels i and j for i ̸= j are shown in Table +I. Note that for i = j, the matrix is orthogonal. + +3 +TABLE I. Coupling coefficients of different spin configurations be- +tween Jacobi channels i and j (i ̸= j). Here, Hc +s,S is the spin func- +tion, s = {0, 2} are alternative spin values of ΩΩ, and S = 3/2 is +the total spin of ΩΩΩ. +Hc=i +0, 3 +2 +Hc=i +2, 3 +2 +Hc=j +0, 3 +2 +− 1 +4 +− +√ +5 +4 +Hc=j +2, 3 +2 +− +√ +5 +4 +3 +4 +It is important to point out that for the ΩΩΩ system, the 5S2 +potential can play a very important role, even more important +than the 1S0 potential. This is because the 5S2 partial wave +is more strongly coupled to the three-body spin-3/2 state than +the 1S0 partial wave. As shown in Table I, the spin coupling +coefficient of different Jacobi channels i and j in the 5S2 par- +tial wave is ⟨Hc=i +2,3/2|Hc=j +2,3/2⟩i̸=j = 3/4 while that in 1S0 is +⟨Hc=i +0,3/2|Hc=j +0,3/2⟩i̸=j = −1/4, which means that in the spin +space, the couplings between channels i and j in the 5S2 par- +tial wave is 9 times larger than that in the 1S0 partial wave. +Once the wave functions are obtained , with either the +lattice QCD or OBE ΩΩ interactions, one can adopt the +GEM [33] to obtain the binding energies and root-mean- +square (RMS) radius of the ΩΩΩ system. +The results for the two-body ΩΩ system are summarized +in Table II, which show that the binding energies and root- +mean-square (RMS) radii obtained with the OBE potentials +are consistent with those of lattice QCD. With both lattice +QCD and OBE potentials, the ΩΩ system can bind with a +binding energy of 1.4+0.9 +−0.4 MeV. The uncertainties are deter- +mined by multiplying a scaling factor to the lattice QCD po- +tential so that the binding energy varies from 1.0 to 2.3 MeV, +consistent with the lattice QCD result 1.6+0.7 +−0.6 MeV [13]. +From the analysis given above, we know that both 1S0 and +5S2 interactions contribute to the 3/2 ΩΩΩ system. Given +that the lattice QCD only provided the 1S0 interaction, we +first consider only the 1S0 two-body interaction and find that +the ΩΩΩ system does not bind. But this result should not be +taken too seriously since the 5S2 partial wave plays an impor- +tant role in the spin configuration (⟨Hc=i +2,3/2|Hc=j +2,3/2⟩i̸=j = 3 +4) +and has a significant correlation with the 1S0 partial wave +(⟨Hc=i +0,3/2|Hc=j +2,3/2⟩i̸=j = +√ +5 +4 ) in the 3-body case. Adding the +synthetic lattice QCD 5S2 potential, we find that the three- +body ΩΩΩ system binds with a binding energy of 3.6+2.6 +−1.2 +MeV and RMS radius of 2.5+0.4 +−0.4 fm. These results are fur- +ther corroborated by the study with the OBE potentials, which +yields a binding energy of 5.8+2.5 +−1.2 MeV and RMS radius +1.9+0.1 +−0.2 fm. +Note that the binding energy per baryon of the ΩΩΩ system +is larger than that of the ΩΩ system, while consequently its +RMS radius is smaller than that of the ΩΩ bound state. This is +understandable because for the ΩΩΩ system the 5S2 potential +plays an important role while only the 1S0 potential is relevant +for the ΩΩ system. +The weights of partial waves and Hamiltonian expectation +values of the predicted ΩΩΩ bound state are given in Table III, +which clearly show that the 5S2 interaction plays a signifi- +cantly important role in the ΩΩΩ system. More specifically, +the weights of the 1S0 and 5S2 partial waves are about 30% +and 70%, respectively. +As we mentioned above, since the Ω baryon is charged, the +impact of the Coulomb interaction is worth discussing. We +find that the Coulomb interaction in this three-body system +affects the binding energy by about 2-3 MeV, but does not +change the conclusion. Considering the Coulomb interaction, +the binding energy and RMS radius of the ΩΩΩ bound state +predicted by the lattice QCD potentials are about 0.7 MeV and +4.5 fm, while for those predicted by the OBE model, they are +2.0 MeV and 2.3 fm, respectively. +It is important to discuss where to search for the predicted +ΩΩ and ΩΩΩ bound states. In Ref. [35], the production yield +of the ΩΩ bound state was estimated using a dynamical co- +alescence mechanism for the relativistic heavy-ion collisions +at √sNN = 200 GeV and 2.76 TeV, which turn out to be +of the order of 10−6. +In Ref. [31], the production yields +of NNΩ and NΩΩ were estimated to be 10−7 and 10−9, +respectively. Comparing these results, one can estimate the +ΩΩΩ production rate for the relativistic heavy-ion collisions +at √sNN = 200 GeV and 2.76 TeV, which is of the order of +10−11. +TABLE II. Binding energies (B.E) and root-mean-square radii (⟨r⟩) +of the ΩΩ and ΩΩΩ bound states obtained with lattice QCD and +OBE potentials (B.E. in MeV and radius ⟨r⟩ in fm.). +ΩΩ(B.E.) +ΩΩ(⟨r⟩) +ΩΩΩ(B.E.) +ΩΩΩ(⟨r⟩) +LQCD +1.41+0.89 +−0.41 +3.45+0.52 +−0.62 +3.55+2.57 +−1.17 +2.48+0.37 +−0.40 +OBE +1.41+0.89 +−0.41 +3.33+0.51 +−0.62 +5.84+2.48 +−1.22 +1.86+0.13 +−0.19 +TABLE III. Weights of the partial waves and Hamiltonian expecta- +tion values (Units in MeV) of the 3 +2 ++ ΩΩΩ bound state. +⟨Ψ +1S0 +3/2 |Ψ +1S0 +3/2 ⟩ +⟨Ψ +5S2 +3/2 |Ψ +5S2 +3/2 ⟩ +⟨T⟩ +⟨V +1S0⟩ +⟨V +5S2⟩ +LQCD +29% +71% +27.77 +−8.80 +−22.53 +OBE +22% +78% +52.72 +−15.71 +−42.84 +Most charming and beautiful tri-baryons.– It is straightfor- +ward to extend the above study to the ΩcccΩccc and ΩbbbΩbbb +systems, for which the lattice QCD simulations already pro- +vided the 1S0 potentials [14, 15] and their OBE counterparts +also exist [16]. Note that in Ref. [15] no analytic form of the +ΩbbbΩbbb potential was provided. We fitted the lattice QCD +potential with a sum of three Gaussian functions as done in +Ref. [14]. We also adopt the transformation introduced above +to derive synthetic lattice QCD potentials for the 5S2 partial +wave. All the lattice QCD potentials and the corresponding + +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-400 +-200 +0 +200 +400 +r [fm] +V [MeV] +OBE 1S0 +OBE 5S2 +LQCD 1S0 +sLQCD 5S2 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1500 +-1000 +-500 +0 +500 +1000 +1500 +r [fm] +V [MeV] +OBE 1S0 +OBE 5S2 +LQCD 1S0 +sLQCD 5S2 +FIG. 2. OBE and lattice QCD potentials of the ΩcccΩccc (top) and +ΩbbbΩbbb (bottom) systems. The blue, orange and green solid lines +denote the 1S0 OBE, the 5S2 OBE, and the 1S0 lattice QCD po- +tentials, respectively. The red dashed lines denote the synthetic 5S2 +lattice QCD potentials. +OBE potentials are shown in Fig. 2. We note that although the +interaction strengths of the lattice QCD potential and those of +the OBE potentials are different, the positions where they be- +come the most attractive are almost the same. The same can +be said about the ΩΩ potentials shown in Fig. 1. Such a co- +incidence indicates that the OBE model must have captured +some essential features of the baryon-baryon potentials. +With the above lattice QCD (including the synthetic 5S2) +and the OBE potentials, we can study the two-body and three- +body systems composed of Ωccc and Ωbbb. As shown in Ta- +ble IV, with only the strong interaction the ΩcccΩccc bound +system can be formed, but it dissolves once the Coulomb +interaction is taken into account. +On the other hand, the +ΩbbbΩbbb system is always bound regardless of the Coulomb +interaction. Furthermore, we note that the results obtained +with the lattice QCD potentials and those with the OBE po- +tentials are similar. Nonetheless, none of the ΩcccΩcccΩccc +and ΩbbbΩbbbΩbbb three-body systems can bind, mainly be- +cause of the much weaker 5S2 interactions, which are non- +trivial predictions of the present work. +TABLE IV. Binding energies (B.E) and root-mean-square radii (⟨r⟩) +of the ΩcccΩccc, and ΩbbbΩbbb bound states obtained with OBE and +LQCD potentials (B.E. in MeV and radius ⟨r⟩ in fm.). NC means that +the Coulomb interaction is not taken into account while C means that +the Coulomb interaction is considered. +ΩcccΩccc +(NC) +ΩcccΩccc +(C) +ΩbbbΩbbb +(NC) +ΩbbbΩbbb +(C) +LQCD +B.E. +5.54 +... +88.7 +79.9 +⟨r⟩ +1.14 +... +0.240 +0.245 +OBE +B.E. +5.52 +... +88.6 +78.4 +⟨r⟩ +1.05 +... +0.198 +0.202 +Summary.— Motivated by the existence of ΩΩ, ΩcccΩccc, and +ΩbbbΩbbb bound states predicted by lattice QCD simulations, +we studied the 3 +2 ++ ΩΩΩ, ΩcccΩcccΩccc, and ΩbbbΩbbbΩbbb +three-body systems with the lattice QCD and OBE potentials. +We found that the ΩΩ, ΩcccΩccc, and ΩbbbΩbbb systems can +also bind with the OBE potentials, with binding energies and +RMS radii consistent with those of lattice QCD simulations. +The repulsive Coulomb interactions plays an important role +in these systems especially in the ΩcccΩccc system, which is +strong enough to break the ΩcccΩccc pair bound by the strong +force. +For the three-body systems, we find that the 5S2 partial +wave plays a very important role in forming the 3 +2 ++ three-body +state. With only the 1S0 lattice QCD potentials, the ΩΩΩ, +ΩcccΩcccΩccc, ΩbbbΩbbbΩbbb three-body systems do not bind. +Supplemented with a synthetic lattice QCD 5S2 potential, the +ΩΩΩ system becomes bound while the ΩcccΩcccΩccc and +ΩbbbΩbbbΩbbb systems remain unbound, mainly due to the +much suppressed attractive 5S2 interaction in the two-body +ΩcccΩccc and ΩbbbΩbbb systems. The results were further cor- +roborated by explicit studies employing the OBE potentials. +To verify the existence of the ΩΩΩ bound state, lattice QCD +studies of the 5S2 interactions of the ΩΩ system will be the +key. We hope that the predicted ΩΩΩ bound state can be +searched for in present and future hadron-hadron colliders. +A particularly interesting discovery of the present work is +that even the two-body interactions are attractive and strong +enough to form two-body bound states, the three-body sys- +tems do not necessarily bind. This is because in three-body +systems, spin-spin interactions can play an important role. +The three highly symmetric systems studied in the present +work provide an ideal platform to understand the relevance +of spin-spin interactions in forming few-body bound states. +Acknowledgement.—This work is partly supported by the Na- +tional Natural Science Foundation of China under Grants +No.11735003, No.11975041, No.11961141004, and the fun- +damental Research Funds for the Central Universities. X.L. +is supported by the China National Funds for Distinguished +Young Scientists under Grant No. 11825503, National Key + +5 +Research and Development Program of China under Con- +tract No. 2020YFA0406400, the 111 Project under Grant No. +B20063, the National Natural Science Foundation of China +under Grant No. 12247101, and the project for top-notch in- +novative talents of Gansu province. Ming-Zhu Liu acknowl- +edges support from the National Natural Science Foundation +of China under Grant No.12105007 and China Postdoctoral +Science Foundation under Grants No. 2022M710317, and No. +2022T150036. Tian-Wei Wu acknowledges support from the +National Natural Science Foundation of China under Grant +No.12147152 and China Postdoctoral Science Foundation un- +der Grant No. 2022M723119. +∗ lisheng.geng@buaa.edu.cn +† xiangliu@lzu.edu.cn +[1] M. Gell-Mann, Phys. Lett. 8, 214 (1964). +[2] G. +Zweig, +“An +SU(3) +model +for +strong +interac- +tion +symmetry +and +its +breaking. +Version +2,” +in +DEVELOPMENTS IN THE QUARK THEORY OF HADRONS. VOL. 1. 1964 - 1978, +edited by D. B. Lichtenberg and S. P. Rosen (1964) pp. 22–101. +[3] V. E. Barnes et al., Phys. Rev. Lett. 12, 204 (1964). +[4] N. Brambilla et al., Eur. Phys. J. C 71, 1534 (2011), +arXiv:1010.5827 [hep-ph]. +[5] X. Liu, Chin. Sci. Bull. 59, 3815 (2014), arXiv:1312.7408 [hep- +ph]. +[6] H.-X. Chen, W. Chen, X. 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9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ∗ and Xiang Liu3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' † 1School of Fundamental Physics and Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Hangzhou Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' UCAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 310024,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 2University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 3School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 4Lanzhou Center for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Gansu 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 5School of Space and Environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beijing 102206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 6School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beijing 102206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 7Peng Huanwu Collaborative Center for Research and Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beijing 100191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 8Beijing Key Laboratory of Advanced Nuclear Materials and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beihang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 102206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 9School of Physics and Microelectronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Zhengzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Zhengzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Henan 450001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 10Research Center for Hadron and CSR Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou University and Institute of Modern Physics of CAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China 11Frontiers Science Center for Rare Isotopes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' China ( Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2023) Motivated by the existence of two-body hadronic molecules composed of ΩΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ΩcccΩccc and ΩbbbΩbbb pre- dicted by lattice QCD simulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' we employ the Gaussian expansion method to study whether three-body systems composed of ΩΩΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb can bind with the two-body 1S0 interactions pro- vided by lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Our results show that none of the three-body systems binds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' On the other hand, we find that supplemented with a synthetic 5S2 potential, the ΩΩΩ system develops a bound state, for which both the 1S0 and 5S2 interactions play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Our predictions are further corroborated by explicit studies employing the one-boson exchange potentials constrained by the lattice QCD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Our studies support the existence of the 3 2 + ΩΩΩ bound state and the non-existence of the 3 2 + ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb bound states, due to the suppressed 5S2 interactions in heavier systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='—The quark model, as a classification scheme for light-flavor hadrons, was proposed by Gell-Mann [1] and Zweig [2] in 1964, which was established when the predicted Ω baryon with the highest strangeness number was observed experimentally [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' It is often viewed as the first stage in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Since 2003, we have witnessed a new stage in hadron physics because of the observations of many new hadronic states such as the charmoniumlike XY Z states and the pentaquark states [4–12], which have stimulated extensive studies, both theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Although re- markable progress has been made, a unified understanding of exotic hadronic states is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' At present, it is widely acknowledged that one should pay more attention to new con- figurations, exotic quantum numbers, and special systems, to better understand the nature of exotic hadrnoic matter and the non-perturbative strong interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In recent years, fully strange and fully heavy di-baryon sys- tems have attracted considerable attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' With increasing computational power, lattice QCD has become the primary force to derive hadron-hadron interactions in a quantitative way from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [13], the authors investigated the ΩΩ interaction in the 1S0 channel, and concluded that there exists a weakly bound state regardless of the Coulomb interaction, which is even shallower than the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [14], the existence of a 1S0 ΩcccΩccc shallow bound state is predicted while it disappears once the Coulomb interaction is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Very recently, the existence of a deeply bound 1S0 ΩbbbΩbbb state was also predicted [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' For the ΩΩ, ΩcccΩccc, and ΩbbbΩbbb systems, some of us developed an extended one-boson-exchange (OBE) model to derive their interactions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [16], and obtained results consistent with those of lattice QCD [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [17], the authors found the existence of fully heavy dibaryon bound states, ΩcccΩccc and ΩbbbΩbbb, in the constituent quark model, while the corre- sponding fully heavy hexaquark states are found to be above the ΩcccΩccc and ΩbbbΩbbb mass thresholds in both the con- stituent quark models [18–20] and the QCD sum rules [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' On the experimental side, studies of fully heavy multi- quarks have made important breakthroughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In 2020, the LHCb Collaboration reported the observation of the first fully heavy tetraquark state, X(6900) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' It was confirmed by the CMS Collaboration with a statistical significance of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4σ, and in addition, two new states X(6600) and X(7200) were observed [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The ATLAS Collaboration further confirmed the discovery of the LHCb Collaboration [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Clearly, the existence of fully heavy multiquark states can be considered as firmly established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' It is natural to expect the existence of tri-baryon systems given the (predicted) existence of di-baryon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We note that experimental and theoretical studies of tribaryon systems other than atomic nuclei and hypernuclei have continued for many years without conclusive results [25–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In this letter, motivated by the remarkable progress achieved on studies of the fully heavy multiquark states from both lattice QCD [13– 15] and experiments [22–24], we study the ΩΩΩ system, as well as the ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Notice that this study is different from all the previous works, where two or three different species of baryons are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The systems we study contain only one species of baryons and therefore have the highest symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' This means that we only need the interactions of a pair of identical baryons and the number of allowed configurations is much reduced as well, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='00630v1 [hep-ph] 2 Jan 2023 2 thus allowing for more robust predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Most strange tribaryon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='—We adopt the Gaussian expansion method (GEM) [32–34] to study the ΩΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' To solve the Schr¨odinger equation with GEM, one needs to derive the two-body interactions and construct the three-body wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The ΩΩ interaction has been derived in lattice QCD [13], where it was shown that the S-wave ΩΩ system can bind with a binding energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6(6)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6 MeV (without taking into account the Coulomb interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In addition to lat- tice QCD, other methods such as the extended one-boson exchange (OBE) model can also provide the ΩΩ interac- tion [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In this work, we utilize both interactions to study the ΩΩΩ three-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The ΩΩ lattice QCD potentials for the 1S0 channel is expressed with three Gaussian functions V 1S0 L (r) = �3 i=1 aie−bir2 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Since the Ω baryon is charged, the Coulomb interaction plays an important role in the ΩΩ and ΩΩΩ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The Coulomb potential between a pair of ΩΩ is VC(r) = −α/r, where α = 1/137 is the electromagnetic fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The lattice QCD simulations only provided the 1S0 poten- tial between the ΩΩ pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' As we see later, the 5S2 potential plays an important role in the three-body system as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In the following, we derive a synthetic “LQCD” 5S2 potential for the ΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' That is to say, we obtain the 5S2 poten- tial from the 1S0 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Assuming that there exists a linear transformation between the 1S0 and 5S2 ΩΩ interactions V ΩΩ 5S2 (r) = βV ΩΩ 1S0 (ωr + γ) + δ, (1) which is a combination of scale and translation transforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Because the potential should approach to zero as r ap- proaches infinity, the vertical translation parameter δ is fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We choose two characteristic points, the minimum and zero points, to fix the other three parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=', V 5S2 ΩΩ (r′ 0) = βV 1S0 ΩΩ (r0), V 5S2 ΩΩ (r′ min) = βV 1S0 ΩΩ (rmin), r0 = ωr′ 0 + γ, rmin = ωr′ min + γ, (2) where r0, rmin (r′ 0, r′ min) are the radial positions where V 1S0 ΩΩ (V 5S1 ΩΩ ) are either vanishing or the most attractive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Using the OBE potentials of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [16], the values of β, ω and γ are determined to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='7619, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='001 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='1395 fm, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' With these parameters, one can obtain a synthetic lat- tice QCD 5S2 potential from the genuine lattice QCD 1S0 po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' These potentials and the corresponding OBE poten- tials of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [16] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' A few remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' First, with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' (1) and the OBE 1S0 potential, the syn- thetic 5S2 potential is nearly identical to that of the OBE 5S2 potential, which shows that the linear transformation can in- deed faithfully relate the 1S0 and 5S2 potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Second, the synthetic 5S2 potential is more short ranged, more repulsive but less attractive compared to the lattice QCD 1S0 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Although less attractive, as we show later, without it, the ΩΩΩ system does not bind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 200 100 0 100 200 r [fm] V [MeV] OBE 1S0 OBE 5S2 LQCD 1S0 sLQCD 5S2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' OBE and lattice QCD potentials for the ΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The blue, orange and green solid lines denote the 1S0 OBE, the 5S2 OBE, and the 1S0 lattice QCD potentials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The red dashed lines denote the synthetic 5S2 lattice QCD potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The ΩΩΩ three-body wave function can be written as a sum of three Jacobi channels ΨJ(⃗r1, ⃗R1) = � Ac αΦc J(⃗r1, ⃗R1), (3) where Ac α is the expansion coefficients, c = 1 − 3 denote the three Jacobi channels, α is the set of quantum numbers characterizing the wave function in each Jacobi channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The wave function of each Jacobi channel reads as Φ1 J(⃗r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ⃗R1) = � [[χ3χ2]s1χ1]S ⊗ [ψl1(⃗r1)φL1(⃗R1)]Λ � J ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Φ2 J(⃗r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ⃗R2) = � [[χ1χ3]s2χ2]S ⊗ [ψl2(⃗r2)φL2(⃗R2)]Λ � J ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Φ3 J(⃗r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ⃗R3) = � [[χ2χ1]s3χ3]S ⊗ [ψl3(⃗r3)φL3(⃗R3)]Λ � J ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' where χi is the spin wave function of the ith particle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Hc s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='S = [[χiχj]sχk]S is the spin wave function of Jacobi channel c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ψ(ri)φ(Ri) is the spatial wave function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' s is the spin of the sub ΩΩ two-body system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' S = 3/2 is the total spin of ΩΩΩ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' li (Li) is the orbit angular momentum corresponding to ri(Ri),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Λ is the total orbit angular momentum built from l and L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' and J is the total angular momentum built from Λ and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Fermi-Dirac statistics dictates that only the 1S0 and 5S2 interactions contribute to the formation of an ΩΩΩ 3 2 + state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The spin coupling coefficients of different spin configurations between Jacobi channels i and j for i ̸= j are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Note that for i = j, the matrix is orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Coupling coefficients of different spin configurations be- tween Jacobi channels i and j (i ̸= j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Here, Hc s,S is the spin func- tion, s = {0, 2} are alternative spin values of ΩΩ, and S = 3/2 is the total spin of ΩΩΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Hc=i 0, 3 2 Hc=i 2, 3 2 Hc=j 0, 3 2 − 1 4 − √ 5 4 Hc=j 2, 3 2 − √ 5 4 3 4 It is important to point out that for the ΩΩΩ system, the 5S2 potential can play a very important role, even more important than the 1S0 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' This is because the 5S2 partial wave is more strongly coupled to the three-body spin-3/2 state than the 1S0 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' As shown in Table I, the spin coupling coefficient of different Jacobi channels i and j in the 5S2 par- tial wave is ⟨Hc=i 2,3/2|Hc=j 2,3/2⟩i̸=j = 3/4 while that in 1S0 is ⟨Hc=i 0,3/2|Hc=j 0,3/2⟩i̸=j = −1/4, which means that in the spin space, the couplings between channels i and j in the 5S2 par- tial wave is 9 times larger than that in the 1S0 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Once the wave functions are obtained , with either the lattice QCD or OBE ΩΩ interactions, one can adopt the GEM [33] to obtain the binding energies and root-mean- square (RMS) radius of the ΩΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The results for the two-body ΩΩ system are summarized in Table II, which show that the binding energies and root- mean-square (RMS) radii obtained with the OBE potentials are consistent with those of lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' With both lattice QCD and OBE potentials, the ΩΩ system can bind with a binding energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The uncertainties are deter- mined by multiplying a scaling factor to the lattice QCD po- tential so that the binding energy varies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='3 MeV, consistent with the lattice QCD result 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6 MeV [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' From the analysis given above, we know that both 1S0 and 5S2 interactions contribute to the 3/2 ΩΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Given that the lattice QCD only provided the 1S0 interaction, we first consider only the 1S0 two-body interaction and find that the ΩΩΩ system does not bind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' But this result should not be taken too seriously since the 5S2 partial wave plays an impor- tant role in the spin configuration (⟨Hc=i 2,3/2|Hc=j 2,3/2⟩i̸=j = 3 4) and has a significant correlation with the 1S0 partial wave (⟨Hc=i 0,3/2|Hc=j 2,3/2⟩i̸=j = √ 5 4 ) in the 3-body case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Adding the synthetic lattice QCD 5S2 potential, we find that the three- body ΩΩΩ system binds with a binding energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='2 MeV and RMS radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' These results are fur- ther corroborated by the study with the OBE potentials, which yields a binding energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='8+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='2 MeV and RMS radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='2 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Note that the binding energy per baryon of the ΩΩΩ system is larger than that of the ΩΩ system, while consequently its RMS radius is smaller than that of the ΩΩ bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' This is understandable because for the ΩΩΩ system the 5S2 potential plays an important role while only the 1S0 potential is relevant for the ΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The weights of partial waves and Hamiltonian expectation values of the predicted ΩΩΩ bound state are given in Table III, which clearly show that the 5S2 interaction plays a signifi- cantly important role in the ΩΩΩ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' More specifically, the weights of the 1S0 and 5S2 partial waves are about 30% and 70%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' As we mentioned above, since the Ω baryon is charged, the impact of the Coulomb interaction is worth discussing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We find that the Coulomb interaction in this three-body system affects the binding energy by about 2-3 MeV, but does not change the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Considering the Coulomb interaction, the binding energy and RMS radius of the ΩΩΩ bound state predicted by the lattice QCD potentials are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='7 MeV and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5 fm, while for those predicted by the OBE model, they are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 MeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='3 fm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' It is important to discuss where to search for the predicted ΩΩ and ΩΩΩ bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [35], the production yield of the ΩΩ bound state was estimated using a dynamical co- alescence mechanism for the relativistic heavy-ion collisions at √sNN = 200 GeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='76 TeV, which turn out to be of the order of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [31], the production yields of NNΩ and NΩΩ were estimated to be 10−7 and 10−9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Comparing these results, one can estimate the ΩΩΩ production rate for the relativistic heavy-ion collisions at √sNN = 200 GeV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='76 TeV, which is of the order of 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Binding energies (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E) and root-mean-square radii (⟨r⟩) of the ΩΩ and ΩΩΩ bound states obtained with lattice QCD and OBE potentials (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' in MeV and radius ⟨r⟩ in fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ΩΩ(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=') ΩΩ(⟨r⟩) ΩΩΩ(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=') ΩΩΩ(⟨r⟩) LQCD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='55+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='57 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='48+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='40 OBE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='84+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='48 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='19 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Weights of the partial waves and Hamiltonian expecta- tion values (Units in MeV) of the 3 2 + ΩΩΩ bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ⟨Ψ 1S0 3/2 |Ψ 1S0 3/2 ⟩ ⟨Ψ 5S2 3/2 |Ψ 5S2 3/2 ⟩ ⟨T⟩ ⟨V 1S0⟩ ⟨V 5S2⟩ LQCD 29% 71% 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='77 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='80 −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='53 OBE 22% 78% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='72 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='71 −42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='84 Most charming and beautiful tri-baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='– It is straightfor- ward to extend the above study to the ΩcccΩccc and ΩbbbΩbbb systems, for which the lattice QCD simulations already pro- vided the 1S0 potentials [14, 15] and their OBE counterparts also exist [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Note that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [15] no analytic form of the ΩbbbΩbbb potential was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We fitted the lattice QCD potential with a sum of three Gaussian functions as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We also adopt the transformation introduced above to derive synthetic lattice QCD potentials for the 5S2 partial wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' All the lattice QCD potentials and the corresponding 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 400 200 0 200 400 r [fm] V [MeV] OBE 1S0 OBE 5S2 LQCD 1S0 sLQCD 5S2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='5 1500 1000 500 0 500 1000 1500 r [fm] V [MeV] OBE 1S0 OBE 5S2 LQCD 1S0 sLQCD 5S2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' OBE and lattice QCD potentials of the ΩcccΩccc (top) and ΩbbbΩbbb (bottom) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The blue, orange and green solid lines denote the 1S0 OBE, the 5S2 OBE, and the 1S0 lattice QCD po- tentials, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The red dashed lines denote the synthetic 5S2 lattice QCD potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' OBE potentials are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We note that although the interaction strengths of the lattice QCD potential and those of the OBE potentials are different, the positions where they be- come the most attractive are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The same can be said about the ΩΩ potentials shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Such a co- incidence indicates that the OBE model must have captured some essential features of the baryon-baryon potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' With the above lattice QCD (including the synthetic 5S2) and the OBE potentials, we can study the two-body and three- body systems composed of Ωccc and Ωbbb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' As shown in Ta- ble IV, with only the strong interaction the ΩcccΩccc bound system can be formed, but it dissolves once the Coulomb interaction is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' On the other hand, the ΩbbbΩbbb system is always bound regardless of the Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Furthermore, we note that the results obtained with the lattice QCD potentials and those with the OBE po- tentials are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Nonetheless, none of the ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb three-body systems can bind, mainly be- cause of the much weaker 5S2 interactions, which are non- trivial predictions of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Binding energies (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E) and root-mean-square radii (⟨r⟩) of the ΩcccΩccc, and ΩbbbΩbbb bound states obtained with OBE and LQCD potentials (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' in MeV and radius ⟨r⟩ in fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' NC means that the Coulomb interaction is not taken into account while C means that the Coulomb interaction is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ΩcccΩccc (NC) ΩcccΩccc (C) ΩbbbΩbbb (NC) ΩbbbΩbbb (C) LQCD B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='54 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='9 ⟨r⟩ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='245 OBE B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='52 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='4 ⟨r⟩ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='202 Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='— Motivated by the existence of ΩΩ, ΩcccΩccc, and ΩbbbΩbbb bound states predicted by lattice QCD simulations, we studied the 3 2 + ΩΩΩ, ΩcccΩcccΩccc, and ΩbbbΩbbbΩbbb three-body systems with the lattice QCD and OBE potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We found that the ΩΩ, ΩcccΩccc, and ΩbbbΩbbb systems can also bind with the OBE potentials, with binding energies and RMS radii consistent with those of lattice QCD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The repulsive Coulomb interactions plays an important role in these systems especially in the ΩcccΩccc system, which is strong enough to break the ΩcccΩccc pair bound by the strong force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' For the three-body systems, we find that the 5S2 partial wave plays a very important role in forming the 3 2 + three-body state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' With only the 1S0 lattice QCD potentials, the ΩΩΩ, ΩcccΩcccΩccc, ΩbbbΩbbbΩbbb three-body systems do not bind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Supplemented with a synthetic lattice QCD 5S2 potential, the ΩΩΩ system becomes bound while the ΩcccΩcccΩccc and ΩbbbΩbbbΩbbb systems remain unbound, mainly due to the much suppressed attractive 5S2 interaction in the two-body ΩcccΩccc and ΩbbbΩbbb systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The results were further cor- roborated by explicit studies employing the OBE potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' To verify the existence of the ΩΩΩ bound state, lattice QCD studies of the 5S2 interactions of the ΩΩ system will be the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' We hope that the predicted ΩΩΩ bound state can be searched for in present and future hadron-hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' A particularly interesting discovery of the present work is that even the two-body interactions are attractive and strong enough to form two-body bound states, the three-body sys- tems do not necessarily bind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' This is because in three-body systems, spin-spin interactions can play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' The three highly symmetric systems studied in the present work provide an ideal platform to understand the relevance of spin-spin interactions in forming few-body bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='—This work is partly supported by the Na- tional Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='11735003, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='11975041, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='11961141004, and the fun- damental Research Funds for the Central Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' is supported by the China National Funds for Distinguished Young Scientists under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 11825503, National Key 5 Research and Development Program of China under Con- tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2020YFA0406400, the 111 Project under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' B20063, the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 12247101, and the project for top-notch in- novative talents of Gansu province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Ming-Zhu Liu acknowl- edges support from the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='12105007 and China Postdoctoral Science Foundation under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2022M710317, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2022T150036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' Tian-Wei Wu acknowledges support from the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content='12147152 and China Postdoctoral Science Foundation un- der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' 2022M723119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} +page_content=' ∗ lisheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtAyT4oBgHgl3EQfvfkj/content/2301.00630v1.pdf'} diff --git a/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/2301.02957v1.pdf.txt b/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/2301.02957v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0eda2e7a3e7c3b66eb12e1856e315e913ee12f2 --- /dev/null +++ b/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/2301.02957v1.pdf.txt @@ -0,0 +1,2061 @@ +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Geophysical and Astrophysical Fluid Dynamics +Vol. 00, No. 00, 00 Month 2023, 1–23 +Semi-Analytical Solutions of Shallow Water Waves +with Idealised Bottom Topographies +CHANG LIU † ‡ ∗ and ANTWAN D. CLARK † +† Department of Applied Mathematics and Statistics, Johns Hopkins University +Baltimore, MD 21218 USA +‡ Department of Physics, University of California, Berkeley +Berkeley, CA 94720 USA +(v4.4 released July 2022) +Analysing two-dimensional shallow water equations with idealised bottom topographies have many ap- +plications in the atmospheric and oceanic sciences; however, restrictive flow pattern assumptions have been +made to achieve explicit solutions. This work employs Adomian decomposition methods (ADMs) to develop +semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters +while capturing the nonlinear phenomenon. Furthermore, we exploit these techniques as reverse engineering +mechanisms to develop key connections between some prevalent ansatz formulations in the open literature +as well as developing new families of exact solutions describing geostrophic inertial oscillations and anticy- +clonic vortices with finite escape times. Our semi-analytical evaluations show the promise of this approach +in terms of providing robust approximations against several oceanic variations and bottom topographies +while also preserving the direct correlation between the physical parameters such as the Froude number, the +bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours. Our numerical +validations provide additional confirmations of this approach while also illustrating that ADMs can also +be used to provide insight and deduce novel solutions that have not been explored, which can be used to +characterize various types of geophysical flows. +Keywords: Adomian decomposition methods; shallow water equations; bottom topographies +1. +Introduction +Analysing two-dimensional shallow water equations has been extensively studied in geophys- +ical fluid dynamics to understand a myriad of atmospheric and oceanic phenomena. Some +examples include understanding the effects of long-term oceanic waves (Pedlosky 2013, Vallis +2017), analyzing the behaviour of oceanic warm-core rings (Cushman-Roisin 1987), investigat- +ing flows in channels and shorelines (Shapiro 1996, Sampson et al. 2005), studying steady-state +flows (Iacono 2005, Sun 2016), and grasping the temporal instability of barotropic zonal flows +(Clark and Herron 2013). These theoretical analyses also serve as a good basis for numer- +ical simulations and validations. For example, the creators of the Shallow Water Analytic +Solutions for Hydraulic and Environmental Studies (SWASHES) software library (Delestre +et al. 2013) incorporated a significant number of theoretical solutions of the shallow water +equations in the open literature, which has been cited by over 200 research papers currently. +Furthermore, several of the solutions in this library are obtained from Thacker (1981) in which +have been widely used to demonstrate the validity and accuracy of several numerical schemes +including finite volume schemes (Gallardo et al. 2007, Bollermann et al. 2011, Nikolos and +Delis 2009) and discontinuous Galerkin methods (Ern et al. 2008, Kesserwani and Liang 2012, +Li et al. 2017, Wintermeyer et al. 2018). Some significant advancements include the original +∗Corresponding author. Email: cliu124@alumni.jh.edu +arXiv:2301.02957v1 [physics.flu-dyn] 8 Jan 2023 + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +2 +C. Liu & A. D. Clark +works of Ball and Thacker who demonstrated that nonlinear oscillations can be modelled as +either low-order polynomials or normal modes (Ball 1963, 1964, 1965, Thacker 1977, 1981). +Researchers also developed elliptical vortex solutions to understand the temporal effects of +oceanic warm-core rings including stationary clockwise rotations (rodons), pulsating circu- +lar eddies (pulsons), and a subclass of these phenomena called pulsrodons (Cushman-Roisin +1987, Cushman-Roisin et al. 1985, Rogers 1989b). Extensions to these approaches have been +made, where some examples include the work of Sachdev et al. (Sachdev et al. 1996) who ex- +tended the approach of (Clarkson and Kruskal 1989) and derived new families of solutions in +paraboloidal basins that provided additional insights in terms of describing flow behaviour due +to deformation modes. Additionally, Matskevich and Chubarov (2019) extended the results +of Ball and Thacker to include the effects of Coriolis forces and bottom friction. Bristeau et +al. (Bristeau et al. 2021) also extended the results of Thacker and introduced two respective +solutions describing velocity distributed along the vertical axis and velocity accounting for +variable density. +Group analysis was also explored. Some pioneering works in this area include that of Curr`o +(Curr`o 1989) and Rodgers (Rogers 1989a) who also advanced the works of Thacker and Ball +and related several forms of the depth function as well as developed invariance theorems. Levi +et al. (Levi et al. 1989) developed symmetry reductions for flows with elliptic and circular bot- +tom topographies. Bila et al. (Bila et al. 2006) derived Lie point symmetries and conservation +laws. Chesnokov (2009) discovered 9-dimensional Lie algebra point symmetries and developed +transformations between rotating and non-rotating cases, which were later used to describe +spatial oscillations in spinning paraboloids (Chesnokov 2011). Some recent advancements in- +clude Meleshko (2020) and Bihlo et al. (Bihlo et al. 2020) who performed group classification +and analysis for zero and constant Coriolis parameters. Meanwhile, Meleshko and Samatova +(2020) performed similar analysis and considered the beta-plane approximation of the Coriolis +parameter and an irregular bottom topography. +However, deriving theoretical solutions to the two-dimensional shallow water equations poses +the following main challenges. First, these efforts involve making specific assumptions regard- +ing the flow conditions which only satisfy specific cases. Some solutions also contain combi- +nations of special functions and integral expressions (Shapiro 1996, Rogers 1989b), which in +turn makes it difficult to determine the correlation between the physical quantities of these +models. Finding invariant solutions via group analysis has the additional advantage of deriv- +ing conservation laws to these equations. However, this approach depends on the construction +of Lie-groups which depend on the problem formulation as well as specific assumptions such +as the Coriolis parameter and bottom topography. Therefore, there is a need to find solutions +that are not only flexible, in terms of relaxing certain limiting assumptions, but also provide +a direct correlation of the physical parameters. +This work applies Adomian decomposition methods (ADMs) (Adomian 1990) to the shal- +low water equations to provide the following main contributions. First, we present the ADM +formulation of the rotating shallow water equations where we also present key connections +between the ansatz formulations in the work of Thacker (1981), Shapiro (1996), Matskevich +and Chubarov (2019). Next, we derive and present some new families of exact solutions, for +flat bottom topographies, that describe inertial oscillations in geostrophic flows and anticy- +clonic vortices with finite escape times. This rest of this paper is organised in the following +manner. Section 2 presents the ADM formulation and initial theoretical formulation of the +problem, where we present the connection to fundamental assumptions on the formulation of +the solutions. Section 3 presents derivations of new families of solutions and their properties. +Section 4 provides numerical experimentation and results. Section 5 provides some concluding +remarks, where we also list some future research directions. + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +3 +2. +Adomian Decomposition Formulation +Figure 1.: Illustration of a thin layer of incompressible flow under the Earth’s rotation +described by rotating shallow-water equations with idealised bottom topography. +The non-dimensional form of the governing equations is defined as +∂u +∂t = − u∂u +∂x − v∂u +∂y − 1 +F 2 +∂h +∂x + ¯fv +∂v +∂t = − u∂v +∂x − v∂v +∂y − 1 +F 2 +∂h +∂y − ¯fu +∂h +∂t = − ∂ +∂x[u(h + D)] − ∂ +∂y[v(h + D)]. +(1) +This is illustrated in Figure 1, where u and v are the flow velocity components, h is the +free surface height, ¯f = fL0/U0 is the dimensionless Coriolis parameter (associated with the +Coriolis force), and F = U0/√gH0 is the Froude number. Here, the spatial variables x, y, l, +and L are normalised by the horizontal length scale L0; h is normalised by a vertical length +scale H0; the horizontal velocities, u and v, are normalised by the characteristic velocity U0; +and time t is normalised by L0/U0. Hence, the dimensionless form of the idealised bottom +topography is defined as +D(x, y) = D0 +� +1 − x2 +L2 − y2 +l2 +� +(2) +where D0 is also normalised by a vertical length scale H0. It is noteworthy to mention that +other bottom topographies can be determined from (2) such as flat bottom (D0 = 0), circular +paraboloid (l = L), and channel (l → ∞ or L → ∞) terrains. Additionally, D(x, y) can +also be used to incorporate linear terms in its description via change of variables (Shapiro +1996, Thacker 1981). The total fluid depth D +h, shown in Figure 1, follows the formulations +of Thacker (1981) and Shapiro (1996) where D + h = 0 represents a moving shoreline and +D +h < 0 represents dry regions. When the moving shoreline is closed, the water mass within +the shoreline is conserved (Thacker 1981, Shapiro 1996). When the moving shoreline is open +such as in tsunami modelling, then water within a bounded domain will have mass exchange +with an infinite mass reservoir. It is also important to mention that our explorations in this + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +4 +C. Liu & A. D. Clark +section consider flow velocities that are linearly varying spatially while the free surface height +either varies linearly or in a quadratic fashion. The initial conditions are given by +u(x, y, 0) = u0(x, y), v(x, y, 0) = v0(x, y), and h(x, y, 0) = h0(x, y). +(3) +Next, u, v, and h are decomposed as follows +u(x, y, t) = +∞ +� +n=0 +un(x, y, t), v(x, y, t) = +∞ +� +n=0 +vn(x, y, t), and h(x, y, t) = +∞ +� +n=0 +hn(x, y, t), +(4) +where the initial components are defined by equation (3). Thus, the recurrence relationships +to equation (1) (for n ≥ 0) are given by +un+1(x, y, t) = L−1 +t +� +−An +� +u, ∂u +∂x +� +− An +� +v, ∂u +∂y +� +− 1 +F 2 +∂hn +∂x + ¯fvn +� +, +vn+1(x, y, t) = L−1 +t +� +−An +� +u, ∂v +∂x +� +− An +� +v, ∂v +∂y +� +− 1 +F 2 +∂hn +∂y − ¯fun +� +, +hn+1(x, y, t) = L−1 +t +� +− ∂ +∂x[An(u, h)] − ∂ +∂y[An(v, h)] − ∂ +∂x[unD] − ∂ +∂y[vnD] +� +, +(5) +where +Lt = ∂(·) +∂t , +L−1 +t += +� t +0 +(·) dτ, +and the Adomian polynomial representing the quadratic nonlinearity is defined as (Adomian +1990, 2013) +An(u, h) = +n +� +j=0 +ujhn−j. +(6) +It is important to note that equation (6) can be used to approximate the quadratic nonlinear +terms, such as uh, as follows +uh = +� ∞ +� +p +up +� � ∞ +� +q +hq +� += +∞ +� +n +An(u, h) +and thus the semi-analytical solution to (1) is expressed via the partial sums +u(x, y, t) = SN(u) = +N +� +n=0 +un, v(x, y, t) = SN(v) = +N +� +n=0 +vn, and h(x, y, t) = SN(h) = +N +� +n=0 +hn. +(7) +Next, the following results connect the properties of the initial conditions to the behaviours +of the true solutions via their partial sums. +Lemma 2.1: +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- +tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal +parabolic topography (2). If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such +that + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +5 +∂2u0(x, y) +∂x2 += ∂2v0(x, y) +∂x2 += ∂3h0(x, y) +∂x3 += 0, +(8) +∂2u0(x, y) +∂y2 += ∂2v0(x, y) +∂y2 += ∂3h0(x, y) +∂y3 += 0, +(9) +and +∂2u0(x, y) +∂xy += ∂2v0(x, y) +∂x∂y += ∂3h0(x, y) +∂x2∂y += ∂3h0(x, y) +∂x∂y2 += 0. +(10) +Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t) also satisfy the same prop- +erty, where +∂2un(x, y, t) +∂x2 += ∂2vn(x, y, t) +∂x2 += ∂3hn(x, y, t) +∂x3 += 0, +(11) +∂2un(x, y, t) +∂y2 += ∂2vn(x, y, t) +∂y2 += ∂3hn(x, y, t) +∂y3 += 0, +(12) +and +∂2un(x, y, t) +∂x∂y += ∂2vn(x, y, t) +∂x∂y += ∂3hn(x, y, t) +∂x2∂y += ∂3hn(x, y, t) +∂x∂y2 += 0 +(13) +for n ∈ N+. +Proof : This is proven via mathematical induction by examining the recursion relationships +for u, v, and h in equation (5). Condition (11) is demonstrated by examining the following +relationships +∂2un+1 +∂x2 += −L−1 +t +� ∂2 +∂x2 +� +An +� +u, ∂u +∂x +� ++ An +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂3hn +∂x3 − ¯f ∂2vn +∂x2 +� +, +(14) +∂2vn+1 +∂x2 += −L−1 +t +� ∂2 +∂x2 +� +An +� +u, ∂v +∂x +� ++ An +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂3hn +∂x2∂y + ¯f ∂2un +∂x2 +� +, +(15) +and +∂3hn+1 +∂x3 += −L−1 +t +� ∂4 +∂x4 [An(u, h)] + +∂4 +∂x3∂y[An(v, h)] + ∂4 +∂x4 [unD] + +∂4 +∂x3∂y[vnD] +� +. +(16) +Therefore, when n = 0 equations (14) - (16) representing the relationship between the initial +and first components for u, v, and h become +∂2u1 +∂x2 = −L−1 +t +� ∂2 +∂x2 +� +u0 +∂u0 +∂x + v0 +∂u0 +∂y +� ++ 1 +F 2 +∂3h0 +∂x3 − ¯f ∂2v0 +∂x2 +� +, +(17) +∂2v1 +∂x2 = −L−1 +t +� ∂2 +∂x2 +� +u0 +∂v0 +∂x + v0 +∂v0 +∂y +� ++ 1 +F 2 +∂3h0 +∂x2∂y + ¯f ∂2u0 +∂x2 +� +, +(18) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +6 +C. Liu & A. D. Clark +and +∂3h1 +∂x3 = −L−1 +t +� ∂4 +∂x4 [u0h0] + +∂4 +∂x3y [v0h0] + ∂4 +∂x4 [u0D] + +∂4 +∂x3∂y [v0D] +� +. +(19) +Employing (8) - (10) it can be shown that equations (17) - (19) reduce to the following +relationship +∂2u1(x, y, t) +∂x2 += ∂2v1(x, y, t) +∂x2 += ∂3h1(x, y, t) +∂x3 += 0. +Continuing this argument for n ∈ N+ yields equation (11). Similar arguments can be made to +produce (12) and (13), respectively. +□ +Theorem 2.2 : +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed +functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal +parabolic topography (2). If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as (8) +- (10), then the solutions of u, v, and h have the same property where +∂2u(x, y, t) +∂x2 += ∂2u(x, y, t) +∂y2 += ∂2u(x, y, t) +∂x∂y += 0, +(20) +∂2v(x, y, t) +∂x2 += ∂2v(x, y, t) +∂y2 += ∂2v(x, y, t) +∂x∂y += 0, +(21) +and +∂3h(x, y, t) +∂x3 += ∂3h(x, y, t) +∂x2∂y += ∂3h(x, y, t) +∂x∂y2 += ∂3h(x, y, t) +∂y3 += 0. +(22) +Consequently, these solutions can be expressed as +u(x, y, t) = ˜u0(t) + ˜ux(t)x + ˜uy(t)y, +(23) +v(x, y, t) = ˜v0(t) + ˜vx(t)x + ˜vy(t)y, +(24) +and +h(x, y, t) = ˜h0(t) + ˜hx(t)x + ˜hy(t)y + 1 +2 +˜hxx(t)x2 + 1 +2 +˜hyy(t)y2 + ˜hxy(t)xy, +(25) +where the coefficients ˜u0(t), ˜ux(t), ˜uy(t), ˜v0(t), ˜vx(t), ˜vy(t), ˜h0(t), ˜hx(t), ˜hy(t), ˜hxx(t), ˜hyy(t), +and ˜hxy(t) are time-dependent. +Proof : Applying Lemma 2.1 to each component in (4) yields (20)-(22). From (20), we observe +that +∂2u(x, y, t) +∂x2 += 0 yields u(x, y, t) = C1(y, t)x + C2(y, t), +where the integration constants, C1(y, t) and C2(y, t), are independent of x. Similarly, we have +∂2u(x, y, t) +∂x∂y += 0 yields C1(y, t) = ˜ux(t) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +7 +and +∂2u(x, y, t) +∂y2 += 0 yields C2(y, t) = ˜uy(t)y + ˜u0(t). +and thus (23) is achieved. Similar arguments can be made to achieve (24) and (25), respectively. +□ +We note the significance of Theorem 2.2. In the works of Thacker (1981), Shapiro (1996), +and Matskevich and Chubarov (2019) equations (23)-(25) were presented as ansatz solutions, +where they were also used to produce the reduced system of shallow water equations to derive +closed-form solutions. This theorem removes these assumptions and provides more insight to +this behaviour by connecting it to the initial conditions (8) - (10). +3. +Novel Exact Solutions for Flat Bottom Topographies with Constant Coriolis Force +Next, we use the ADM construction to derive new families of solutions and their properties +that describe other geophysical flows such as inertial oscillations and anticyclonic vortices +which have a profound effect on oceanic and atmospheric dynamics Vallis (2017), Kafiabad +et al. (2021). Here, we consider flows over flat bottom topologies where D0 = 0 in (2) with +constant Coriolis parameter ( ¯f ̸= 0). +3.1. +Inertial Oscillations in Geostrophic Flows +For these types of flows, our analysis considers the following initial conditions. +• Condition I +u0(x, y) = v0(x, y) = 0, h0(x, y) = ηxx + ηyy, +(26) +• Condition II +u0(x, y) = v0(x, y) = 0, h0(x, y) = ηxx, +(27) +• Condition III +u0(x, y) = v0(x, y) = 0, h0(x, y) = ηyy, +(28) +where ¯f ̸= 0 is the constant Coriolis parameter, and ηx and ηy are the respective constant +free surface gradients in the x and y directions. We note that the behaviour of the initial +conditions (26) - (28) affect the decomposition of the decomposed functions of u, v, and h as +presented in the following lemma. +Lemma 3.1: +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- +tions of u, v, and h such that their relationship is defined by (5) (for n ∈ N). If D = 0 and +the initial conditions u0(x, y), v0(x, y), h0(x, y) satisfy the following properties +∂u0(x, y) +∂x += ∂v0(x, y) +∂x += ∂2h0(x, y) +∂x2 += 0, +(29) +∂u0(x, y) +∂y += ∂v0(x, y) +∂y += ∂2h0(x, y) +∂y2 += 0, +(30) +and + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +8 +C. Liu & A. D. Clark +∂2h0(x, y) +∂x∂y += 0. +(31) +Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t) also satisfy the property +that +∂un(x, y, t) +∂x += ∂vn(x, y, t) +∂x += ∂hn(x, y, t) +∂x += 0 +(32) +and +∂un(x, y, t) +∂y += ∂vn(x, y, t) +∂y += ∂hn(x, y, t) +∂y += 0 +(33) +for n ∈ N+. +Proof : This is proven via mathematical induction by examining the recursion relationships +for u, v, and h in (5). Condition (32) is demonstrated by examining the following relationships +∂un+1 +∂x += −L−1 +t +� ∂ +∂x +� +An +� +u, ∂u +∂x +� ++ An +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂2hn +∂x2 − ¯f ∂vn +∂x +� +, +(34) +∂vn+1 +∂x += −L−1 +t +� ∂ +∂x +� +An +� +u, ∂v +∂x +� ++ An +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂2hn +∂x∂y + ¯f ∂un +∂x +� +, +(35) +and +∂hn+1 +∂x += −L−1 +t +� ∂2 +∂x2 [An(u, h)] + +∂2 +∂x∂y[An(v, h)] +� +. +(36) +Therefore, when n = 0, equations (34) - (36) representing the relationship between the initial +and first components for u, v, and h become +∂u1 +∂x = −L−1 +t +� ∂ +∂x +� +A0 +� +u, ∂u +∂x +� ++ A0 +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂2h0 +∂x2 − ¯f ∂v0 +∂x +� +, +(37) +∂v1 +∂x = −L−1 +t +� ∂ +∂x +� +A0 +� +u, ∂v +∂x +� ++ A0 +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂2h0 +∂x∂y + ¯f ∂u0 +∂x +� +, +(38) +and +∂h1 +∂x = −L−1 +t +� ∂2 +∂x2 [A0(u, h)] + +∂2 +∂x∂y[A0(v, h)] +� +. +(39) +Employing (29) - (31) it can be shown that equations (37) - (39) reduce to the following +relationship +∂u1(x, y, t) +∂x += ∂v1(x, y, t) +∂x += ∂h1(x, y, t) +∂x += 0, +and continuing this argument for n ∈ N+ yields equation (32). Following similar arguments +yields (33). +□ + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +9 +From this, the behaviour of uniform u, v over space, and planar free surface h with constant +spatial gradients over time can be summarised in the following theorem. +Theorem 3.2 : +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed +functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N). If D = 0 and +the initial conditions u0(x, y), v0(x, y), h0(x, y) satisfy the properties defined in (29) - (31), +then the solutions u, v, and h have the following properties +∂u(x, y, t) +∂x += ∂u(x, y, t) +∂y += 0, +(40) +∂v(x, y, t) +∂x += ∂v(x, y, t) +∂y += 0, +(41) +∂h(x, y, t) +∂x += ∂h(x, y, 0) +∂x +, +∂h(x, y, t) +∂y += ∂h(x, y, 0) +∂y +, +(42) +and +∂2h(x, y, t) +∂x2 += ∂2h(x, y, t) +∂x∂y += ∂2h(x, y, t) +∂y2 += 0. +(43) +Additionally, u, v, and h are reduced to the following forms +u(x, y, t) = ˜u0(t), +(44) +v(x, y, t) = ˜v0(t), +(45) +and +h(x, y, t) = ˜h0(t) + ˜hxx + ˜hyy, +(46) +where the coefficients ˜u0(t), ˜v0(t), and ˜h0(t) are time-dependent, while ˜hx and ˜hy are constants. +Additionally, (44) - (46) satisfy the reduced system of equations +d +dt ˜u0(t) = − 1 +F 2 ˜hx + ¯f˜v0(t), +d +dt˜v0(t) = − 1 +F 2 ˜hy − ¯f ˜u0(t), +d +dt +˜h0(t) = − ˜u0(t)˜hx − ˜v0(t)˜hy. +(47) +Proof : Applying Lemma 3.1 to each component in (4) yields (40)-(43). From (40), we observe +that +∂u(x, y, t) +∂x += 0 yields u(x, y, t) = C1(y, t), +where the integration constants, C1(y, t), are independent of x. Similarly, we have + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +10 +C. Liu & A. D. Clark +∂u(x, y, t) +∂y += 0 yields C1(y, t) = ˜u0(t) +and thus (44) is achieved. Similar arguments can be made to achieve (45) and (46), respectively. +Substituting (44) - (46) into (1) achieves the reduced system of equations (47), which completes +the proof. +□ +Hence, we have the following results for inertial oscillations for geostrophic flows. +Theorem 3.3 : +Given inertial oscillations over flat bottom topographies with constant Cori- +olis parameter ¯f ̸= 0, where the initial behaviour is defined by (26). The solutions u, v, and +h are expressed as +u(x, y, t) = − ηx +¯fF 2 sin +� ¯ft +� +− ηy +¯fF 2 +� +1 − cos +� ¯ft +�� +, +(48) +v(x, y, t) = +ηx +¯fF 2 +� +1 − cos +� ¯ft +�� +− ηy +¯fF 2 sin( ¯ft), +(49) +and +h(x, y, t) = +η2 +x +¯f2F 2 +� +1 − cos +� ¯ft +�� ++ xηx + +η2 +y +¯f2F 2 +� +1 − cos +� ¯ft +�� ++ ηyy +(50) +where ηx and ηy are the constant free surface gradients in the x and y directions, respectively. +Proof : The initial conditions (26) satisfy (29) - (31). Therefore, the sequence of decomposed +functions {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} satisfy (32) and (33) for n ∈ N+ which sat- +isfies Lemma 3.1 and consequently Theorem 3.2. Examining the system of reduced equations +(47), the initial conditions (26) also produce the following reduced relationships: ˜hx = ηx, +˜hy = ηy, and ˜u(t = 0) = ˜v(t = 0) = ˜h0(t = 0) = 0. Solving this reduced system achieves (48) +- (50) which proves the theorem. +□ +Corollary 3.4: +Given inertial oscillations over flat bottom topographies with constant Cori- +olis parameter ¯f ̸= 0. +(i) If the initial behaviour is defined by (27), then the solutions u, v, and h are expressed as +u(x, y, t) = − ηx +¯fF 2 sin +� ¯ft +� +, v(x, y, t) = +ηx +¯fF 2 +� +1 − cos +� ¯ft +�� +, +(51) +and +h(x, y, t) = +η2 +x +¯f2F 2 +� +1 − cos +� ¯ft +�� ++ xηx. +(52) +(ii) If the initial behaviour is defined by (28), then the solutions u, v, and h are expressed as +u(x, y, t) = − ηy +¯fF 2 +� +1 − cos +� ¯ft +�� +, v(x, y, t) = − ηy +¯fF 2 sin +� ¯ft +� +, +(53) +and +h(x, y, t) = +η2 +y +¯f2F 2 +� +1 − cos +� ¯ft +�� ++ ηyy. +(54) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +11 +ηx and ηy are the constant free surface gradients in the x and y directions, respectively. +Proof : This is a special case of Theorem 3.3 for ηx = 0 and ηy = 0, respectively. +□ +Theorem 3.3 and Corollary 3.4 show the explicit relationship between the these types of flows +with respect to the constant Coriolis parameter, the free surface gradients, and the Froude +number where the inertial oscillation frequency is defined by the constant Coriolis parameter +¯f. These results also demonstrate that these oscillations are based on the magnitude of the +free surface gradients that depend on the initial behaviour and the geostrophic flows, which +are consistent with the results of (Vallis 2017). Moreover, Theorem 3.3 describes these types +of oscillations as the interaction between the geostrophic flow fluctuations and the free surface +gradients, where Corollary 3.4 considers cases when these gradients are negligible in the x and +y directions. +3.2. +Anticyclonic Vortices with Finite Escape Times +For these types of flows our analysis considers the following initial conditions +• Condition IV +u0(x, y) = ¯fy, v0(x, y) = 0, h0(x, y) = h0, +(55) +• Condition V +u0(x, y) = ¯fy, v0(x, y) = − ¯fx + ¯fy, h0(x, y) = h0, +(56) +• Condition VI +u0(x, y) = 0, v0(x, y) = − ¯fx, h0(x, y) = h0, +(57) +• Condition VII +u0(x, y) = ¯fx + ¯fy, v0(x, y) = − ¯fx, h0(x, y) = h0, +(58) +where h0 is the constant free surface height. These describe anticyclonic vortices for the initial +vorticity is proportional to the negative constant Coriolis parameter. The behaviour of the +initial conditions (55) - (58) affect the decomposition of the decomposed functions of u, v, +and h as presented in the following lemmas. +Lemma 3.5: +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- +tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom +topography D = 0. If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such that +u0(x, y) = ¯fy, +(59) +∂2v0(x, y) +∂x2 += ∂h0(x, y) +∂x += 0, +(60) +∂2v0(x, y) +∂y2 += ∂h0(x, y) +∂y += 0, +(61) +and +∂2v0(x, y) +∂x∂y += 0. +(62) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +12 +C. Liu & A. D. Clark +Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t), for n ∈ N+ satisfy +un(x, y, t) = 0, +(63) +∂2vn(x, y, t) +∂x2 += ∂hn(x, y, t) +∂x += 0, +(64) +∂2vn(x, y, t) +∂y2 += ∂hn(x, y, t) +∂y += 0, +(65) +and +∂2vn(x, y, t) +∂x∂y += 0. +(66) +Proof : This is proven via mathematical induction by examining the recursion relationships +for u, v, and h in equation (5). Condition (63) is demonstrated by examining +un+1 = −L−1 +t +�� +An +� +u, ∂u +∂x +� ++ An +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂hn +∂x − ¯f vn +� +, +(67) +In the case of n = 0 and using (59) - (62), it reduces to +u1 = −L−1 +t +�� +A0 +� +u, ∂u +∂x +� ++ A0 +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂h0 +∂x − ¯f v0 +� += −L−1 +t +� +A0 +� +v, ∂u +∂y +� +− ¯f v0 +� += −L−1 +t +� +v0 +∂u0 +∂y − ¯f v0 +� += 0, +and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (63). Condition (64) is +demonstrated by examining the following relationships +∂2vn+1 +∂x2 += −L−1 +t +� ∂2 +∂x2 +� +An +� +u, ∂v +∂x +� ++ An +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂3hn +∂x2∂y + ¯f ∂2un +∂x2 +� +, +(68) +and +∂hn+1 +∂x += −L−1 +t +� ∂2 +∂x2 [An(u, h)] + +∂2 +∂x∂y[An(v, h)] +� +. +(69) +Therefore, when n = 0, equations (68) - (69) representing the relationship between the initial +and first components for v and h become +∂2v1 +∂x2 = −L−1 +t +� ∂2 +∂x2 +� +A0 +� +u, ∂v +∂x +� ++ A0 +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂3h0 +∂x2∂y + ¯f ∂2u0 +∂x2 +� +, +(70) +and + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +13 +∂h1 +∂x = −L−1 +t +� ∂2 +∂x2 [A0(u, h)] + +∂2 +∂x∂y[A0(v, h)] +� +. +(71) +Employing (59) - (62), it can be shown that equations (70) - (71) reduce to the following +relationship +∂2v1(x, y, t) +∂x2 += ∂h1(x, y, t) +∂x += 0, +and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (64). Following similar +arguments yields (65) and (66). +□ +Lemma 3.6: +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- +tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom +topography D = 0. If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such that +v0(x, y, t) = − ¯fx, +(72) +∂2u0(x, y) +∂x2 += ∂h0(x, y) +∂x += 0, +(73) +∂2u0(x, y) +∂y2 += ∂h0(x, y) +∂y += 0, +(74) +and +∂2u0(x, y) +∂x∂y += 0. +(75) +Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t), for n ∈ N+ satisfy the +property +vn(x, y, t) = 0, +(76) +∂2un(x, y, t) +∂x2 += ∂hn(x, y, t) +∂x += 0, +(77) +∂2un(x, y, t) +∂y2 += ∂hn(x, y, t) +∂y += 0, +(78) +and +∂2un(x, y, t) +∂x∂y += 0. +(79) +Proof : This is proven via mathematical induction by examining the recursion relationships +for u, v, and h in equation (5). Condition (76) is demonstrated by examining the following +relationships + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +14 +C. Liu & A. D. Clark +vn+1 = −L−1 +t +�� +An +� +u, ∂v +∂x +� ++ An +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂hn +∂y + ¯f un +� +, +(80) +At n = 0, we have +v1 = −L−1 +t +�� +A0 +� +u, ∂v +∂x +� ++ A0 +� +v, ∂v +∂y +�� ++ 1 +F 2 +∂h0 +∂y + ¯f u0 +� += −L−1 +t +� +A0 +� +u, ∂v +∂x +� ++ ¯f u0 +� += −L−1 +t +� +−u0 ¯f + ¯f u0 +� += 0. +Employing a similar argument for n = 1, 2, . . . , n − 1, we have (76). Equation (77) is demon- +strated by examining the following +∂2un+1 +∂x2 += −L−1 +t +� ∂2 +∂x2 +� +An +� +u, ∂u +∂x +� ++ An +� +v, ∂u +∂y +�� ++ 1 +F 2 +∂3hn +∂x3 − ¯f ∂2vn +∂x2 +� +, +(81) +and +∂hn+1 +∂x += −L−1 +t +� ∂2 +∂x2 [An(u, h)] + +∂2 +∂x∂y[An(v, h)] +� +. +(82) +Therefore, when n = 0, equations (81) - (82) representing the relationship between the initial +and first components for u and h become +∂2u1 +∂x2 = −L−1 +t +� ∂2 +∂x2 +� +u0 +∂u0 +∂x + v0 +∂u0 +∂y +� ++ 1 +F 2 +∂3h0 +∂x3 − ¯f ∂2v0 +∂x2 +� +, +(83) +and +∂h1 +∂x = −L−1 +t +� ∂2 +∂x2 [A0(u, h)] + +∂2 +∂x∂y[A0(v, h)] +� +. +(84) +Employing (72) - (75), it can be shown that equations (83) - (84) reduce to the following +relationship +∂2u1(x, y, t) +∂x2 += ∂h1(x, y, t) +∂x += 0, +and continuing this argument for n = {1, 2, . . . , n − 1} yields equation (77). Following similar +arguments yields (78) and (79). +□ +Therefore, the behaviour of u, v, and h can be summarised in the following theorem. +Theorem 3.7 : +Given a flat bottom topography, let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} +be the sequence of decomposed functions of u, v, and h, defined by (5) (for n ∈ N). If the +initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as (59) - (62), then the solutions of +u, v, and h have the same property where +u(x, y, t) = ¯fy, +(85) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +15 +∂2v(x, y, t) +∂x2 += ∂2v(x, y, t) +∂y2 += ∂2v(x, y, t) +∂x∂y += 0, +(86) +and +∂h(x, y, t) +∂x += ∂h(x, y, t) +∂y += 0. +(87) +Consequently, these solutions can be expressed as +u(x, y, t) = ¯fy, +(88) +v(x, y, t) = ˜v0(t) + ˜vx(t)x + ˜vy(t)y, +(89) +and +h(x, y, t) = ˜h0(t) +(90) +where the coefficients ˜v0(t), ˜vx(t), ˜vy(t), and ˜h0(t) are time-dependent that also satisfy the +following reduced system of equations +d +dt˜v0(t) = − ˜v0(t)˜vy(t) +d +dt˜vx(t) = − ˜vx(t)˜vy(t) +d +dt˜vy(t) = − ¯f˜vx(t) − ˜vy(t)2 − ¯f2 +d +dt +˜h0(t) = − ˜h0(t)˜vy(t). +(91) +Proof : Applying Lemma 3.5 to each component in (4) yields (85)-(87). From (86), we observe +that +∂2v(x, y, t) +∂x2 += 0 yields v(x, y, t) = C1(y, t)x + C2(y, t), +where the integration constants, C1(y, t) and C2(y, t), are independent of x. Similarly, we have +∂2v(x, y, t) +∂x∂y += 0 yields C1(y, t) = ˜vx(t) +and +∂2v(x, y, t) +∂y2 += 0 yields C2(y, t) = ˜vy(t)y + ˜v0(t). +and thus (89) is achieved. Similar arguments can be made to achieve (88) and (90), respectively. +The reduced system of equations (91) is obtained via substituting (88) - (90) into (1). +□ +Theorem 3.8 : +Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed +functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +16 +C. Liu & A. D. Clark +bottom topography D = 0. If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as +(72) - (75), then the solutions of u, v, and h have the same property where +∂2u(x, y, t) +∂x2 += ∂2u(x, y, t) +∂y2 += ∂2u(x, y, t) +∂x∂y += 0, +(92) +v(x, y, t) = − ¯fx, +(93) +and +∂h(x, y, t) +∂x += ∂h(x, y, t) +∂y += 0. +(94) +Consequently, these solutions can be expressed as +u(x, y, t) = ˜u0(t) + ˜ux(t)x + ˜uy(t)y, +(95) +v(x, y, t) = − ¯fx, +(96) +and +h(x, y, t) = ˜h0(t) +(97) +where the coefficients ˜u0(t), ˜ux(t), ˜uy(t), and ˜h0(t), are time-dependent. These coefficients +satisfying +d +dt ˜u0(t) = − ˜u0(t)˜ux(t) +d +dt ˜ux(t) = − ˜ux(t)2 + ¯f ˜uy(t) − ¯f2 +d +dt ˜uy(t) = − ˜uy(t)˜ux(t) +d +dt +˜h0(t) = − ˜h0(t)˜ux(t). +(98) +Proof : Applying Lemma 3.6 to each component in (4) yields (92)-(94). From (92), we observe +that +∂2u(x, y, t) +∂x2 += 0 yields u(x, y, t) = C1(y, t)x + C2(y, t), +where the integration constants, C1(y, t) and C2(y, t), are independent of x. Similarly, we have +∂2u(x, y, t) +∂x∂y += 0 yields C1(y, t) = ˜ux(t) +and +∂2u(x, y, t) +∂y2 += 0 yields C2(y, t) = ˜uy(t)y + ˜u0(t). +and thus (95) is achieved. Similar arguments can be made to achieve (96) and (97). The +reduced equations (98) is obtained by substituting (95)-(97) into (1). +□ + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +17 +Therefore, the following results describe closed-form solutions for anticyclonic vortices with +finite escape times. +Theorem 3.9 : +For any flows over flat bottom topographies (D = 0) with a constant Coriolis +parameter ( ¯f ̸= 0) and initial constant free surface height (h0), the solutions u, v, and h with +respect to their corresponding initial conditions are defined as follows. +(i) If the initial behaviour is defined by (55) then +u(x, y, t) = ¯fy, v(x, y, t) = − ¯fy tan( ¯ft), h(x, y, t) = h0 sec( ¯ft). +(99) +(ii) If the initial behaviour is defined by (56) then +u(x, y, t) = ¯f y, +v(x, y, t) = ¯fy sec( ¯ft) − ¯fy tan( ¯ft) + x +� +− d +dt tan( ¯ft) + d +dt sec( ¯ft) +� +, +h(x, y, t) =h0 +¯f +� d +dt tan( ¯ft) − d +dt sec( ¯ft) +� +. +(100) +Furthermore, these solutions describe anticyclonic vortices with finite escape times that are +based on the initial zonal velocity being represented as u(x, y, 0) = u0(x, y) = ¯fy. +Proof : Equations (55) and (56) satisfy Theorem 3.7, where these flows can be represented +by (91). The initial conditions (55) require +˜v0(t = 0) = ˜vx(t = 0) = ˜vy(t = 0) = 0 +(101) +and +˜h0(t = 0) = h0. +(102) +Similarly, the initial conditions (56) require +˜v0(t = 0) = 0, ˜vx(t = 0) = − ¯f, ˜vy(t = 0) = ¯f, +(103) +and +˜h0(t = 0) = h0. +(104) +Solving (91) with the initial conditions, defined by (101) - (104), achieves (99) and (100). +□ +Theorem 3.10 : +For any flows over flat bottom topographies (D = 0) with a constant Cori- +olis parameter ( ¯f ̸= 0) and initial constant free surface height (h0 ̸= 0), the solutions u, v, +and h with respect to their corresponding initial conditions are defined as follows. +(i) If the initial behaviour is defined by (57) then +u(x, y, t) = − ¯fx tan( ¯ft), v(x, y, t) = − ¯fx, h(x, y, t) = h0 sec( ¯ft). +(105) +(ii) If the initial behaviour is defined by (58) then + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +18 +C. Liu & A. D. Clark +u(x, y, t) = ¯fx sec( ¯ft) − ¯fx tan( ¯ft) + y +� d +dt tan( ¯ft) − d +dt sec( ¯ft) +� +, +v(x, y, t) = − ¯fx, and h(x, y, t) = h0 +¯f +� d +dt tan( ¯ft) − d +dt sec( ¯ft) +� +. +(106) +Furthermore, these solutions describe anticyclonic vortices with finite escape times that are +based on the initial meridional velocity being represented as v(x, y, 0) = v0(x, y) = − ¯fx. +Proof : Equations (57) and (58) satisfy Theorem 3.8, where these flows can be represented +by (98). The initial conditions (57) require +˜u0(t = 0) = ˜ux(t = 0) = ˜uy(t = 0) = 0 +(107) +and +˜h0(t = 0) = h0. +(108) +Similarly, the initial conditions (58) require +˜u0(t = 0) = 0, ˜ux(t = 0) = ˜uy(t = 0) = ¯f, +(109) +and +˜h0(t = 0) = h0. +(110) +Solving (98) with the initial conditions, defined by (107) - (110), achieves (105) and (106). □ +Theorems 3.9 and 3.10 show that the flow velocity components directly depend only on the +constant Coriolis parameter whereas the free surface height depends on both the constant +Coriolis parameter and the initial free surface height. Since these solutions become infinite as +¯f → ∞, these results also represent anticyclonic vortices with finite escape times that rotate +faster and are more unstable than cyclonic ones which is consistent with previous observations +(Tsang and Dritschel 2015, McKiver 2020). These solutions also consider the nonlinear bal- +ance between the inertial and Coriolis terms in the momentum portion of the shallow water +equations, which is important to understand irregularities between cyclonic and anticyclonic +vortices which also improves previous results using quasi-geostrophic approximations (Val- +lis 2019, McKiver 2020), linear stability analysis techniques (Clark and Herron 2013), and +numerical approaches (Tsang and Dritschel 2015). +4. +Numerical Validation and Results +Numerical validation is provided via examining the convergence and accuracy of the partial +sums of u, v, and h (given by SN(u), SN(v), and SN(h)) against the governing equations (1), +the exact solutions (u, v, and h), and numerical solutions (ˆu, ˆv, and ˆh) via the relative integral +squared error defined as +E(N) = +� Lx +−Lx +� Ly +−Ly +� T +0 e(N; x, y, t) dt dx dy +� Lx +−Lx +� Ly +−Ly +� T +0 (u2 + v2 + h2) dt dx dy +, +(111) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +19 +where Lx = 1, Ly = 1, and T = 1. The convergence Ec(N) is measured by evaluating (111) +with +e(N; x, y, t) = +� +∂SN(u) +∂t ++ SN(u)∂SN(u) +∂x ++ SN(v)∂SN(u) +∂y ++ 1 +F 2 +∂SN(h) +∂x +− ¯fSN(v) +�2 ++ +� +∂SN(v) +∂t ++ SN(u)∂SN(v) +∂x ++ SN(v)∂SN(v) +∂y ++ 1 +F 2 +∂SN(h) +∂y ++ ¯fSN(u) +�2 ++ +� +∂SN(h) +∂t ++ ∂ +∂x[SN(u)(SN(h) + D)] + ∂ +∂y[SN(v)(SN(h) + D)] +�2 +. +(112) +Eex(N) is the accuracy of the partial sums of u, v, and h against the exact solutions ˆEex +which is measured via evaluating (111) with +e(N; x, y, t) = (SN(u) − u)2 + (SN(v) − v)2 + (SN(h) − h)2 . +(113) +ˆE(N) is the accuracy of the numerical solutions against the partial sums of u, v, and h which +is measured via evaluating (111) with +e(N; x, y, t) = (SN(u) − ˆu)2 + (SN(v) − ˆv)2 + +� +SN(h) − ˆh +�2 +. +(114) +ˆEex is the accuracy between the numerical and exact solutions, which is measured via evalu- +ating (111) with +e(N; x, y, t) = (u − ˆu)2 + (v − ˆv)2 + +� +h − ˆh +�2 +. +(115) +In all evaluations, we follow Matskevich and Chubarov (2019) where F = 1 represents the +characteristic velocity as U0 = √gH0. The summaries of all parameters used for our evalua- +tions are listed in Table 1 below. Equation (111) is discretised with spatial grid spacings of +∆x = 0.1 and ∆y = 0.1 and a temporal grid spacing of ∆t = 0.1. Numerical implementa- +tions (ˆu, ˆv, and ˆh) are done using the large-particle method as outlined by Matskevich and +Chubarov (2019). +Table 1.: Summary of evaluation parameters, initial conditions, and applicable exact +solutions used to validate Conditions I-VII. +Condition +F +¯f +D0 +L +l +Other Parameters +Exact solutions +I +1 +0.5 +0 +- +- +ηx = 10−4 +Theorem 3.3 +II +1 +0.5 +0 +- +- +ηy = 10−4 +Corollary 3.4(i) +III +1 +0.5 +0 +- +- +ηx = ηy = 10−4 +Corollary 3.4(ii) +IV +1 +0.5 +0 +- +- +h0 = 10−4 +Theorem 3.9(i) +V +1 +0.5 +0 +- +- +h0 = 10−4 +Theorem 3.9(ii) +VI +1 +0.5 +0 +- +- +h0 = 10−4 +Theorem 3.10(i) +VII +1 +0.5 +0 +- +- +h0 = 10−4 +Theorem 3.10(ii) + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +20 +C. Liu & A. D. Clark +4.1. +Results +Table 2 presents a summary of the convegence and accuracy results, where the partial sums +(for N = 2, 4 and 6) was used to assess the level of convergence. We note the convergence +trend where the relative error margins stabilise between O +� +10−11� +and O +� +10−6� +at N = 6, +which indicate that the Adomian approximations of up to six terms in its partial sum yield +effective and robust estimates for Conditions I-VII. This is further validated when examining +the accuracy of these partial sums with the numerical solutions, where the accuracies range +between O +� +10−6� +and O +� +10−4� +. We also note the comparisons between the explicit solutions +generated for Conditions I-VII and the numerical solutions, where these deviations are also +miniscule. +Table 2.: Summary of convergence trend (for N = 2, 4 and 6) and accuracy (for N = 6) via +integral squared error E(N) calculations for Conditions I-VII. +Ec(N = 2) +Ec(N = 4) +Ec(N = 6) +Eex(N = 6) +ˆE(N = 6) +ˆEex +Minimum +3.3 × 10−3 +1.5 × 10−6 +8.2 × 10−11 +1.6 × 10−12 +3.1 × 10−6 +3.1 × 10−6 +Maximum +6.5 × 10−3 +2.5 × 10−4 +7.0 × 10−6 +1.3 × 10−7 +2.1 × 10−4 +2.1 × 10−4 +Figures 2 through 3 present the behaviour of the ADM partial sums of u, v, and h (for +N = 6) along with Conditions II and IV (Section 3) are used as examples. In each case +we note the direct relationship between the initial conditions (Figures 2-3 part (a)) and a +temporal snapshot of the behaviour of the corresponding partial sums at t = 1 (Figures 2-3 +part (b)), which illustrates the velocity vector field u = ⟨u (x, y, 1) , v (x, y, 1)⟩ over the contour +representing the free surface height h (x, y, 1). Figure 2 (part (a)) shows the initial zero velocity +over a parabolic mound of constant free surface height η = 10−4, which corresponds to the +initial conditions represented by (27). Figure 2 (part b) confirms the temporal behaviour +where we note the behaviour of u over the contour, which is analogous to the exact solutions +described in equations (51) and (52). However, in Figure 2 (b) we also observe the rotating +velocity field u over the contour illustrating the behaviour of inertial geostrophic oscillations. +These effects are not only driven by the pressure gradient due to variations in the free surface +height but also due to the Coriolis force, which are also noticed analytically when constructing +the ADM decompositions. These confirmations continue in Figure 3, where part (a) illustrates +the behaviour of the initial velocity u0 = ⟨u (x, y, 0) , v (x, y, 0)⟩ with respect to the initial free +surface height h0 = h (x, y, 0). Figure 3 also shows the correlation between the initial conditions +and analytical solutions to Condition IV while also illustrating the effects of anticyclonic +vortices with finite escape time as shown in Figure 3 (b). Specifically, we note the clockwise +orientation of u that is consistent with the behaviour of anticyclonic vortices which are valid +for t ∈ +� +0, π/ +� +2 ¯f +�� +. +5. +Discussion +This work employs Adomian decomposition methods (ADMs) to the shallow water equations, +where we made the following main contributions. First, we used these methods as reverse +engineering mechanisms to develop theoretical connections between the ansatz formulations +of previous works, such as Thacker (1981), Shapiro (1996) and Matskevich and Chubarov +(2019), as well as develop a connection to the corresponding reduced systems of shallow +water equations. Furthermore, we developed some novel families of closed-form solutions that +respectively describe inertial oscillations and anticyclonic vortices with finite escape times over +flat bottom topographies. We perform various numerical experiments against several cases that + +January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +Semi-Analytical Solutions of Shallow Water Waves +21 +(a) +(b) +Figure 2.: Velocity vector field u (arrow) and free surface height h (contour) behaviour for +Case II: (a) initial condition at t = 0 and (b) partial sum approximation based on ADM +(with N = 6) at t = 1. Parameters used include F = 1, ¯f = 0.5, D0 = 0, and ηx = 10−4. +(a) +(b) +Figure 3.: Velocity vector field u (arrow) and free surface height h (contour) behaviour for +Case IV: (a) initial condition at t = 0, (b) partial sum approximation based on ADM (with +N = 6) at t = 1. Parameters used include F = 1, ¯f = 0.5, D0 = 0, and h0 = 10−4. +yielded relative errors between O +� +10−6� +and O +� +10−4� +. Our numerical visualizations further +demonstrate the validity of our approach, which illustrate the consistency with the dynamic +behaviour for several scenarios while also preserving the correlation between the physical +parameters. +Our study establishes the flexibility of these methods in terms of not only preserving the +correlation of parameters with respect to the overall nonlinear physical behaviour but also +alleviating the need to make restrictive assumptions like those based on the overall flow +behaviour. Moreover, we illustrate that these techniques can be used to analytically deduce +other aspects of shallow water phenomenon such as the characteristics of initial flows in which, +to the best of our knowledge, this work is the first to explore these concepts. Therefore, some +avenues of future work include extending these techniques to understand the implications of +external forces such as the effects of bottom friction which are applicable to understanding +various coastal effects such as impacts from tsunamis. Another area of research is extending +this framework to analyse practical bottom topographies and shocks, which will consider + +X10-4 +1 +5 +0.5 +0 +0 +9 +-0.5 +-1 +5 +-1 +-0.5 +0 +0.5 +1X10-4 +5 +0.5 +0 +9 +-0.5 +-0.5 +0 +0.5 +1X10-4 +5 +4 +0.5 +3 +0 +9 +2 +-0.5 +-0.5 +0 +0.5 +1X10-4 +5 +4 +0.5 +3 +0 +9 +2 +-0.5 +-0.5 +0 +0.5 +1January 10, 2023 +Geophysical and Astrophysical Fluid Dynamics +main +22 +REFERENCES +bottom terrains that extend beyond those of parabolic shapes. +Disclosure Statement +No potential conflict of interest was reported by the author(s). +Article Word Count +6,645 words +References +Adomian, G., A review of the decomposition method and some recent results for nonlinear equations. 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Journal of Fluid Mechanics, 2015, 762, 196–231. +Vallis, G.K., Atmospheric and oceanic fluid dynamics, 2017 (Cambridge University Press). +Vallis, G.K., Essentials of Atmospheric and Oceanic Dynamics, 2019 (Cambridge University Press). +Wintermeyer, N., Winters, A.R., Gassner, G.J. and Warburton, T., An entropy stable discontinuous Galerkin +method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. +Journal of Computational Physics, 2018, 375, 447–480. + diff --git a/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/load_file.txt b/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0941b9160461144ce12643ad0daaeeedf69680ba --- /dev/null +++ b/Y9E1T4oBgHgl3EQfJwPp/content/tmp_files/load_file.txt @@ -0,0 +1,598 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf,len=597 +page_content='January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Geophysical and Astrophysical Fluid Dynamics Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 00, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 00, 00 Month 2023, 1–23 Semi-Analytical Solutions of Shallow Water Waves with Idealised Bottom Topographies CHANG LIU † ‡ ∗ and ANTWAN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' CLARK † † Department of Applied Mathematics and Statistics, Johns Hopkins University Baltimore, MD 21218 USA ‡ Department of Physics, University of California, Berkeley Berkeley, CA 94720 USA (v4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4 released July 2022) Analysing two-dimensional shallow water equations with idealised bottom topographies have many ap- plications in the atmospheric and oceanic sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' however, restrictive flow pattern assumptions have been made to achieve explicit solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' This work employs Adomian decomposition methods (ADMs) to develop semi-analytical formulations of these problems that preserve the direct correlation of the physical parameters while capturing the nonlinear phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Furthermore, we exploit these techniques as reverse engineering mechanisms to develop key connections between some prevalent ansatz formulations in the open literature as well as developing new families of exact solutions describing geostrophic inertial oscillations and anticy- clonic vortices with finite escape times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Our semi-analytical evaluations show the promise of this approach in terms of providing robust approximations against several oceanic variations and bottom topographies while also preserving the direct correlation between the physical parameters such as the Froude number, the bottom topography, the Coriolis parameter, as well as the flow and free surface behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Our numerical validations provide additional confirmations of this approach while also illustrating that ADMs can also be used to provide insight and deduce novel solutions that have not been explored, which can be used to characterize various types of geophysical flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Keywords: Adomian decomposition methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' shallow water equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' bottom topographies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Introduction Analysing two-dimensional shallow water equations has been extensively studied in geophys- ical fluid dynamics to understand a myriad of atmospheric and oceanic phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Some examples include understanding the effects of long-term oceanic waves (Pedlosky 2013, Vallis 2017), analyzing the behaviour of oceanic warm-core rings (Cushman-Roisin 1987), investigat- ing flows in channels and shorelines (Shapiro 1996, Sampson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2005), studying steady-state flows (Iacono 2005, Sun 2016), and grasping the temporal instability of barotropic zonal flows (Clark and Herron 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These theoretical analyses also serve as a good basis for numer- ical simulations and validations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' For example, the creators of the Shallow Water Analytic Solutions for Hydraulic and Environmental Studies (SWASHES) software library (Delestre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2013) incorporated a significant number of theoretical solutions of the shallow water equations in the open literature, which has been cited by over 200 research papers currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Furthermore, several of the solutions in this library are obtained from Thacker (1981) in which have been widely used to demonstrate the validity and accuracy of several numerical schemes including finite volume schemes (Gallardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2007, Bollermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2011, Nikolos and Delis 2009) and discontinuous Galerkin methods (Ern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2008, Kesserwani and Liang 2012, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2017, Wintermeyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Some significant advancements include the original ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Email: cliu124@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='jh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='02957v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='flu-dyn] 8 Jan 2023 January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark works of Ball and Thacker who demonstrated that nonlinear oscillations can be modelled as either low-order polynomials or normal modes (Ball 1963, 1964, 1965, Thacker 1977, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Researchers also developed elliptical vortex solutions to understand the temporal effects of oceanic warm-core rings including stationary clockwise rotations (rodons), pulsating circu- lar eddies (pulsons), and a subclass of these phenomena called pulsrodons (Cushman-Roisin 1987, Cushman-Roisin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 1985, Rogers 1989b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Extensions to these approaches have been made, where some examples include the work of Sachdev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (Sachdev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 1996) who ex- tended the approach of (Clarkson and Kruskal 1989) and derived new families of solutions in paraboloidal basins that provided additional insights in terms of describing flow behaviour due to deformation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Additionally, Matskevich and Chubarov (2019) extended the results of Ball and Thacker to include the effects of Coriolis forces and bottom friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Bristeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (Bristeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2021) also extended the results of Thacker and introduced two respective solutions describing velocity distributed along the vertical axis and velocity accounting for variable density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Group analysis was also explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Some pioneering works in this area include that of Curr`o (Curr`o 1989) and Rodgers (Rogers 1989a) who also advanced the works of Thacker and Ball and related several forms of the depth function as well as developed invariance theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Levi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (Levi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 1989) developed symmetry reductions for flows with elliptic and circular bot- tom topographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Bila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (Bila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2006) derived Lie point symmetries and conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Chesnokov (2009) discovered 9-dimensional Lie algebra point symmetries and developed transformations between rotating and non-rotating cases, which were later used to describe spatial oscillations in spinning paraboloids (Chesnokov 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Some recent advancements in- clude Meleshko (2020) and Bihlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (Bihlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2020) who performed group classification and analysis for zero and constant Coriolis parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Meanwhile, Meleshko and Samatova (2020) performed similar analysis and considered the beta-plane approximation of the Coriolis parameter and an irregular bottom topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' However, deriving theoretical solutions to the two-dimensional shallow water equations poses the following main challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' First, these efforts involve making specific assumptions regard- ing the flow conditions which only satisfy specific cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Some solutions also contain combi- nations of special functions and integral expressions (Shapiro 1996, Rogers 1989b), which in turn makes it difficult to determine the correlation between the physical quantities of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Finding invariant solutions via group analysis has the additional advantage of deriv- ing conservation laws to these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' However, this approach depends on the construction of Lie-groups which depend on the problem formulation as well as specific assumptions such as the Coriolis parameter and bottom topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Therefore, there is a need to find solutions that are not only flexible, in terms of relaxing certain limiting assumptions, but also provide a direct correlation of the physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' This work applies Adomian decomposition methods (ADMs) (Adomian 1990) to the shal- low water equations to provide the following main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' First, we present the ADM formulation of the rotating shallow water equations where we also present key connections between the ansatz formulations in the work of Thacker (1981), Shapiro (1996), Matskevich and Chubarov (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Next, we derive and present some new families of exact solutions, for flat bottom topographies, that describe inertial oscillations in geostrophic flows and anticy- clonic vortices with finite escape times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' This rest of this paper is organised in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Section 2 presents the ADM formulation and initial theoretical formulation of the problem, where we present the connection to fundamental assumptions on the formulation of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Section 3 presents derivations of new families of solutions and their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Section 4 provides numerical experimentation and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Section 5 provides some concluding remarks, where we also list some future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Adomian Decomposition Formulation Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=': Illustration of a thin layer of incompressible flow under the Earth’s rotation described by rotating shallow-water equations with idealised bottom topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The non-dimensional form of the governing equations is defined as ∂u ∂t = − u∂u ∂x − v∂u ∂y − 1 F 2 ∂h ∂x + ¯fv ∂v ∂t = − u∂v ∂x − v∂v ∂y − 1 F 2 ∂h ∂y − ¯fu ∂h ∂t = − ∂ ∂x[u(h + D)] − ∂ ∂y[v(h + D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (1) This is illustrated in Figure 1, where u and v are the flow velocity components, h is the free surface height, ¯f = fL0/U0 is the dimensionless Coriolis parameter (associated with the Coriolis force), and F = U0/√gH0 is the Froude number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Here, the spatial variables x, y, l, and L are normalised by the horizontal length scale L0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' h is normalised by a vertical length scale H0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' the horizontal velocities, u and v, are normalised by the characteristic velocity U0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' and time t is normalised by L0/U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Hence, the dimensionless form of the idealised bottom topography is defined as D(x, y) = D0 � 1 − x2 L2 − y2 l2 � (2) where D0 is also normalised by a vertical length scale H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' It is noteworthy to mention that other bottom topographies can be determined from (2) such as flat bottom (D0 = 0), circular paraboloid (l = L), and channel (l → ∞ or L → ∞) terrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Additionally, D(x, y) can also be used to incorporate linear terms in its description via change of variables (Shapiro 1996, Thacker 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The total fluid depth D +h, shown in Figure 1, follows the formulations of Thacker (1981) and Shapiro (1996) where D + h = 0 represents a moving shoreline and D +h < 0 represents dry regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' When the moving shoreline is closed, the water mass within the shoreline is conserved (Thacker 1981, Shapiro 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' When the moving shoreline is open such as in tsunami modelling, then water within a bounded domain will have mass exchange with an infinite mass reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' It is also important to mention that our explorations in this January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark section consider flow velocities that are linearly varying spatially while the free surface height either varies linearly or in a quadratic fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The initial conditions are given by u(x, y, 0) = u0(x, y), v(x, y, 0) = v0(x, y), and h(x, y, 0) = h0(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (3) Next, u, v, and h are decomposed as follows u(x, y, t) = ∞ � n=0 un(x, y, t), v(x, y, t) = ∞ � n=0 vn(x, y, t), and h(x, y, t) = ∞ � n=0 hn(x, y, t), (4) where the initial components are defined by equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' the recurrence relationships to equation (1) (for n ≥ 0) are given by un+1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' t) = L−1 t � −An � u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' ∂u ∂x � − An � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' ∂u ∂y � − 1 F 2 ∂hn ∂x + ¯fvn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' vn+1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' t) = L−1 t � −An � u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' ∂v ∂x � − An � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' ∂v ∂y � − 1 F 2 ∂hn ∂y − ¯fun � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' hn+1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' t) = L−1 t � − ∂ ∂x[An(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' h)] − ∂ ∂y[An(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' h)] − ∂ ∂x[unD] − ∂ ∂y[vnD] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (5) where Lt = ∂(·) ∂t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' L−1 t = � t 0 (·) dτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' and the Adomian polynomial representing the quadratic nonlinearity is defined as (Adomian 1990,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 2013) An(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' h) = n � j=0 ujhn−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (6) It is important to note that equation (6) can be used to approximate the quadratic nonlinear terms, such as uh, as follows uh = � ∞ � p up � � ∞ � q hq � = ∞ � n An(u, h) and thus the semi-analytical solution to (1) is expressed via the partial sums u(x, y, t) = SN(u) = N � n=0 un, v(x, y, t) = SN(v) = N � n=0 vn, and h(x, y, t) = SN(h) = N � n=0 hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (7) Next, the following results connect the properties of the initial conditions to the behaviours of the true solutions via their partial sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1: Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal parabolic topography (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such that January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 5 ∂2u0(x, y) ∂x2 = ∂2v0(x, y) ∂x2 = ∂3h0(x, y) ∂x3 = 0, (8) ∂2u0(x, y) ∂y2 = ∂2v0(x, y) ∂y2 = ∂3h0(x, y) ∂y3 = 0, (9) and ∂2u0(x, y) ∂xy = ∂2v0(x, y) ∂x∂y = ∂3h0(x, y) ∂x2∂y = ∂3h0(x, y) ∂x∂y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (10) Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t) also satisfy the same prop- erty, where ∂2un(x, y, t) ∂x2 = ∂2vn(x, y, t) ∂x2 = ∂3hn(x, y, t) ∂x3 = 0, (11) ∂2un(x, y, t) ∂y2 = ∂2vn(x, y, t) ∂y2 = ∂3hn(x, y, t) ∂y3 = 0, (12) and ∂2un(x, y, t) ∂x∂y = ∂2vn(x, y, t) ∂x∂y = ∂3hn(x, y, t) ∂x2∂y = ∂3hn(x, y, t) ∂x∂y2 = 0 (13) for n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition (11) is demonstrated by examining the following relationships ∂2un+1 ∂x2 = −L−1 t � ∂2 ∂x2 � An � u, ∂u ∂x � + An � v, ∂u ∂y �� + 1 F 2 ∂3hn ∂x3 − ¯f ∂2vn ∂x2 � , (14) ∂2vn+1 ∂x2 = −L−1 t � ∂2 ∂x2 � An � u, ∂v ∂x � + An � v, ∂v ∂y �� + 1 F 2 ∂3hn ∂x2∂y + ¯f ∂2un ∂x2 � , (15) and ∂3hn+1 ∂x3 = −L−1 t � ∂4 ∂x4 [An(u, h)] + ∂4 ∂x3∂y[An(v, h)] + ∂4 ∂x4 [unD] + ∂4 ∂x3∂y[vnD] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (16) Therefore, when n = 0 equations (14) - (16) representing the relationship between the initial and first components for u, v, and h become ∂2u1 ∂x2 = −L−1 t � ∂2 ∂x2 � u0 ∂u0 ∂x + v0 ∂u0 ∂y � + 1 F 2 ∂3h0 ∂x3 − ¯f ∂2v0 ∂x2 � , (17) ∂2v1 ∂x2 = −L−1 t � ∂2 ∂x2 � u0 ∂v0 ∂x + v0 ∂v0 ∂y � + 1 F 2 ∂3h0 ∂x2∂y + ¯f ∂2u0 ∂x2 � , (18) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark and ∂3h1 ∂x3 = −L−1 t � ∂4 ∂x4 [u0h0] + ∂4 ∂x3y [v0h0] + ∂4 ∂x4 [u0D] + ∂4 ∂x3∂y [v0D] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (19) Employing (8) - (10) it can be shown that equations (17) - (19) reduce to the following relationship ∂2u1(x, y, t) ∂x2 = ∂2v1(x, y, t) ∂x2 = ∂3h1(x, y, t) ∂x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Continuing this argument for n ∈ N+ yields equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similar arguments can be made to produce (12) and (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2 : Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given an ideal parabolic topography (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as (8) (10), then the solutions of u, v, and h have the same property where ∂2u(x, y, t) ∂x2 = ∂2u(x, y, t) ∂y2 = ∂2u(x, y, t) ∂x∂y = 0, (20) ∂2v(x, y, t) ∂x2 = ∂2v(x, y, t) ∂y2 = ∂2v(x, y, t) ∂x∂y = 0, (21) and ∂3h(x, y, t) ∂x3 = ∂3h(x, y, t) ∂x2∂y = ∂3h(x, y, t) ∂x∂y2 = ∂3h(x, y, t) ∂y3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (22) Consequently, these solutions can be expressed as u(x, y, t) = ˜u0(t) + ˜ux(t)x + ˜uy(t)y, (23) v(x, y, t) = ˜v0(t) + ˜vx(t)x + ˜vy(t)y, (24) and h(x, y, t) = ˜h0(t) + ˜hx(t)x + ˜hy(t)y + 1 2 ˜hxx(t)x2 + 1 2 ˜hyy(t)y2 + ˜hxy(t)xy, (25) where the coefficients ˜u0(t), ˜ux(t), ˜uy(t), ˜v0(t), ˜vx(t), ˜vy(t), ˜h0(t), ˜hx(t), ˜hy(t), ˜hxx(t), ˜hyy(t), and ˜hxy(t) are time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 to each component in (4) yields (20)-(22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' From (20), we observe that ∂2u(x, y, t) ∂x2 = 0 yields u(x, y, t) = C1(y, t)x + C2(y, t), where the integration constants, C1(y, t) and C2(y, t), are independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similarly, we have ∂2u(x, y, t) ∂x∂y = 0 yields C1(y, t) = ˜ux(t) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 7 and ∂2u(x, y, t) ∂y2 = 0 yields C2(y, t) = ˜uy(t)y + ˜u0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' and thus (23) is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similar arguments can be made to achieve (24) and (25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ We note the significance of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' In the works of Thacker (1981), Shapiro (1996), and Matskevich and Chubarov (2019) equations (23)-(25) were presented as ansatz solutions, where they were also used to produce the reduced system of shallow water equations to derive closed-form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' This theorem removes these assumptions and provides more insight to this behaviour by connecting it to the initial conditions (8) - (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Novel Exact Solutions for Flat Bottom Topographies with Constant Coriolis Force Next, we use the ADM construction to derive new families of solutions and their properties that describe other geophysical flows such as inertial oscillations and anticyclonic vortices which have a profound effect on oceanic and atmospheric dynamics Vallis (2017), Kafiabad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Here, we consider flows over flat bottom topologies where D0 = 0 in (2) with constant Coriolis parameter ( ¯f ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Inertial Oscillations in Geostrophic Flows For these types of flows, our analysis considers the following initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition I u0(x, y) = v0(x, y) = 0, h0(x, y) = ηxx + ηyy, (26) Condition II u0(x, y) = v0(x, y) = 0, h0(x, y) = ηxx, (27) Condition III u0(x, y) = v0(x, y) = 0, h0(x, y) = ηyy, (28) where ¯f ̸= 0 is the constant Coriolis parameter, and ηx and ηy are the respective constant free surface gradients in the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' We note that the behaviour of the initial conditions (26) - (28) affect the decomposition of the decomposed functions of u, v, and h as presented in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1: Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- tions of u, v, and h such that their relationship is defined by (5) (for n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If D = 0 and the initial conditions u0(x, y), v0(x, y), h0(x, y) satisfy the following properties ∂u0(x, y) ∂x = ∂v0(x, y) ∂x = ∂2h0(x, y) ∂x2 = 0, (29) ∂u0(x, y) ∂y = ∂v0(x, y) ∂y = ∂2h0(x, y) ∂y2 = 0, (30) and January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark ∂2h0(x, y) ∂x∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (31) Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t) also satisfy the property that ∂un(x, y, t) ∂x = ∂vn(x, y, t) ∂x = ∂hn(x, y, t) ∂x = 0 (32) and ∂un(x, y, t) ∂y = ∂vn(x, y, t) ∂y = ∂hn(x, y, t) ∂y = 0 (33) for n ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition (32) is demonstrated by examining the following relationships ∂un+1 ∂x = −L−1 t � ∂ ∂x � An � u, ∂u ∂x � + An � v, ∂u ∂y �� + 1 F 2 ∂2hn ∂x2 − ¯f ∂vn ∂x � , (34) ∂vn+1 ∂x = −L−1 t � ∂ ∂x � An � u, ∂v ∂x � + An � v, ∂v ∂y �� + 1 F 2 ∂2hn ∂x∂y + ¯f ∂un ∂x � , (35) and ∂hn+1 ∂x = −L−1 t � ∂2 ∂x2 [An(u, h)] + ∂2 ∂x∂y[An(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (36) Therefore, when n = 0, equations (34) - (36) representing the relationship between the initial and first components for u, v, and h become ∂u1 ∂x = −L−1 t � ∂ ∂x � A0 � u, ∂u ∂x � + A0 � v, ∂u ∂y �� + 1 F 2 ∂2h0 ∂x2 − ¯f ∂v0 ∂x � , (37) ∂v1 ∂x = −L−1 t � ∂ ∂x � A0 � u, ∂v ∂x � + A0 � v, ∂v ∂y �� + 1 F 2 ∂2h0 ∂x∂y + ¯f ∂u0 ∂x � , (38) and ∂h1 ∂x = −L−1 t � ∂2 ∂x2 [A0(u, h)] + ∂2 ∂x∂y[A0(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (39) Employing (29) - (31) it can be shown that equations (37) - (39) reduce to the following relationship ∂u1(x, y, t) ∂x = ∂v1(x, y, t) ∂x = ∂h1(x, y, t) ∂x = 0, and continuing this argument for n ∈ N+ yields equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Following similar arguments yields (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 9 From this, the behaviour of uniform u, v over space, and planar free surface h with constant spatial gradients over time can be summarised in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2 : Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If D = 0 and the initial conditions u0(x, y), v0(x, y), h0(x, y) satisfy the properties defined in (29) - (31), then the solutions u, v, and h have the following properties ∂u(x, y, t) ∂x = ∂u(x, y, t) ∂y = 0, (40) ∂v(x, y, t) ∂x = ∂v(x, y, t) ∂y = 0, (41) ∂h(x, y, t) ∂x = ∂h(x, y, 0) ∂x , ∂h(x, y, t) ∂y = ∂h(x, y, 0) ∂y , (42) and ∂2h(x, y, t) ∂x2 = ∂2h(x, y, t) ∂x∂y = ∂2h(x, y, t) ∂y2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (43) Additionally, u, v, and h are reduced to the following forms u(x, y, t) = ˜u0(t), (44) v(x, y, t) = ˜v0(t), (45) and h(x, y, t) = ˜h0(t) + ˜hxx + ˜hyy, (46) where the coefficients ˜u0(t), ˜v0(t), and ˜h0(t) are time-dependent, while ˜hx and ˜hy are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Additionally, (44) - (46) satisfy the reduced system of equations d dt ˜u0(t) = − 1 F 2 ˜hx + ¯f˜v0(t), d dt˜v0(t) = − 1 F 2 ˜hy − ¯f ˜u0(t), d dt ˜h0(t) = − ˜u0(t)˜hx − ˜v0(t)˜hy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (47) Proof : Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 to each component in (4) yields (40)-(43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' From (40), we observe that ∂u(x, y, t) ∂x = 0 yields u(x, y, t) = C1(y, t), where the integration constants, C1(y, t), are independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similarly, we have January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark ∂u(x, y, t) ∂y = 0 yields C1(y, t) = ˜u0(t) and thus (44) is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similar arguments can be made to achieve (45) and (46), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Substituting (44) - (46) into (1) achieves the reduced system of equations (47), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Hence, we have the following results for inertial oscillations for geostrophic flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 : Given inertial oscillations over flat bottom topographies with constant Cori- olis parameter ¯f ̸= 0, where the initial behaviour is defined by (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The solutions u, v, and h are expressed as u(x, y, t) = − ηx ¯fF 2 sin � ¯ft � − ηy ¯fF 2 � 1 − cos � ¯ft �� , (48) v(x, y, t) = ηx ¯fF 2 � 1 − cos � ¯ft �� − ηy ¯fF 2 sin( ¯ft), (49) and h(x, y, t) = η2 x ¯f2F 2 � 1 − cos � ¯ft �� + xηx + η2 y ¯f2F 2 � 1 − cos � ¯ft �� + ηyy (50) where ηx and ηy are the constant free surface gradients in the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : The initial conditions (26) satisfy (29) - (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Therefore, the sequence of decomposed functions {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} satisfy (32) and (33) for n ∈ N+ which sat- isfies Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 and consequently Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Examining the system of reduced equations (47), the initial conditions (26) also produce the following reduced relationships: ˜hx = ηx, ˜hy = ηy, and ˜u(t = 0) = ˜v(t = 0) = ˜h0(t = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Solving this reduced system achieves (48) (50) which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4: Given inertial oscillations over flat bottom topographies with constant Cori- olis parameter ¯f ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (i) If the initial behaviour is defined by (27), then the solutions u, v, and h are expressed as u(x, y, t) = − ηx ¯fF 2 sin � ¯ft � , v(x, y, t) = ηx ¯fF 2 � 1 − cos � ¯ft �� , (51) and h(x, y, t) = η2 x ¯f2F 2 � 1 − cos � ¯ft �� + xηx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (52) (ii) If the initial behaviour is defined by (28), then the solutions u, v, and h are expressed as u(x, y, t) = − ηy ¯fF 2 � 1 − cos � ¯ft �� , v(x, y, t) = − ηy ¯fF 2 sin � ¯ft � , (53) and h(x, y, t) = η2 y ¯f2F 2 � 1 − cos � ¯ft �� + ηyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (54) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 11 ηx and ηy are the constant free surface gradients in the x and y directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : This is a special case of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 for ηx = 0 and ηy = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4 show the explicit relationship between the these types of flows with respect to the constant Coriolis parameter, the free surface gradients, and the Froude number where the inertial oscillation frequency is defined by the constant Coriolis parameter ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These results also demonstrate that these oscillations are based on the magnitude of the free surface gradients that depend on the initial behaviour and the geostrophic flows, which are consistent with the results of (Vallis 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Moreover, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 describes these types of oscillations as the interaction between the geostrophic flow fluctuations and the free surface gradients, where Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4 considers cases when these gradients are negligible in the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Anticyclonic Vortices with Finite Escape Times For these types of flows our analysis considers the following initial conditions Condition IV u0(x, y) = ¯fy, v0(x, y) = 0, h0(x, y) = h0, (55) Condition V u0(x, y) = ¯fy, v0(x, y) = − ¯fx + ¯fy, h0(x, y) = h0, (56) Condition VI u0(x, y) = 0, v0(x, y) = − ¯fx, h0(x, y) = h0, (57) Condition VII u0(x, y) = ¯fx + ¯fy, v0(x, y) = − ¯fx, h0(x, y) = h0, (58) where h0 is the constant free surface height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These describe anticyclonic vortices for the initial vorticity is proportional to the negative constant Coriolis parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The behaviour of the initial conditions (55) - (58) affect the decomposition of the decomposed functions of u, v, and h as presented in the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5: Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom topography D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such that u0(x, y) = ¯fy, (59) ∂2v0(x, y) ∂x2 = ∂h0(x, y) ∂x = 0, (60) ∂2v0(x, y) ∂y2 = ∂h0(x, y) ∂y = 0, (61) and ∂2v0(x, y) ∂x∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (62) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t), for n ∈ N+ satisfy un(x, y, t) = 0, (63) ∂2vn(x, y, t) ∂x2 = ∂hn(x, y, t) ∂x = 0, (64) ∂2vn(x, y, t) ∂y2 = ∂hn(x, y, t) ∂y = 0, (65) and ∂2vn(x, y, t) ∂x∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (66) Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition (63) is demonstrated by examining un+1 = −L−1 t �� An � u, ∂u ∂x � + An � v, ∂u ∂y �� + 1 F 2 ∂hn ∂x − ¯f vn � , (67) In the case of n = 0 and using (59) - (62), it reduces to u1 = −L−1 t �� A0 � u, ∂u ∂x � + A0 � v, ∂u ∂y �� + 1 F 2 ∂h0 ∂x − ¯f v0 � = −L−1 t � A0 � v, ∂u ∂y � − ¯f v0 � = −L−1 t � v0 ∂u0 ∂y − ¯f v0 � = 0, and continuing this argument for n = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' , n − 1} yields equation (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition (64) is demonstrated by examining the following relationships ∂2vn+1 ∂x2 = −L−1 t � ∂2 ∂x2 � An � u, ∂v ∂x � + An � v, ∂v ∂y �� + 1 F 2 ∂3hn ∂x2∂y + ¯f ∂2un ∂x2 � , (68) and ∂hn+1 ∂x = −L−1 t � ∂2 ∂x2 [An(u, h)] + ∂2 ∂x∂y[An(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (69) Therefore, when n = 0, equations (68) - (69) representing the relationship between the initial and first components for v and h become ∂2v1 ∂x2 = −L−1 t � ∂2 ∂x2 � A0 � u, ∂v ∂x � + A0 � v, ∂v ∂y �� + 1 F 2 ∂3h0 ∂x2∂y + ¯f ∂2u0 ∂x2 � , (70) and January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 13 ∂h1 ∂x = −L−1 t � ∂2 ∂x2 [A0(u, h)] + ∂2 ∂x∂y[A0(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (71) Employing (59) - (62), it can be shown that equations (70) - (71) reduce to the following relationship ∂2v1(x, y, t) ∂x2 = ∂h1(x, y, t) ∂x = 0, and continuing this argument for n = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' , n − 1} yields equation (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Following similar arguments yields (65) and (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='6: Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed func- tions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat bottom topography D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined such that v0(x, y, t) = − ¯fx, (72) ∂2u0(x, y) ∂x2 = ∂h0(x, y) ∂x = 0, (73) ∂2u0(x, y) ∂y2 = ∂h0(x, y) ∂y = 0, (74) and ∂2u0(x, y) ∂x∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (75) Then the higher order components un(x, y, t), vn(x, y, t), hn(x, y, t), for n ∈ N+ satisfy the property vn(x, y, t) = 0, (76) ∂2un(x, y, t) ∂x2 = ∂hn(x, y, t) ∂x = 0, (77) ∂2un(x, y, t) ∂y2 = ∂hn(x, y, t) ∂y = 0, (78) and ∂2un(x, y, t) ∂x∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (79) Proof : This is proven via mathematical induction by examining the recursion relationships for u, v, and h in equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition (76) is demonstrated by examining the following relationships January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark vn+1 = −L−1 t �� An � u, ∂v ∂x � + An � v, ∂v ∂y �� + 1 F 2 ∂hn ∂y + ¯f un � , (80) At n = 0, we have v1 = −L−1 t �� A0 � u, ∂v ∂x � + A0 � v, ∂v ∂y �� + 1 F 2 ∂h0 ∂y + ¯f u0 � = −L−1 t � A0 � u, ∂v ∂x � + ¯f u0 � = −L−1 t � −u0 ¯f + ¯f u0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Employing a similar argument for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' , n − 1, we have (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Equation (77) is demon- strated by examining the following ∂2un+1 ∂x2 = −L−1 t � ∂2 ∂x2 � An � u, ∂u ∂x � + An � v, ∂u ∂y �� + 1 F 2 ∂3hn ∂x3 − ¯f ∂2vn ∂x2 � , (81) and ∂hn+1 ∂x = −L−1 t � ∂2 ∂x2 [An(u, h)] + ∂2 ∂x∂y[An(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (82) Therefore, when n = 0, equations (81) - (82) representing the relationship between the initial and first components for u and h become ∂2u1 ∂x2 = −L−1 t � ∂2 ∂x2 � u0 ∂u0 ∂x + v0 ∂u0 ∂y � + 1 F 2 ∂3h0 ∂x3 − ¯f ∂2v0 ∂x2 � , (83) and ∂h1 ∂x = −L−1 t � ∂2 ∂x2 [A0(u, h)] + ∂2 ∂x∂y[A0(v, h)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (84) Employing (72) - (75), it can be shown that equations (83) - (84) reduce to the following relationship ∂2u1(x, y, t) ∂x2 = ∂h1(x, y, t) ∂x = 0, and continuing this argument for n = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' , n − 1} yields equation (77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Following similar arguments yields (78) and (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Therefore, the behaviour of u, v, and h can be summarised in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='7 : Given a flat bottom topography, let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed functions of u, v, and h, defined by (5) (for n ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as (59) - (62), then the solutions of u, v, and h have the same property where u(x, y, t) = ¯fy, (85) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 15 ∂2v(x, y, t) ∂x2 = ∂2v(x, y, t) ∂y2 = ∂2v(x, y, t) ∂x∂y = 0, (86) and ∂h(x, y, t) ∂x = ∂h(x, y, t) ∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (87) Consequently, these solutions can be expressed as u(x, y, t) = ¯fy, (88) v(x, y, t) = ˜v0(t) + ˜vx(t)x + ˜vy(t)y, (89) and h(x, y, t) = ˜h0(t) (90) where the coefficients ˜v0(t), ˜vx(t), ˜vy(t), and ˜h0(t) are time-dependent that also satisfy the following reduced system of equations d dt˜v0(t) = − ˜v0(t)˜vy(t) d dt˜vx(t) = − ˜vx(t)˜vy(t) d dt˜vy(t) = − ¯f˜vx(t) − ˜vy(t)2 − ¯f2 d dt ˜h0(t) = − ˜h0(t)˜vy(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (91) Proof : Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 to each component in (4) yields (85)-(87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' From (86), we observe that ∂2v(x, y, t) ∂x2 = 0 yields v(x, y, t) = C1(y, t)x + C2(y, t), where the integration constants, C1(y, t) and C2(y, t), are independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similarly, we have ∂2v(x, y, t) ∂x∂y = 0 yields C1(y, t) = ˜vx(t) and ∂2v(x, y, t) ∂y2 = 0 yields C2(y, t) = ˜vy(t)y + ˜v0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' and thus (89) is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similar arguments can be made to achieve (88) and (90), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The reduced system of equations (91) is obtained via substituting (88) - (90) into (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='8 : Let {un(x, y, t)}, {vn(x, y, t)}, {hn(x, y, t)} be the sequence of decomposed functions of u, v, and h, where their relationship is defined by (5) (for n ∈ N) given a flat January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark bottom topography D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' If the initial conditions u0(x, y), v0(x, y), h0(x, y) are defined as (72) - (75), then the solutions of u, v, and h have the same property where ∂2u(x, y, t) ∂x2 = ∂2u(x, y, t) ∂y2 = ∂2u(x, y, t) ∂x∂y = 0, (92) v(x, y, t) = − ¯fx, (93) and ∂h(x, y, t) ∂x = ∂h(x, y, t) ∂y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (94) Consequently, these solutions can be expressed as u(x, y, t) = ˜u0(t) + ˜ux(t)x + ˜uy(t)y, (95) v(x, y, t) = − ¯fx, (96) and h(x, y, t) = ˜h0(t) (97) where the coefficients ˜u0(t), ˜ux(t), ˜uy(t), and ˜h0(t), are time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These coefficients satisfying d dt ˜u0(t) = − ˜u0(t)˜ux(t) d dt ˜ux(t) = − ˜ux(t)2 + ¯f ˜uy(t) − ¯f2 d dt ˜uy(t) = − ˜uy(t)˜ux(t) d dt ˜h0(t) = − ˜h0(t)˜ux(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (98) Proof : Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='6 to each component in (4) yields (92)-(94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' From (92), we observe that ∂2u(x, y, t) ∂x2 = 0 yields u(x, y, t) = C1(y, t)x + C2(y, t), where the integration constants, C1(y, t) and C2(y, t), are independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similarly, we have ∂2u(x, y, t) ∂x∂y = 0 yields C1(y, t) = ˜ux(t) and ∂2u(x, y, t) ∂y2 = 0 yields C2(y, t) = ˜uy(t)y + ˜u0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' and thus (95) is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Similar arguments can be made to achieve (96) and (97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The reduced equations (98) is obtained by substituting (95)-(97) into (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 17 Therefore, the following results describe closed-form solutions for anticyclonic vortices with finite escape times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='9 : For any flows over flat bottom topographies (D = 0) with a constant Coriolis parameter ( ¯f ̸= 0) and initial constant free surface height (h0), the solutions u, v, and h with respect to their corresponding initial conditions are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (i) If the initial behaviour is defined by (55) then u(x, y, t) = ¯fy, v(x, y, t) = − ¯fy tan( ¯ft), h(x, y, t) = h0 sec( ¯ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (99) (ii) If the initial behaviour is defined by (56) then u(x, y, t) = ¯f y, v(x, y, t) = ¯fy sec( ¯ft) − ¯fy tan( ¯ft) + x � − d dt tan( ¯ft) + d dt sec( ¯ft) � , h(x, y, t) =h0 ¯f � d dt tan( ¯ft) − d dt sec( ¯ft) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (100) Furthermore, these solutions describe anticyclonic vortices with finite escape times that are based on the initial zonal velocity being represented as u(x, y, 0) = u0(x, y) = ¯fy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : Equations (55) and (56) satisfy Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='7, where these flows can be represented by (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The initial conditions (55) require ˜v0(t = 0) = ˜vx(t = 0) = ˜vy(t = 0) = 0 (101) and ˜h0(t = 0) = h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (102) Similarly, the initial conditions (56) require ˜v0(t = 0) = 0, ˜vx(t = 0) = − ¯f, ˜vy(t = 0) = ¯f, (103) and ˜h0(t = 0) = h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (104) Solving (91) with the initial conditions, defined by (101) - (104), achieves (99) and (100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='10 : For any flows over flat bottom topographies (D = 0) with a constant Cori- olis parameter ( ¯f ̸= 0) and initial constant free surface height (h0 ̸= 0), the solutions u, v, and h with respect to their corresponding initial conditions are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (i) If the initial behaviour is defined by (57) then u(x, y, t) = − ¯fx tan( ¯ft), v(x, y, t) = − ¯fx, h(x, y, t) = h0 sec( ¯ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (105) (ii) If the initial behaviour is defined by (58) then January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark u(x, y, t) = ¯fx sec( ¯ft) − ¯fx tan( ¯ft) + y � d dt tan( ¯ft) − d dt sec( ¯ft) � , v(x, y, t) = − ¯fx, and h(x, y, t) = h0 ¯f � d dt tan( ¯ft) − d dt sec( ¯ft) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (106) Furthermore, these solutions describe anticyclonic vortices with finite escape times that are based on the initial meridional velocity being represented as v(x, y, 0) = v0(x, y) = − ¯fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Proof : Equations (57) and (58) satisfy Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='8, where these flows can be represented by (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The initial conditions (57) require ˜u0(t = 0) = ˜ux(t = 0) = ˜uy(t = 0) = 0 (107) and ˜h0(t = 0) = h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (108) Similarly, the initial conditions (58) require ˜u0(t = 0) = 0, ˜ux(t = 0) = ˜uy(t = 0) = ¯f, (109) and ˜h0(t = 0) = h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (110) Solving (98) with the initial conditions, defined by (107) - (110), achieves (105) and (106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' □ Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='10 show that the flow velocity components directly depend only on the constant Coriolis parameter whereas the free surface height depends on both the constant Coriolis parameter and the initial free surface height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Since these solutions become infinite as ¯f → ∞, these results also represent anticyclonic vortices with finite escape times that rotate faster and are more unstable than cyclonic ones which is consistent with previous observations (Tsang and Dritschel 2015, McKiver 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These solutions also consider the nonlinear bal- ance between the inertial and Coriolis terms in the momentum portion of the shallow water equations, which is important to understand irregularities between cyclonic and anticyclonic vortices which also improves previous results using quasi-geostrophic approximations (Val- lis 2019, McKiver 2020), linear stability analysis techniques (Clark and Herron 2013), and numerical approaches (Tsang and Dritschel 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Numerical Validation and Results Numerical validation is provided via examining the convergence and accuracy of the partial sums of u, v, and h (given by SN(u), SN(v), and SN(h)) against the governing equations (1), the exact solutions (u, v, and h), and numerical solutions (ˆu, ˆv, and ˆh) via the relative integral squared error defined as E(N) = � Lx −Lx � Ly −Ly � T 0 e(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' x, y, t) dt dx dy � Lx −Lx � Ly −Ly � T 0 (u2 + v2 + h2) dt dx dy , (111) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 19 where Lx = 1, Ly = 1, and T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The convergence Ec(N) is measured by evaluating (111) with e(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' x, y, t) = � ∂SN(u) ∂t + SN(u)∂SN(u) ∂x + SN(v)∂SN(u) ∂y + 1 F 2 ∂SN(h) ∂x − ¯fSN(v) �2 + � ∂SN(v) ∂t + SN(u)∂SN(v) ∂x + SN(v)∂SN(v) ∂y + 1 F 2 ∂SN(h) ∂y + ¯fSN(u) �2 + � ∂SN(h) ∂t + ∂ ∂x[SN(u)(SN(h) + D)] + ∂ ∂y[SN(v)(SN(h) + D)] �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (112) Eex(N) is the accuracy of the partial sums of u, v, and h against the exact solutions ˆEex which is measured via evaluating (111) with e(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' x, y, t) = (SN(u) − u)2 + (SN(v) − v)2 + (SN(h) − h)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (113) ˆE(N) is the accuracy of the numerical solutions against the partial sums of u, v, and h which is measured via evaluating (111) with e(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' x, y, t) = (SN(u) − ˆu)2 + (SN(v) − ˆv)2 + � SN(h) − ˆh �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (114) ˆEex is the accuracy between the numerical and exact solutions, which is measured via evalu- ating (111) with e(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' x, y, t) = (u − ˆu)2 + (v − ˆv)2 + � h − ˆh �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (115) In all evaluations, we follow Matskevich and Chubarov (2019) where F = 1 represents the characteristic velocity as U0 = √gH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' The summaries of all parameters used for our evalua- tions are listed in Table 1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Equation (111) is discretised with spatial grid spacings of ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 and ∆y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 and a temporal grid spacing of ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Numerical implementa- tions (ˆu, ˆv, and ˆh) are done using the large-particle method as outlined by Matskevich and Chubarov (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=': Summary of evaluation parameters, initial conditions, and applicable exact solutions used to validate Conditions I-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Condition F ¯f D0 L l Other Parameters Exact solutions I 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 ηx = 10−4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 II 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 ηy = 10−4 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4(i) III 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 ηx = ηy = 10−4 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='4(ii) IV 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 h0 = 10−4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='9(i) V 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 h0 = 10−4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='9(ii) VI 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 h0 = 10−4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='10(i) VII 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 h0 = 10−4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='10(ii) January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Liu & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Clark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Results Table 2 presents a summary of the convegence and accuracy results, where the partial sums (for N = 2, 4 and 6) was used to assess the level of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' We note the convergence trend where the relative error margins stabilise between O � 10−11� and O � 10−6� at N = 6, which indicate that the Adomian approximations of up to six terms in its partial sum yield effective and robust estimates for Conditions I-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' This is further validated when examining the accuracy of these partial sums with the numerical solutions, where the accuracies range between O � 10−6� and O � 10−4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' We also note the comparisons between the explicit solutions generated for Conditions I-VII and the numerical solutions, where these deviations are also miniscule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=': Summary of convergence trend (for N = 2, 4 and 6) and accuracy (for N = 6) via integral squared error E(N) calculations for Conditions I-VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Ec(N = 2) Ec(N = 4) Ec(N = 6) Eex(N = 6) ˆE(N = 6) ˆEex Minimum 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 × 10−6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='2 × 10−11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='6 × 10−12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 × 10−6 Maximum 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 × 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='0 × 10−6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='3 × 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='1 × 10−4 Figures 2 through 3 present the behaviour of the ADM partial sums of u, v, and h (for N = 6) along with Conditions II and IV (Section 3) are used as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' In each case we note the direct relationship between the initial conditions (Figures 2-3 part (a)) and a temporal snapshot of the behaviour of the corresponding partial sums at t = 1 (Figures 2-3 part (b)), which illustrates the velocity vector field u = ⟨u (x, y, 1) , v (x, y, 1)⟩ over the contour representing the free surface height h (x, y, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Figure 2 (part (a)) shows the initial zero velocity over a parabolic mound of constant free surface height η = 10−4, which corresponds to the initial conditions represented by (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Figure 2 (part b) confirms the temporal behaviour where we note the behaviour of u over the contour, which is analogous to the exact solutions described in equations (51) and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' However, in Figure 2 (b) we also observe the rotating velocity field u over the contour illustrating the behaviour of inertial geostrophic oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These effects are not only driven by the pressure gradient due to variations in the free surface height but also due to the Coriolis force, which are also noticed analytically when constructing the ADM decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' These confirmations continue in Figure 3, where part (a) illustrates the behaviour of the initial velocity u0 = ⟨u (x, y, 0) , v (x, y, 0)⟩ with respect to the initial free surface height h0 = h (x, y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Figure 3 also shows the correlation between the initial conditions and analytical solutions to Condition IV while also illustrating the effects of anticyclonic vortices with finite escape time as shown in Figure 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Specifically, we note the clockwise orientation of u that is consistent with the behaviour of anticyclonic vortices which are valid for t ∈ � 0, π/ � 2 ¯f �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Discussion This work employs Adomian decomposition methods (ADMs) to the shallow water equations, where we made the following main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' First, we used these methods as reverse engineering mechanisms to develop theoretical connections between the ansatz formulations of previous works, such as Thacker (1981), Shapiro (1996) and Matskevich and Chubarov (2019), as well as develop a connection to the corresponding reduced systems of shallow water equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Furthermore, we developed some novel families of closed-form solutions that respectively describe inertial oscillations and anticyclonic vortices with finite escape times over flat bottom topographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' We perform various numerical experiments against several cases that January 10, 2023 Geophysical and Astrophysical Fluid Dynamics main Semi-Analytical Solutions of Shallow Water Waves 21 (a) (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=': Velocity vector field u (arrow) and free surface height h (contour) behaviour for Case II: (a) initial condition at t = 0 and (b) partial sum approximation based on ADM (with N = 6) at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Parameters used include F = 1, ¯f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5, D0 = 0, and ηx = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' (a) (b) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=': Velocity vector field u (arrow) and free surface height h (contour) behaviour for Case IV: (a) initial condition at t = 0, (b) partial sum approximation based on ADM (with N = 6) at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Parameters used include F = 1, ¯f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5, D0 = 0, and h0 = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' yielded relative errors between O � 10−6� and O � 10−4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Our numerical visualizations further demonstrate the validity of our approach, which illustrate the consistency with the dynamic behaviour for several scenarios while also preserving the correlation between the physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Our study establishes the flexibility of these methods in terms of not only preserving the correlation of parameters with respect to the overall nonlinear physical behaviour but also alleviating the need to make restrictive assumptions like those based on the overall flow behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Moreover, we illustrate that these techniques can be used to analytically deduce other aspects of shallow water phenomenon such as the characteristics of initial flows in which, to the best of our knowledge, this work is the first to explore these concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Therefore, some avenues of future work include extending these techniques to understand the implications of external forces such as the effects of bottom friction which are applicable to understanding various coastal effects such as impacts from tsunamis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content=' Another area of research is extending this framework to analyse practical bottom topographies and shocks, which will consider X10-4 1 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} +page_content='5 0 0 9 0.' metadata={'source': 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Computational Physics, 2018, 375, 447–480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9E1T4oBgHgl3EQfJwPp/content/2301.02957v1.pdf'} diff --git a/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/2301.03187v1.pdf.txt b/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/2301.03187v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..08b8390fdb6d48d40dfe1defb2ea810ab27fd344 --- /dev/null +++ b/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/2301.03187v1.pdf.txt @@ -0,0 +1,2369 @@ +arXiv:2301.03187v1 [eess.SY] 9 Jan 2023 +Modeling of Four-Winged Micro +Ornithopters Inspired by Dragonflies +Oussama Sifour ∗ Soulaimane Berkane ∗∗ +Abdelhamid Tayebi ∗∗∗ +∗ Department of Computer Science and Engineering, University of +Quebec in Outaouais, Gatineau, QC, Canada. (e-mail: sifo01@uqo.ca). +∗∗ Department of Computer Science and Engineering, University of +Quebec in Outaouais, Gatineau, QC, Canada. (e-mail: +Soulaimane.Berkane@uqo.ca) +∗∗∗ Department of Electrical Engineering, Lakehead University, +Thunder Bay, ON P7B 5E1, Canada. (e-mail: atayebi@lakeheadu.ca) +Abstract: In this paper, we present a full dynamical model of a four-winged micro ornithopter +inspired by a dragonfly-type insect. The micro ornithopter is modeled as four articulated rigid +body components (wings) connected to the main body via spherical joints. The dynamical model +is derived using Lagrangian mechanics with intrinsic global coordinates, without relying on the +common assumptions that neglect the wings-body interactions. Furthermore, the aerodynamic +forces are modeled under the quasi-steady motion assumption without restricting the flapping +frequency to be relatively high. This provides a full and elegant four-winged micro ornithopter +model that captures the interaction between the body and the wings while avoiding the +complexities and singularities associated with other coordinate representations (e.g., Euler +angles). Simulation studies of the inertial effects of the relative motion between the different +parts of the multibody system shows the importance of considering the forces and torques, +resulting from the wings-body interaction, in motion generation of these insects. +Keywords: Flapping Wing Unmanned Aerial Vehicles, Dragonfly, Ornithopter, Lagrangian +Mechanics, Global Coordinates. +1. INTRODUCTION +Flapping wing unmanned aerial vehicles (FWUAVs), com- +monly known as ornithopters, have received an increasing +attention over the last decade. The recent developments +are encouraging, but the field is still in its infancy and +considerable research efforts are needed to efficiently de- +ploy these bio-inspired platforms. FWUAVs are particu- +larly suitable for micro aerial vehicles (MAVs) applications +where classical ground and aerial vehicles are inefficient, +such as in search and rescue missions where these minia- +ture autonomous vehicle can, for instance, fly through +cracks in concrete to search for earthquake victims. In +fact, FWUAVs enjoy some interesting features that can- +not be found in fixed-wing and rotary-wing vehicles, such +as energy efficiency, agility and miniaturization capacity. +This has motivated to study the aerodynamics and flight +mechanics of insects and birds for clues that would help in +the design of FWUAVs. +The development of FWUAVs is often inspired from real +birds and insects flights (Breugel et al., 2008). One of the +best flyers in the insect world is the dragonfly–a four- +wing insect that has been extensively studied due to its +astonishing flight capabilities. For example, using only its +own body muscles, the dragonfly can accelerate up to 3g +and reach a speed of 10m/s (May, 1991; Ruppell, 1989), +and is capable of generating an instantaneous lift five times +greater than its weight (Reavis and Luttges, 1988). Each of +its wings can be actuated independently (Alexander, 1984; +Azuma and Watanabe, 1988), which provides additional +degrees of freedom helping the insect to perform agile +and complex flight maneuvers. The difficulty in modeling +and controlling FWUAVS is mainly due to the aeroelastic +phenomena and the intrinsic coupling between the wings +and the main body of the vehicle. Most of the existing +mathematical models in the literature, for insect-inspired +vehicles, rely on simplifying assumptions such as large flap- +ping frequency, small body-wing mass ratio and neglecting +the aerodynamic couplings between the different parts of +the vehicle (Taha et al., 2012; Sun, 2014; Deng et al., 2006). +The resulting simplified models are often similar to single +rigid-body systems with forces and torques depending on +the flapping parameters. The coupling between the dif- +ferent parts of the multibody system (wings, tail, main +body, etc), is particularly important for FWUAVs with +large wings and relatively low flapping frequency. Some +attempts to capture these coupling effects have been made +in (Sridhar et al., 2020; Sun et al., 2007). It is known that +insects generate aerodynamic forces through wings motion +(flapping). These aerodynamic forces are often modeled +using a quasi-steady assumption (the forces generated by a +flapping wing are equal to the forces generated by the fixed +wing at the same instantaneous velocity and attitude of the +wings blade). This approach was mainly based on fixed- +wing theory, but it has been used as a simple but robust +modeling tool for flapping wings for several decades (Weis- +Fogh, 1973; Ellington, 1984). Later on, this approach was + +refined by the introduction of the lift and drag coefficients +of the wings in flapping motion (Dickinson et al., 1999; +Usherwood and Ellington, 2002) to make it applicable +for flapping wing vehicles. However, even at the fastest +flight speeds, the quasi-steady aerodynamic interpretation +seems inadequate to explain the extra lift produced by real +insects flights. The importance of unsteady aerodynamic +mechanisms for flapping insects flights has become widely +recognized. Some numerical simulations of unsteady insect +flight aerodynamics based on the finite element solution +of the Navier–Stokes equations gave accurate results for +the estimated aerodynamic forces (Sanders and Verhulst, +1985). However, their implementation is unsuitable for +control purposes since they require high processing power +and, contrary to quasi-steady aerodynamics, they cannot +be formulated as a function of the flapping parameters. In +this work, using Langrange formalism, we propose a high- +fidelity dynamical model for the dragonfly-like ornithopter, +without relying on the high flapping frequency assumption +and taking into consideration all the interaction forces and +torques inherent to this multibody system. The dynamical +model uses intrinsic global coordinates, mainly attitudes +on the Special Orthogonal group of rotations, which avoids +the complexities and singularities associated with other +coordinate representations (e.g., Euler angles). +2. NOTATION +We denote by R the set of reals and by N the set of +natural numbers. We denote by Rn the n−dimensional +Euclidean space. We use ∥x∥ to denote the Euclidean +norm of a vector x ∈ Rn. We denote by F = {c, x, y, z} +an Euclidean 3-dimensional frame with center at c ∈ +R3 and axes {x, y, z} defining an orthonormal basis of +R3. For a vector x ∈ Rn, we denote by sgn(x) := +[sgn(x1), · · · , sgn(xn)]⊤ the element-wise signum function +of x. The special orthogonal group of order three is denoted +by SO(3) := +� +A ∈ R3×3 : det(A) = 1, AA⊤ = I +� +, where I +is the 3 × 3 identity matrix. The unit vector ei denotes +the ith column of the identity matrix I. The set so(3) := +� +Ω ∈ R3×3 : Ω⊤ = −Ω +� +denotes the Lie algebra of SO(3). +For each z ∈ Rn \ {0}, we denote by P(z) := I − zz⊤∥z∥−2 +the orthogonal projection operator. For x, y ∈ R3, the map +ˆ. : R3 → so(3) is defined as +ˆx := + + +0 +−x3 +x2 +x3 +0 +−x1 +−x2 +x1 +0 + + . +Then, we have ˆxy := x × y where × is the vector cross- +product on R3. Any rotation matrix A ∈ SO(3) can be +parameterized by a unit vector u ∈ R3 and an angle θ ∈ R +through the exponential map (Rodrigues formula) +A = exp (θˆu) := I + sin(θ)ˆu + (1 − cos(θ))ˆu2. +3. MECHANICAL CONFIGURATION +Let us consider a flapping wing micro aerial vehicle that +can translate and rotate in three dimensions as multiple +rigid bodies (2 fore-wings and 2 hind-wings) connected +to the main body via spherical joints that constrain the +five rigid bodies to remain in contact, see Fig. 1. We +define six Euclidean frames: an inertial frame of reference +FI += {0, rIx, rIy, rIz}, a body-attached frame FB += +{OB, rbx, rby, rbz} for the main body, and a wing-attached +frame Fi = {Oi, rix, riy, riz}, i ∈ {1, · · · , 4}, where OB is +the center of mass of the main body, and Oi is the center +of the joint connecting the the ith wing to the body. We +use index 1 for the right fore-wing, 2 for the left fore-wing, +3 for the right hind-wing, and 4 for the left hind-wing. +Let AB ∈ SO(3) denote the attitude matrix from the +body-attached frame FB to the inertial frame FI, and +let ΩB ∈ R3 represent the angular velocity of the body- +attached frame FB with respect to the inertial frame FI +expressed in FB. Let Ai ∈ SO(3) denote the attitude +matrix from the ith wing-attached frame Fi to the body- +attached frame FB and let Ωi ∈ R3 represent its angular +velocity with respect to FB expressed in Fi. Now, let µi, +with i ∈ {1, . . . , 4}, represent the vector from the origin +of the body-attached frame FB to the connection joint +(center of frame Fi) expressed in FB. Let κi represent +the vector from the joint that connects the wing and the +main body (center of frame Fi) to the wing center of mass +expressed in Fi. Finally, let p ∈ R3 be the position vector +of the origin of FB expressed in FI. +p FB +FI +Right Fore-wing (B1) +Main Body +F1 +e2 +e1 +e3 +r1y +r1x +µ1 +Left Fore-wing (B2) +F2 +Connection Joint +Right Hind-wing (B3) +Left Hind-wing (B4) +c3(r) +dr +κ2 +qLE +2 +qTE +2 +qLE +4 +qTE +4 +r +cf +3,r +Fig. 1. A schematic diagram of a four-winged ornithopter. +Finally, the overall configuration of this multi-body system +is denoted by g := (gB, gw), where gB := (p, AB) ∈ +R3 × SO(3) denotes the main body pose and gw +:= +(A1, A2, A3, A4) ∈ SO(3)4 represents the four wings con- +figurations (attitudes). The group velocity is denoted by +ξ := (ξB, ξw), where ξB := ( ˙p, ΩB) ∈ R6 is the main body +group velocity and ξw := (Ω1, Ω2, Ω3, Ω4) ∈ R12 represents +the wings group velocity. +4. QUASI-STEADY AERODYNAMICS +Flapping-wings flights are more complicated than fixed- +wings flights because of the structural movement and +the resulting unsteady fluid dynamics. In conventional +airplanes with fixed wings, the forward motion relative +to the air causes the wings to produce lift. However, in +a flapping-wings flight, the wings not only move forward +relative to the air but also flap up and down and perform +rotations around their roots. In this section, we will need +to specify the geometry of the wings, as the resultant forces +will depend on the shape of the wing. Then, we will use +blade-element theory (Ellington, 1984) under the quasi- +steady aerodynamics assumption to express the forces and +torques generated through flapping motion. We rely on the +quasi-steady assumption to simplify the forces and torques +calculation. Intuitively, this assumption implies that these + +forces and torques are equivalent to those generated in +steady motion at the same instantaneous velocity and the +same angle of attack. +4.1 Wing Geometry +The dragonfly’s wings are responsible for its incredible +flight performance and force generation. The shape of +the fore-wing and hind-wing of the dragonfly are often +approximated by an elliptical function (tear-drop shape), +see (Weis-Fogh, 1973). However, as it is shown in (Faisal +and A.Filippone, 2016), a more realistic representation +of the wings’ shape gives a better approximation of the +aerodynamic forces generated by a real dragonfly, than +the traditional tear-drop shape representation. A better +approximation of the dragonfly’s wings geometry can be +obtained via a set of polynomials derived from the analysis +of real photography images. We assume that each wing is a +flat plate lying on the rix − riy plane of Fi (no component +on riz). Each wing’s shape is described by two polynomial +functions that represent the leading (upper bound) and +the trailing (lower bound) edge, see Fig. 1: +qLE +i +(r) := +j=n +� +j=0 +λLE +ij rj, +qT E +i +(r) := +j=n +� +j=0 +λT E +ij rj, +(1) +where r is the argument along the riy axis and the qLE +i +(r) +and qT E +i +(r) are the corresponding points locations of the +trailing and leading edges, respectively, along the rix axis. +The polynomials coefficients λLE +ij +and λT E +ij +are calculated +by fitting the points from the contour lines of the wings +(obtained from real photography images) with polynomials +of degree n to minimize the mean squared errors while +avoiding overfitting, see Section 6 for a numerical example. +4.2 Aerodynamic Forces +To inspect the quasi-steady aerodynamic forces, let dr be +an infinitesimal wing segment, parallel to the rix axis of the +wing frame, and located at a distance r from the wing root +(Fig. 1), which is measured along the riy axis of the wing +frame. Let its chord length be defined as ci(r) := qLE +i +(r)− +qT E +i +(r) where qLE +i +(·) and qT E +i +(·) are defined as in (1). +Lemma 1. Let νi(r, γ) := (qLE +i +(r)−γci(r))e1+(−1)i+1re2, +with γ ∈ [0, 1], be the Fi-coordinates of an arbitrary point +on the chord. Then, the velocity of this point with respect +to FI, expressed in Fi, is given by +Wi,r,γ(g, ξ) = (ABAi)⊤ ˙p + A⊤ +i ˆΩB(µi + Aiνi(r, γ)) ++ˆΩiνi(r, γ). +(2) +The first term of Wi,r,γ(g, ξ) is due to main body’s trans- +lation motion, the second term is due the main body’s +rotation, and the last term is due to the wing’s flapping. +The proof of this lemma is given in appendix A. Now, we +proceed as in (Sridhar et al., 2020) to determine the angle +of attack. As per the blade-element theory (Ellington, +1984), the aerodynamic force generated by the infinites- +imal chord depends only on the rix, and riz components. +Therefore, we project the above velocity on the plane +rix − riz to obtain the effective velocity: +W i,r,γ(g, ξ) := P(e2)Wi,r,γ(g, ξ). +(3) +The state-dependent angle of attack of the chord, denoted +αi,r(g, ξ), is the angle between the chord line (from the +leading edge to the trailing edge), and the above velocity +W i,r,γ 1 . Since the angle of attack changes slightly chord- +wise, we will define the angle of attack as the angle between +the chord line and the velocity of the center of the chord +(γ = 1/2), and it is given by +αi,r(g, ξ) := cos−1 + +e⊤ +1 W i,r, 1 +2 (g, ξ) +���W i,r, 1 +2 (g, ξ) +��� + + . +(4) +We consider the location of the state-dependent aerody- +namic center, denoted cf +i,r(g, ξ), as a function of the angle +of attack αi,r, and it is given by +cf +i,r(g, ξ) := νi(r, γac(αi,r(g, ξ)). +(5) +where γac(·) : [0, π] → [0, 1] maps the angle of attack to the +position of the aerodynamic center along the chord. The +effective velocity of the aerodynamic center will be denoted +by W +ac +i,r(g, ξ) := W i,r,γ(g, ξ) with γ = γac(αi,r(g, ξ)). The +magnitude of the lift and drag forces generated by the +infinitesimal wing segment are given by +∥dLi,r(g, ξ)∥ = 1 +2ρ∥W +ac +i,r∥2CL(αi,r)ci(r)dr, +(6) +∥dDi,r(g, ξ)∥ = 1 +2ρ∥W +ac +i,r∥2CD(αi,r)ci(r)dr, +(7) +where ρ ∈ R is the atmospheric density and CL, CD : +[0, π] → R are the lift and drag coefficients, which depend +on the angle of attack αi,r. Following the same steps as +(Sridhar et al., 2020), one can determine the infinitesimal +lift and drag forces as follows. The direction of the lift is +normal to both the velocity W +ac +i,r and the wing span-wise +direction riy. As such, the direction of the lift is along +±e2 × W +ac +i,r in Fi. Thus, we multiply the lift magnitude +by the unit vector (e2 × W +ac +i,r)∥W +ac +i,r∥−1. To solve the sign +ambiguity induced by the flapping motion (up-stroke and +down-stroke), we consider the four cases shown in Fig. 2. +dLi +rix +riz +W ac +i,r +Chord +case 1 +dLi +rix +riz +W ac +i,r +Chord +case 2 +dLi +rix +riz +W ac +i,r +Chord +case 3 +dLi +rix +riz +W ac +i,r +Chord +case 4 +Fig. 2. Different cases of the direction of the lift with +respect to the direction of W +ac +i,r. Figure inspired from +(Sridhar et al., 2020). +From Fig. 2, one can notice that the direction of the lift +force is in the direction of e2 × W +ac +i,r if the first and third +components of W +ac +i,r have the same sign (case 1 and case +3), otherwise it is in the direction of −e2 × W +ac +i,r (case 2 +and case 4). The direction of the drag force, however, is +opposite to W +ac +i,r. Finally, the corresponding aerodynamic +forces and torques generated by the infinitesimal wing +segment can be expressed in Fi as follows: +1 We might drop the arguments of a given function whenever clear +from context. + +dLi,r(g, ξ) = 1 +2 ρCL(αi,r)ci(r) ��W +ac +i,r +�� sign( ¯wi +rx ¯wi +rz)(e2 × W +ac +i,r)dr, +dDi,r(g, ξ) = − 1 +2ρCD(αi,r)ci(r) +��W +ac +i,r +�� W +ac +i,rdr, +where ¯wi +rx and ¯wi +rz are the first and third components of +W +ac +i,r, respectively. The infinitesimal lift and drag forces, +which are applied at the aerodynamic center, also generate +the following infinitesimal torque about the wing root +dMi,r(g, ξ) = cf +i,r(g, ξ) × (dLi,r + dDi,r) . +(8) +The total lift Li(g, ξ), drag Di(g, ξ), and torque Mi(g, ξ) +generated on the ith wing, are obtained by integrating the +above infinitesimal expressions span-wise for r ∈ [0, li], +where li > 0 is the wing’s length. +Li(g, ξ) := +� li +0 +dLi,r(g, ξ), +(9) +Di(g, ξ) := +� li +0 +dDi,r(g, ξ), +(10) +Mi(g, ξ) := +� li +0 +dMi,r(g, ξ). +(11) +Compared with the other models considering uniform +forces over the wing (Lai et al., 2005), this approach +captures the span-wide variations of the aerodynamic +forces, which are critical for FWUAVs with large wings +flapping at a relatively low frequency. +5. DRAGONFLY MODELING USING +LAGRANGE-D’ALEMBERT PRINCIPLE +In this section, we will derive the expressions of the kinetic +and potential energies. We will use them along with the +aerodynamic forces, detailed in Section 4, to derive a com- +plete dynamical model for the four-winged ornithopter, us- +ing the Lagrange-D’Alembert principle (Lagrange, 1788). +5.1 Kinetic and Potential Energies +The kinetic and the potential energies of the complete +multibody system, denoted respectively by T and U, are +the sum of the kinetic and potential energies of each rigid- +body. They are given by +T = +� +j∈{B,1,2,3,4} +Tj, +U = +� +j∈{B,1,2,3,4} +Uj. +(12) +The kinetic and the potential energies are explicitly ex- +pressed in the following proposition. +Proposition 1. The total kinetic energy can be expressed +as follows: +T (g, ξ) = 1 +2 +4 +� +i=1 + + +˙p +ΩB +Ωi + + +⊤ +Ji(g) + + +˙p +ΩB +Ωi + + , +(13) +U(g) = − +� +i∈{B,1,2,3,4} +mige⊤ +3 (p + AB (µi + Aiκi)) , +(14) +with µB = κB = 0 and Ji(·) is the symmetric matrix +Ji(g) := + + +� 1 +4mB + mi +� +I +⋆ +⋆ +mi +� +ˆµi + � +Aiκi +� +A⊤ +B +J[22] +i +⋆ +miˆκiA⊤ +i A⊤ +B +JiA⊤ +i + miˆκ⊤ +i A⊤ +i ˆµi Ji + + , +where +J[22] +i +:= +� +AiJiA⊤ +i −miˆµ2 +i +miˆµ⊤ +i � +Aiκi+mi � +Aiκiˆµ⊤ +i + 1 +4JB +� +, +mB and mi are the mass of the main body and the con- +nected bodies respectively, g represents the acceleration of +gravity, JB ∈ R3×3 represents the constant inertia matrix +of the main body about FB, and Ji ∈ R3×3 represents the +inertia matrix of the ith body about Fi. +The proof of this proposition is given in appendix B. +5.2 Lagrange-D’Alembert principle +This principle consists of a modification of Hamilton’s +principle to incorporate the effects of external forces. +These external forces may or may not be derivable from +a potential. This modification states that the infinitesimal +variation of the integral action of T − U over a fixed time +period equals the work, denoted δW, done by the external +forces, corresponding to an infinitesimal variation of the +configuration, during this same time period (also known +as the virtual work of the external forces). Formally, for +any t0 ≥ 0 and tf ≥ t0, we have +� tf +t0 +δ(T (t) − U(t))dt = +� tf +t0 +δW(t)dt. +(15) +This version of the variational principle requires determin- +ing the virtual work that corresponds to an infinitesimal +variation of the configuration. Let L := T − U represent +the Lagrangian and let Fi := Li + Di represent the sum of +forces acting on the ith wing, and let τi ∈ R3 be the control +torque exerted at the joint connecting the ith wing to the +body, expressed in the body-attached frame. Following the +developments in (Lee et al., 2018), and using abj ∈ R3 and +aij ∈ R3 to denote the jth column of AB ∈ SO(3) and +the jth column of Ai ∈ SO(3), respectively, for i = 1, 2, 3, +the Lagrange-d’Alembert principle leads to the equations +stated in the following proposition: +Proposition 2. The Lagrange-d’Alembert principle (15) +leads to the following equations: +d +dt +�∂L +∂ ˙p +� +− ∂L +∂p = +4 +� +i=1 +ABAiFi, +(16) +d +dt +� ∂L +∂ΩB +� ++ ˆΩB +∂L +∂ΩB ++ +3 +� +j=1 +ˆabj +∂L +∂abj += +4 +� +i=1 +ˆµiAiFi − +4 +� +i=1 +τi, +(17) +d +dt +� ∂L +∂Ωi +� ++ ˆΩi +∂L +∂Ωi ++ +3 +� +j=1 +ˆaij +∂L +∂aij += Mi + A⊤ +i τi. +(18) +The proof is given in appendix C. +5.3 Full Dynamical Model +In this subsection, we use proposition 3 and the expression +of L = T − U to derive a dynamical model for the +dragonfly. Using the kinetic and potential energies in +the previous Lagrange-D’Alembert equations, and defining +τ = (τ1, τ2, τ3, τ4) ∈ R12, one can derive the dynamical +model as follows: +C(g) ˙ξ + D(g, ξ)ξ = Fa (g, ξ) + Hc (g) τ + Fg(g). +(19) +with D(g, ξ) := S (ξ) C(g) + N(g, ξ), such that S (ξ) := +diag +� +03×3 ˆΩB ˆΩ1 ˆΩ2 ˆΩ3 ˆΩ4 +� +and C(g) is a 18×18 matrix +given by + +C(g) := + + +mI +4 +� +i=1 +J[12] +i +J[13] +1 +J[13] +2 +J[13] +3 +J[13] +4 +4 +� +i=1 +J[21] +i +4 +� +i=1 +J[22] +i +J[23] +1 +J[23] +2 +J[23] +3 +J[23] +4 +J[31] +1 +J[32] +1 +J[33] +1 +03×3 03×3 03×3 +J[31] +2 +J[32] +2 +03×3 J[33] +2 +03×3 03×3 +J[31] +3 +J[32] +3 +03×3 03×3 J[33] +3 +03×3 +J[31] +4 +J[32] +4 +03×3 03×3 03×3 J[33] +4 + + +, +(20) +where m := � +i∈{B,1,··· ,4} mi, and J[jk] +i +denotes the matrix +block at the jth line and the kth row of the matrix Ji. The +3 × 3 block components of the 18 × 18 matrix N(g, ξ) are +given below: +N11 = 0, +N12 = +4 +� +i=1 +−miAB ˆΩB +� +ˆµi + � +Aiκi +� +− miAB � +Ai ˆΩiκi +N1(k+2) = −mkAB +�ˆΩBAk + Ak ˆΩk +� +ˆκk, +N21 = +4 +� +i=1 +−mi +� +ˆµi + � +Aiκi +� ˆΩBA⊤ +B + mi +� +� +ˆµiΩB + +� +� +AiκiΩB +� +A⊤ +B +N22 = +4 +� +i=1 +Ai ˆΩiJiA⊤ +i − AiJi ˆΩiA⊤ +i − mi +� +ˆµi � +Ai ˆΩiκi + � +Ai ˆΩiκiˆµi +� +, +N2(k+2) = Ak ˆΩkJk − mk ˆµkAk ˆΩkˆκk, +N31 = +4 +� +i=1 +−miˆκi +� +A⊤ +i ˆΩB + ˆΩiA⊤ +i +� +A⊤ +B + miˆκiAi ˆΩBA⊤ +B ++ mi � +ˆκiΩiA⊤ +i A⊤ +B, +N32 = +4 +� +i=1 +−Ji ˆΩiA⊤ +i + miˆκi ˆΩiA⊤ +i ˆµi − +� +JiA⊤ +i ΩBA⊤ +i +− miˆκiA⊤ +i +�ˆΩB ˆµi − � +ˆµiΩB +� +, +Nj(k+2) = � +A⊤ +k ΩB +⊤ +Jk + mk +� +A⊤ +k ˆµkΩBˆκ⊤ +k , +with k ∈ {1, · · · , 4} and j ∈ {3, · · · , 6}. The aerodynamic, +gravitational, and torque control forces are given by +Fa := + + +4 +� +i=1 +ABAiFi +4 +� +i=1 +ˆµiAiFi +M1 +M2 +M3 +M4 + + +, Hc(g) := + + +0 +0 +0 +0 +−I −I −I −I +A⊤ +1 +0 +0 +0 +0 +A⊤ +2 +0 +0 +0 +0 +A⊤ +3 +0 +0 +0 +0 +A⊤ +4 + + +, +Fg := + + +mge3 +4 +� +i=1 +mig +� +ˆµi + � +Aiκi +� +A⊤ +Be3 +m1gˆκ1 +� +A⊤ +1 A⊤ +Be3 +� +m2gˆκ2 +� +A⊤ +2 A⊤ +Be3 +� +m3gˆκ3 +� +A⊤ +3 A⊤ +Be3 +� +m4gˆκ4 +� +A⊤ +4 A⊤ +Be3 +� + + +. +(21) +This is a complete dynamical model, that provides the +position, orientation, and wing dynamics of a four-winged +ornithopter, with input torques applied at the joints con- +necting the wings to the body. +Torque-controlled +model (19) +τ +(g, ξ) +Fig. 3. Torque-controlled model (19). +6. REDUCED DYNAMICAL MODEL VIA WINGS +KINEMATICS ASSIGNMENT +Through a particular choice of the control inputs, one +can generate wings motions mimicking real flapping-wing +animals flights. This leads to a simplified model where +the control inputs are the parameters related to the wings +kinematics. +6.1 Desired Wing Flapping Kinematics +Berman and Wang (2007) proposed a wing kinematic +configuration for insects using Euler angles. This config- +uration minimizes energy in hovering flights, and captures +several qualitative aspects of observed real insect flights. +It can also be applied to other flight maneuvers (e.g., +accelerating in different directions) as manipulating the +three angles parameters generate aerodynamic forces in +different directions. In the approach of Berman and Wang +(2007), each wing’s attitude is given by three Euler angles: +the flapping angle φi(t), the deviation angle ψi(t), and the +pitching angle θi(t), about the stroke frame. The stroke +frame, denoted by FiS = {Oi, six, siy, siz}, is obtained by +rotating the wing frame Fi by a fixed angle βi ∈ R about +the body-frame y-axis. The corresponding wing attitude +for this kinematic configuration is taken from (Sridhar +et al., 2020), and it is given by +Ai = exp (βiˆe2) exp((−1)i+1φiˆe1) exp((−1)iψiˆe3) exp (θiˆe2) , +with i ∈ {1, . . ., 4}. The flapping angle is given by +φi(t) = +φim +sin−1 φiK +sin−1 (φiK cos(2πft + φia)) + φi0, +(22) +where f ∈ R represents the flapping frequency in Hz, +φim ∈ R is the amplitude, φia ∈ R is the phase, φi0 ∈ R +is the offset, and 0 < φiK ≤ 1 determines the waveform +shape (sinusoidal if φiK → 0, triangular if φiK → 1). +The pitch angle is given by the following function: +θi(t) = +θim +tanh θiC +tanh (θiC sin (2πft + θia)) + θi0, +(23) +where θim ∈ R is the amplitude, θi0 ∈ R is the offset, +θiC ∈ (0, ∞) determines the waveform (sinusoidal when +θiC → 0, step function when θiC → ∞), and θia ∈ (−π, π) +is the phase offset. The value of θiC is related to the +duration of wing pitch reversal. Finally, the deviation angle +is given by +ψi(t) = ψim cos (2πψiNft + ψia) + ψi0, +(24) +where ψim ∈ R is the amplitude, ψi0 ∈ R is the offset, +and the parameter ψia ∈ (−π, π) is the phase offset. The +parameter ψiN ∈ {1, 2}, where ψiN = 1 corresponds to +one oscillation per flapping period, and ψiN = 2 is for a +figure-eight motion. + +Fig. 4. Flapping, pitching, and deviation angles. Positive +angles are measured from FiS (in blue) to Fi (in red). +For specific wings kinematics, we need to assign 13 pa- +rameters per wing (51 parameters). These parameters are +constrained for the dragonfly as in Table 1, see (Faisal and +A.Filippone, 2016). +Parameter +range +f : flapping frequency +30.0 − 45.00 Hz +φim : flapping amplitude +30.0◦ − 60.0◦ +ψim : deviation amplitude +1.0◦ − 20.0◦ +θim : pitching amplitude +1.0◦ − 90.0◦ +φi0 : flapping offset +−30.0◦ − 30.0◦ +ψi0 : deviation offset +5.0◦ − 30.0◦ +θi0 : pitching offset +−90.0◦ − 90.0◦ +ψia : deviation phase +−180.0◦ − 180.0◦ +φia : flapping phase +−180.0◦ − 180.0◦ +θia : pitching phase +−180.0◦ − 180.0◦ +φiK : waveform shape +0.01 − 1.00 +θiC : waveform shape +0.01 − 5.00 +βi : stroke plane angle +5.0◦ − 30.0◦ +Table 1. Parameters range for dragonfly wing +kinematics. +The attitude kinematics of each wing is given by +˙Ai = Ai ˆΩi, +(25) +with i ∈ {1, · · · , 4}. We consider the angular velocities +of the wings that are obtained from the time-derivatives +of the Euler-angles equations (22)-(24) as follows (Sridhar +et al., 2020): +Ωi = + + +(−1)i+1 cos ψi cos θi 0 (−1)i+1 sin θi +sin ψi +1 +0 +(−1)i+1 cos ψi sin θi 0 +(−1)i cos θi + + + + +˙φi +˙θi +˙ψi + + . +(26) +6.2 Reduced Dynamical Model +Let (gd +w, ξd +w) be a desired time-varying wings kinematics +generated according to equations (25)-(26), using a given +parameters set Θ := (f, βi, φmi, ψmi, · · · ) as described in +Table II. Now, according to equation (18), the control +torque τi is written as: +τi = Ai + + d +dt +� ∂L +∂Ωi +� ++ ˆΩi +∂L +∂Ωi ++ +3 +� +j=1 +ˆaij +∂L +∂aij +− Mi + + , +:= T(gB, ξB, gw, ξw). +(27) +Now assuming that the inner loop is fast enough such that +(gw, ξw) ≈ (gd +w, ξd +w), the control torques can be written as +τi ≈ T(gB, ξB, gd +w, ξd +w) =: Tt +Θ(gB, ξB), +(28) +where we have used the fact that (gd +w, ξd +w) is a function of +parameters Θ and time t. Replacing this torque expression +in the second equation of Proposition 2, will allow us to +write the translational, and rotational dynamics of the +main body as follows: +Ct +Θ(gB) ˙ξB + Dt +Θ(gB, ξB)ξB + Vt +Θ (gB) = Ft +Θ (gB, ξB) . +(29) +The matrices Ct +Θ, Dt +Θ, Vt +Θ and Ft +Θ and the development of +this model are detailed in appendix D. +Controller of +wing +Torque +model (19) +τ +(gB, ξB) +(gw, ξw) +(gd +w, ξd +w) +Wing +(25) − (26) +Θ +kinematics +controlled +dynamics +Fig. 5. Parameters-controlled model. +Grouping the forces and torques of the same nature to- +gether is another way to make this model ideal for design- +ing control laws, and investigating wing-body interaction. +First, we write the aerodynamic forces and torques, which +can be considered as control inputs in a tracking scenario. +Second, we consider the inertial forces and torques of the +wings due to body acceleration and rotation. Finally, we +consider the inertial forces and torques of the wings due +to wing motion (flapping). The model is given by + + + + + + + + + + + +˙p = v, +˙AB = AB ˆΩB, +m˙v = mge3 + Fc + FB + Fw, +JB ˙ΩB = ˆΩBJBΩB + Γc + ΓB + Γw, +(30) +where v is the linear velocity of the body in the inertial +frame, Fc and Γc are the aerodynamic forces and torques +that can be used as a control inputs, Fw and Γw are the +inertial forces and torques due to the flapping motion of +the wings, FB and ΓB are the inertial forces and torques +due to the translational and rotational motion of the body. +The expression of these forces are given in appendix F. +In most cases, the inertial forces due to the wing motion +are neglected because of the wing mass mi being too +small compared to the body mass mB. In the case of +the dragonfly, the wing mass mi represents around 3.5% +of the body mass mB. Moreover, neglecting both the +inertial forces and torques due to the wing motion and +body motion FB, Fw, ΓB and Γw results in the widely used +model of a flying rigid body (Padfield, 1996), used in (Deng +et al., 2006; Xiong and Sun, 2008; Dietl and Garcia, 2008) +to model insects flights. In the present work, we will take +a closer look into the importance of these forces in near- +hover scenarios. +7. NUMERICAL RESULTS +In this section, we will investigate the effect of the inertial +forces in a near-hover scenario. We will compare the +trajectories performed by the dragonfly when neglecting +the inertial forces versus when considering them. We will +also compare the magnitudes of these inertial forces. +First, it is necessary to describe the wings’ shape, using the +polynomial functions qLE +i +and qT E +i +, since the aerodynamic +forces depend on these polynomials, as shown in Section +4. Let us consider Fig. 6 which shows an image of a +dead dragonfly insect. The polynomials’ coefficients in +equation (1) are calculated by fitting the points from the + +r2y +S2yr1c +0 +&'S1cS2y +12ycontour lines of the wings acquired by the algorithm in +(Seo et al., 2016) with n = 7. The polynomials’ coefficients +are provided in the following table, and the polynomial +functions are plotted in Fig. 6 +j +λLE +1j +λT E +1j +λLE +3j +λT E +3j +0 +-0.873 +2.133 +-0.214 +0.183 +1 +5.648 +-12.12 +1.8389 +-1.372 +2 +-13.85 +26.20 +-6.153 +3.574 +3 +15.16 +-26.25 +9.981 +-3.016 +4 +-6.111 +-11.47 +-7.857 +-2.210 +5 +-0.579 +-0.728 +2.625 +5.627 +6 +1.789 +-0.429 +0.051 +-3.164 +7 +0.122 +-0.096 +0.140 +-0.082 +R2 +0.006 +0.028 +0.006 +0.012 +Fig. 6. Image of a dragonfly with coefficients of wings +polynomials. +Next, we take the location of the aerodynamic center cf +i,r +similar to the location taken in (Dickson et al., 2006), i.e., +the function γac(αi,r) in equation (5) is given by +γac(α) := 0.82|α| +π + 0.05. +(31) +The lift and drag coefficients CL(αi,r), and CD(αi,r) are +functions of αi,r, and are taken similar to the coefficients +proposed in (Dickinson et al., 1999): +CL(αi,r) = 0.225 + 1.58 sin(2.13α◦ +i,r − 7.20), +(32) +CD(αi,r) = 1.92 − 1.55 cos(2.04α◦ +i,r − 9.82). +(33) +with α◦ +i,r = 180αi,r/π if αi,r ≤ π/2 and α◦ +i,r = 180(π − +αi,r)/π otherwise. To make the dragonfly perform a near- +to-hover flight, we will need to find suitable parameters +for the dragonfly’s wings. This is challenging due to +the complexity of the dynamics, and the relatively large +number of the wing’s kinematics parameters. This problem +is addressed by performing an optimization to minimize +a cost function, which is the error of the position and +the velocity of the dragonfly from a reference hovering +position. More specifically, this problem is formulated as +follows. We define the cost +J = w1 +� Tf +t0 +∥p − pref∥2dt + w2 +� Tf +t0 +∥ ˙p∥2dt, +where w1, w2 ∈ R, Tf = 10T is the flight time, T = +1/f is the flapping period, and pref = [0 0 2]⊤ is the +hovering position reference. This is to minimize the error +of the position and velocity to ensure that the dragonfly +is hovering at 2m altitude. Next, as in (Tejaswi et al., +2021; Sridhar et al., 2020), we assume that the dragonfly’s +body exhibits a pitching motion (oscillating around e2), +i.e., AB(t) = exp(ΦB(t)ˆe2), with +ΦB(t) = ΦBm cos (2πft + ΦBa) + ΦB0, +where ΦBm, ΦBa and ΦB0 are some parameters to be +determined hereafter. Replacing AB and ˆΩB = A⊤ +B ˙AB +in (30), the trajectory p(t) can be obtained by integrat- +ing the translational dynamics (first and third equations +of (30)). Finally, the optimization parameter is ¯Θ := +(Θ, ΦBm, ΦBa, ΦB0). The constraints on the parameters +are given in Table 1, and the morphological parameters of +the dragonfly like JB, Ji, µi are taken to be similar to those +of an actual insect, and are presented in appendix E. This +problem is carried out with the Genetic Algorithm opti- +mization (Anweer et al., 2020) implemented in MATLAB. +The corresponding optimized parameters are summarized +in Table 2, and the resulting trajectory is shown in Fig. 8. +Parameter +Fore-wing1,2 +Hind-wing3,4 +f +35.6476 Hz +35.6476 Hz +φim +58.42◦ +32.48◦ +ψim +11.16◦ +4.26◦ +θim +1.43◦ +37.18◦ +φi0 +4.64◦ +28.49◦ +ψi0 +26.49◦ +20.29◦ +θi0 +−35.24◦ +−1.83◦ +φia +0◦ +92.56◦ +ψia +−40.10◦ +29.37◦ +θia +−98.82◦ +−138.47◦ +φiK +0.533 +0.895 +θiC +2.394 +1.613 +βi +10.95◦ +23.53◦ +ΦBm +0.37◦ +ΦBa +−5.72◦ +ΦB0 +0.434◦ +Table 2. Optimized parameters. +0 +2 +4 +6 +8 +10 +0 +1 +2 +3 +4 +Fig. 7. Forces magnitude comparison. +0 +5 +10 +-1 +-0.5 +0 +0.5 +1 +10-3 +0 +5 +10 +-1 +-0.5 +0 +0.5 +1 +10-3 +0 +5 +10 +2 +2.1 +2.2 +2.3 +2.4 +Fig. 8. Position comparison for the full model in (30) (red), +when neglecting FB and Fw (black), when neglecting +FB only (dashed green), and when neglecting Fw only +(dashed blue). +From Fig. 8, one can observe that neglecting the inertial +forces due to body motion FB, results in a closer trajectory + +lcm +LE +LE +qi +q3(dashed green line) to the full model trajectory (red line). +This result is due to the small magnitude of the inertial +forces due to body motion (∥FB∥) as it is shown in Fig. 7 +in green lines. However, neglecting the inertial forces due +to the wing motion, results in a trajectory (dashed blue +line in Fig. 8) relatively far from the full model trajectory +(red line), and closer to the flying rigid body model (Deng +et al. (2006)) trajectory (black line). This result is due to +the relatively important magnitude of the inertial forces +due to the wing motion (∥Fw∥), as it is shown in Fig. 7 in +blue line. Fig.8 and Fig.7 illustrate the importance of the +wing-body interaction. The inertial forces caused by the +wing motion are found to be quite important and should +not be neglected. The inertial forces caused by the body +motion are found to be negligible in this scenario, due to +the slow body motion compared with the wing motion. +Considering the flying rigid body dynamical model as the +one in (Deng et al., 2006), which is obtained by neglecting +both inertial forces due to the body and wing motion, +resulted in a different trajectory of the dragonfly. This +trajectory is significantly far from its actual trajectory in +view of the dragonfly’s size. +8. CONCLUSION +A full high-fidelity dynamical model for a four-winged +micro ornithopter is presented. The ornithopter is modeled +as four rigid bodies (wings) connected to a main rigid +body via spherical joints. Lagrangian formalism is used +to derive the dynamical model using intrinsic global coor- +dinates, without neglecting the wing masses. Furthermore, +the quasi-steady aerodynamics assumption is used to cal- +culate the span-wise aerodynamic forces generated by in- +finitesimal wing chords, in contrast to considering uniform +aerodynamic forces over the wing, and without relying on +the high flapping frequency assumption. Assuming some +desired parameterized rotational kinematics (inspired from +real insects), the presented model is simulated in a near- +hover scenario by formulating and solving numerically an +optimization problem. The simulation demonstrated the +importance of the inertial forces and wing-body interac- +tions on the ornithopter’s dynamics. +ACKNOWLEDGEMENTS +This work was supported by the National Sciences and +Engineering Research Council of Canada (NSERC), under +the grants NSERC-DG RGPIN 2020-06270 and NSERC- +DG RGPIN-2020-04759. +REFERENCES +Alexander, D. 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(2016). Fast contour-tracing algorithm based on a +pixel-following method for image sensors. Sensors. doi: +10.3390/s16030353. +Sridhar, M., Kang, C., and Lee, T. (2020). +Geometric +formulation for the dynamics of monarch butterfly with +the effects of abdomen undulation. +In AIAA Scitech +2020 Forum, 1962. +Sun, M. (2014). +Insect flight dynamics: Stability and +control. Reviews of Modern Physics, 86(2), 615. +Sun, M., Wang, J., and Xiong, Y. (2007). Dynamic flight +stability of hovering insects. Acta Mech Sin. +Taha, H.E., Hajj, M.R., and Nayfeh, A.H. (2012). Flight +dynamics and control of flapping-wing mavs: a review. +Nonlinear Dynamics, 70(2), 907–939. +Tejaswi, K., Kang, C.K., and Lee, T. (2021). Dynamics +and control of a flapping wing uav with abdomen +undulation inspired by monarch butterfly. +In 2021 +American Control Conference (ACC), 66–71. + +Usherwood, J. and Ellington, C. (2002). The aerodynamics +of revolving wings. i. model hawkmoth wing. Journal of +Experimental Biology, 205, 1547–1564. +Weis-Fogh, T. (1973). Quick estimates of flight fitness in +hovering animals, including novel mechanisms for lift +production. Journal of Experimental Biology, 59, 169– +230. +Xiong, Y. and Sun, M. (2008). Dynamic flight stability of +a bumblebee in forward flight. Acta Mechanica Sinica, +24, 25–36. doi:10.1007/s10409-007-0121-2. +Appendix A. PROOF OF LEMMA 1 +Let us consider the root of one of the wings that is located +at p + ABµi with respect to the inertial frame. Thus, the +FI-coordinates of an arbitrary point on the chord are given +by +˜νi,r,γ(g, ξ) := p + ABµi + ABAiνi(r, γ). +(A.1) +Therefore, its linear velocity in FI is given by +d +dt ˜νi,r,γ(g, ξ) = ˙p + AB ˆΩBµi + AB ˆΩBAiνi(r, γ) +(A.2) ++ ABAi ˆΩiνi(r, γ), +which is expressed in Fi by left-multiplying A⊤ +i A⊤ +B as: +Wi,r,γ(g, ξ) : = A⊤ +i A⊤ +B +d +dt ˜νi,r,γ(g, ξ) += (ABAi)⊤ ˙p + A⊤ +i ˆΩB(µi + Aiνi) + ˆΩiνi. +(A.3) +Appendix B. PROOF OF PROPOSITION 1 +The kinetic energy of the main body is the sum of the +translational and rotational kinetic energies of the main +body and is given by +TB = 1 +2mB∥ ˙p∥2 + 1 +2Ω⊤ +BJBΩB. +(B.1) +The kinetic energy of the each wing is the sum of the +translational and rotational kinetic energies and is given +by +Ti = 1 +2mi∥ ˙κI +i ∥2 + 1 +2ΩI⊤ +i +� +Ji + miˆκ2 +i +� +ΩI +i , +(B.2) +where κI +i is the position of the center of mass of the ith +wing in the inertial frame and is given by +κI +i = p + ABµi + ABAiκi, +(B.3) +and ΩI +i is the angular velocity of the ith wing with respect +to FI expressed in Fi and is given by +ΩI +i = Ωi + A⊤ +i ΩB. +(B.4) +The velocity of the center of mass of the ith wing is given +by +˙κI +i = ˙p + AB ˆΩB (µi + Aiκi) + ABAi ˆΩiκi. +(B.5) +Substituting the expressions of TB and Ti in (12), we +obtain the following expression of the total kinetic energy +T =1 +2mB∥ ˙p∥2 + 1 +2ΩB +⊤JBΩB + 1 +2 +� +i∈{1,...,4} +mi +� +∥ ˙p∥2− +2 ˙p⊤AB +� +ˆµi + � +Aiκi +� +ΩB − 2 ˙p⊤ABAiˆκiΩi− +Ω⊤ +B +� +ˆµi + � +Aiκi +�2 +ΩB + 2Ω⊤ +i ˆκ⊤ +i A⊤ +i +� +ˆµi + � +Aiκi +� +ΩB +� ++ +1 +2 +� +i∈{1,...,4} +� +Ω⊤ +BAi +� +Ji + miˆκ2 +i +� +A⊤ +i ΩB+ +2Ω⊤ +i +� +Ji + miˆκ2 +i +� +A⊤ +i ΩB + Ω⊤ +i JiΩi +� +, +(B.6) +which can be rearranged as in (13). The potential energy +of the main body is given by +UB = −mBge⊤ +3 p, +(B.7) +and the potential energy of each connected body Bi, i ∈ +{1, . . ., 4} is given by +Ui = −mige⊤ +3 (p + AB (µi + Aiκi)) . +(B.8) +The sum of potential energies gives (14). +Appendix C. PROOF OF PROPOSITION 2 +The left hand side of the equation δS = � tf +t0 δ(T − U)dt +can be written as follows: +δS = +� tf +t0 +� +∂L +∂ ˙p δ ˙p + ∂L +∂p δp + ∂L +∂ΩB +δΩB ++ ∂L +∂AB +δAB + +4 +� +i=1 +� ∂L +∂Ωi +δΩi + ∂L +∂Ai +δAi +� � +dt +Replacing the infinitesimal variations of the orientation +and the angular velocity by (Lee et al., 2018) +δAB = AB ˆη, +δΩB = ˙η + ˆΩBη, +δAi = Ai ˆηi, +δΩi = ˙ηi + ˆΩiηi. +(C.1) +where η : [t0, tf] → R3 is an infinitesimal variation that +vanishes at t0 and tf. The left-hand side of the equation +(15) will result in the following equation: +δS = +� tf +t0 +� � d +dt +�∂L +∂ ˙p +� +− ∂L +∂p +� +δp+ +� +d +dt +� ∂L +∂ΩB +� ++ ˆΩB +∂L +∂ΩB ++ +3 +� +i=1 +ˆabi +∂L +∂abi +� +η+ +4 +� +i=1 + + d +dt +� ∂L +∂Ωi +� ++ ˆΩi +∂L +∂Ωi ++ +3 +� +j=1 +ˆaij +∂L +∂aij + + ηi +� +dt. +(C.2) +The right hand side of (15) can be developed by consid- +ering an infinitesimal aerodynamic force dFi := dLi + +dDi ∈ R3 acting at the aerodynamic center location of +cf +i,r ∈ R3 of the wings. The location of the aerodynamic +center is given by ˜cf +i,r = p+ABµi+ABAicf +i,r in the inertial +frame. Thus, the corresponding virtual work due to the +aerodynamic force generated on one wing is given by +δWi = +� +Bi +(ABAidFi)⊤ δ +� +p + ABµi + ABAicf +i,r +� +. + +Replacing the infinitesimal variations by their equations +given in (C.1), the virtual work on one wing is given by +δWi = +� +ABAi +� +Bi +dFi +�⊤ +(δp + AB ˆηµi) ++ +�� +Bi +cf +i,r × dFi +�⊤ +ηi. +We replace dFi and the integral over the wing body surface +by its formula, to obtain +δWi = +� +ABAi +� li +0 +(dLi + dDi) +�⊤ +(δp + AB ˆηµi) ++ +�� li +0 +cf +i,r × (dLi + dDi) +�⊤ +ηi. +(C.3) +Next, we consider the virtual work due to the exerted +control torques on the joints, which is given by A⊤ +i τi in +the wing frame. There will be a reactive torque, namely +(−τi) exerted to the body. The total corresponding virtual +work will be +δWτ = +4 +� +i=1 +� +(A⊤ +i τi)⊤ηi + (−τi)⊤η +� +. +(C.4) +Now, we replace the aerodynamic forces and torques given +by (9)-(11) in (C.3), in view of the fact that δW = δWτ + +�4 +i=1 δWi, the right-hand side of (15) yields +� tf +t0 +δWdt = +� tf +t0 +� � +AB +4 +� +i=1 +AiFi +�⊤ +δp ++ +� 4 +� +i=1 +(µi × AiFi − τi) +�⊤ +η + +4 +� +i=1 +� +Mi + A⊤ +i τi +�⊤ ηi +� +dt. +The Lagrange-D’Alembert equations are obtained by +grouping the terms from the right-hand side, and the left- +hand side that are multiplied by the linearly independent +infinitesimal variations δx, δη and δηi. Then, invoking +Hamilton’s principle that states that δS = 0, for all +possible variations with fixed endpoints, and using the +fact that the infinitesimal variations vanish at t0 and tf, +one obtains the Lagrange-D’Alembert equations given in +(16)-(18) as a result of the value inside of the integral in +equations (C.2), and (D.1) being equal to zero +Appendix D. THE REDUCED DYNAMICAL MODEL +To obtain the reduced dynamical model presented in (29), +first, we will proceed by writing the matrices C, N and +S, and the vectors Fu := Hcτ, Fg and Fa, presented in +equation (19) as follows: +C = +� ˜C11 ˜C12 +˜C21 ˜C22 +� +, +N = +� ˜N11 ˜N12 +˜N21 ˜N22, +� +, +S = +� ˜S11 ˜S12 +˜S21 ˜S22 +� +Fa = +� ˜Fa1 +˜Fa2 +� +, +Fu = +� ˜Fu1 +˜Fu1 +� +, +Fg = +� ˜Fg1 +˜Fg1 +� +, +where the matrices ˜C11, ˜N11, ˜S11 ∈ R6×6, and ˜C12, ˜N12, +˜S12 ∈ R6×12, and ˜C21, ˜N21, ˜S21 ∈ R12×6, and ˜C22, ˜N22, +˜S22 ∈ R12×12. The vectors +˜Fa1, ˜Fu1, +˜Fg1 ∈ R6, and +Parameter +Numerical Value +ρ +1.2kg/m3 +mB +5.4483e−5kg +m1,2 +1.9069e−6kg +m3,4 +2.3126e−6kg +l1,2 +0.0185m +l3,4 +0.025m +Table D.1. Wings morphology parameters. +˜Fa2, ˜Fu2, ˜Fg2 ∈ R12. Moreover, from the expression of +Fu := Hcτ, one can easily notice the following relationship +˜Fu1 = +� 0 +0 +0 +0 +−A1 −A2 −A3 A4 +� +˜Fu2 =: K ˜Fu2. +(D.1) +Replacing ˜Fu2 with the wings dynamics given by the 3th, +. . . , 6th rows of (19), one can write the translational and +rotational dynamics of the main body as follows: +� +˜C11 − K ˜C21 +� +˙ξB = +� +K ˜S22 ˜C21 − ˜S11 ˜C11 +� +ξB ++ +� +K ˜N21 − ˜N11 +� +ξB + +� +K ˜C22 − ˜C12 +� +˙ξw ++ +� +K ˜S22 ˜C22 − ˜S11 ˜C12 +� +ξw + +� +K ˜N22 − ˜N12 +� +ξw ++ ˜Fa1 + ˜Fg1 − K +� +˜Fa2 + ˜Fg2 +� +. +(D.2) +Consequently, the matrices CΘ +t +and DΘ +t +and the vectors +VΘ +t , FΘ +t in (29) are given by +Ct +Θ(gB) = +� +˜C11 − K ˜C21 +� +, +Dt +Θ(gB, ξB) = +� � +K ˜S22 ˜C21 − ˜S11 ˜C11 + K ˜N21 − ˜N11 +� ++ + + +0 +� +j∈{1,...,4} +Aj +4 +� +i=1 +� +� +JiΩiA⊤ +i − mi � +ˆκiΩiA⊤ +i ˆµi +� + + ++ + + +0 +4 +� +i=1 +miAB � +AiˆκiΩi + + + + + +0 +4 +� +i=1 +− � +J[23] +i +Ωi + + +� +Vt +Θ (gB) = +� +K ˜C22 − ˜C12 +� +˙ξw + +� +K ˜S22 ˜C22 − ˜Nw +� +ξw+ +˜Fg1 − K ˜Fg2 +Ft +Θ (gB, ξB) = ˜Fa1 − K ˜Fa2. +with ˜Nw defined as follows +˜Nw := +� +m1ABA1 ˆΩ1ˆκ1 miABAi ˆΩiˆκi miABAi ˆΩiˆκi miABAi ˆΩiˆκi +N23 +N24 +N25 +N26 +� +. +Appendix E. DRAGONFLY MORPHOLOGY +The dragonfly morphological parameters are calculated +from the dead dragonfly shown in Fig. +6. First, we will +suppose that the main body is a cylinder, with a diameter +of DB = 7.6mm and a height of HB = 25mm. Using +this assumption, and the fact that the distance between +the hind and fore wings’ roots is 5.42mm, one can easily +calculate JB and µi as follows: + +JB = 10−8 + + +0.0109 +0 +0 +0 +0.2847 +0 +0 +0 +0.2847 + + , +(E.1) +µ1 = 10−3 [2.71 3.8 0]⊤ , +µ2 = 10−3 [2.71 −3.8 0]⊤ , +µ3 = 10−3 [−2.71 3.8 0]⊤ , +µ4 = 10−3 [−2.71 −3.8 0]⊤ . +(E.2) +Next, using the wings’ polynomials qLE +i +and qT E +i +, one can +directly calculate the wings parameters Ji, κi, mi and li as +follows: +κ1,2 = 10−3 � +7.978 (−1)i+14.975 0 +�⊤ , +κ3,4 = 10−3 � +1.175 (−1)i+15.89 0 +�⊤ , +(E.3) +J1,2 = 10−3 + + +0.2303 +(−1)i+10.1902 +0 +(−1)i+10.1902 +0.1731 +0 +0 +0 +0.4034 + + , +J3,4 = 10−3 + + +0.4164 +(−1)i+10.0693 +0 +(−1)i+10.0693 +0.0326 +0 +0 +0 +0.4489 + + . +Appendix F. THE CONTROL DYNAMICAL MODEL +The control dynamical model, presented in (30), is ob- +tained by considering the aerodynamic forces and torques +as control inputs denoted by FC and ΓC. The terms that +represent the inertial forces and torques due to the body +motion are grouped and denoted by FB and ΓB i.e. the +terms that are multiplied by ˙ξB and ξB. Finally, the terms +that are multiplied by ˙ξw and ξw alone, represent the forces +and torques due to the wing motion. Consequently, the +forces and torques expressions are given by +Fc = +4 +� +i=1 +ABAiFi, +(F.1) +Fw = +4 +� +i=1 +� +2miAB ˆΩBAiˆκiΩi + miAiˆκi ˙Ωi+ +miAi ˆΩiˆκiΩi +� +(F.2) +FB = +4 +� +i=1 +� +miAB +� +ˆµi + � +Aiκi +� +˙ΩB+ +miAB ˆΩB +� +ˆµi + � +Aiκi +� +ΩB +� +, +(F.3) +Γc = +4 +� +i=1 +(ˆµiAiFi + AiMi) . +(F.4) +Γw = − +4 +� +i=1 +� +Ai ˆΩiJiA⊤ +i + AiJi ˆΩ⊤ +i A⊤ +i + mi +� +ˆµ⊤ +i � +Ai ˆΩiκi + � +Ai ˆΩiκi +⊤ +ˆµi +� +− miAiˆκi ˆΩiA⊤ +i ˆµi +� +ΩB ++ +4 +� +i=1 +� +Ai � +A⊤ +i ΩBJi + mi � +A⊤ +i ˆµiΩBˆκi + mi ˆΩB ˆµiAiˆκi + ˆΩBAiJi +� +Ωi + +4 +� +i=1 +� +miAi ˆΩiˆκ⊤ +i A⊤ +i ˆµiΩB +− (2AiJi − miˆµiAiˆκi) ˙Ωi +� +. +(F.5) +ΓB = − +4 +� +i=1 +mi +� +ˆµi + 2 � +Aiκi +� +A⊤ +B ¨p − +4 +� +i=1 +� +miˆµ⊤ +i ˆµi + 2AiJiA⊤ +i + mi +� +ˆµ⊤ +i � +Aiκi + 2 � +Aiκi +⊤ˆµi +� � +˙ΩB +− +4 +� +i=1 +� +− mi +� +ˆµi + � +Aiκi +� +ˆΩBA⊤ +B + mi +� +� +ˆµiΩB + � +� +AiκiΩB +� +A⊤ +B + mi ˆΩB +� +ˆµi + +ˆ +Aiκi +� +A⊤ +B +� +˙p +− +4 +� +i=1 +� +− Ai � +JiA⊤ +i ΩBA⊤ +i − miAiˆκiA⊤ +i +� +ˆΩB ˆµi − Ai� +ˆµiΩB +� ++ mi ˆΩB +� +ˆµi + � +Aiκi +� +A⊤ +B +� +ΩB + +4 +� +i=1 +migˆµiA⊤ +Be3. +(F.6) + diff --git a/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/load_file.txt b/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..56b0f2b275d34adf72eaf7ccac76a4774b7bfa31 --- /dev/null +++ b/Z9E1T4oBgHgl3EQfcwRV/content/tmp_files/load_file.txt @@ -0,0 +1,786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf,len=785 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='03187v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='SY] 9 Jan 2023 Modeling of Four-Winged Micro Ornithopters Inspired by Dragonflies Oussama Sifour ∗ Soulaimane Berkane ∗∗ Abdelhamid Tayebi ∗∗∗ ∗ Department of Computer Science and Engineering, University of Quebec in Outaouais, Gatineau, QC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (e-mail: sifo01@uqo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ∗∗ Department of Computer Science and Engineering, University of Quebec in Outaouais, Gatineau, QC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (e-mail: Soulaimane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='Berkane@uqo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='ca) ∗∗∗ Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (e-mail: atayebi@lakeheadu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='ca) Abstract: In this paper, we present a full dynamical model of a four-winged micro ornithopter inspired by a dragonfly-type insect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The micro ornithopter is modeled as four articulated rigid body components (wings) connected to the main body via spherical joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The dynamical model is derived using Lagrangian mechanics with intrinsic global coordinates, without relying on the common assumptions that neglect the wings-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Furthermore, the aerodynamic forces are modeled under the quasi-steady motion assumption without restricting the flapping frequency to be relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This provides a full and elegant four-winged micro ornithopter model that captures the interaction between the body and the wings while avoiding the complexities and singularities associated with other coordinate representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', Euler angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Simulation studies of the inertial effects of the relative motion between the different parts of the multibody system shows the importance of considering the forces and torques, resulting from the wings-body interaction, in motion generation of these insects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Keywords: Flapping Wing Unmanned Aerial Vehicles, Dragonfly, Ornithopter, Lagrangian Mechanics, Global Coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' INTRODUCTION Flapping wing unmanned aerial vehicles (FWUAVs), com- monly known as ornithopters, have received an increasing attention over the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The recent developments are encouraging, but the field is still in its infancy and considerable research efforts are needed to efficiently de- ploy these bio-inspired platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' FWUAVs are particu- larly suitable for micro aerial vehicles (MAVs) applications where classical ground and aerial vehicles are inefficient, such as in search and rescue missions where these minia- ture autonomous vehicle can, for instance, fly through cracks in concrete to search for earthquake victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In fact, FWUAVs enjoy some interesting features that can- not be found in fixed-wing and rotary-wing vehicles, such as energy efficiency, agility and miniaturization capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This has motivated to study the aerodynamics and flight mechanics of insects and birds for clues that would help in the design of FWUAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The development of FWUAVs is often inspired from real birds and insects flights (Breugel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' One of the best flyers in the insect world is the dragonfly–a four- wing insect that has been extensively studied due to its astonishing flight capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' For example, using only its own body muscles, the dragonfly can accelerate up to 3g and reach a speed of 10m/s (May, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Ruppell, 1989), and is capable of generating an instantaneous lift five times greater than its weight (Reavis and Luttges, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Each of its wings can be actuated independently (Alexander, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Azuma and Watanabe, 1988), which provides additional degrees of freedom helping the insect to perform agile and complex flight maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The difficulty in modeling and controlling FWUAVS is mainly due to the aeroelastic phenomena and the intrinsic coupling between the wings and the main body of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Most of the existing mathematical models in the literature, for insect-inspired vehicles, rely on simplifying assumptions such as large flap- ping frequency, small body-wing mass ratio and neglecting the aerodynamic couplings between the different parts of the vehicle (Taha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Sun, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The resulting simplified models are often similar to single rigid-body systems with forces and torques depending on the flapping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The coupling between the dif- ferent parts of the multibody system (wings, tail, main body, etc), is particularly important for FWUAVs with large wings and relatively low flapping frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Some attempts to capture these coupling effects have been made in (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' It is known that insects generate aerodynamic forces through wings motion (flapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' These aerodynamic forces are often modeled using a quasi-steady assumption (the forces generated by a flapping wing are equal to the forces generated by the fixed wing at the same instantaneous velocity and attitude of the wings blade).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This approach was mainly based on fixed- wing theory, but it has been used as a simple but robust modeling tool for flapping wings for several decades (Weis- Fogh, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Ellington, 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Later on, this approach was refined by the introduction of the lift and drag coefficients of the wings in flapping motion (Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Usherwood and Ellington, 2002) to make it applicable for flapping wing vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' However, even at the fastest flight speeds, the quasi-steady aerodynamic interpretation seems inadequate to explain the extra lift produced by real insects flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The importance of unsteady aerodynamic mechanisms for flapping insects flights has become widely recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Some numerical simulations of unsteady insect flight aerodynamics based on the finite element solution of the Navier–Stokes equations gave accurate results for the estimated aerodynamic forces (Sanders and Verhulst, 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' However, their implementation is unsuitable for control purposes since they require high processing power and, contrary to quasi-steady aerodynamics, they cannot be formulated as a function of the flapping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In this work, using Langrange formalism, we propose a high- fidelity dynamical model for the dragonfly-like ornithopter, without relying on the high flapping frequency assumption and taking into consideration all the interaction forces and torques inherent to this multibody system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The dynamical model uses intrinsic global coordinates, mainly attitudes on the Special Orthogonal group of rotations, which avoids the complexities and singularities associated with other coordinate representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', Euler angles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' NOTATION We denote by R the set of reals and by N the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We denote by Rn the n−dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We use ∥x∥ to denote the Euclidean norm of a vector x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We denote by F = {c, x, y, z} an Euclidean 3-dimensional frame with center at c ∈ R3 and axes {x, y, z} defining an orthonormal basis of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' For a vector x ∈ Rn, we denote by sgn(x) := [sgn(x1), · · · , sgn(xn)]⊤ the element-wise signum function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The special orthogonal group of order three is denoted by SO(3) := � A ∈ R3×3 : det(A) = 1, AA⊤ = I � , where I is the 3 × 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The unit vector ei denotes the ith column of the identity matrix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The set so(3) := � Ω ∈ R3×3 : Ω⊤ = −Ω � denotes the Lie algebra of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' For each z ∈ Rn \\ {0}, we denote by P(z) := I − zz⊤∥z∥−2 the orthogonal projection operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' For x, y ∈ R3, the map ˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' : R3 → so(3) is defined as ˆx := \uf8ee \uf8f0 0 −x3 x2 x3 0 −x1 −x2 x1 0 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Then, we have ˆxy := x × y where × is the vector cross- product on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Any rotation matrix A ∈ SO(3) can be parameterized by a unit vector u ∈ R3 and an angle θ ∈ R through the exponential map (Rodrigues formula) A = exp (θˆu) := I + sin(θ)ˆu + (1 − cos(θ))ˆu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' MECHANICAL CONFIGURATION Let us consider a flapping wing micro aerial vehicle that can translate and rotate in three dimensions as multiple rigid bodies (2 fore-wings and 2 hind-wings) connected to the main body via spherical joints that constrain the five rigid bodies to remain in contact, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We define six Euclidean frames: an inertial frame of reference FI = {0, rIx, rIy, rIz}, a body-attached frame FB = {OB, rbx, rby, rbz} for the main body, and a wing-attached frame Fi = {Oi, rix, riy, riz}, i ∈ {1, · · · , 4}, where OB is the center of mass of the main body, and Oi is the center of the joint connecting the the ith wing to the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We use index 1 for the right fore-wing, 2 for the left fore-wing, 3 for the right hind-wing, and 4 for the left hind-wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let AB ∈ SO(3) denote the attitude matrix from the body-attached frame FB to the inertial frame FI, and let ΩB ∈ R3 represent the angular velocity of the body- attached frame FB with respect to the inertial frame FI expressed in FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let Ai ∈ SO(3) denote the attitude matrix from the ith wing-attached frame Fi to the body- attached frame FB and let Ωi ∈ R3 represent its angular velocity with respect to FB expressed in Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Now, let µi, with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' , 4}, represent the vector from the origin of the body-attached frame FB to the connection joint (center of frame Fi) expressed in FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let κi represent the vector from the joint that connects the wing and the main body (center of frame Fi) to the wing center of mass expressed in Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, let p ∈ R3 be the position vector of the origin of FB expressed in FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' p FB FI Right Fore-wing (B1) Main Body F1 e2 e1 e3 r1y r1x µ1 Left Fore-wing (B2) F2 Connection Joint Right Hind-wing (B3) Left Hind-wing (B4) c3(r) dr κ2 qLE 2 qTE 2 qLE 4 qTE 4 r cf 3,r Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' A schematic diagram of a four-winged ornithopter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, the overall configuration of this multi-body system is denoted by g := (gB, gw), where gB := (p, AB) ∈ R3 × SO(3) denotes the main body pose and gw := (A1, A2, A3, A4) ∈ SO(3)4 represents the four wings con- figurations (attitudes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The group velocity is denoted by ξ := (ξB, ξw), where ξB := ( ˙p, ΩB) ∈ R6 is the main body group velocity and ξw := (Ω1, Ω2, Ω3, Ω4) ∈ R12 represents the wings group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' QUASI-STEADY AERODYNAMICS Flapping-wings flights are more complicated than fixed- wings flights because of the structural movement and the resulting unsteady fluid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In conventional airplanes with fixed wings, the forward motion relative to the air causes the wings to produce lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' However, in a flapping-wings flight, the wings not only move forward relative to the air but also flap up and down and perform rotations around their roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In this section, we will need to specify the geometry of the wings, as the resultant forces will depend on the shape of the wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Then, we will use blade-element theory (Ellington, 1984) under the quasi- steady aerodynamics assumption to express the forces and torques generated through flapping motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We rely on the quasi-steady assumption to simplify the forces and torques calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Intuitively, this assumption implies that these forces and torques are equivalent to those generated in steady motion at the same instantaneous velocity and the same angle of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1 Wing Geometry The dragonfly’s wings are responsible for its incredible flight performance and force generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The shape of the fore-wing and hind-wing of the dragonfly are often approximated by an elliptical function (tear-drop shape), see (Weis-Fogh, 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' However, as it is shown in (Faisal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='Filippone, 2016), a more realistic representation of the wings’ shape gives a better approximation of the aerodynamic forces generated by a real dragonfly, than the traditional tear-drop shape representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' A better approximation of the dragonfly’s wings geometry can be obtained via a set of polynomials derived from the analysis of real photography images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We assume that each wing is a flat plate lying on the rix − riy plane of Fi (no component on riz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Each wing’s shape is described by two polynomial functions that represent the leading (upper bound) and the trailing (lower bound) edge, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 1: qLE i (r) := j=n � j=0 λLE ij rj, qT E i (r) := j=n � j=0 λT E ij rj, (1) where r is the argument along the riy axis and the qLE i (r) and qT E i (r) are the corresponding points locations of the trailing and leading edges, respectively, along the rix axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The polynomials coefficients λLE ij and λT E ij are calculated by fitting the points from the contour lines of the wings (obtained from real photography images) with polynomials of degree n to minimize the mean squared errors while avoiding overfitting, see Section 6 for a numerical example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2 Aerodynamic Forces To inspect the quasi-steady aerodynamic forces, let dr be an infinitesimal wing segment, parallel to the rix axis of the wing frame, and located at a distance r from the wing root (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 1), which is measured along the riy axis of the wing frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let its chord length be defined as ci(r) := qLE i (r)− qT E i (r) where qLE i (·) and qT E i (·) are defined as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let νi(r, γ) := (qLE i (r)−γci(r))e1+(−1)i+1re2, with γ ∈ [0, 1], be the Fi-coordinates of an arbitrary point on the chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Then, the velocity of this point with respect to FI, expressed in Fi, is given by Wi,r,γ(g, ξ) = (ABAi)⊤ ˙p + A⊤ i ˆΩB(µi + Aiνi(r, γ)) +ˆΩiνi(r, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (2) The first term of Wi,r,γ(g, ξ) is due to main body’s trans- lation motion, the second term is due the main body’s rotation, and the last term is due to the wing’s flapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The proof of this lemma is given in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Now, we proceed as in (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020) to determine the angle of attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' As per the blade-element theory (Ellington, 1984), the aerodynamic force generated by the infinites- imal chord depends only on the rix, and riz components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Therefore, we project the above velocity on the plane rix − riz to obtain the effective velocity: W i,r,γ(g, ξ) := P(e2)Wi,r,γ(g, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (3) The state-dependent angle of attack of the chord, denoted αi,r(g, ξ), is the angle between the chord line (from the leading edge to the trailing edge), and the above velocity W i,r,γ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Since the angle of attack changes slightly chord- wise, we will define the angle of attack as the angle between the chord line and the velocity of the center of the chord (γ = 1/2), and it is given by αi,r(g, ξ) := cos−1 \uf8eb \uf8ede⊤ 1 W i,r, 1 2 (g, ξ) ���W i,r, 1 2 (g, ξ) ��� \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (4) We consider the location of the state-dependent aerody- namic center, denoted cf i,r(g, ξ), as a function of the angle of attack αi,r, and it is given by cf i,r(g, ξ) := νi(r, γac(αi,r(g, ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (5) where γac(·) : [0, π] → [0, 1] maps the angle of attack to the position of the aerodynamic center along the chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The effective velocity of the aerodynamic center will be denoted by W ac i,r(g, ξ) := W i,r,γ(g, ξ) with γ = γac(αi,r(g, ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The magnitude of the lift and drag forces generated by the infinitesimal wing segment are given by ∥dLi,r(g, ξ)∥ = 1 2ρ∥W ac i,r∥2CL(αi,r)ci(r)dr, (6) ∥dDi,r(g, ξ)∥ = 1 2ρ∥W ac i,r∥2CD(αi,r)ci(r)dr, (7) where ρ ∈ R is the atmospheric density and CL, CD : [0, π] → R are the lift and drag coefficients, which depend on the angle of attack αi,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Following the same steps as (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020), one can determine the infinitesimal lift and drag forces as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The direction of the lift is normal to both the velocity W ac i,r and the wing span-wise direction riy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' As such, the direction of the lift is along ±e2 × W ac i,r in Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Thus, we multiply the lift magnitude by the unit vector (e2 × W ac i,r)∥W ac i,r∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' To solve the sign ambiguity induced by the flapping motion (up-stroke and down-stroke), we consider the four cases shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' dLi rix riz W ac i,r Chord case 1 dLi rix riz W ac i,r Chord case 2 dLi rix riz W ac i,r Chord case 3 dLi rix riz W ac i,r Chord case 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Different cases of the direction of the lift with respect to the direction of W ac i,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Figure inspired from (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 2, one can notice that the direction of the lift force is in the direction of e2 × W ac i,r if the first and third components of W ac i,r have the same sign (case 1 and case 3), otherwise it is in the direction of −e2 × W ac i,r (case 2 and case 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The direction of the drag force, however, is opposite to W ac i,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, the corresponding aerodynamic forces and torques generated by the infinitesimal wing segment can be expressed in Fi as follows: 1 We might drop the arguments of a given function whenever clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' dLi,r(g, ξ) = 1 2 ρCL(αi,r)ci(r) ��W ac i,r �� sign( ¯wi rx ¯wi rz)(e2 × W ac i,r)dr, dDi,r(g, ξ) = − 1 2ρCD(αi,r)ci(r) ��W ac i,r �� W ac i,rdr, where ¯wi rx and ¯wi rz are the first and third components of W ac i,r, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The infinitesimal lift and drag forces, which are applied at the aerodynamic center, also generate the following infinitesimal torque about the wing root dMi,r(g, ξ) = cf i,r(g, ξ) × (dLi,r + dDi,r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (8) The total lift Li(g, ξ), drag Di(g, ξ), and torque Mi(g, ξ) generated on the ith wing, are obtained by integrating the above infinitesimal expressions span-wise for r ∈ [0, li], where li > 0 is the wing’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Li(g, ξ) := � li 0 dLi,r(g, ξ), (9) Di(g, ξ) := � li 0 dDi,r(g, ξ), (10) Mi(g, ξ) := � li 0 dMi,r(g, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (11) Compared with the other models considering uniform forces over the wing (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2005), this approach captures the span-wide variations of the aerodynamic forces, which are critical for FWUAVs with large wings flapping at a relatively low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' DRAGONFLY MODELING USING LAGRANGE-D’ALEMBERT PRINCIPLE In this section, we will derive the expressions of the kinetic and potential energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We will use them along with the aerodynamic forces, detailed in Section 4, to derive a com- plete dynamical model for the four-winged ornithopter, us- ing the Lagrange-D’Alembert principle (Lagrange, 1788).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1 Kinetic and Potential Energies The kinetic and the potential energies of the complete multibody system, denoted respectively by T and U, are the sum of the kinetic and potential energies of each rigid- body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' They are given by T = � j∈{B,1,2,3,4} Tj, U = � j∈{B,1,2,3,4} Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (12) The kinetic and the potential energies are explicitly ex- pressed in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The total kinetic energy can be expressed as follows: T (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ξ) = 1 2 4 � i=1 \uf8ee \uf8f0 ˙p ΩB Ωi \uf8f9 \uf8fb ⊤ Ji(g) \uf8ee \uf8f0 ˙p ΩB Ωi \uf8f9 \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (13) U(g) = − � i∈{B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4} mige⊤ 3 (p + AB (µi + Aiκi)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (14) with µB = κB = 0 and Ji(·) is the symmetric matrix Ji(g) := \uf8ee \uf8ef\uf8f0 � 1 4mB + mi � I ⋆ ⋆ mi � ˆµi + � Aiκi � A⊤ B J[22] i ⋆ miˆκiA⊤ i A⊤ B JiA⊤ i + miˆκ⊤ i A⊤ i ˆµi Ji \uf8f9 \uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' where J[22] i := � AiJiA⊤ i −miˆµ2 i +miˆµ⊤ i � Aiκi+mi � Aiκiˆµ⊤ i + 1 4JB � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' mB and mi are the mass of the main body and the con- nected bodies respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' g represents the acceleration of gravity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' JB ∈ R3×3 represents the constant inertia matrix of the main body about FB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and Ji ∈ R3×3 represents the inertia matrix of the ith body about Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The proof of this proposition is given in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2 Lagrange-D’Alembert principle This principle consists of a modification of Hamilton’s principle to incorporate the effects of external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' These external forces may or may not be derivable from a potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This modification states that the infinitesimal variation of the integral action of T − U over a fixed time period equals the work, denoted δW, done by the external forces, corresponding to an infinitesimal variation of the configuration, during this same time period (also known as the virtual work of the external forces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Formally, for any t0 ≥ 0 and tf ≥ t0, we have � tf t0 δ(T (t) − U(t))dt = � tf t0 δW(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (15) This version of the variational principle requires determin- ing the virtual work that corresponds to an infinitesimal variation of the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let L := T − U represent the Lagrangian and let Fi := Li + Di represent the sum of forces acting on the ith wing, and let τi ∈ R3 be the control torque exerted at the joint connecting the ith wing to the body, expressed in the body-attached frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Following the developments in (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2018), and using abj ∈ R3 and aij ∈ R3 to denote the jth column of AB ∈ SO(3) and the jth column of Ai ∈ SO(3), respectively, for i = 1, 2, 3, the Lagrange-d’Alembert principle leads to the equations stated in the following proposition: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The Lagrange-d’Alembert principle (15) leads to the following equations: d dt �∂L ∂ ˙p � − ∂L ∂p = 4 � i=1 ABAiFi, (16) d dt � ∂L ∂ΩB � + ˆΩB ∂L ∂ΩB + 3 � j=1 ˆabj ∂L ∂abj = 4 � i=1 ˆµiAiFi − 4 � i=1 τi, (17) d dt � ∂L ∂Ωi � + ˆΩi ∂L ∂Ωi + 3 � j=1 ˆaij ∂L ∂aij = Mi + A⊤ i τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (18) The proof is given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3 Full Dynamical Model In this subsection, we use proposition 3 and the expression of L = T − U to derive a dynamical model for the dragonfly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Using the kinetic and potential energies in the previous Lagrange-D’Alembert equations, and defining τ = (τ1, τ2, τ3, τ4) ∈ R12, one can derive the dynamical model as follows: C(g) ˙ξ + D(g, ξ)ξ = Fa (g, ξ) + Hc (g) τ + Fg(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (19) with D(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ξ) := S (ξ) C(g) + N(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ξ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' such that S (ξ) := diag � 03×3 ˆΩB ˆΩ1 ˆΩ2 ˆΩ3 ˆΩ4 � and C(g) is a 18×18 matrix given by C(g) := \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 mI 4 � i=1 J[12] i J[13] 1 J[13] 2 J[13] 3 J[13] 4 4 � i=1 J[21] i 4 � i=1 J[22] i J[23] 1 J[23] 2 J[23] 3 J[23] 4 J[31] 1 J[32] 1 J[33] 1 03×3 03×3 03×3 J[31] 2 J[32] 2 03×3 J[33] 2 03×3 03×3 J[31] 3 J[32] 3 03×3 03×3 J[33] 3 03×3 J[31] 4 J[32] 4 03×3 03×3 03×3 J[33] 4 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (20) where m := � i∈{B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4} mi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and J[jk] i denotes the matrix block at the jth line and the kth row of the matrix Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The 3 × 3 block components of the 18 × 18 matrix N(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ξ) are given below: N11 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N12 = 4 � i=1 −miAB ˆΩB � ˆµi + � Aiκi � − miAB � Ai ˆΩiκi N1(k+2) = −mkAB �ˆΩBAk + Ak ˆΩk � ˆκk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N21 = 4 � i=1 −mi � ˆµi + � Aiκi � ˆΩBA⊤ B + mi � � ˆµiΩB + � � AiκiΩB � A⊤ B N22 = 4 � i=1 Ai ˆΩiJiA⊤ i − AiJi ˆΩiA⊤ i − mi � ˆµi � Ai ˆΩiκi + � Ai ˆΩiκiˆµi � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N2(k+2) = Ak ˆΩkJk − mk ˆµkAk ˆΩkˆκk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N31 = 4 � i=1 −miˆκi � A⊤ i ˆΩB + ˆΩiA⊤ i � A⊤ B + miˆκiAi ˆΩBA⊤ B + mi � ˆκiΩiA⊤ i A⊤ B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N32 = 4 � i=1 −Ji ˆΩiA⊤ i + miˆκi ˆΩiA⊤ i ˆµi − � JiA⊤ i ΩBA⊤ i − miˆκiA⊤ i �ˆΩB ˆµi − � ˆµiΩB � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Nj(k+2) = � A⊤ k ΩB ⊤ Jk + mk � A⊤ k ˆµkΩBˆκ⊤ k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' with k ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 4} and j ∈ {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The aerodynamic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' gravitational,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and torque control forces are given by Fa := \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 4 � i=1 ABAiFi 4 � i=1 ˆµiAiFi M1 M2 M3 M4 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Hc(g) := \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 −I −I −I −I A⊤ 1 0 0 0 0 A⊤ 2 0 0 0 0 A⊤ 3 0 0 0 0 A⊤ 4 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fg := \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 mge3 4 � i=1 mig � ˆµi + � Aiκi � A⊤ Be3 m1gˆκ1 � A⊤ 1 A⊤ Be3 � m2gˆκ2 � A⊤ 2 A⊤ Be3 � m3gˆκ3 � A⊤ 3 A⊤ Be3 � m4gˆκ4 � A⊤ 4 A⊤ Be3 � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (21) This is a complete dynamical model, that provides the position, orientation, and wing dynamics of a four-winged ornithopter, with input torques applied at the joints con- necting the wings to the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Torque-controlled model (19) τ (g, ξ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Torque-controlled model (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' REDUCED DYNAMICAL MODEL VIA WINGS KINEMATICS ASSIGNMENT Through a particular choice of the control inputs, one can generate wings motions mimicking real flapping-wing animals flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This leads to a simplified model where the control inputs are the parameters related to the wings kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1 Desired Wing Flapping Kinematics Berman and Wang (2007) proposed a wing kinematic configuration for insects using Euler angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This config- uration minimizes energy in hovering flights, and captures several qualitative aspects of observed real insect flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' It can also be applied to other flight maneuvers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', accelerating in different directions) as manipulating the three angles parameters generate aerodynamic forces in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In the approach of Berman and Wang (2007), each wing’s attitude is given by three Euler angles: the flapping angle φi(t), the deviation angle ψi(t), and the pitching angle θi(t), about the stroke frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The stroke frame, denoted by FiS = {Oi, six, siy, siz}, is obtained by rotating the wing frame Fi by a fixed angle βi ∈ R about the body-frame y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The corresponding wing attitude for this kinematic configuration is taken from (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020), and it is given by Ai = exp (βiˆe2) exp((−1)i+1φiˆe1) exp((−1)iψiˆe3) exp (θiˆe2) , with i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The flapping angle is given by φi(t) = φim sin−1 φiK sin−1 (φiK cos(2πft + φia)) + φi0, (22) where f ∈ R represents the flapping frequency in Hz, φim ∈ R is the amplitude, φia ∈ R is the phase, φi0 ∈ R is the offset, and 0 < φiK ≤ 1 determines the waveform shape (sinusoidal if φiK → 0, triangular if φiK → 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The pitch angle is given by the following function: θi(t) = θim tanh θiC tanh (θiC sin (2πft + θia)) + θi0, (23) where θim ∈ R is the amplitude, θi0 ∈ R is the offset, θiC ∈ (0, ∞) determines the waveform (sinusoidal when θiC → 0, step function when θiC → ∞), and θia ∈ (−π, π) is the phase offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The value of θiC is related to the duration of wing pitch reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, the deviation angle is given by ψi(t) = ψim cos (2πψiNft + ψia) + ψi0, (24) where ψim ∈ R is the amplitude, ψi0 ∈ R is the offset, and the parameter ψia ∈ (−π, π) is the phase offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The parameter ψiN ∈ {1, 2}, where ψiN = 1 corresponds to one oscillation per flapping period, and ψiN = 2 is for a figure-eight motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Flapping, pitching, and deviation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Positive angles are measured from FiS (in blue) to Fi (in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' For specific wings kinematics, we need to assign 13 pa- rameters per wing (51 parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' These parameters are constrained for the dragonfly as in Table 1, see (Faisal and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='Filippone, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Parameter range f : flapping frequency 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0 − 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='00 Hz φim : flapping amplitude 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ ψim : deviation amplitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ θim : pitching amplitude 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ φi0 : flapping offset −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ ψi0 : deviation offset 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ θi0 : pitching offset −90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ ψia : deviation phase −180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ φia : flapping phase −180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ θia : pitching phase −180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ φiK : waveform shape 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='01 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='00 θiC : waveform shape 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='01 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='00 βi : stroke plane angle 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ − 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0◦ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Parameters range for dragonfly wing kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The attitude kinematics of each wing is given by ˙Ai = Ai ˆΩi, (25) with i ∈ {1, · · · , 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We consider the angular velocities of the wings that are obtained from the time-derivatives of the Euler-angles equations (22)-(24) as follows (Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020): Ωi = \uf8ee \uf8f0 (−1)i+1 cos ψi cos θi 0 (−1)i+1 sin θi sin ψi 1 0 (−1)i+1 cos ψi sin θi 0 (−1)i cos θi \uf8f9 \uf8fb \uf8ee \uf8ef\uf8f0 ˙φi ˙θi ˙ψi \uf8f9 \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (26) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2 Reduced Dynamical Model Let (gd w, ξd w) be a desired time-varying wings kinematics generated according to equations (25)-(26), using a given parameters set Θ := (f, βi, φmi, ψmi, · · · ) as described in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Now, according to equation (18), the control torque τi is written as: τi = Ai \uf8eb \uf8ed d dt � ∂L ∂Ωi � + ˆΩi ∂L ∂Ωi + 3 � j=1 ˆaij ∂L ∂aij − Mi \uf8f6 \uf8f8 , := T(gB, ξB, gw, ξw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (27) Now assuming that the inner loop is fast enough such that (gw, ξw) ≈ (gd w, ξd w), the control torques can be written as τi ≈ T(gB, ξB, gd w, ξd w) =: Tt Θ(gB, ξB), (28) where we have used the fact that (gd w, ξd w) is a function of parameters Θ and time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Replacing this torque expression in the second equation of Proposition 2, will allow us to write the translational, and rotational dynamics of the main body as follows: Ct Θ(gB) ˙ξB + Dt Θ(gB, ξB)ξB + Vt Θ (gB) = Ft Θ (gB, ξB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (29) The matrices Ct Θ, Dt Θ, Vt Θ and Ft Θ and the development of this model are detailed in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Controller of wing Torque model (19) τ (gB, ξB) (gw, ξw) (gd w, ξd w) Wing (25) − (26) Θ kinematics controlled dynamics Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Parameters-controlled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Grouping the forces and torques of the same nature to- gether is another way to make this model ideal for design- ing control laws, and investigating wing-body interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' First, we write the aerodynamic forces and torques, which can be considered as control inputs in a tracking scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Second, we consider the inertial forces and torques of the wings due to body acceleration and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, we consider the inertial forces and torques of the wings due to wing motion (flapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The model is given by \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˙p = v, ˙AB = AB ˆΩB, m˙v = mge3 + Fc + FB + Fw, JB ˙ΩB = ˆΩBJBΩB + Γc + ΓB + Γw, (30) where v is the linear velocity of the body in the inertial frame, Fc and Γc are the aerodynamic forces and torques that can be used as a control inputs, Fw and Γw are the inertial forces and torques due to the flapping motion of the wings, FB and ΓB are the inertial forces and torques due to the translational and rotational motion of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The expression of these forces are given in appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In most cases, the inertial forces due to the wing motion are neglected because of the wing mass mi being too small compared to the body mass mB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In the case of the dragonfly, the wing mass mi represents around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5% of the body mass mB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Moreover, neglecting both the inertial forces and torques due to the wing motion and body motion FB, Fw, ΓB and Γw results in the widely used model of a flying rigid body (Padfield, 1996), used in (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Xiong and Sun, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Dietl and Garcia, 2008) to model insects flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' In the present work, we will take a closer look into the importance of these forces in near- hover scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we will investigate the effect of the inertial forces in a near-hover scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We will compare the trajectories performed by the dragonfly when neglecting the inertial forces versus when considering them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We will also compare the magnitudes of these inertial forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' First, it is necessary to describe the wings’ shape, using the polynomial functions qLE i and qT E i , since the aerodynamic forces depend on these polynomials, as shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Let us consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6 which shows an image of a dead dragonfly insect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=" The polynomials’ coefficients in equation (1) are calculated by fitting the points from the r2y S2yr1c 0 &'S1cS2y 12ycontour lines of the wings acquired by the algorithm in (Seo et al." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2016) with n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The polynomials’ coefficients are provided in the following table, and the polynomial functions are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6 j λLE 1j λT E 1j λLE 3j λT E 3j 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='873 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='183 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='648 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8389 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='372 2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='85 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='153 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='574 3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='16 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='25 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='981 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='016 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='111 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='857 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='210 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='728 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='625 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='627 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='051 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='164 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='082 R2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='012 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Image of a dragonfly with coefficients of wings polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Next, we take the location of the aerodynamic center cf i,r similar to the location taken in (Dickson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2006), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', the function γac(αi,r) in equation (5) is given by γac(α) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='82|α| π + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (31) The lift and drag coefficients CL(αi,r), and CD(αi,r) are functions of αi,r, and are taken similar to the coefficients proposed in (Dickinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 1999): CL(αi,r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='225 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='58 sin(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='13α◦ i,r − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='20), (32) CD(αi,r) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='92 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='55 cos(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='04α◦ i,r − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (33) with α◦ i,r = 180αi,r/π if αi,r ≤ π/2 and α◦ i,r = 180(π − αi,r)/π otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' To make the dragonfly perform a near- to-hover flight, we will need to find suitable parameters for the dragonfly’s wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This is challenging due to the complexity of the dynamics, and the relatively large number of the wing’s kinematics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This problem is addressed by performing an optimization to minimize a cost function, which is the error of the position and the velocity of the dragonfly from a reference hovering position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' More specifically, this problem is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We define the cost J = w1 � Tf t0 ∥p − pref∥2dt + w2 � Tf t0 ∥ ˙p∥2dt, where w1, w2 ∈ R, Tf = 10T is the flight time, T = 1/f is the flapping period, and pref = [0 0 2]⊤ is the hovering position reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This is to minimize the error of the position and velocity to ensure that the dragonfly is hovering at 2m altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Next, as in (Tejaswi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Sridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020), we assume that the dragonfly’s body exhibits a pitching motion (oscillating around e2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', AB(t) = exp(ΦB(t)ˆe2), with ΦB(t) = ΦBm cos (2πft + ΦBa) + ΦB0, where ΦBm, ΦBa and ΦB0 are some parameters to be determined hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Replacing AB and ˆΩB = A⊤ B ˙AB in (30), the trajectory p(t) can be obtained by integrat- ing the translational dynamics (first and third equations of (30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, the optimization parameter is ¯Θ := (Θ, ΦBm, ΦBa, ΦB0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The constraints on the parameters are given in Table 1, and the morphological parameters of the dragonfly like JB, Ji, µi are taken to be similar to those of an actual insect, and are presented in appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This problem is carried out with the Genetic Algorithm opti- mization (Anweer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2020) implemented in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The corresponding optimized parameters are summarized in Table 2, and the resulting trajectory is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Parameter Fore-wing1,2 Hind-wing3,4 f 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='6476 Hz 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='6476 Hz φim 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='42◦ 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='48◦ ψim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='16◦ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='26◦ θim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='43◦ 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='18◦ φi0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='64◦ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='49◦ ψi0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='49◦ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='29◦ θi0 −35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='24◦ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='83◦ φia 0◦ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='56◦ ψia −40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='10◦ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='37◦ θia −98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='82◦ −138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='47◦ φiK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='895 θiC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='394 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='613 βi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='95◦ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='53◦ ΦBm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='37◦ ΦBa −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='72◦ ΦB0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='434◦ Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Optimized parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 0 2 4 6 8 10 0 1 2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Forces magnitude comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 0 5 10 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5 1 10-3 0 5 10 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5 1 10-3 0 5 10 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Position comparison for the full model in (30) (red), when neglecting FB and Fw (black), when neglecting FB only (dashed green), and when neglecting Fw only (dashed blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 8, one can observe that neglecting the inertial forces due to body motion FB, results in a closer trajectory lcm LE LE qi q3(dashed green line) to the full model trajectory (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This result is due to the small magnitude of the inertial forces due to body motion (∥FB∥) as it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 7 in green lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' However, neglecting the inertial forces due to the wing motion, results in a trajectory (dashed blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 8) relatively far from the full model trajectory (red line), and closer to the flying rigid body model (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (2006)) trajectory (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This result is due to the relatively important magnitude of the inertial forces due to the wing motion (∥Fw∥), as it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 7 in blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='7 illustrate the importance of the wing-body interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The inertial forces caused by the wing motion are found to be quite important and should not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The inertial forces caused by the body motion are found to be negligible in this scenario, due to the slow body motion compared with the wing motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Considering the flying rigid body dynamical model as the one in (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2006), which is obtained by neglecting both inertial forces due to the body and wing motion, resulted in a different trajectory of the dragonfly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' This trajectory is significantly far from its actual trajectory in view of the dragonfly’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' CONCLUSION A full high-fidelity dynamical model for a four-winged micro ornithopter is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The ornithopter is modeled as four rigid bodies (wings) connected to a main rigid body via spherical joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Lagrangian formalism is used to derive the dynamical model using intrinsic global coor- dinates, without neglecting the wing masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Furthermore, the quasi-steady aerodynamics assumption is used to cal- culate the span-wise aerodynamic forces generated by in- finitesimal wing chords, in contrast to considering uniform aerodynamic forces over the wing, and without relying on the high flapping frequency assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Assuming some desired parameterized rotational kinematics (inspired from 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Journal of Experimental Biology, 59, 169– 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Xiong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Dynamic flight stability of a bumblebee in forward flight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Acta Mechanica Sinica, 24, 25–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1007/s10409-007-0121-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' PROOF OF LEMMA 1 Let us consider the root of one of the wings that is located at p + ABµi with respect to the inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Thus, the FI-coordinates of an arbitrary point on the chord are given by ˜νi,r,γ(g, ξ) := p + ABµi + ABAiνi(r, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) Therefore, its linear velocity in FI is given by d dt ˜νi,r,γ(g, ξ) = ˙p + AB ˆΩBµi + AB ˆΩBAiνi(r, γ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) + ABAi ˆΩiνi(r, γ), which is expressed in Fi by left-multiplying A⊤ i A⊤ B as: Wi,r,γ(g, ξ) : = A⊤ i A⊤ B d dt ˜νi,r,γ(g, ξ) = (ABAi)⊤ ˙p + A⊤ i ˆΩB(µi + Aiνi) + ˆΩiνi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3) Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' PROOF OF PROPOSITION 1 The kinetic energy of the main body is the sum of the translational and rotational kinetic energies of the main body and is given by TB = 1 2mB∥ ˙p∥2 + 1 2Ω⊤ BJBΩB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) The kinetic energy of the each wing is the sum of the translational and rotational kinetic energies and is given by Ti = 1 2mi∥ ˙κI i ∥2 + 1 2ΩI⊤ i � Ji + miˆκ2 i � ΩI i , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) where κI i is the position of the center of mass of the ith wing in the inertial frame and is given by κI i = p + ABµi + ABAiκi, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3) and ΩI i is the angular velocity of the ith wing with respect to FI expressed in Fi and is given by ΩI i = Ωi + A⊤ i ΩB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4) The velocity of the center of mass of the ith wing is given by ˙κI i = ˙p + AB ˆΩB (µi + Aiκi) + ABAi ˆΩiκi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5) Substituting the expressions of TB and Ti in (12), we obtain the following expression of the total kinetic energy T =1 2mB∥ ˙p∥2 + 1 2ΩB ⊤JBΩB + 1 2 � i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=',4} mi � ∥ ˙p∥2− 2 ˙p⊤AB � ˆµi + � Aiκi � ΩB − 2 ˙p⊤ABAiˆκiΩi− Ω⊤ B � ˆµi + � Aiκi �2 ΩB + 2Ω⊤ i ˆκ⊤ i A⊤ i � ˆµi + � Aiκi � ΩB � + 1 2 � i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=',4} � Ω⊤ BAi � Ji + miˆκ2 i � A⊤ i ΩB+ 2Ω⊤ i � Ji + miˆκ2 i � A⊤ i ΩB + Ω⊤ i JiΩi � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='6) which can be rearranged as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The potential energy of the main body is given by UB = −mBge⊤ 3 p, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='7) and the potential energy of each connected body Bi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 4} is given by Ui = −mige⊤ 3 (p + AB (µi + Aiκi)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8) The sum of potential energies gives (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' PROOF OF PROPOSITION 2 The left hand side of the equation δS = � tf t0 δ(T − U)dt can be written as follows: δS = � tf t0 � ∂L ∂ ˙p δ ˙p + ∂L ∂p δp + ∂L ∂ΩB δΩB + ∂L ∂AB δAB + 4 � i=1 � ∂L ∂Ωi δΩi + ∂L ∂Ai δAi � � dt Replacing the infinitesimal variations of the orientation and the angular velocity by (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=', 2018) δAB = AB ˆη, δΩB = ˙η + ˆΩBη, δAi = Ai ˆηi, δΩi = ˙ηi + ˆΩiηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) where η : [t0, tf] → R3 is an infinitesimal variation that vanishes at t0 and tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The left-hand side of the equation (15) will result in the following equation: δS = � tf t0 � � d dt �∂L ∂ ˙p � − ∂L ∂p � δp+ � d dt � ∂L ∂ΩB � + ˆΩB ∂L ∂ΩB + 3 � i=1 ˆabi ∂L ∂abi � η+ 4 � i=1 \uf8eb \uf8ed d dt � ∂L ∂Ωi � + ˆΩi ∂L ∂Ωi + 3 � j=1 ˆaij ∂L ∂aij \uf8f6 \uf8f8 ηi � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) The right hand side of (15) can be developed by consid- ering an infinitesimal aerodynamic force dFi := dLi + dDi ∈ R3 acting at the aerodynamic center location of cf i,r ∈ R3 of the wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The location of the aerodynamic center is given by ˜cf i,r = p+ABµi+ABAicf i,r in the inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Thus, the corresponding virtual work due to the aerodynamic force generated on one wing is given by δWi = � Bi (ABAidFi)⊤ δ � p + ABµi + ABAicf i,r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Replacing the infinitesimal variations by their equations given in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1), the virtual work on one wing is given by δWi = � ABAi � Bi dFi �⊤ (δp + AB ˆηµi) + �� Bi cf i,r × dFi �⊤ ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' We replace dFi and the integral over the wing body surface by its formula, to obtain δWi = � ABAi � li 0 (dLi + dDi) �⊤ (δp + AB ˆηµi) + �� li 0 cf i,r × (dLi + dDi) �⊤ ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3) Next, we consider the virtual work due to the exerted control torques on the joints, which is given by A⊤ i τi in the wing frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' There will be a reactive torque, namely (−τi) exerted to the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The total corresponding virtual work will be δWτ = 4 � i=1 � (A⊤ i τi)⊤ηi + (−τi)⊤η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4) Now, we replace the aerodynamic forces and torques given by (9)-(11) in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3), in view of the fact that δW = δWτ + �4 i=1 δWi, the right-hand side of (15) yields � tf t0 δWdt = � tf t0 � � AB 4 � i=1 AiFi �⊤ δp + � 4 � i=1 (µi × AiFi − τi) �⊤ η + 4 � i=1 � Mi + A⊤ i τi �⊤ ηi � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The Lagrange-D’Alembert equations are obtained by grouping the terms from the right-hand side, and the left- hand side that are multiplied by the linearly independent infinitesimal variations δx, δη and δηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Then, invoking Hamilton’s principle that states that δS = 0, for all possible variations with fixed endpoints, and using the fact that the infinitesimal variations vanish at t0 and tf, one obtains the Lagrange-D’Alembert equations given in (16)-(18) as a result of the value inside of the integral in equations (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2), and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) being equal to zero Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' THE REDUCED DYNAMICAL MODEL To obtain the reduced dynamical model presented in (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' we will proceed by writing the matrices C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N and S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and the vectors Fu := Hcτ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fg and Fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' presented in equation (19) as follows: C = � ˜C11 ˜C12 ˜C21 ˜C22 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' N = � ˜N11 ˜N12 ˜N21 ˜N22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' S = � ˜S11 ˜S12 ˜S21 ˜S22 � Fa = � ˜Fa1 ˜Fa2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fu = � ˜Fu1 ˜Fu1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Fg = � ˜Fg1 ˜Fg1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' where the matrices ˜C11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜N11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜S11 ∈ R6×6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and ˜C12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜N12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜S12 ∈ R6×12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and ˜C21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜N21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜S21 ∈ R12×6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' and ˜C22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜N22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜S22 ∈ R12×12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The vectors ˜Fa1, ˜Fu1, ˜Fg1 ∈ R6, and Parameter Numerical Value ρ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2kg/m3 mB 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4483e−5kg m1,2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='9069e−6kg m3,4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3126e−6kg l1,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0185m l3,4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='025m Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Wings morphology parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' ˜Fa2, ˜Fu2, ˜Fg2 ∈ R12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Moreover, from the expression of Fu := Hcτ, one can easily notice the following relationship ˜Fu1 = � 0 0 0 0 −A1 −A2 −A3 A4 � ˜Fu2 =: K ˜Fu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) Replacing ˜Fu2 with the wings dynamics given by the 3th, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' , 6th rows of (19), one can write the translational and rotational dynamics of the main body as follows: � ˜C11 − K ˜C21 � ˙ξB = � K ˜S22 ˜C21 − ˜S11 ˜C11 � ξB + � K ˜N21 − ˜N11 � ξB + � K ˜C22 − ˜C12 � ˙ξw + � K ˜S22 ˜C22 − ˜S11 ˜C12 � ξw + � K ˜N22 − ˜N12 � ξw + ˜Fa1 + ˜Fg1 − K � ˜Fa2 + ˜Fg2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) Consequently, the matrices CΘ t and DΘ t and the vectors VΘ t , FΘ t in (29) are given by Ct Θ(gB) = � ˜C11 − K ˜C21 � , Dt Θ(gB, ξB) = � � K ˜S22 ˜C21 − ˜S11 ˜C11 + K ˜N21 − ˜N11 � + \uf8ee \uf8ef\uf8f0 0 � j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=',4} Aj 4 � i=1 � � JiΩiA⊤ i − mi � ˆκiΩiA⊤ i ˆµi � \uf8f9 \uf8fa\uf8fb + \uf8ee \uf8ef\uf8f0 0 4 � i=1 miAB � AiˆκiΩi \uf8f9 \uf8fa\uf8fb + \uf8ee \uf8ef\uf8f0 0 4 � i=1 − � J[23] i Ωi \uf8f9 \uf8fa\uf8fb � Vt Θ (gB) = � K ˜C22 − ˜C12 � ˙ξw + � K ˜S22 ˜C22 − ˜Nw � ξw+ ˜Fg1 − K ˜Fg2 Ft Θ (gB, ξB) = ˜Fa1 − K ˜Fa2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' with ˜Nw defined as follows ˜Nw := � m1ABA1 ˆΩ1ˆκ1 miABAi ˆΩiˆκi miABAi ˆΩiˆκi miABAi ˆΩiˆκi N23 N24 N25 N26 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' DRAGONFLY MORPHOLOGY The dragonfly morphological parameters are calculated from the dead dragonfly shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' First, we will suppose that the main body is a cylinder, with a diameter of DB = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='6mm and a height of HB = 25mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Using this assumption, and the fact that the distance between the hind and fore wings’ roots is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='42mm, one can easily calculate JB and µi as follows: JB = 10−8 \uf8ee \uf8f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0109 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2847 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2847 \uf8f9 \uf8fb , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) µ1 = 10−3 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8 0]⊤ , µ2 = 10−3 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='71 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8 0]⊤ , µ3 = 10−3 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8 0]⊤ , µ4 = 10−3 [−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='71 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='8 0]⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) Next, using the wings’ polynomials qLE i and qT E i , one can directly calculate the wings parameters Ji, κi, mi and li as follows: κ1,2 = 10−3 � 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='978 (−1)i+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='975 0 �⊤ , κ3,4 = 10−3 � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='175 (−1)i+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='89 0 �⊤ , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3) J1,2 = 10−3 \uf8ee \uf8f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2303 (−1)i+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1902 0 (−1)i+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1731 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4034 \uf8f9 \uf8fb , J3,4 = 10−3 \uf8ee \uf8f0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4164 (−1)i+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0693 0 (−1)i+10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='0326 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4489 \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' THE CONTROL DYNAMICAL MODEL The control dynamical model, presented in (30), is ob- tained by considering the aerodynamic forces and torques as control inputs denoted by FC and ΓC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' The terms that represent the inertial forces and torques due to the body motion are grouped and denoted by FB and ΓB i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' the terms that are multiplied by ˙ξB and ξB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Finally, the terms that are multiplied by ˙ξw and ξw alone, represent the forces and torques due to the wing motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' Consequently, the forces and torques expressions are given by Fc = 4 � i=1 ABAiFi, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='1) Fw = 4 � i=1 � 2miAB ˆΩBAiˆκiΩi + miAiˆκi ˙Ωi+ miAi ˆΩiˆκiΩi � (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='2) FB = 4 � i=1 � miAB � ˆµi + � Aiκi � ˙ΩB+ miAB ˆΩB � ˆµi + � Aiκi � ΩB � , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='3) Γc = 4 � i=1 (ˆµiAiFi + AiMi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='4) Γw = − 4 � i=1 � Ai ˆΩiJiA⊤ i + AiJi ˆΩ⊤ i A⊤ i + mi � ˆµ⊤ i � Ai ˆΩiκi + � Ai ˆΩiκi ⊤ ˆµi � − miAiˆκi ˆΩiA⊤ i ˆµi � ΩB + 4 � i=1 � Ai � A⊤ i ΩBJi + mi � A⊤ i ˆµiΩBˆκi + mi ˆΩB ˆµiAiˆκi + ˆΩBAiJi � Ωi + 4 � i=1 � miAi ˆΩiˆκ⊤ i A⊤ i ˆµiΩB − (2AiJi − miˆµiAiˆκi) ˙Ωi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='5) ΓB = − 4 � i=1 mi � ˆµi + 2 � Aiκi � A⊤ B ¨p − 4 � i=1 � miˆµ⊤ i ˆµi + 2AiJiA⊤ i + mi � ˆµ⊤ i � Aiκi + 2 � Aiκi ⊤ˆµi � � ˙ΩB − 4 � i=1 � − mi � ˆµi + � Aiκi � ˆΩBA⊤ B + mi � � ˆµiΩB + � � AiκiΩB � A⊤ B + mi ˆΩB � ˆµi + ˆ Aiκi � A⊤ B � ˙p − 4 � i=1 � − Ai � JiA⊤ i ΩBA⊤ i − miAiˆκiA⊤ i � ˆΩB ˆµi − Ai� ˆµiΩB � + mi ˆΩB � ˆµi + � Aiκi � A⊤ B � ΩB + 4 � i=1 migˆµiA⊤ Be3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} +page_content='6)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf'} diff --git a/ZtAzT4oBgHgl3EQf2f4s/content/2301.01814v1.pdf b/ZtAzT4oBgHgl3EQf2f4s/content/2301.01814v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ccf542dfe46b3085c048ac0187acb795a25992a2 --- /dev/null +++ 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Several recent studies show the success of deep learning +on text processing. Convolutional neural network (CNN), as a popular deep learning model, has shown remarkable success in the task of +text classification. In this paper, new baseline models have been studied for text classification using CNN. In these models, documents are +fed to the network as a three-dimensional tensor representation to provide sentence-level analysis. Applying such a method enables the +models to take advantage of the positional information of the sentences in the text. Besides, analysing adjacent sentences allows extracting +additional features. The proposed models have been compared with the state-of-the-art models using several datasets. The results have +shown that the proposed models have better performance, particularly in the longer documents. + +Keywords:Text Classification, Deep Learning, Convolutional Neural Network, Natural Language Processing +1. Introduction +In recent years, the production of unstructured texts (documents) has grown exponentially. Unstructured texts can +be found everywhere, e.g., emails, social media, chat conversations, comments and websites. Although text data can +be a rich source of information, it is hard to extract value from this type of unstructured data. +Text classification is a fundamental task in natural language processing (NLP). The task is the process of assigning a +class label from a set of predefined classes to a given text according to its content, and has many applications such as +sentiment analysis (Pang, Lee, and Vaithyanathan 2002) , spam detection (Jindal and Liu 2007) and topic +categorization (Blei 2012). +Text classification can be done manually or automatically. Despite the manual method is more accurate, it is very +costly and time consuming. Therefore, to provide scalability, several machine learning, NLP and other techniques are +used for automatic text classification. +Supervised learning is a machine learning task of learning a function (classifier) using pre-labeled samples as a +training dataset (Russell and Norvig 2010). A key step in supervised learning is feature extraction. Traditional +machine learning methods represent text with hand-crafted methods, e.g., n-grams (Wang and Manning 2012). +Recently, deep learning methods have been used for automatic feature extraction, including convolutional neural +networks (CNNs) (LeCun et al. 1989), recurrent neural networks (RNNs) (Lipton 2015) and particularly long short- +term memory (LSTM) (Hochreiter and Schmidhuber 1997). +In this paper, we present a new baseline model for text classification using CNN. In this model, documents are fed to +the network as a three-dimensional tensor representation to provide sentence-level analysis. +The paper is structured as follows. The previous works have been summarized in the next section. The details of the +proposed methods are described in section 3. We have evaluated our approach on several benchmark datasets. The +experimental results are presented in section 4. Finally, the paper concludes with future research directions in section +5. + + + + + + + + + +2. Related works + +Different approaches have been proposed for text classification. Initial approaches were based on the classical +machine learning techniques, which followed two stages, i.e., extracting hand-crafted features and classifying the +documents. Typical features include bag-of-words (BoW), n-grams, and their TF-IDF* (Zhang, Zhao, and LeCun 2015). +Alternatively, several recent studies show the success of deep learning on text classification. As the neural networks +receive their inputs numerically, word embeddings, e.g., word2vec (Mikolov et al. 2013) or GloVe (Pennington, Socher, +and Manning 2014), are usually used to represent words as a numerical vectors by capturing the +similarities/regularities between words. +There are variety of deep learning models for text classification. Due to the sequential nature of textual data, recurrent +neural networks (RNN), including long short-term memory (LSTM) and gated recurrent units (GRU) (Cho et al. 2014) +have been widely used in text processing. For example, in (Yogatama et al. 2017) authors examined generative and +discriminative LSTM models for text classification. They found that although the generative models perform better +than BoW, they have a higher asymptotic error rates than discriminative RNN-based models. +Another popular model is CNN, which originally invented for computer vision (LeCun et al. 1998). Subsequently, CNN +models have been applied in NLP and have achieved excellent results (Collobert et al. 2011). Many researchers have +worked on the effective use of CNNs in text classification since a single layer word-level CNN was successfully used in +sentence classification with a pre-trained word embeddings (Kim 2014). The proposed method in (Zhang, Zhao, and +LeCun 2015) was the first attempt to perform text classification entirely at the character-level, and reported +competitive results. Their models use 70 characters by one-hot encoding, including 26 English letters, 10 digits, 33 +other characters and the new line character. (Conneau et al. 2017) adopted very deep convolutional networks, i.e., +ResNet (He et al. 2015), to the character-level text classification. +Some researchers tried to improve performance of the models by applying extra mechanisms. Attention is one of the +most effective mechanism that selects significant information to achieve superior results (Ashish et al. 2017). Deep +neural networks with attention mechanism can yield better results. Some of the remarkable examples include source- +target attention and self-attention (Lin et al. 2017). Particularly, two-level attention mechanism, i.e., word attention +and sentence attention, was developed on GRU by (Yang et al. 2016) for document classification. In (Wang, Huang, +and Deng 2018), authors used dense connections with multi-scale feature attention in order to produce variable n- +gram features. Since this paper aims to present a new baseline model, employing such mechanisms has been avoided. + +3. Method + +In this section, we describe the architecture of proposed Sentence-Level Convolutional Neural Network (SLCNN) for +classifying the documents. The key idea of the model is that using positional information of each sentence in the +document may improve the performance of the classifier. Furthermore, analysing adjacent sentences allows +extracting some extra features, e.g., writing style features, which can be useful in some applications, such as spam +review detection and fake news detection. Hence, we present two baseline models based on the CNN architecture for +the text classification task. For this purpose, we introduce a three-dimensional representation of documents to enable +sentence-level analysis. The pre-processing phase and the architecture of the SLCNN and its variant SLCNN+V are +explained in the following subsections. + +3-1. Pre-processing + +During the pre-processing phase, the documents are cleaned by removing some unimportant characters, like the html +tags and the punctuations. Then all words are normalized by converting to their lowercase forms. After that, as the +most important step, each document is transformed into a three-dimensional tensor, illustrated in Figure 1. As shown +in the figure, the sentences of the document form the first dimension of the tensor. In the same way, the words of the +sentences shape the second dimension, while the third dimension represents the word vectors of the words. The pre- +trained word embeddings, e.g., word2vec and GloVe, could be used for representing the word vectors. +Since, the input size of the network must be fixed, and according to different size of both the texts and the sentences, +we consider two thresholds, one for the number of sentences in the documents, Td, and another for the number of + + + +* Term Frequency–Inverse Document Frequency + + +words in the sentences, Ts. The documents and the sentences longer than the thresholds would be cropped and shorter +ones would be padded by zeros. + + +Figure 1- Shape of the converted documents. +After some statistical analysis on the datasets in our experiments, as well as considering the structure of the SLCNN, +we chose Ts=46. In the same way, the threshold for the number of sentences in the documents is calculated by the +following equation: + +𝑇𝑑 = ⌈𝜇 + 1.5 𝜎⌉ (1) + +where µ is the average number of sentences in the documents, and 𝜎 is the standard deviation. As a result, the outlier +sizes are ignored to prevent model from constructing very large and sparse tensors. The relevant statistical data is +provided in section 4. + + +Figure 2- The architecture of the proposed models. The dashed block (VCB) is used only in SLCNN+V. + +3-2. The Architecture + +The architecture of the proposed models is illustrated in Figure 2. Overall, in the input layer, the documents are +provided in the form of the 3D tensor, introduced in section 3-1. After that, using four horizontal convolutional blocks +(HCB), one feature per filter is extracted for each sentence individually. In other words, one feature vector for each +sentence is provided just before the fully-connected layers with the size equal to the number of filters. In this way, in +addition to the word-level features, the positional information of the sentences is also used in the learning process. +Moreover, as mentioned before, analysing of adjacent sentences can extract some useful features. For this purpose, +the second model (SLCNN+V) is created by adding a vertical convolutional block (VCB) before fully-connected layers. +Finally, there are two fully-connected (dense) layers which end to the output layer. + + +S1 +S2 +S3 +... +STa +W1W2 +W3 +WTsOutput: Ta × 22 +Output: Ta ×4 +Output:(Td-2)/2× 1 +×PI:andano +1 +IBlock +Block +Block +Block +H +B +B +Convolutional +Convolutional +Convolutional +Convolutional +Fully-Connected +-- +- +- +Output +-- +- +Horizontal +Horizontal +Horizontal +Horizontal +Vertical +Ful +Input:Ta×46×d +工 +工 +工 + + +Figure 3- The convolutional blocks. k is the number of filters. (a) HCB and (b) VCB. +Looking at the details of the convolutional blocks, as shown in Figure 3, there are two sequential convolution layers, +each one followed by a Rectified Linear Unit (ReLU) activation function, f(x)= max (0, x). A convolution operation +consists of a filter 𝑤 ∈ ℝ𝑠×𝑡×𝑑, which is applied to each possible window of s×t features from its input feature map, X, +to produce a new feature map by equation 3: + +𝑋 = [ +𝑥1,1 +𝑥1,2 +⋯ +𝑥2,1 +𝑥2,2 +⋯ +𝑥1,𝑛 +𝑥2,𝑛 +⋮ + ⋮ + +𝑥𝑚,1 +𝑥𝑚,2 +⋯ +⋮ +𝑥𝑚,𝑛 +] (2) + +𝑥̃𝑖,𝑗 = 𝑓(𝑤. 𝑥𝑖,𝑗:𝑖+𝑠−1,𝑗+𝑡−1 + 𝑏) (3) + +where xi,j:y,z is the concatenation of features within the specified interval, b ∊ ℝ is a bias term and f is a non-linear +function such as the ReLU. For the HCB, we consider s=1 and t=2, and for the VCB s=2 and t=1. It should be noted that, +in the first convolution layer of the first HCB, d (the third dimension of the filters) is equal to the size of the word +vectors, and in other cases d=1. At the end of the blocks, there is a max-pooling operation, with the pooling size = 2, +that is applied over the generated intermediate feature map to select the maximum value from any two adjacent +features as a more important feature. The new feature map is calculated by following equations: + +𝑥̃𝑖,𝑗 = {max{ 𝑥𝑖,2𝑗−1, 𝑥𝑖,2𝑗} , 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐻𝐶𝐵 +max{ 𝑥2𝑖−1,𝑗, 𝑥2𝑖,𝑗} , 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑉𝐶𝐵 (4) + +The process of extracting one feature from one filter was described. The model uses multiple filters to obtain multiple +features. The final extracted features are passed to the fully-connected layers that end to a softmax output layer which +is the probability distribution over labels. For regularization, a dropout module (Hinton et al. 2012) is employed after +each fully-connected layer. + +4. Experiments + +4-1. Experimental settings + +The Natural Language Toolkit (NLTK) was used in order to tokenize words and sentences. In the input layer, as +mentioned before, pre-trained word-embeddings are used to convert the words into the corresponding word vectors. +We used 100-dimensional GloVe in our experiments. Out-Of-Vocabulary (OOV) words were initialized from a uniform +distribution with range [-0.01, 0.01]. We set number of filters to 128 for all the convolutional blocks. Also, we +considered two different sizes for fully-connected layers, shown in Table 1. Both the dropout rates were set to 0.5. +The model’s parameters were trained by the Adam Optimizer (Kingma and Ba 2014), with the initial learning rate of +0.001. The model has been implemented using Keras and run for 50 epochs. + + + +k, max-pool, (1,2) +k, max-pool, (2,1) +ReLU +ReLU +k, Conv, (1,2,d) +k, Conv, (2,1) +ReLU +ReLU +k, Conv, (1,2,d) +k, Conv, (2,1) +(a) +(b) +Table 1- Fully-connected layers in our experiments. +Layers +Small +Large +Fully-connected 1 +512 +1024 +Fully-connected 2 +512 +1024 +Output +Depends on the problem + + +4-2. Benchmark Datasets + +We utilized six datasets covering different classification tasks compiled by (Zhang, Zhao, and LeCun 2015). General +specifications are presented in Table 2. All data are evenly distributed across class labels. AG and DBPedia are news +and ontology classification datasets, respectively. Yelp and Amazon are sentiment classification datasets, where ‘.P’ +(Polarity) in the dataset names indicates that the labels are binary while ‘.F’ (Full) means that the labels refer to the +number of stars. + +Table 2- Datasets in our experiments. +Datasets +AG News +DBPedia +Yelp.P +Yelp.F +Amazon.P +Amazon.F +# of training samples +120k +560k +560k +650k +3600k +3000k +# of test samples +7.6k +70k +38k +50k +400k +650k +# of classes +4 +14 +2 +5 +2 +5 + + +Some of the statistical information extracted from the datasets, after the pre-processing step, is summarized in Table +3. As presented in the table, by considering Ts=46, the proportions of cropped sentences are between 2 and 2.9 +percent, that shows the length of sentences in the different datasets are almost similar. By contrast, the number of +sentences of the documents in the different datasets are quite different. By utilizing Equation 1, Td for AG News, +DBPedia, Amazon and Yelp are equal to 4, 6, 10 and 20 respectively. Also, the proportions of cropped documents, +using relevant Td, are 0.4, 3, 3.6 and 6 percent for AG News, Amazon, DBPedia and Yelp respectively, which means that +the variance of the number of sentences in the documents of Yelp is greater than others. + +Table 3- The statistical information of the datasets. +Statistics +AG +DBPedia +Yelp.P +Yelp.F +Amazon.P +Amazon.F +# of sentences +164k +1505k +5082k +5958k +18654k +16986k +Cropped sentences (%) +2 +2.9 +2.6 +2.6 +2.4 +2.5 +Cropped documents (%) +0.4 +3.6 +6 +6 +3.1 +3 +Documents that contain cropped sentences (%) +2.5 +6.9 +16.1 +16.4 +10.3 +10.9 +# of sentences in the longest text +15 +25 +141 +151 +85 +99 +# of words in the longest sentence +135 +1302 +1104 +1175 +522 +520 +Vocab size +62k +786k +283k +311k +1546k +1464k +Td +4 +6 +20 +20 +10 +10 +# of trainable parameters in SLCNN small +783k +920k +1831k +1832k +1176k +1177k +# of trainable parameters in SLCNN large +1835k +2107k +3930k +3933k +2619k +2622k +# of trainable parameters in SLCNN+V small +653k +723k +1176k +1177k +848k +850k +# of trainable parameters in SLCNN+V large +1508k +1649k +2554k +2557k +1899k +1902k +Training time for a single epoch (s) +10 +51 +150 +170 +510 +440 + +4-3. Results + +We compared our models with several popular base models, e.g., linear models (Zhang, Zhao, and LeCun 2015), RNN- +based model, i.e., Discriminative-LSTM (Yogatama et al. 2017), and CNN-based models including classical word-level +CNN (Kim 2014), character-level CNN (Zhang, Zhao, and LeCun 2015), very deep CNN (Conneau et al. 2017) and CNN + + + +with fastText embedding (Joulin et al. 2017). Since our aim was to provide new baseline models, and using other +mechanisms, such as the attention, has been avoided, therefore such models have been excluded from the comparison. +The results are listed in Table 4 based on accuracy. Overall, it can be seen that the proposed models have +outperformed all the models in half of the datasets, DBPedia, Yelp.P and Yelp.F. Especially, the improvement is +significant in Yelp datasets, i.e., around 2 percent in Yelp.P and around 5 percent in Yelp.F compared to character- +level and word-level CNNs. In terms of Amazon datasets, the SLCNN+V was ranked third after VDCNN and character- +level CNN with around 94 and 58.1 percent in Amazon.P and Amazon.F, respectively. +If we look at AG News, despite competitive results with other CNN models, n-grams and Discriminative-LSTM have +achieved better results. One of the main reasons we can mention is the number of sentences in the documents. So that +the proposed models perform better in documents with large number of sentences, i.e., Yelp. Another reason that +hinders better performance in Amazon datasets is the very high vocabulary size (see Table 3), since we used the word +embedding with just over 1M vocabularies in our experiments. + +Table 4- Test accuracy (%) of all the models on the datasets. Results marked with * are reported in (Wang, Huang, and Deng 2018) and others are +reprinted from the references. +Models +AG +DBPedia +Yelp.P +Yelp.F +Amazon.P Amazon.F +Linear +Bag of Words (Zhang et al. 2015) +88.81 +96.61 +92.24 +57.99 +90.40 +54.64 +n-grams (Zhang et al. 2015) +92.04 +98.63 +95.64 +56.26 +92.02 +54.27 +n-grams TFIDF (Zhang et al. 2015) +92.36 +98.69 +95.44 +54.80 +91.54 +52.44 +CNN +Char-level CNN small (Zhang et al. 2015) +84.35 +98.02 +93.47 +59.16 +94.50 +59.47 +Char-level CNN large (Zhang et al. 2015) +87.18 +98.27 +94.11 +60.38 +94.49 +58.69 +VDCNN- 29 layers (Conneau et al. 2017) +91.27 +98.71 +95.72 +64.26 +95.69 +63.00 +Word-level CNN (Kim 2014)* +91.60 +98.60 +93.50 +61.00 +- +57.40 +fastText (Joulin et al. 2017) +91.50 +98.10 +93.80 +60.40 +91.20 +55.80 +RNN +Discriminative-LSTM (Yogatama et al. 2017) +92.10 +98.70 +92.60 +59.60 +- +- +Ours +SLCNN small +91.22 +98.75 +96.03 +64.67 +93.87 +58.03 +SLCNN large +91.26 +98.76 +96.01 +64.56 +93.93 +58.02 +SLCNN+V small +91.45 +98.73 +96.09 +64.46 +93.91 +58.11 +SLCNN+V large +91.39 +98.76 +96.07 +64.39 +93.94 +58.05 + + +5. Conclusion and future works + +This paper offers new baseline models for text classification using a sentence-level CNN. The key idea is representing +the documents as a 3D tensor to enable the models to sentence-level analysis. The proposed models have been +compared with the state-of-the-art models using several datasets. The results have shown that the proposed models +have better performance, particularly in the longer documents. +As future works, the attention mechanism will be utilized in the proposed models in order to improve the overall +performance. Also, we will work on sentence standardization. We believe that applying a standard form of sentences +enables the proposed models to use compositional methods (with different 3D filters), due to the 3D structure of the +input tensor. +REFERENCES +Ashish, Vaswani, Shazeer Noam, Parmar Niki, Uszkoreit Jakob, Jones Llion, Gomez Aidan N, Kaiser Lukasz, and Polosukhin Illia. 2017. "Attention is +All you Need." 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Montreal, Canada: MIT Press. + + diff --git a/d9FJT4oBgHgl3EQf_y11/content/tmp_files/load_file.txt b/d9FJT4oBgHgl3EQf_y11/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a3f036fda2f42d1649463464c9c461a1f3f8ea48 --- /dev/null +++ b/d9FJT4oBgHgl3EQf_y11/content/tmp_files/load_file.txt @@ -0,0 +1,550 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf,len=549 +page_content='Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' E mail address: jarrahi@znu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='ir SLCNN: Sentence Level Convolutional Neural Network for Text Classification Ali Jarrahiα , Leila Safariα , Ramin Mousaα αUniversity of Zanjan, Zanjan , Iran A B S T R A C T Text classification is a fundamental task in natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Several recent studies show the success of deep learning on text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Convolutional neural network (CNN), as a popular deep learning model, has shown remarkable success in the task of text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In this paper, new baseline models have been studied for text classification using CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In these models, documents are fed to the network as a three-dimensional tensor representation to provide sentence-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Applying such a method enables the models to take advantage of the positional information of the sentences in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Besides, analysing adjacent sentences allows extracting additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The proposed models have been compared with the state-of-the-art models using several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The results have shown that the proposed models have better performance, particularly in the longer documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Keywords:Text Classification, Deep Learning, Convolutional Neural Network, Natural Language Processing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Introduction In recent years, the production of unstructured texts (documents) has grown exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Unstructured texts can be found everywhere, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', emails, social media, chat conversations, comments and websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Although text data can be a rich source of information, it is hard to extract value from this type of unstructured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Text classification is a fundamental task in natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The task is the process of assigning a class label from a set of predefined classes to a given text according to its content, and has many applications such as sentiment analysis (Pang, Lee, and Vaithyanathan 2002) , spam detection (Jindal and Liu 2007) and topic categorization (Blei 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Text classification can be done manually or automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Despite the manual method is more accurate, it is very costly and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Therefore, to provide scalability, several machine learning, NLP and other techniques are used for automatic text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Supervised learning is a machine learning task of learning a function (classifier) using pre-labeled samples as a training dataset (Russell and Norvig 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' A key step in supervised learning is feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Traditional machine learning methods represent text with hand-crafted methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', n-grams (Wang and Manning 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Recently, deep learning methods have been used for automatic feature extraction, including convolutional neural networks (CNNs) (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 1989), recurrent neural networks (RNNs) (Lipton 2015) and particularly long short- term memory (LSTM) (Hochreiter and Schmidhuber 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In this paper, we present a new baseline model for text classification using CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In this model, documents are fed to the network as a three-dimensional tensor representation to provide sentence-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The previous works have been summarized in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The details of the proposed methods are described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' We have evaluated our approach on several benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The experimental results are presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Finally, the paper concludes with future research directions in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Related works Different approaches have been proposed for text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Initial approaches were based on the classical machine learning techniques, which followed two stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', extracting hand-crafted features and classifying the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Typical features include bag-of-words (BoW), n-grams, and their TF-IDF* (Zhang, Zhao, and LeCun 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Alternatively, several recent studies show the success of deep learning on text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' As the neural networks receive their inputs numerically, word embeddings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', word2vec (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2013) or GloVe (Pennington, Socher, and Manning 2014), are usually used to represent words as a numerical vectors by capturing the similarities/regularities between words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' There are variety of deep learning models for text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Due to the sequential nature of textual data, recurrent neural networks (RNN), including long short-term memory (LSTM) and gated recurrent units (GRU) (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2014) have been widely used in text processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' For example, in (Yogatama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) authors examined generative and discriminative LSTM models for text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' They found that although the generative models perform better than BoW, they have a higher asymptotic error rates than discriminative RNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Another popular model is CNN, which originally invented for computer vision (LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Subsequently, CNN models have been applied in NLP and have achieved excellent results (Collobert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Many researchers have worked on the effective use of CNNs in text classification since a single layer word-level CNN was successfully used in sentence classification with a pre-trained word embeddings (Kim 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The proposed method in (Zhang, Zhao, and LeCun 2015) was the first attempt to perform text classification entirely at the character-level, and reported competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Their models use 70 characters by one-hot encoding, including 26 English letters, 10 digits, 33 other characters and the new line character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) adopted very deep convolutional networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015), to the character-level text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Some researchers tried to improve performance of the models by applying extra mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Attention is one of the most effective mechanism that selects significant information to achieve superior results (Ashish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Deep neural networks with attention mechanism can yield better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Some of the remarkable examples include source- target attention and self-attention (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Particularly, two-level attention mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', word attention and sentence attention, was developed on GRU by (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2016) for document classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In (Wang, Huang, and Deng 2018), authors used dense connections with multi-scale feature attention in order to produce variable n- gram features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Since this paper aims to present a new baseline model, employing such mechanisms has been avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Method In this section, we describe the architecture of proposed Sentence-Level Convolutional Neural Network (SLCNN) for classifying the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The key idea of the model is that using positional information of each sentence in the document may improve the performance of the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Furthermore, analysing adjacent sentences allows extracting some extra features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', writing style features, which can be useful in some applications, such as spam review detection and fake news detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Hence, we present two baseline models based on the CNN architecture for the text classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' For this purpose, we introduce a three-dimensional representation of documents to enable sentence-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The pre-processing phase and the architecture of the SLCNN and its variant SLCNN+V are explained in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Pre processing During the pre-processing phase, the documents are cleaned by removing some unimportant characters, like the html tags and the punctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Then all words are normalized by converting to their lowercase forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' After that, as the most important step, each document is transformed into a three-dimensional tensor, illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' As shown in the figure, the sentences of the document form the first dimension of the tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In the same way, the words of the sentences shape the second dimension, while the third dimension represents the word vectors of the words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The pre- trained word embeddings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', word2vec and GloVe, could be used for representing the word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Since, the input size of the network must be fixed, and according to different size of both the texts and the sentences, we consider two thresholds, one for the number of sentences in the documents, Td, and another for the number of Term Frequency–Inverse Document Frequency words in the sentences, Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The documents and the sentences longer than the thresholds would be cropped and shorter ones would be padded by zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Figure 1- Shape of the converted documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' After some statistical analysis on the datasets in our experiments, as well as considering the structure of the SLCNN, we chose Ts=46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In the same way, the threshold for the number of sentences in the documents is calculated by the following equation: 𝑇𝑑 = ⌈𝜇 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='5 𝜎⌉ (1) where µ is the average number of sentences in the documents, and 𝜎 is the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' As a result, the outlier sizes are ignored to prevent model from constructing very large and sparse tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The relevant statistical data is provided in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Figure 2- The architecture of the proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The dashed block (VCB) is used only in SLCNN+V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The Architecture The architecture of the proposed models is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Overall, in the input layer, the documents are provided in the form of the 3D tensor, introduced in section 3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' After that, using four horizontal convolutional blocks (HCB), one feature per filter is extracted for each sentence individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In other words, one feature vector for each sentence is provided just before the fully-connected layers with the size equal to the number of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In this way, in addition to the word-level features, the positional information of the sentences is also used in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Moreover, as mentioned before, analysing of adjacent sentences can extract some useful features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' For this purpose, the second model (SLCNN+V) is created by adding a vertical convolutional block (VCB) before fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Finally, there are two fully-connected (dense) layers which end to the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' S1 S2 S3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' STa W1W2 W3 WTsOutput: Ta × 22 Output: Ta ×4 Output:(Td 2)/2× 1 ×PI:andano 1 IBlock Block Block Block H B B Convolutional Convolutional Convolutional Convolutional Fully Connected Output Horizontal Horizontal Horizontal Horizontal Vertical Ful Input:Ta×46×d 工 工 工 Figure 3- The convolutional blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' k is the number of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' (a) HCB and (b) VCB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Looking at the details of the convolutional blocks, as shown in Figure 3, there are two sequential convolution layers, each one followed by a Rectified Linear Unit (ReLU) activation function, f(x)= max (0, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' A convolution operation consists of a filter 𝑤 ∈ ℝ𝑠×𝑡×𝑑, which is applied to each possible window of s×t features from its input feature map, X, to produce a new feature map by equation 3: 𝑋 = [ 𝑥1,1 𝑥1,2 ⋯ 𝑥2,1 𝑥2,2 ⋯ 𝑥1,𝑛 𝑥2,𝑛 ⋮ ⋮ 𝑥𝑚,1 𝑥𝑚,2 ⋯ ⋮ 𝑥𝑚,𝑛 ] (2) 𝑥̃𝑖,𝑗 = 𝑓(𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 𝑥𝑖,𝑗:𝑖+𝑠−1,𝑗+𝑡−1 + 𝑏) (3) where xi,j:y,z is the concatenation of features within the specified interval, b ∊ ℝ is a bias term and f is a non-linear function such as the ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' For the HCB, we consider s=1 and t=2, and for the VCB s=2 and t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' It should be noted that, in the first convolution layer of the first HCB, d (the third dimension of the filters) is equal to the size of the word vectors, and in other cases d=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' At the end of the blocks, there is a max-pooling operation, with the pooling size = 2, that is applied over the generated intermediate feature map to select the maximum value from any two adjacent features as a more important feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The new feature map is calculated by following equations: 𝑥̃𝑖,𝑗 = {max{ 𝑥𝑖,2𝑗−1, 𝑥𝑖,2𝑗} , 𝑓𝑜𝑟 𝑡ℎ𝑒 𝐻𝐶𝐵 max{ 𝑥2𝑖−1,𝑗, 𝑥2𝑖,𝑗} , 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑉𝐶𝐵 (4) The process of extracting one feature from one filter was described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The model uses multiple filters to obtain multiple features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The final extracted features are passed to the fully-connected layers that end to a softmax output layer which is the probability distribution over labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' For regularization, a dropout module (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2012) is employed after each fully-connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Experiments 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Experimental settings The Natural Language Toolkit (NLTK) was used in order to tokenize words and sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In the input layer, as mentioned before, pre-trained word-embeddings are used to convert the words into the corresponding word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' We used 100-dimensional GloVe in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Out-Of-Vocabulary (OOV) words were initialized from a uniform distribution with range [-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' We set number of filters to 128 for all the convolutional blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Also, we considered two different sizes for fully-connected layers, shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Both the dropout rates were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The model’s parameters were trained by the Adam Optimizer (Kingma and Ba 2014), with the initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The model has been implemented using Keras and run for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' k, max-pool, (1,2) k, max-pool, (2,1) ReLU ReLU k, Conv, (1,2,d) k, Conv, (2,1) ReLU ReLU k, Conv, (1,2,d) k, Conv, (2,1) (a) (b) Table 1- Fully-connected layers in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Layers Small Large Fully-connected 1 512 1024 Fully-connected 2 512 1024 Output Depends on the problem 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Benchmark Datasets We utilized six datasets covering different classification tasks compiled by (Zhang, Zhao, and LeCun 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' General specifications are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' All data are evenly distributed across class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' AG and DBPedia are news and ontology classification datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Yelp and Amazon are sentiment classification datasets, where ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P’ (Polarity) in the dataset names indicates that the labels are binary while ‘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F’ (Full) means that the labels refer to the number of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Table 2 Datasets in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Datasets AG News DBPedia Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F # of training samples 120k 560k 560k 650k 3600k 3000k # of test samples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='6k 70k 38k 50k 400k 650k # of classes 4 14 2 5 2 5 Some of the statistical information extracted from the datasets, after the pre-processing step, is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' As presented in the table, by considering Ts=46, the proportions of cropped sentences are between 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='9 percent, that shows the length of sentences in the different datasets are almost similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' By contrast, the number of sentences of the documents in the different datasets are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' By utilizing Equation 1, Td for AG News, DBPedia, Amazon and Yelp are equal to 4, 6, 10 and 20 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Also, the proportions of cropped documents, using relevant Td, are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='4, 3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='6 and 6 percent for AG News, Amazon, DBPedia and Yelp respectively, which means that the variance of the number of sentences in the documents of Yelp is greater than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Table 3- The statistical information of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Statistics AG DBPedia Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F # of sentences 164k 1505k 5082k 5958k 18654k 16986k Cropped sentences (%) 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='5 Cropped documents (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='6 6 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='1 3 Documents that contain cropped sentences (%) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Results We compared our models with several popular base models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', linear models (Zhang, Zhao, and LeCun 2015), RNN- based model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', Discriminative-LSTM (Yogatama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017), and CNN-based models including classical word-level CNN (Kim 2014), character-level CNN (Zhang, Zhao, and LeCun 2015), very deep CNN (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) and CNN with fastText embedding (Joulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Since our aim was to provide new baseline models, and using other mechanisms, such as the attention, has been avoided, therefore such models have been excluded from the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The results are listed in Table 4 based on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Overall, it can be seen that the proposed models have outperformed all the models in half of the datasets, DBPedia, Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P and Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Especially, the improvement is significant in Yelp datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', around 2 percent in Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P and around 5 percent in Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F compared to character- level and word-level CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' In terms of Amazon datasets, the SLCNN+V was ranked third after VDCNN and character- level CNN with around 94 and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='1 percent in Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P and Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' If we look at AG News, despite competitive results with other CNN models, n-grams and Discriminative-LSTM have achieved better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' One of the main reasons we can mention is the number of sentences in the documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' So that the proposed models perform better in documents with large number of sentences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=', Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Another reason that hinders better performance in Amazon datasets is the very high vocabulary size (see Table 3), since we used the word embedding with just over 1M vocabularies in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Table 4- Test accuracy (%) of all the models on the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Results marked with * are reported in (Wang, Huang, and Deng 2018) and others are reprinted from the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Models AG DBPedia Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='P Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='F Linear Bag of Words (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='81 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='61 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='24 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='99 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='40 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='64 n-grams (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='04 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='63 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='64 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='26 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='02 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='27 n-grams TFIDF (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='36 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='69 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='44 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='80 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='54 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='44 CNN Char-level CNN small (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='35 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='02 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='47 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='16 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='50 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='47 Char-level CNN large (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='18 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='27 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='11 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='38 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='49 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='69 VDCNN- 29 layers (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='27 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='71 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='72 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='26 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='69 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='00 Word-level CNN (Kim 2014)* 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='60 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='60 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='50 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='00 - 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='40 fastText (Joulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='50 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='10 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='80 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='40 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='20 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='80 RNN Discriminative-LSTM (Yogatama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='10 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='70 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='60 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='60 - - Ours SLCNN small 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='22 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='75 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='03 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='67 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='87 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='03 SLCNN large 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='26 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='76 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='01 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='56 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='93 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='02 SLCNN+V small 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='45 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='73 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='09 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='46 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='91 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='11 SLCNN+V large 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='39 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='76 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='07 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='39 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='94 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Conclusion and future works This paper offers new baseline models for text classification using a sentence-level CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The key idea is representing the documents as a 3D tensor to enable the models to sentence-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The proposed models have been compared with the state-of-the-art models using several datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' The results have shown that the proposed models have better performance, particularly in the longer documents.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Yang, Zichao, Diyi Yang, Chris Dyer, Xiaodong He, Alex Smola, and Eduard Hovy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' "Hierarchical Attention Networks for Document Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='" In, 1480-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' San Diego, California: Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Yogatama, Dani, Chris Dyer, Wang Ling, and Phil Blunsom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' "Generative and Discriminative Text Classification with Recurrent Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='" In arXiv e-prints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Zhang, Xiang, Junbo Zhao, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' "Character-level convolutional networks for text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content='" In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 1, 649-57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} +page_content=' Montreal, Canada: MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FJT4oBgHgl3EQf_y11/content/2301.11696v1.pdf'} diff --git a/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/2301.02055v1.pdf.txt b/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/2301.02055v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e21f88817ac21c6bea3418789c4a694dbac4e506 --- /dev/null +++ b/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/2301.02055v1.pdf.txt @@ -0,0 +1,2152 @@ +An adaptive solution strategy for Richards’ equation +Jakob S. Stokke1, Koondanibha Mitra2, Erlend Storvik1,3, Jakub W. +Both1, and Florin A. Radu1 +1Center for Modeling of Coupled Subsurface Dynamics, Department of Mathematics, +University of Bergen, Norway +2Computational Mathematics group, Hasselt University, Belgium +3Department of Computer science, Electrical engineering and Mathematical sciences, Western +Norway University of Applied Sciences, Norway +January 6, 2023 +Abstract +Flow in variably saturated porous media is typically modelled by the Richards +equation, a nonlinear elliptic-parabolic equation which is notoriously challenging to +solve numerically. In this paper, we propose a robust and fast iterative solver for +Richards’ equation. The solver relies on an adaptive switching algorithm, based +on rigorously derived a posteriori indicators, between two linearization methods: +L-scheme and Newton. Although a combined L-scheme/Newton strategy was in- +troduced previously in [1], here, for the first time we propose a reliable and robust +criteria for switching between these schemes. The performance of the solver, which +can be in principle applied to any spatial discretization and linearization methods, +is illustrated through several numerical examples. +Keywords: Iterative linearization, Adaptivity, L-scheme, Newton’s method, Richards’ +equation, Nonlinear degenerate diffusion +1 +Introduction +In this paper, we consider the pressure head ψ based formulation of the Richards equation +∂tθ(ψ) − ∇ · [K(θ(ψ))∇(ψ + z))] = f, +(1) +where θ : R → [0, 1] is the water content, K is the rank 2 permeability tensor of the +porous medium, z is the height against the gravitational direction, and f is a source/sink +term. Richards’ equation is used to model the flow of water in saturated/unsaturated +porous media. It is a highly nonlinear and degenerate elliptic-parabolic equation which +makes solving it a very challenging task, see e.g. the review work of [2]. We refer to [3] +for the existence and uniqueness of a weak solution of Richards’ equation. +There are plenty of works regarding discretization of Richards’ equation. Due to the +low regularity of solutions of (1), see [4], generally, a backward Euler (implicit) scheme +(3) is employed to discretize it in time, see e.g. [1, 5]. Regarding spatial discretization +we mention continuous Galerkin finite elements [6, 7], mixed or expanded mixed finite +1 +arXiv:2301.02055v1 [math.NA] 5 Jan 2023 + +elements [8, 9, 10, 11, 12], finite volumes [13, 14] (see also the recent review [15]), or +multipoint flux approximation (MPFA) [16]. Regardless of the choice of the spatial dis- +cretization method, one has to solve at each time step a nonlinear, finite-dimensional +problem. In this paper, we will focus on how to efficiently solve these problems using +iterative linearization techniques. +The main iterative linearization methods used for this type of nonlinear problem are +the Newton method, Picard or modified Picard, L-scheme, the Jaeger-Kacur method, +or combinations of them. Perhaps the most common choice is the Newton method [17, +18] which converges quadratically provided the initial guess is close enough to the final +solution. For a r-H¨older continuous θ′ function (r ∈ (0, 1]) and the initial guess equal to +the solution of the previous time step, it was shown in [10] that the Newton scheme is +(1 + r)th order convergent if +τ ≤ Cθ +2+r +r +m hd, +(2) +where τ > 0 is the time step size, h > 0 the mesh size, d ∈ N the spatial dimension, C > 0 +a constant which depends on the domain and the nonlinearities, and θm := inf θ′ ≥ 0. +However, for simulations in 2 or 3 dimensions, condition (2) is quite restrictive partic- +ularly if the mesh size h is small, or if the problem is degenerate (θm = 0). This fact +is corroborated by numerical simulations in [1, 19] which show that the Newton method +fails to converge in many such cases. One can improve the robustness of Newton method +by using a damped version of it. Line search, variable switching [20] or trust-regions +techniques [21] are examples of such. Alternatively, one can increase the robustness of +Newton’s method by performing first a few fixed-point iterations. This was proposed in +[17, 18] by using the Picard method and in [1] by using the L-scheme. Nevertheless, the +switching between the schemes was not based on an a posteriori indicator, but done in a +heuristic manner. +The other linearization schemes are fixed-point type schemes, typically more robust, +however only linearly convergent. It has been shown in [22, 13] that the Picard method +does not perform well for Richards’ equation. A modified Picard method was proposed +in [22]. The modified Picard coincides with Newton’s method for the case of a constant +permeability, therefore it inherits robustness problems. The L-scheme, first proposed in +[23, 24, 1], is a stabilized Picard method and it was designed to be unconditionally con- +verging irrespective of the choice of the initial guess even in degenerate settings and for +larger time steps. The L-scheme (see Definition 2.3) uses a global constant as a stabiliza- +tion coefficient, does not involve the computation of any derivatives, and thus, is not only +more stable but also consumes less computational time per iteration due to easier assem- +bly of the stiffness matrices which are better conditioned. Numerical results in [1, 19] +clearly demonstrate this. However, they also reveal that the L-scheme converges consid- +erably slower in terms of number of iterations compared to the Newton scheme and at a +linear rate. Furthermore, its overall performance strongly depends on the careful choice +of a tuning parameter; despite theoretical stability, an improper choice may effectively +result in stagnation. The sensitivity of the performance of the L-scheme with respect to +the stabilization can be significantly relaxed when combining the L-scheme with Anderson +acceleration [25]. Indeed, for Richards equation extended to deformable porous media and +solved by an L-scheme, it has been demonstrated that, first, the stabilization parameter +can be chosen outside the theoretical range, and second, the non-degenerate convergence +can be retained in case of previous divergence or accelerated, as also discussed from a the- +oretical perspective [26]. Similar stabilizing properties of the Anderson acceleration have +2 + +been also discussed for general fixed-point methods [27, 28]. Other fixed point iterations +schemes include J¨ager-Kacˇur scheme [29] which converges unconditionally albeit slowly, +and is more computationally expensive than the L-scheme per iteration, see Table 1. The +modified L-scheme, proposed in [19], shows stability similar to the L-scheme while having +much faster convergence rates (scaling with τ); yet, the convergence is still linear. +In this paper, we investigate a hybrid strategy, dynamically switching between the +L-scheme and Newton’s method. This utilizes the advantages of both methods: the un- +conditional stability of the L-scheme, and the quadratic convergence of Newton’s method +when close to the exact solution. +The crucial difference to previous works on hybrid +approaches, e.g. [1, 17], is the adaptive nature of the switch between both linearization +methods. A switch from the L-scheme to Newton’s method is performed when the iterate +is sufficiently close to the solution. This finally allows us to balance robustness and speed. +The main challenge in implementing this strategy originates from deriving a rigorous +switching criteria between the schemes. Since, the a priori estimates, such as the ones pro- +vided in [10], involve unknown constants and assume the worst-case scenario, we pursue +an a posteriori estimate-based approach here instead. A rigorous and efficient a posteri- +ori estimator for the fully degenerate Richards equation involving linearization errors was +derived in [30] in the continuous space-time setting. For the time-discrete problem (3), a +robust, efficient, and reliable estimator was derived in [31] using an orthogonal decompo- +sition result dividing the total error into a discretization and a linearization component. +Furthermore, its effectiveness was demonstrated numerically. These papers serve as the +main inspirations in deriving the a posteriori based switching criteria in Section 3 and an +adaptive L-scheme algorithm in Appendix A. Nevertheless, since we are only interested +in computing the linearization error component, the computation of equilibrated flux will +be avoided wherever possible. +The paper is organized as follows. In Section 2, we introduce the mathematical no- +tation, state the assumptions, define the fully-discrete solution, and elaborate on differ- +ent linearization methods. In Section 3, the adaptive switching algorithm is developed. +Firstly, a concept of linearization error is introduced along with the derivation of a predic- +tive indicator for linearization error of the next iteration. The adaptive algorithm com- +pares the linearization error with the estimator to determine the exact switching points. +In Section 4, three numerical test cases (partially saturated, degenerate, and realistic +benchmarks) are presented which illustrate the robustness and computational efficiency +of the adaptive scheme compared to the standard Newton’s method or the L-scheme. +Section 5 contains the conclusions of this work. The paper ends with two appendices, one +concerning an adaptive L-scheme and the other on the details of the computation of the +equilibrated flux. +2 +Mathematical and numerical formulation +We consider Richards’ equation in the space-time domain G = Ω × [0, T], where Ω is +a bounded domain in Rd with a Lipschitz continuous boundary ∂Ω, and T > 0. Let +(·, ·) and ∥ · ∥ be the inner product and norm of the square-integrable functions in Ω, i.e. +L2(Ω), respectively. Moreover, using common notation from functional analysis, H1(Ω) +represents the Sobolev space of functions with first-order weak derivatives in L2(Ω), and +H1 +0(Ω) its subspace containing functions with vanishing trace at the boundary. +Assumption 1. For the material properties θ and K, and source term f in (1), the +following assumptions are made: +3 + +(a) The saturation function θ(·) is Lipschitz continuous and monotonically increasing +with Lθ and θm ≥ 0 being the Lipschitz constant and the lower bound for the deriva- +tive, respectively. +(b) The permeability tensor K : [0, 1] → Rd×d satisfies the uniform (pseudo) ellipticity +condition, i.e., for constants κM > κm ≥ 0, +κm|z|2 ≤ zT K z ≤ κM|z|2, +∀ z ∈ Rd. +Moreover, (K ◦ θ) is Lipschitz continuous, with Lipschitz constant Lκ. +(c) The source function satisfies f ∈ C(0, T; L2(Ω)). +Note that these assumptions are consistent with the commonly used Brooks-Corey +[32] and van Genuchten [33] parametrizations of the functions θ and K. +2.1 +Time-discretization: Backward Euler +To discretize the Richards equation in time we consider the backward-Euler time dis- +cretization of (1). For this implicit scheme, no CFL conditions need to be satisfied for +stability (thus avoiding restrictions on the time step size). Moreover, it does not require +higher-order time regularity (unlike the Crank-Nicholson scheme) to converge to the time- +continuous solutions. We subdivide the time-interval [0, T] uniformly N times with time +step size τ = T/N and discrete time steps tn = τn, where n ∈ {1, ..., N}. Then, we look +for a sequence {ψn}N +n=1 of functions in Ω, satisfying the time-discrete system +θ(ψn) − θ(ψn−1) +τ +− ∇ · [K(θ(ψn))∇(ψn + z))] = f(tn). +(3) +Denoting f(tn) by f n subsequently, a more precise and general definition of the weak +solutions of (3) is given below. For simplicity, we assume homogeneous Dirichlet boundary +condition although our results are valid for Dirichlet and Neumann boundary conditions +in general. +Definition 2.1 (Backward Euler time-discretization of (1)). Let ψ0 ∈ L2(Ω) be given. +Then the sequence {ψn}N +n=1 ⊂ H1 +0(Ω) is the backward Euler solution of (1) if for all +n ∈ {1, ..., N}, and v ∈ H1 +0(Ω), +1 +τ (θ(ψn) − θ(ψn−1), v) + (K(θ(ψn))∇(ψn + z), ∇v) = (f n, v). +(4) +2.2 +Space-discretization: Continuous Galerkin finite elements +We consider the finite element method to discretize (4) further in space. Let Th be a +triangulation of Ω into closed d-simplices, where h := maxE∈Th (diam(E)) denotes the +mesh size. Assuming Ω is a polygon, the Galerkin finite element space is +Vh = +� +vh ∈ H1 +0(Ω)| vh|E ∈ Pp(E), T ∈ Th +� +, +(5) +where Pp(E) denotes the space of p-order polynomials on E, p ∈ N. Then, the fully +discrete Galerkin formulation of Richards’ equation reads +Definition 2.2 (Fully discrete solution of (1)). Let ψ0 +h := ψ0 ∈ L2(Ω). Then the sequence +{ψn +h}N +n=1 ⊂ Vh is the fully discrete solution of (1) if for all n ∈ {1, ..., N}, and vh ∈ Vh, +(θ(ψn +h) − θ(ψn−1 +h +), vh) + τ(K(θ(ψn +h))∇(ψn +h + z), ∇vh) = τ(f n, vh). +(6) +4 + +2.3 +Iterative linearization schemes +To obtain the solution of the nonlinear problem (6) an iterative linearization scheme is +generally employed. To investigate the trade-off between the stability and speed of such +schemes, we focus on two linearization strategies that will be representatives of linearly +and quadratically convergent methods with convergence meant in the L2 sense. +2.3.1 +Linearly convergent schemes: The L-scheme +Where the quadratically convergent Newton method utilizes a proper first-order Taylor +expansion of the nonlinear terms in (6), the linearly convergent methods that we consider +here, only exploit an expansion of the monotone components, i.e. the nonlinear saturation +function. Moreover, the expansion does not need to be exact. Consider the following +scheme: Given ψn−1 +h +, ψn,j−1 +h +∈ Vh, find ψn,j +h +∈ Vh such that +(L(ψn,j−1 +h +)(ψn,j +h +− ψn,j−1 +h +), vh) + τ(K(θ(ψn,j−1 +h +))∇(ψn,j +h ++ z), ∇vh) += τ(f n, vh) − (θ(ψn,j−1 +h +) − θ(ψn−1 +h +), vh), +(7) +for all vh ∈ Vh, where L : R → [0, ∞) is a predetermined positive weight function, and +j ∈ N is the iteration index. Observe that, provided κm > 0 in Assumption 1, the problem +above is linear, monotone, and Lipschitz with respect to ψn,j +h , and hence a unique weak +solution of (7) exists. Moreover, if the iteration converges, i.e. if ψn,j +h +→ ψn +h strongly in +H1 +0(Ω), then ψn +h indeed solves (6). There can be many different choices of the function L +which leads to different linearization schemes, see Table 1. For the rest of this paper, we +mainly focus on the case when L is constant which leads to the widely studied L-scheme. +Definition 2.3 (L-scheme). Let ψn−1 +h +, ψn,0 +h +∈ L2(Ω) and L > 0 be given. Then the L- +scheme solves for the sequence {ψn,j +h }j∈N ⊂ Vh which satisfies for all iteration indices +j ∈ N, and vh ∈ Vh +L((ψn,j +h +− ψn,j−1 +h +), vh) + τ(K(θ(ψn,j−1 +h +))∇(ψn,j +h ++ z), ∇vh) += τ(f n, vh) − (θ(ψn,j−1 +h +) − θ(ψn−1 +h +), vh). +(8) +Different choices of L and the resulting schemes are listed below +Scheme +L(ψ) +Picard +0 +Modified Picard [22] +θ′(ψ) +J¨ager-Kacˇur [29] +supξ∈R +θ(ξ)−θ(ψ) +ξ−ψ +L-scheme [23, 24, 1] +L > 0 constant +Modified L-scheme [19] +θ′(ψ) + Mτ, M > 0 constant +Table 1: Different linearly convergent schemes (7) defined along with their linearization +weight function L. +Remark 1 (Non-constant L for heterogeneous media). For the L-scheme, L might not +necessarily be a constant, but can be a function of the spatial variable x. This would +be typically the case for heterogeneous media. All the proofs can be adapted to include +a spatially dependent L, see [34] where this was done for a splitting scheme for Biot +equations. +5 + +It has been shown in [1, Theorem 1] that if L ≥ 1 +2 supξ∈R θ′(ξ), then the L-scheme +iterations converge irrespective of the initial guess under minor restrictions on the time +step size τ and independent of the mesh size. However, numerical results in [1, 19] reveal +that the convergence of the L-scheme can be relatively slow, depending on the choice +of the stabilization parameter L, see please the Appendix A for an adaptive L-scheme. +One can enhance the convergence speed by computing L using the previous iterates and +derivatives. In general, taking L as the Jacobian matrix, would lead to Newton method, +this is the reason one can interpret the L-scheme also as a modified Newton method. +This is exploited in the modified Picard scheme, first proposed in [22], uses L(ψn,j−1) = +θ′(ψn,j−1), complying with the first-order Taylor series expansion θ(ψn,j) ≈ θ(ψn,j−1) + +θ′(ψn,j−1)(ψn,j − ψn,j−1). As a result, if converging it requires fewer iterations compared +to the L-scheme although the convergence is still linear. Nevertheless, this choice of the +L function may lead to divergence of the scheme for larger time step sizes, as predicted in +[10] and observed numerically in [1, 19]. In an attempt to resolve this issue, a modified L- +scheme was proposed in [19] that inherits the characteristics of both the L-scheme (except +that it is using derivatives and the linear systems are not necessarily well conditioned) +and the Picard scheme. The modified L-scheme exhibits increased stability compared to +the Picard scheme while retaining its speed. However, the modified L-scheme converges +unconditionally under the additional restriction that ψn,0 +h += ψn−1 +h +and the discrete time- +derivative (ψn +h − ψn−1 +h +)/τ is in L∞(Ω). Since the objective of this paper is to start the +linearization iterations with a stable scheme, and then switch to a quadratically converging +scheme when its convergence can be guaranteed, the rest of the study will be with respect +to the L-scheme which is arguably the most stable among the schemes presented in Table 1 +and the cheapest in terms of computing time per iteration (due to well-conditioned linear +systems and not involving derivatives). Nonetheless, we remark that our methodology +generalizes to all other linearly converging iterative methods. +Remark 2 (Generality of the results). Although the analysis of Section 3 primarily fo- +cuses on the switching between L-scheme and the Newton method, the same techniques +can be directly extended to cover switching between the schemes in Table 1 and Newton. +Moreover, the L-adaptive strategy in Appendix A can be extended to the modified L-scheme +(see Table 1) to select the parameter M > 0 adaptively. +2.3.2 +Quadratically convergent scheme: The Newton method +The Newton method uses the first order Taylor series expansions of all the nonlinear +functions in (1) to ensure quadratic rates of convergence. +Definition 2.4 (The Newton method). Let ψn−1 +h +, ψn,0 +h +∈ L2(Ω) be given. Then the Newton +method solves for the sequence {ψn,j +h }j∈N ⊂ Vh which satisfies for all iteration indices +j ∈ N, and vh ∈ Vh +(θ′(ψn,j−1 +h +)(ψn,j +h +− ψn,j−1 +h +), vh) + τ(K(θ(ψn,j−1 +h +))∇(ψn,j−1 +h ++ z), ∇vh) ++ τ +� +(K ◦ θ)′(ψn,j−1 +h +)∇(ψn,j−1 +h ++ z)(ψn,j +h +− ψn,j−1 +h +), ∇vh +� += τ(f n, vh) − (θ(ψn,j−1 +h +) − θ(ψn−1 +h +), vh). +(9) +However, this comes at the cost of decreased numerical stability as discussed in Sec- +tion 1. +In the next section we combine the L-scheme and the Newton method in a +consistent manner in order to obtain a linerization strategy that is both stable and fast. +6 + +3 +A posteriori estimate based adaptive switching be- +tween L-scheme and Newton +In this section, we develop the switching algorithm between L-scheme and the Newton +method using a posteriori error analysis. For comparing the errors between different lin- +earization schemes we introduce a uniform notion of linearization errors ηlin in Section 3.1 +based on arguments in [31]. The idea behind the adaptive algorithm is to start with +the L-scheme and derive an estimator ηL→N in Section 3.2 that predicts from the jth and +(j − 1)th iterate the linearization error for the next iteration if done using the Newton +scheme. If the error is predicted to decrease, then the iteration switches to Newton. Then +another estimator ηN→L is derived in Section 3.3 which predicts the linearization error of +the next step of the Newton iteration. The algorithm switches back to the L-scheme in +case the error is predicted to increase. In fact, we go one step further in Appendix A and +derive an estimator ηL→L to predict if the L-scheme itself will converge and to tune the +value of L accordingly. Finally, the full algorithm is laid out in Section 3.4 based on these +estimators. +L-scheme +Initial guess +ηL→N < +ηlin +Newton +ηN→L < +ηlin +NO +YES +YES +NO +Figure 1: Flowchart of Adaptive switching algortihm between L-scheme and Newton’s +method. +3.1 +Linearization errors and iteration-dependent energy norms +In [31] it is shown that the total numerical error corresponding to a finite element-based +linearization scheme can be orthogonally decomposed into a discretization component and +a linearization component if the errors are computed using an iteration-dependent energy +norm (for linearly convergent schemes in Table 1 this is just the energy norm invoked +by the symmetric bilinear form associated with the unknown ψn,j +h +in (7)). Here, we are +only interested in the linearization component which is defined as the difference between +successive iterates in the aforementioned energy norm, i.e., +ηj +lin := +������ψn,j +h +− ψn,j−1 +h +������ +L,ψn,j−1 +h +, +(10) +where |||·|||L,ψn,j−1 +h +represents the particular H1 equivalent-norm defined using the iterate +ψn,j−1 +h +and associated with the linearization scheme denoted by L. The fully computable +estimator ηj +lin encapsulates the entirety of the linearization error, as shown in Section 5 of +[31], and hence, will be used as its sole measure in the subsequent sections. We mention +explicitly the energy norms of the two schemes that are discussed: With reference to +Definition 2.3, the energy norm for L-scheme is defined as +|||ξ|||L,ψn,j−1 +h +:= +�� +Ω +Lξ2 + τ +���K(θ(ψn,j−1 +h +)) +1 +2∇ξ +��� +2� 1 +2 +(11) +7 + +for all ξ ∈ H1 +0(Ω), and with reference to Definition 2.4 the norm for the Newton method +is +|||ξ|||N,ψn,j−1 +h +:= +�� +Ω +θ′(ψn,j−1 +h +) ξ2 + τ|K(θ(ψn,j−1 +h +)) +1 +2∇ξ|2 +� 1 +2 +. +(12) +3.2 +L-scheme to Newton switching estimate +For some i ∈ N, let the sequence {ψn,j +h }i +j=1 ⊂ Vh be obtained using the L-scheme (8), +and in the (i + 1)th-iteration we want to test for switching to the Newton scheme. Let +˜ψn,i+1 +h +∈ Vh be the solution of the Newton scheme (9) having ψn,i +h +as the previous iterate. +In this section, we will assume the following: +Assumption 2 (Convection term is not dominant). For a given i ∈ N, there exists a +constant Ci +N ∈ [0, 2) such that +τ|K(θ(ψn,i +h ))− 1 +2(K ◦ θ)′(ψn,i +h )∇(ψn,i +h + z)|2 ≤ (Ci +N)2θ′(ψn,i +h ), +(13) +a.e. in Ω. +The assumption above is also required to show the coercivity of the linear problem +(9) for j = i + 1, and hence, to show the existence of solution ˜ψn,i+1 +h +. Observe that, since +ψn,i +h +is known, the constant Ci +N is fully computable. Additionally, it is smaller than 2 if +the numerical flux is bounded, and τ is small. Notably, the estimate holds even in the +degenerate case when θ′(ψn,i +h ) = 0, since the left-hand side has (θ′(ψn,i +h ))2. To cover the +degenerate case, we also introduce the concept of an equilibrated flux. +Definition 3.1 (Equilibrated flux σi +L for degenerate regions). For a pre-determined ϵ > 0, +let T i,ϵ +deg := {K ∈ Th : inf θ′(ψn,i +h ) < ϵ in K}. +Let Πh : L2(Ω) → Pp(Th) be the Pp +projection operator, i.e. (Πhu, vh) = (u, vh) for all u ∈ L2(Ω) and vh ∈ Pp(Th). Moreover, +let RTp(Th) be the pth-order Raviart-Thomas space on Th, i.e., σ ∈ RTp(Th) implies +σ|K ∈ (Pp(K))d + xPp(K) for all K ∈ Th. Then, we define σi +L ∈ RTp(Th) ∩ H(div, Ω) +as +∇ · σi +L = +� +1 +τ Πh(L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +))) +in T i,ϵ +deg, +0 +otherwise . +(14) +We defer to Appendix B for discussions on how to compute σi +L in practice. Then, we +have the following result. +Proposition 1 (Error control of L-scheme to Newton switching step). For a given +ψn,0 +h , ψn−1 +h +∈ Vh, let {ψn,j +h }i +j=1 ⊂ Vh solve (8) for some i ∈ N. Let ˜ψn,i+1 +h +∈ Vh be the +solution of (9) with the previous iterate ψn,i +h . Recall Definition 3.1. Then, under the +Assumptions 1–2, one has +��� +��� +��� ˜ψn,i+1 +h +− ψn,i +h +��� +��� +��� +N,ψn,i +h +≤ ηi +L→N, +where, +ηi +L→N := +2 +2−Ci +N +�� +ηi,poten +L→N +�2 + τ +� +ηi,flux +L→N,2 +�2� 1 +2 +8 + +with +ηi,poten +L→N +:= +���θ′(ψn,i +h )− 1 +2 � +L +� +ψn,i +h − ψn,i−1 +h +� +− +� +θ(ψn,i +h ) − θ(ψn,i−1 +h +) +����� +Th\T i,ϵ +deg +, +ηi,flux +L→N := +���K(θ(ψn,i +h ))− 1 +2 �� +K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)) +� +∇ +� +ψn,i +h + z +� ++ σi +L +���� . +Proof. Observe from (9) that δψi+1 +h +:= ˜ψn,i+1 +h +− ψn,i +h +∈ Vh satisfies +(θ′(ψn,i +h )δψi+1 +h +, vh) + τ(K(θ(ψn,i +h ))∇δψi+1 +h +, ∇vh) ++ τ +� +(K ◦ θ)′(ψn,i +h )∇(ψn,i +h + z) δψi+1 +h +, ∇vh +� += τ(f n, vh) − (θ(ψn,i +h ) − θ(ψn−1 +h +), vh) − τ(K(θ(ψn,i +h ))∇ψi +h, ∇vh), +(15) +for all vh ∈ Vh. Inserting the test function vh = δψi+1 +h +in (15), one has +������δψi+1 +h +������2 +N,ψn,i +h +(12) += +� +Ω +� +θ′(ψn,i +h )|δψi+1 +h +|2 + τ|K(θ(ψn,i +h )) +1 +2∇δψi+1 +h +|2� +(15) += −τ +� +(K ◦ θ)′(ψn,i +h )∇(ψn,i +h + z) δψi+1 +h +, ∇δψi+1 +h +� +� +�� +� +=:T1 ++ τ(f n, δψi+1 +h +) − (θ(ψn,i +h ) − θ(ψn−1 +h +), δψi+1 +h +) − τ(K(θ(ψn,i +h ))∇(ψn,i +h + z), ∇δψi+1 +h +) +� +�� +� +=:T2 +. +(16a) +Calling σi = (K ◦ θ)′(ψn,i +h )∇(ψn,i +h + z) for brevity, we estimate that +T1 := −τ(σiδψi+1 +h +, ∇δψi+1 +h +) +≤ +� +τ +� +Ω +|K(θ(ψn,i +h ))− 1 +2σi|2(δψi+1 +h +)2 +� 1 +2 � +τ +� +Ω +|K(θ(ψn,i +h )) +1 +2∇δψi+1 +h +|2 +� 1 +2 +(13) +≤ Ci +N +�� +Ω +θ′(ψn,i +h )(δψi+1 +h +)2 +� 1 +2 � +τ +� +Ω +|K(θ(ψn,i +h )) +1 +2∇δψi+1 +h +|2 +� 1 +2 +≤ Ci +N +2 +� +Ω +� +θ′(ψn,i +h )|δψi+1 +h +|2 + τ|K(θ(ψn,i +h )) +1 +2∇δψi+1 +h +|2� += Ci +N +2 +������δψi+1 +h +������2 +N,ψn,i +h . +(16b) +For estimating the last term, we observe from the divergence theorem that +− (σi +L, ∇δψi+1 +h +) = (∇ · σi +L, δψi+1 +h +) +(17) += +1 +τ (Πh(L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +))), δψi+1 +h +)T i,ϵ +deg += 1 +τ (L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)), δψi+1 +h +)T i,ϵ +deg +The last equality follows from the definition of the projection operator Πh and δψi+1 +h +∈ +9 + +Vh ⊂ Pp(Th). Using this result, along with (8) and δψi+1 +h +∈ Vh, one has +T2 := τ(f n, δψi+1 +h +) − (θ(ψn,i +h ) − θ(ψn−1 +h +), δψi+1 +h +) − τ(K(θ(ψn,i +h ))∇ψi +h, ∇δψi+1 +h +) +(8) += (L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)), δψi+1 +h +) +− τ((K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)))∇(ψn,i +h + z), ∇δψi+1 +h +) += (L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)), δψi+1 +h +) + τ(σi +L, ∇δψi+1 +h +) +− τ((K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)))∇(ψn,i +h + z) + σi +L, ∇δψi+1 +h +) += (L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)), δψi+1 +h +)Th\T i,ϵ +deg +− τ((K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)))∇(ψn,i +h + z) + σi +L, ∇δψi+1 +h +) +(14) +≤ (θ′(ψn,i +h )− 1 +2(L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +))), θ′(ψn,i +h ) +1 +2δψi+1 +h +)Th\T i,ϵ +deg ++ τ[ηi,flux +L→N ] ∥K(ψn,i +h ) +1 +2∇δψi+1 +h +∥ +≤ [ηi,poten +L→N +] · ∥θ′(ψn,i +h ) +1 +2δψi+1 +h +∥ + √τ [ηi,flux +L→N ] · √τ∥K(ψn,i +h ) +1 +2∇δψi+1 +h +∥. +(16c) +Combining (16), using the Cauchy-Schwarz inequality along with the definition of ηi +L→N, +one has the estimate. +3.3 +Newton to L-scheme switching estimate +Assuming that the L-scheme converges unconditionally, after switching to Newton we +would want to switch back to the L-scheme only if linearization error of the Newton +scheme increases with iterations. Similar to before, we can estimate if this is going to +happen in the (i + 1)th-step, purely from the iterates up to the ith-step. For this purpose, +we introduce another equilibrated flux. +Definition 3.2 (Equilibrated flux σi +L for degenerate regions (Newton scheme)). Recalling +Definition 3.1, we define σi +N ∈ RTp(Th) ∩ H(div, Ω) as +∇ · σi +N = +� +1 +τ Πh(θ′(ψn,i +h )(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +))) +in T i,ϵ +deg, +0 +otherwise . +(17) +The corresponding result mirroring Proposition 1 is +Proposition 2 (Error control of Newton to Newton step). For a given ψn,0 +h , ψn−1 +h +∈ Vh, +let {ψn,j +h }i+1 +j=1 ⊂ Vh solve (9) for some i ∈ N. Then, under Assumptions 1–2, one has +������ψn,i+1 +h +− ψn,i +h +������ +N,ψn,i +h +≤ ηi +N→L, +where +ηi +N→L := +2 +(2−Ci +N) +� +[ηi,poten +N→L +]2 + τ[ηi,flux +N→L ]2� 1 +2 +with +ηi,poten +N→L +:= ∥θ′(ψn,i +h )− 1 +2(θ′(ψn,i−1 +h +)(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)))∥Th\T i,ϵ +deg, +ηi,flux +N→L := +������ +� +(K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)))∇(ψn,i +h + z) +−(K ◦ θ)′(ψn,i−1 +h +)(ψn,i +h − ψn,i−1 +h +)∇(ψn,i−1 +h ++ z)) +� +K(θ(ψn,i +h ))− 1 +2 ++K(θ(ψn,i +h ))− 1 +2σi +N +������ +. +10 + +The proof is identical to the proof of Proposition 1 and hence is left for the avid reader. +Remark 3 (Effectivity of the estimators ηi +L→N and ηi +N→L). The estimators ηi +L→N and ηi +N→L +predict the linearization error ηi+1 +lin +of the (i + 1)th iteration if done using the Newton +scheme (9). In the cases where the iteration is done indeed using the Newton scheme, the +sharpness of the estimate can be measured using the effectivity index, i.e., if (i + 1)th +iteration is Newton then +(Eff. Ind.)i := +� +ηi +L→N/ηi+1 +lin +if ith iteration is L-scheme, +ηi +N→L/ηi+1 +lin +if ith iteration is Newton. +(18) +Observe that it is always greater than 1 due to Propositions 1 and 2 and an effectivity +index close to 1 implies a sharp estimate. The estimators are expected to be quite accurate +since mainly the Cauchy-Schwarz inequality is used to derive them, except for estimate +(16b) where the term T1 is bounded above using the global approximation in Assumption +2. This expected sharpness is shown to be the case through the numerical experiments of +Section 4, see in particular Figures 5 and 8 +3.4 +A-posteriori estimate based adaptive linearization algorithm +With the above estimates in mind, we propose a switching algorithm between the L- +scheme and the Newton method. The linearization scheme used at iteration j = i + 1 +should be Newton if the linearization error, predicted by the estimators ηi +L→N and ηi +N→L, +is smaller than the linearization error ηi +lin of the ith step, see (10). However, to optimize +the algorithm we take a few numerical considerations into account first. +3.4.1 +Computational considerations +To speed up the computations of this switching criteria, we make a few more reductions +• [Equilibrated flux] If the saturated domain is much smaller than the unsaturated +domain, then we take σi +L = σi +N = 0. +• [Switching condition] The condition ηi +L→N ≤ ηi +lin might be difficult to satisfy if +the estimators are not sharp (see Remark 3), and even when it is satisfied it might +require large values of i. Hence, to expedite the switching between L-scheme and +Newton, we will use the criteria ηi +L→N < Ctol ηi +lin for a constant Ctol > 1. +3.4.2 +Adaptive linearization algorithm +Under these considerations we propose the following adaptive algorithm: +11 + +Algorithm 1 L-scheme/Newton a-posteriori switching +Require: ψn,0 ∈ L2(Ω) as initial guess. +Ensure: Scheme= L-scheme , Ctol = 1.5 +for i=1,2,.. do +if Scheme= L-scheme then +Compute iterate using L-scheme , i.e., (8) +if Ci +N ≥ 2 then continue. +else if ηi +L→N ≤ Ctolηi +lin then +Set Scheme= Newton +else +Compute iterate using Newton , i.e., (9) +if ηi +N→L > ηi +lin then +Set Scheme= L-scheme +Remark 4 (Combining L-scheme adaptivity). In Appendix A, we further propose an +algorithm to adaptively select L in order to expedite the convergence of the L-scheme. This +can directly be implemented in conjunction to Algorithm 1 to improve the convergence speed +of the composite scheme. Nevertheless, we have refrained from combining these schemes +for the ease of presentation. +Remark 5 (Computational cost of the estimators). In the non-degenerate case, the quan- +tities Ci +N, ηi +L→N and ηi +N→L, can be directly computed from the iterates ψn,i +h +and ψn,i−1 +h +by +inserting σi +L = σi +N = 0, see Propositions 1 and 2. Hence, the cost of computing the +estimators is small in comparison to the cost of the iterations. Since the L-scheme iter- +ations are less expensive than the Newton iterations, the L/N scheme generally performs +better or similarly to the Newton scheme time-wise. This is evident from the numerical +experiments, e.g. see Figure 3b. In the degenerate case, global computation are required +for computing σi +L and σi +N if they are used. We discuss the computation of these equili- +brated fluxes in Appendix B and their computation can be made relatively inexpensive by +precomputing the associated stiffness matrices. The computational cost for the estimators +can be reduced even further by evaluating them only for selected iterations. Nevertheless, +we do not pursue this option for the sake of simplicity. +4 +Numerical results +In this section, we perform several numerical examples that demonstrate the robustness +and efficiency of the proposed algorithm for switching between Newton’s method and the +L-scheme. This is done through careful comparison between the switching algorithm, +hereafter called the L/N-scheme, the standard Newton method and the L-scheme. It is +important to note that the L-scheme includes a tuning parameter that significantly affects +the performance of the method. As a remedy, we choose two different values, L1 and L2 +in the performance comparison. Here, L1 is a quasi-optimal choice of tuning parameter +and will be defined for each specific subproblem, see Table 2, and L2 = sup {θ′ (ψ)}. For +the L/N-scheme, L1 is always chosen for the L-scheme iterations. +To measure the performance of each separate method, we examine both the number +of iterations and computational time that they require to satisfy the stopping criterion +������ψn,j +h +− ψn,j−1 +h +������ +L,ψn,j−1 +h +< 10−7, +12 + +where |||·|||L,ψn,j−1 +h +is the iteration and linearization-dependent energy norm for the pressure +head, with L ∈ {L, N}. Here, the computational time covers the entire simulations and all +experiments were performed on an Acer Swift 3, with an Intel core i7-1165G7-processor. +In total, three different test cases for the numerical experiments are considered: +• Test case 1: The first test case is taken from [35], although it is modified in the sense +that we disregard surfactant transport. Here, the flow is always partially saturated. +• Test case 2: The second test case can be found in [1], and it considers extrac- +tion/injection above the water table. +• Test case 3: The final test case is a known benchmark problem that is studied in +[1, 36, 37, 38]. +Here, a time-dependent Dirichlet boundary condition is used to +describe the recharge of a groundwater reservoir from a drainage trench. +For all test cases, the van Genuchten-Mualem parametrization [33] is used to describe +the relation between the saturation, the pressure head and the permeability, +θ(ψ) = +� +� +� +θR + (θS − θR) +� +1 +1+(−αψ)n +� n−1 +n , +ψ ≤ 0, +θS, +ψ > 0, +K(Θ(ψ)) = +� +� +� +� +� +Ks (Θ(ψ)) +1 +2 +� +1 − +� +1 − Θ(ψ) +n +n−1 +� n−1 +n �2 +, +ψ ≤ 0, +Ks, +ψ > 0. +(19) +Here, +Θ(ψ) = θ(ψ) − θR +θS − θR +, +with θS and θR being the water volume and the residual water content respectively, Ks +the hydraulic conductivity of the fully saturated porous medium, and α and n soil related +parameters. +In all of the test-cases, triangular linear conforming finite elements with mesh diameter +h are applied together with the implicit Euler time-discretization with time step size τ, as +described in Sections 2.1 and 2.2. The mesh diameter h and time step size τ vary between +the different experiments and will be specified for each individual experiment. We note +that the numerical experiments are expected to perform equivalently for other spatial +discretization methods such as the Raviart-Thomas mixed finite elements or discontinuous +Galerkin finite elements. +The finite element implementation is Python based and uses the simulation toolbox +PorePy [39] for grid management. It is available for download at https://github.com/ +MrShuffle/RichardsEquation/releases/tag/v1.0.1. +13 + +Parameters +Test case 1 +Test case 2 +Test case 3 +van Genuchten-Mualem +θR +0.026 +0.026 +0.131 +θS +0.42 +0.42 +0.396 +KS +0.12 +0.12 +4.96 · 10−2 +α +0.551 +0.95 +0.423 +n +2.9 +2.9 +2.06 +L-scheme +L1 +0.1 +0.15 +3.501·10−3 +L2 = Lθ +0.136 +0.2341 +4.501·10−3 +Table 2: Parameter values for all test cases. The parameters are presented in column +format, where each column corresponds to the parameters for the specified test case. +4.1 +Test case 1: Strictly unsaturated medium +In this test case, we consider a strictly unsaturated porous medium, and use the van +Genuchten-Mualem parametrization that is described by parameters from Table 2. The +test case is heavily inspired by [35], and the domain is given by Ω = Ω1 ∪ Ω2, where +Ω1 = [0, 1] × [0, 1/4] and Ω2 = [0, 1] × (1/4, 1]. We consider the time interval [0, T], where +T = τ varies with choice of time step size τ, as we only take one time step. As initial +condition, we choose the pressure head +ψ0(x, z) = +� +−z − 1/4 +(x, z) ∈ Ω1 +−4 +(x, z) ∈ Ω2, +where x represents the positional variable in the horizontal direction and z in the vertical +direction. A Dirichlet boundary condition is imposed at the top boundary that complies +with the initial condition. For the rest of the boundary no-flow boundary conditions are +used, and the following source term is applied +f(x, z) = +� +0 +(x, z) ∈ Ω1 +0.06 cos +� 4 +3π(z) +� +sin (x) +(x, z) ∈ Ω2. +The solution after one time step with time step size τ = 1, is given in Figure 2. +Figure 2: Test case 1: Strictly unsaturated medium. Pressure head ψ at final time T = 1. +14 + +1.0 +1.35 +1.95 +0.8 +2.55 +0.6 +3.15 +3.75 +0.4 +4.35 +4.95 +0.2 +5.55 +6.15 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +104.1.1 +Comparison of convergence properties. +Here, we discuss the performance and convergence properties of the newly proposed L/N- +scheme and compare it to the Newton method and the L-scheme. +In Figure 3a, the +number of iterations for different choices of the mesh size parameters, with time step +size τ = 0.01 are presented. As expected the L-scheme is robust and converges in each +scenario, for both L1 and L2. Newton’s method, however, only converges for sufficiently +coarse meshes. Yet, when converging, it converges in fewer iterations than the L-scheme. +Finally, the hybrid L/N method converges in as few if not fewer iterations as the Newton +method (when it converges) and converges robustly, and in far fewer iterations than the +L-scheme for the other mesh sizes. +Furthermore, a similar experiment is performed for a fixed mesh size h = +√ +2/40, and +varying time step sizes, see Figure 4a. For larger time step sizes the Newton method +diverges, while the other methods converge robustly. Again the L/N-scheme converges +with the performance expected of Newton’s method, in addition to being as robust as +the L-scheme. We highlight the enormous difference in the number of iterations for the +largest time step size τ = 1 in Figure 4a. +0 +20 +40 +60 +80 +0 +10 +20 +30 +40 +24 +24 +25 +25 +25 +26 +26 +26 +33 +34 +35 +35 +35 +35 +35 +36 +(3/8) +(3/7) +(2/7) +(2/6) +(1/7) +(1/5) +(1/4) +(1/4) +8 +7 +5 +5 +√ +2/h +Number of Iterations +(a) Total number of iterations. The numbers +in the red parentheses correspond to (number +of L-scheme iterations/number of Newton it- +erations). +20 +30 +40 +50 +60 +70 +80 +0 +500 +1,000 +1,500 +332 +442 +577 +772 +991 +1230 +469 +599 +832 +1032 +1373 +1666 +518 +373 +275 +193 +153 +91 +104 +151 +√ +2/h +CPU time [s] +L1 +L2 +Newton +L/N +(b) Computational time in seconds. +Figure 3: Test case 1: Strictly unsaturated medium. Performance metrics for all lineariza- +tion schemes for fixed τ = 0.01 and varying mesh size. +Then, the performance of the linearization schemes is compared in terms of computa- +tional time, cf. Figure 3b and Figure 4b. One can observe virtually the same performance +for the hybrid method as for Newton’s method when the latter converges. The former in +fact is sometimes slightly faster, due to each L-scheme iteration being slightly less expen- +sive than a Newton iteration, see Remark 6. In addition, the hybrid method continues to +show the same performance for the cases in which Newton’s method does not converge. +Finally, Figure 3b shows that, for all meshes, the computational time of the L-schemes is +consistent with the reported numbers of iterations in Figure 3a with L1 being the fastest. +Although it uses more than double the computational time of the L/N-scheme. +Overall, the newly proposed L/N-scheme shows the best performance. It is as fast as +Newton’s method when it converges, and is significantly more robust. +15 + +0.001 +0.01 +0.1 +1 +0 +10 +20 +30 +40 +55 +84 +114 +(1/7) +(1/7) +(1/7) +(3/8) +(3/8) +(3/8) +(1/7) +(1/7) +(1/7) +(1/4) +(1/4) +(1/4) +τ +Number of iterations +L/N +Newton +L1 +L2 +(a) Number of iterations for different time step +sizes. +0.001 +0.01 +0.1 +1 +0 +500 +1,000 +1,500 +2,000 +τ +CPU time [s] +L/N +Newton +L1 +L2 +(b) Total computational time in seconds for dif- +ferent time step sizes. +Figure 4: Test case 1: Strictly unsaturated medium: Performance comparison for all of +the linearization schemes for different time step sizes and fixed mesh size h = +√ +2/40. +Remark 6 (Computational time per iteration). It is known that condition numbers for +matrices coming from systems linearized by Newton’s method are higher than for those +linearized by the L-scheme [1]. Therefore, each iteration of Newton’s method, when im- +plemented without preconditioning, takes more time than each L-scheme iteration. +Remark 7 (Computational time for the coarsest mesh). The computational times of the +coarsest meshes are omitted due to the use of multiprocessing in the implementations. +This causes the most time consuming part to be the spawn process of the local assembly +on each element. As a result, the computational times for the coarsest meshes are very +similar for all the linearization methods. +4.1.2 +Switching characteristics +Finally, the dynamic switch between the L-scheme and Newton’s method is inspected in +further detail. In Figure 5, the evolution of the indicators for the switch is displayed for a +fixed mesh and time step size. The example particularly demonstrates the ability of the +hybrid method to switch back and forth between both linearizations before switching fully +to Newton. In addition, the final number of L-scheme iterations is kept at its minimum. +The plot also shows the effectivity indices introduced in (18) and discussed in Remark 3. +The effectivity index is greater than 1 in all cases, which validates Propositions 1 and 2 +and it stays between 1.27 to 2.3, implying that the estimators ηi +L→N and ηi +N→L are sharp. +4.2 +Test case 2: Variably saturated medium +The example parameters are as in Table 2, Test case 3. We consider a variably saturated +medium, Ω = Ωgw ∪ Ωvad, where the groundwater zone is Ωgw = [0, 1] × [0, 1/4) and a +vadoze zone is Ωvad = [0, 1] × [1/4, 1]. Here, we consider the time interval [0, T], where +T = 0.01 and we only take one time step with τ = 0.01. As initial condition, we choose +16 + +0 +2 +4 +6 +8 +10 +12 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +Iteration number +ηi +N→L/ηi +lin +ηi +L→N/ηi +lin +(a) Evolution of switching indicators for L/N- +scheme where the dashed line is Ctol = 1.5. +The L/N-scheme oscillates between the lin- +earization strategies, but eventually recovers. +0 +2 +4 +6 +8 +10 +12 +1 +1.5 +2 +2.5 +Iteration number +(Eff. Ind.)i +(b) Efficency index. Notice that the iterations +correspond to the ones in Figure 5(a), and that +only the ones where the Newton method is +performed are counted, i.e., iteration 1,3 and +5 are removed. +Figure 5: Test case 1: Strictly unsaturated medium. Evolution of switching indicators +for the L/N-scheme and efficiency indices (18) for the Newton iterations (see Remark 3). +Here, the mesh size is h = +√ +2/80 and time step size τ = 0.01. +the pressure head +ψ0(x, z) = +� +−z + 1/4 +(x, z) ∈ Ωgw +−3 +(x, z) ∈ Ωvad, +where x represents the positional variable in the horizontal direction and z in the vertical +direction. +On the surface a constant Dirichlet boundary condition is imposed, being +equal to the initial condition at all times. For the rest of the boundary no-flow boundary +conditions are used. We apply the following source term +f(x, z) = +� +0 +(x, z) ∈ Ωgw +0.006 cos +� 4 +3π(z − 1) +� +sin (2πx) +(x, z) ∈ Ωvad. +After one time step the pressure head profile is given in Figure 6. +Figure 6: Test case 2: Variably saturated medium: Pressure head profile at T = 0.01. +17 + +1.0 +0.16 +0.48 +0.8 +0.80 +1.12 +0.6 +1.44 +1.76 +0.4 +2.08 +0.2 +2.40 +2.72 +0.0 +t0'E- +0.0 +0.2 +0.4 +0.6 +0.8 +100 +20 +40 +60 +80 +0 +20 +40 +60 +(4/6) +(5/5) +(5/5) +(8/3) +(6/3) +(8/4) +(5/4) +(43) +(39) +(38) +(36) +(36) +(35) +(35) +(43) +(43) +(58) +(57) +(55) +(56) +(54) +(51) +(66) +(67) +√ +2/h +Number of iterations +(a) Total number of iterations. +The numbers +in the red parentheses correspond to (number +of L-scheme iterations/number of Newton itera- +tions). +20 +30 +40 +50 +60 +70 +80 +0 +500 +1,000 +1,500 +2,000 +2,500 +(484) +(395) +(304) +(273) +(173) +(169) +(1808) +(1442) +(1058) +(828) +(616) +(477) +(2676) +(2141) +(1614) +(1269) +(939) +(700) +√ +2/h +CPU time [s] +L1 +L2 +L/N +(b) Computational time in seconds. +Figure 7: Test case 2: Variably saturated medium: Performance metrics for all lineariza- +tion schemes for fixed τ = 0.01 and varying mesh size. +4.2.1 +Comparison of convergence properties. +The iteration count for the second test case for different mesh sizes and fixed time step +for all linearization schemes is illustrated in Figure 7a. Again the L-scheme converges in +every case. However, Newton’s method does not converge for any mesh size. The hybrid +method needs the fewest number of iterations, which shows that the dynamic switch is +successful. +The CPU time performance of the linearization schemes is compared in Figure 7b. +Both versions of the L-scheme takes computational times consistent with the number of +iterations, with the simulations with the parameter L1 being less expensive. However, +the L-scheme (using L1) requires approximately 373% of the computational time of the +hybrid method including the computation of the switching indicators. In addition, the +benefit of a few additional L-scheme iterations further decreases the computational time +of the hybrid method. +4.2.2 +Switching characteristics +We also give a more in-depth look to the dynamic switch between the Newton’s method +and the L-scheme. In Figure 8, the evolution of the switching indicators is shown for a +fixed time step and a fixed mesh size. After 8 L-scheme iterations the switching indicator +ηL→N becomes lower than Ctol and then Newton’s method converges. From Figure 7a +the number of L-scheme iterations required before the switching indicator becomes small +enough to switch to Newton’s method varies with the mesh size. Note that for the coarsest +mesh no switch to Newton’s method happens. +18 + +0 +2 +4 +6 +8 +10 +12 +10−8 +10−6 +10−4 +10−2 +100 +102 +Iteration number +ηi +N→L/ηi +lin +ηi +L→N/ηi +lin +(Eff. Ind.)i +Figure 8: Test case 2: Variably saturated medium: Evolution of switching indicators +for L/N-scheme for fixed h = +√ +2/50 and τ = 0.01. +The dashed line is Ctol = 1.5, +the switching criterion from L-scheme to Newton’s method. The effectivity indices (18) +corresponding to the Newton iterations are also plotted and they remain below 2.8. +4.3 +Test case 3: Benchmark problem +Here, we consider a known benchmark problem [38], also used e.g. in [1], which models +the recharge of a groundwater reservoir from a drainage trench in two spatial dimensions. +The domain Ω ⊂ R2 represents a vertical segment of the subsurface. One portion of +the right side of the domain is fixed by a constant Dirichlet boundary condition. +A +time-dependent Dirichlet boundary condition on parts of the upper boundary is used to +mimic the drainage trench. No-flow conditions are utilized on the remaining parts of the +boundary. The used parameters are given in Table 2 Test case 3, corresponding to silt +loam. The geometry is given by +Ω = [0, 2] × [0, 3], +ΓD1 = [0, 1] × (3), +ΓD2 = (2) × [0, 1], +ΓN = Ω\ {ΓD1 ∪ ΓD2} , +and the initial pressure head distribution and boundary conditions are +ψ(0, x, z) = 1 − z +ψ(t, x, z) = +� +� +� +� +� +−2 + 35.2t, +if t ≤ +1 +16, +on ΓD1, +0.2, +if t > +1 +16, +on ΓD1, +1 − z, +on ΓD2, +− K(θ(ψ(t, x, z)))∇(ψ(t, x, z) + z) · ν = 0, +on ΓN, +where ν is the outward normal vector. The solution is computed over 9 timesteps, where +the time unit is in days, with time step size τ = 1/48 and with a regular mesh consisting +19 + +of 2501 nodes. The pressure head profile at the final time for the L/N-scheme is shown +in Figure 9. +Figure 9: Test case 3: Benchmark problem: Pressure head profile at 4.5 hours. +No. Itr +CPU time [s] +L1 +274 +6136 +L2 +330 +7356 +Newton +39 +980 +L/N +(10/30) +1021 +Table 3: Test case 3: Benchmark problem: Performance metrics for 2501 nodes. +4.3.1 +Comparison of convergence properties. +The performance of all schemes for test case 3 is displayed in Table 3. All schemes converge +for this example. The Newton method requires the least amount of iterations. However, +the hybrid method only needs one more iteration. Both uses significantly less iterations +than the L-schemes. For all time steps except one, only one L-scheme iteration is needed +per time step, which indicates a successful dynamic switch for almost all time steps. +The computational time for the L-schemes is much higher than both Newton’s method +and the hybrid method, which is consistent with the expense per iteration discussed in +Remark 6. More significantly, the L/N-scheme performs almost the same as Newton’s +method. +5 +Conclusions +In this paper, we considered solving Richards’ equation, which models the flow of water +through saturated/unsaturated porous media (soil). After applying backward Euler time- +discretization and continuous Galerkin finite element space-discretization to Richards’ +equation, to solve the resulting nonlinear finite-dimensional problem we developed a hy- +brid iterative linearization strategy that combines the L-scheme with the Newton method. +20 + +3.0 +0.96 +2.5 +0.64 +0.32 +2.0 +0.00 +0.32 +1.5 +0.64 +10 +0.96 +1.28 +0.5 +1.60 +0.0 - +1.92 +0.0 +0.5 +1.0 +1.5 +2.0The idea behind this is to use the robust, but only first-order convergent L-scheme to +stabilize the quadratically convergent Newton method. The switching between the two +schemes is done in an adaptive manner using a posteriori indicators which predict the +linearization error of the next iteration using a concept of iteration-dependent energy +norms. After each iteration, it is checked whether the Newton method is predicted to +decrease the linearization error of the next iteration. If so, then the Newton method is +used, otherwise, the iteration is done using the L-scheme. The hybrid scheme is now +robust, but still quadratically convergent after switching to the Newton scheme. +The performance of the hybrid scheme is tested on illustrative, realistic numerical +examples which reveal that the scheme is as robust as the L-scheme and it converges in +cases where Newton fails. Moreover, in cases when Newton converges, the hybrid scheme +takes roughly the same amount of iterations and computational time and is considerably +faster than even the optimized L-scheme. Lastly, we comment that the scheme is quite +general as it can, in principle, be extended to other spatial discretization and linearization +methods. +Appendix A +An adaptive L-scheme +As discussed in Sections 1 and 2.3.1, the L-scheme converges unconditionally provided that +L ≥ 1 +2 supξ∈R θ′(ξ) and the time step size τ is smaller than a constant independent of the +mesh size. However, numerical results in [1] suggest that the optimal rate of convergence +of the L-scheme is obtained for a considerably smaller L although convergence cannot +always be guaranteed for such values. Hence, to speed up the computations, it is possible +to start the iterations with a smaller value of L and then use the a posteriori estimates to +decide if L is to be increased or not. Analogous to Propositions 1 and 2 we state a result +that allows us to do this rigorously. +Proposition 3 (Error control of L-scheme). For a given ψn,0 +h , ψn−1 +h +∈ Vh, let {ψn,j +h }i+1 +j=1 ⊂ +Vh solve (8) for some i ∈ N. Then under Assumption 1, +������ψn,i+1 +h +− ψn,i +h +������ +L,ψn,i +h +≤ ηi +L→L, +where +ηi +L→L := +� +[ηi,poten +L→L +]2 + τ[ηi,flux +L→L ]2� 1 +2 +with +ηi,poten +L→L +:= ∥L− 1 +2(L(ψn,i +h − ψn,i−1 +h +) − (θ(ψn,i +h ) − θ(ψn,i−1 +h +)))∥, +ηi,flux +L→L := +���(K(θ(ψn,i +h )) − K(θ(ψn,i−1 +h +)))K(θ(ψn,i +h ))− 1 +2∇(ψn,i +h + z) +��� . +The detailed proof is again omitted. Observe that for the estimate above, neither +Assumption 2 nor any separate treatment of the degenerate domains is required. +A.1 +L-adaptive algorithm +Based on Proposition 3, we propose an algorithm that selects optimal L-values adaptively. +21 + +Algorithm 2 The L-adaptive scheme +Require: ψn,0 ∈ L2(Ω) as initial guess, LM := supψ∈R θ′(ψ), and Lm := LM/8 +Ensure: CL→L = +√ +2, L = Lm +for i=1,2,.. do +Compute iterate using L-scheme, i.e., (8) +if ηi +L→L > ηi +lin then +Replace Lm = L, L = min(CL→LL, LM), and continue. +else if ηj +L→L > 0.8 ηj +lin for j ∈ {i, i − 1, i − 2} then +Replace L = max(0.9L, 1.1Lm) and continue. +A.2 +Numerical result +0 +10 +20 +30 +40 +50 +60 +10−1 +100 +101 +Iteration number +ηi +L→L/ηi +lin +Figure 10: Test case 1: Strictly unsaturated medium: L-scheme with L-adaptivity and +initial stabilization parameter L0 = L2/8, h = +√ +2/40 and τ = 1. +In Figure 10 we show a result where the L-adaptive scheme is superior to a fixed L- +approach. In this case, Lθ/2 is too small for convergence due to a large time step size. +Compared with fixed L1 with the same mesh size and time step size, see Figure 4, the +number of iterations is improved by 20. For smaller time steps, the numerical results +reveal that Algorithm 2 results in roughly the same number of iterations compared to +a fixed and optimized L = L1 lesser than Lθ. But in all examples considered, it uses +fewer iterations than simply choosing L = L2 = Lθ. The advantage of such an adaptive +technique is that an optimization study of L does not need to be conducted prior to the +simulation. However, since the L-adaptive strategy does not significantly improve the +behavior of the L-scheme over the optimized L = L1, we refrained from including it in +Algorithm 1 for the sake of simplicity. +Appendix B +Computation of equilibrated flux +Recalling Definitions 3.1 and 3.2, let us propose a simple algorithm to compute an +equilibrated flux σh ∈ RTp(Th) ∩ H(div, Ω) satisfying ∇ · σh = Πhf in T i,ϵ +deg, and +∇ · σh = 0 otherwise, where f ∈ L2(Ω). Defining Qh := RTp(Th) ∩ H(div, Ω) and +22 + +˜Vh := {vh ∈ Pp(Th)| Tr∂Ω(vh) = 0}, we seek a pair (σh, rh) ∈ Qh × ˜Vh that satisfies the +mixed finite element problem, +(K(1)−1σh, qh) = (rh, ∇ · qh), +∀ qh ∈ Qh, +(20a) +(∇ · σh, vh) = (f, vh), +∀ vh ∈ ˜Vh. +(20b) +The advantage of this flux is that it minimizes ∥K(1)− 1 +2σh∥ which appears in the estimates +in Propositions 1 and 2. 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Knabner, Finite Element Simulation of Saturated-Unsaturated Flow Through +Porous Media, Birkh¨auser Boston, 1987, Ch. 6, pp. 83–93. +[37] R. Haverkamp, M. Vauclin, J. Touma, P. J. Wierenga, G. Vachaud, A Comparison of +Numerical Simulation Models For One-Dimensional Infiltration, Soil Science Society +of America Journal 41 (2) (1977) 285–294. +25 + +[38] E. +Schneid, +Hybrid-Gemischte +Finite-Elemente-Diskretisierung +der +Richards- +Gleichung, +Naturwissenschaftliche Fakult¨at der Friedrich-Alexander-Universit¨at +Erlangen-N¨urnberg, 2000. +[39] E. Keilegavlen, R. Berge, A. Fumagalli, M. Starnoni, I. Stefansson, J. Varela, I. Berre, +Porepy: an open-source software for simulation of multiphysics processes in fractured +porous media, Computational geosciences 25 (1) (2021) 243–265. +26 + diff --git a/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/load_file.txt b/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a3b4822b7e1e93c56c4587328f7c7d81407353b --- /dev/null +++ b/ddA0T4oBgHgl3EQfG_-E/content/tmp_files/load_file.txt @@ -0,0 +1,871 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf,len=870 +page_content='An adaptive solution strategy for Richards’ equation Jakob S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Stokke1, Koondanibha Mitra2, Erlend Storvik1,3, Jakub W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Both1, and Florin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Radu1 1Center for Modeling of Coupled Subsurface Dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Department of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' University of Bergen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Norway 2Computational Mathematics group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Hasselt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Belgium 3Department of Computer science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Electrical engineering and Mathematical sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Western Norway University of Applied Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Norway January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 2023 Abstract Flow in variably saturated porous media is typically modelled by the Richards equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' a nonlinear elliptic-parabolic equation which is notoriously challenging to solve numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In this paper, we propose a robust and fast iterative solver for Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The solver relies on an adaptive switching algorithm, based on rigorously derived a posteriori indicators, between two linearization methods: L-scheme and Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Although a combined L-scheme/Newton strategy was in- troduced previously in [1], here, for the first time we propose a reliable and robust criteria for switching between these schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The performance of the solver, which can be in principle applied to any spatial discretization and linearization methods, is illustrated through several numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Keywords: Iterative linearization, Adaptivity, L-scheme, Newton’s method, Richards’ equation, Nonlinear degenerate diffusion 1 Introduction In this paper, we consider the pressure head ψ based formulation of the Richards equation ∂tθ(ψ) − ∇ · [K(θ(ψ))∇(ψ + z))] = f, (1) where θ : R → [0, 1] is the water content, K is the rank 2 permeability tensor of the porous medium, z is the height against the gravitational direction, and f is a source/sink term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Richards’ equation is used to model the flow of water in saturated/unsaturated porous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It is a highly nonlinear and degenerate elliptic-parabolic equation which makes solving it a very challenging task, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' the review work of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We refer to [3] for the existence and uniqueness of a weak solution of Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' There are plenty of works regarding discretization of Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Due to the low regularity of solutions of (1), see [4], generally, a backward Euler (implicit) scheme (3) is employed to discretize it in time, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Regarding spatial discretization we mention continuous Galerkin finite elements [6, 7], mixed or expanded mixed finite 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='02055v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='NA] 5 Jan 2023 elements [8, 9, 10, 11, 12], finite volumes [13, 14] (see also the recent review [15]), or multipoint flux approximation (MPFA) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Regardless of the choice of the spatial dis- cretization method, one has to solve at each time step a nonlinear, finite-dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In this paper, we will focus on how to efficiently solve these problems using iterative linearization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The main iterative linearization methods used for this type of nonlinear problem are the Newton method, Picard or modified Picard, L-scheme, the Jaeger-Kacur method, or combinations of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Perhaps the most common choice is the Newton method [17, 18] which converges quadratically provided the initial guess is close enough to the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a r-H¨older continuous θ′ function (r ∈ (0, 1]) and the initial guess equal to the solution of the previous time step, it was shown in [10] that the Newton scheme is (1 + r)th order convergent if τ ≤ Cθ 2+r r m hd, (2) where τ > 0 is the time step size, h > 0 the mesh size, d ∈ N the spatial dimension, C > 0 a constant which depends on the domain and the nonlinearities, and θm := inf θ′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, for simulations in 2 or 3 dimensions, condition (2) is quite restrictive partic- ularly if the mesh size h is small, or if the problem is degenerate (θm = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This fact is corroborated by numerical simulations in [1, 19] which show that the Newton method fails to converge in many such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' One can improve the robustness of Newton method by using a damped version of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Line search, variable switching [20] or trust-regions techniques [21] are examples of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Alternatively, one can increase the robustness of Newton’s method by performing first a few fixed-point iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This was proposed in [17, 18] by using the Picard method and in [1] by using the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nevertheless, the switching between the schemes was not based on an a posteriori indicator, but done in a heuristic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The other linearization schemes are fixed-point type schemes, typically more robust, however only linearly convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It has been shown in [22, 13] that the Picard method does not perform well for Richards’ equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A modified Picard method was proposed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The modified Picard coincides with Newton’s method for the case of a constant permeability, therefore it inherits robustness problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The L-scheme, first proposed in [23, 24, 1], is a stabilized Picard method and it was designed to be unconditionally con- verging irrespective of the choice of the initial guess even in degenerate settings and for larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The L-scheme (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3) uses a global constant as a stabiliza- tion coefficient, does not involve the computation of any derivatives, and thus, is not only more stable but also consumes less computational time per iteration due to easier assem- bly of the stiffness matrices which are better conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Numerical results in [1, 19] clearly demonstrate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, they also reveal that the L-scheme converges consid- erably slower in terms of number of iterations compared to the Newton scheme and at a linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Furthermore, its overall performance strongly depends on the careful choice of a tuning parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' despite theoretical stability, an improper choice may effectively result in stagnation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The sensitivity of the performance of the L-scheme with respect to the stabilization can be significantly relaxed when combining the L-scheme with Anderson acceleration [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Indeed, for Richards equation extended to deformable porous media and solved by an L-scheme, it has been demonstrated that, first, the stabilization parameter can be chosen outside the theoretical range, and second, the non-degenerate convergence can be retained in case of previous divergence or accelerated, as also discussed from a the- oretical perspective [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Similar stabilizing properties of the Anderson acceleration have 2 been also discussed for general fixed-point methods [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Other fixed point iterations schemes include J¨ager-Kacˇur scheme [29] which converges unconditionally albeit slowly, and is more computationally expensive than the L-scheme per iteration, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The modified L-scheme, proposed in [19], shows stability similar to the L-scheme while having much faster convergence rates (scaling with τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' yet, the convergence is still linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In this paper, we investigate a hybrid strategy, dynamically switching between the L-scheme and Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This utilizes the advantages of both methods: the un- conditional stability of the L-scheme, and the quadratic convergence of Newton’s method when close to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The crucial difference to previous works on hybrid approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' [1, 17], is the adaptive nature of the switch between both linearization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A switch from the L-scheme to Newton’s method is performed when the iterate is sufficiently close to the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This finally allows us to balance robustness and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The main challenge in implementing this strategy originates from deriving a rigorous switching criteria between the schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Since, the a priori estimates, such as the ones pro- vided in [10], involve unknown constants and assume the worst-case scenario, we pursue an a posteriori estimate-based approach here instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A rigorous and efficient a posteri- ori estimator for the fully degenerate Richards equation involving linearization errors was derived in [30] in the continuous space-time setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the time-discrete problem (3), a robust, efficient, and reliable estimator was derived in [31] using an orthogonal decompo- sition result dividing the total error into a discretization and a linearization component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Furthermore, its effectiveness was demonstrated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' These papers serve as the main inspirations in deriving the a posteriori based switching criteria in Section 3 and an adaptive L-scheme algorithm in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nevertheless, since we are only interested in computing the linearization error component, the computation of equilibrated flux will be avoided wherever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Section 2, we introduce the mathematical no- tation, state the assumptions, define the fully-discrete solution, and elaborate on differ- ent linearization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Section 3, the adaptive switching algorithm is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Firstly, a concept of linearization error is introduced along with the derivation of a predic- tive indicator for linearization error of the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The adaptive algorithm com- pares the linearization error with the estimator to determine the exact switching points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Section 4, three numerical test cases (partially saturated, degenerate, and realistic benchmarks) are presented which illustrate the robustness and computational efficiency of the adaptive scheme compared to the standard Newton’s method or the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Section 5 contains the conclusions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The paper ends with two appendices, one concerning an adaptive L-scheme and the other on the details of the computation of the equilibrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 2 Mathematical and numerical formulation We consider Richards’ equation in the space-time domain G = Ω × [0, T], where Ω is a bounded domain in Rd with a Lipschitz continuous boundary ∂Ω, and T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let (·, ·) and ∥ · ∥ be the inner product and norm of the square-integrable functions in Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' L2(Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, using common notation from functional analysis, H1(Ω) represents the Sobolev space of functions with first-order weak derivatives in L2(Ω), and H1 0(Ω) its subspace containing functions with vanishing trace at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the material properties θ and K, and source term f in (1), the following assumptions are made: 3 (a) The saturation function θ(·) is Lipschitz continuous and monotonically increasing with Lθ and θm ≥ 0 being the Lipschitz constant and the lower bound for the deriva- tive, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (b) The permeability tensor K : [0, 1] → Rd×d satisfies the uniform (pseudo) ellipticity condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', for constants κM > κm ≥ 0, κm|z|2 ≤ zT K z ≤ κM|z|2, ∀ z ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, (K ◦ θ) is Lipschitz continuous, with Lipschitz constant Lκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (c) The source function satisfies f ∈ C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Note that these assumptions are consistent with the commonly used Brooks-Corey [32] and van Genuchten [33] parametrizations of the functions θ and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Time-discretization: Backward Euler To discretize the Richards equation in time we consider the backward-Euler time dis- cretization of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For this implicit scheme, no CFL conditions need to be satisfied for stability (thus avoiding restrictions on the time step size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, it does not require higher-order time regularity (unlike the Crank-Nicholson scheme) to converge to the time- continuous solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We subdivide the time-interval [0, T] uniformly N times with time step size τ = T/N and discrete time steps tn = τn, where n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, we look for a sequence {ψn}N n=1 of functions in Ω, satisfying the time-discrete system θ(ψn) − θ(ψn−1) τ − ∇ · [K(θ(ψn))∇(ψn + z))] = f(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (3) Denoting f(tn) by f n subsequently, a more precise and general definition of the weak solutions of (3) is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For simplicity, we assume homogeneous Dirichlet boundary condition although our results are valid for Dirichlet and Neumann boundary conditions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 (Backward Euler time-discretization of (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ψ0 ∈ L2(Ω) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then the sequence {ψn}N n=1 ⊂ H1 0(Ω) is the backward Euler solution of (1) if for all n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', N}, and v ∈ H1 0(Ω), 1 τ (θ(ψn) − θ(ψn−1), v) + (K(θ(ψn))∇(ψn + z), ∇v) = (f n, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Space-discretization: Continuous Galerkin finite elements We consider the finite element method to discretize (4) further in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let Th be a triangulation of Ω into closed d-simplices, where h := maxE∈Th (diam(E)) denotes the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Assuming Ω is a polygon, the Galerkin finite element space is Vh = � vh ∈ H1 0(Ω)| vh|E ∈ Pp(E), T ∈ Th � , (5) where Pp(E) denotes the space of p-order polynomials on E, p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, the fully discrete Galerkin formulation of Richards’ equation reads Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 (Fully discrete solution of (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ψ0 h := ψ0 ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then the sequence {ψn h}N n=1 ⊂ Vh is the fully discrete solution of (1) if for all n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', N}, and vh ∈ Vh, (θ(ψn h) − θ(ψn−1 h ), vh) + τ(K(θ(ψn h))∇(ψn h + z), ∇vh) = τ(f n, vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (6) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3 Iterative linearization schemes To obtain the solution of the nonlinear problem (6) an iterative linearization scheme is generally employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' To investigate the trade-off between the stability and speed of such schemes, we focus on two linearization strategies that will be representatives of linearly and quadratically convergent methods with convergence meant in the L2 sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Linearly convergent schemes: The L-scheme Where the quadratically convergent Newton method utilizes a proper first-order Taylor expansion of the nonlinear terms in (6), the linearly convergent methods that we consider here, only exploit an expansion of the monotone components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' the nonlinear saturation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, the expansion does not need to be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Consider the following scheme: Given ψn−1 h , ψn,j−1 h ∈ Vh, find ψn,j h ∈ Vh such that (L(ψn,j−1 h )(ψn,j h − ψn,j−1 h ), vh) + τ(K(θ(ψn,j−1 h ))∇(ψn,j h + z), ∇vh) = τ(f n, vh) − (θ(ψn,j−1 h ) − θ(ψn−1 h ), vh), (7) for all vh ∈ Vh, where L : R → [0, ∞) is a predetermined positive weight function, and j ∈ N is the iteration index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Observe that, provided κm > 0 in Assumption 1, the problem above is linear, monotone, and Lipschitz with respect to ψn,j h , and hence a unique weak solution of (7) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, if the iteration converges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' if ψn,j h → ψn h strongly in H1 0(Ω), then ψn h indeed solves (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' There can be many different choices of the function L which leads to different linearization schemes, see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the rest of this paper, we mainly focus on the case when L is constant which leads to the widely studied L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3 (L-scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ψn−1 h , ψn,0 h ∈ L2(Ω) and L > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then the L- scheme solves for the sequence {ψn,j h }j∈N ⊂ Vh which satisfies for all iteration indices j ∈ N, and vh ∈ Vh L((ψn,j h − ψn,j−1 h ), vh) + τ(K(θ(ψn,j−1 h ))∇(ψn,j h + z), ∇vh) = τ(f n, vh) − (θ(ψn,j−1 h ) − θ(ψn−1 h ), vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (8) Different choices of L and the resulting schemes are listed below Scheme L(ψ) Picard 0 Modified Picard [22] θ′(ψ) J¨ager-Kacˇur [29] supξ∈R θ(ξ)−θ(ψ) ξ−ψ L-scheme [23, 24, 1] L > 0 constant Modified L-scheme [19] θ′(ψ) + Mτ, M > 0 constant Table 1: Different linearly convergent schemes (7) defined along with their linearization weight function L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 1 (Non-constant L for heterogeneous media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the L-scheme, L might not necessarily be a constant, but can be a function of the spatial variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This would be typically the case for heterogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' All the proofs can be adapted to include a spatially dependent L, see [34] where this was done for a splitting scheme for Biot equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 5 It has been shown in [1, Theorem 1] that if L ≥ 1 2 supξ∈R θ′(ξ), then the L-scheme iterations converge irrespective of the initial guess under minor restrictions on the time step size τ and independent of the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, numerical results in [1, 19] reveal that the convergence of the L-scheme can be relatively slow, depending on the choice of the stabilization parameter L, see please the Appendix A for an adaptive L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' One can enhance the convergence speed by computing L using the previous iterates and derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In general, taking L as the Jacobian matrix, would lead to Newton method, this is the reason one can interpret the L-scheme also as a modified Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This is exploited in the modified Picard scheme, first proposed in [22], uses L(ψn,j−1) = θ′(ψn,j−1), complying with the first-order Taylor series expansion θ(ψn,j) ≈ θ(ψn,j−1) + θ′(ψn,j−1)(ψn,j − ψn,j−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As a result, if converging it requires fewer iterations compared to the L-scheme although the convergence is still linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nevertheless, this choice of the L function may lead to divergence of the scheme for larger time step sizes, as predicted in [10] and observed numerically in [1, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In an attempt to resolve this issue, a modified L- scheme was proposed in [19] that inherits the characteristics of both the L-scheme (except that it is using derivatives and the linear systems are not necessarily well conditioned) and the Picard scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The modified L-scheme exhibits increased stability compared to the Picard scheme while retaining its speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, the modified L-scheme converges unconditionally under the additional restriction that ψn,0 h = ψn−1 h and the discrete time- derivative (ψn h − ψn−1 h )/τ is in L∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Since the objective of this paper is to start the linearization iterations with a stable scheme, and then switch to a quadratically converging scheme when its convergence can be guaranteed, the rest of the study will be with respect to the L-scheme which is arguably the most stable among the schemes presented in Table 1 and the cheapest in terms of computing time per iteration (due to well-conditioned linear systems and not involving derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nonetheless, we remark that our methodology generalizes to all other linearly converging iterative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 2 (Generality of the results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Although the analysis of Section 3 primarily fo- cuses on the switching between L-scheme and the Newton method, the same techniques can be directly extended to cover switching between the schemes in Table 1 and Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, the L-adaptive strategy in Appendix A can be extended to the modified L-scheme (see Table 1) to select the parameter M > 0 adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Quadratically convergent scheme: The Newton method The Newton method uses the first order Taylor series expansions of all the nonlinear functions in (1) to ensure quadratic rates of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 (The Newton method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ψn−1 h , ψn,0 h ∈ L2(Ω) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then the Newton method solves for the sequence {ψn,j h }j∈N ⊂ Vh which satisfies for all iteration indices j ∈ N, and vh ∈ Vh (θ′(ψn,j−1 h )(ψn,j h − ψn,j−1 h ), vh) + τ(K(θ(ψn,j−1 h ))∇(ψn,j−1 h + z), ∇vh) + τ � (K ◦ θ)′(ψn,j−1 h )∇(ψn,j−1 h + z)(ψn,j h − ψn,j−1 h ), ∇vh � = τ(f n, vh) − (θ(ψn,j−1 h ) − θ(ψn−1 h ), vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (9) However, this comes at the cost of decreased numerical stability as discussed in Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In the next section we combine the L-scheme and the Newton method in a consistent manner in order to obtain a linerization strategy that is both stable and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 6 3 A posteriori estimate based adaptive switching be- tween L-scheme and Newton In this section, we develop the switching algorithm between L-scheme and the Newton method using a posteriori error analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For comparing the errors between different lin- earization schemes we introduce a uniform notion of linearization errors ηlin in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 based on arguments in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The idea behind the adaptive algorithm is to start with the L-scheme and derive an estimator ηL→N in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 that predicts from the jth and (j − 1)th iterate the linearization error for the next iteration if done using the Newton scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' If the error is predicted to decrease, then the iteration switches to Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then another estimator ηN→L is derived in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3 which predicts the linearization error of the next step of the Newton iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The algorithm switches back to the L-scheme in case the error is predicted to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In fact, we go one step further in Appendix A and derive an estimator ηL→L to predict if the L-scheme itself will converge and to tune the value of L accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Finally, the full algorithm is laid out in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 based on these estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' L-scheme Initial guess ηL→N < ηlin Newton ηN→L < ηlin NO YES YES NO Figure 1: Flowchart of Adaptive switching algortihm between L-scheme and Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Linearization errors and iteration-dependent energy norms In [31] it is shown that the total numerical error corresponding to a finite element-based linearization scheme can be orthogonally decomposed into a discretization component and a linearization component if the errors are computed using an iteration-dependent energy norm (for linearly convergent schemes in Table 1 this is just the energy norm invoked by the symmetric bilinear form associated with the unknown ψn,j h in (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, we are only interested in the linearization component which is defined as the difference between successive iterates in the aforementioned energy norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', ηj lin := ������ψn,j h − ψn,j−1 h ������ L,ψn,j−1 h , (10) where |||·|||L,ψn,j−1 h represents the particular H1 equivalent-norm defined using the iterate ψn,j−1 h and associated with the linearization scheme denoted by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The fully computable estimator ηj lin encapsulates the entirety of the linearization error, as shown in Section 5 of [31], and hence, will be used as its sole measure in the subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We mention explicitly the energy norms of the two schemes that are discussed: With reference to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3, the energy norm for L-scheme is defined as |||ξ|||L,ψn,j−1 h := �� Ω Lξ2 + τ ���K(θ(ψn,j−1 h )) 1 2∇ξ ��� 2� 1 2 (11) 7 for all ξ ∈ H1 0(Ω), and with reference to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 the norm for the Newton method is |||ξ|||N,ψn,j−1 h := �� Ω θ′(ψn,j−1 h ) ξ2 + τ|K(θ(ψn,j−1 h )) 1 2∇ξ|2 � 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 L-scheme to Newton switching estimate For some i ∈ N, let the sequence {ψn,j h }i j=1 ⊂ Vh be obtained using the L-scheme (8), and in the (i + 1)th-iteration we want to test for switching to the Newton scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ˜ψn,i+1 h ∈ Vh be the solution of the Newton scheme (9) having ψn,i h as the previous iterate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In this section, we will assume the following: Assumption 2 (Convection term is not dominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a given i ∈ N, there exists a constant Ci N ∈ [0, 2) such that τ|K(θ(ψn,i h ))− 1 2(K ◦ θ)′(ψn,i h )∇(ψn,i h + z)|2 ≤ (Ci N)2θ′(ψn,i h ), (13) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The assumption above is also required to show the coercivity of the linear problem (9) for j = i + 1, and hence, to show the existence of solution ˜ψn,i+1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Observe that, since ψn,i h is known, the constant Ci N is fully computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Additionally, it is smaller than 2 if the numerical flux is bounded, and τ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Notably, the estimate holds even in the degenerate case when θ′(ψn,i h ) = 0, since the left-hand side has (θ′(ψn,i h ))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' To cover the degenerate case, we also introduce the concept of an equilibrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 (Equilibrated flux σi L for degenerate regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a pre-determined ϵ > 0, let T i,ϵ deg := {K ∈ Th : inf θ′(ψn,i h ) < ϵ in K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let Πh : L2(Ω) → Pp(Th) be the Pp projection operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (Πhu, vh) = (u, vh) for all u ∈ L2(Ω) and vh ∈ Pp(Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, let RTp(Th) be the pth-order Raviart-Thomas space on Th, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', σ ∈ RTp(Th) implies σ|K ∈ (Pp(K))d + xPp(K) for all K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, we define σi L ∈ RTp(Th) ∩ H(div, Ω) as ∇ · σi L = � 1 τ Πh(L(ψn,i h − ψn,i−1 h ) − (θ(ψn,i h ) − θ(ψn,i−1 h ))) in T i,ϵ deg, 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (14) We defer to Appendix B for discussions on how to compute σi L in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Proposition 1 (Error control of L-scheme to Newton switching step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a given ψn,0 h , ψn−1 h ∈ Vh, let {ψn,j h }i j=1 ⊂ Vh solve (8) for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Let ˜ψn,i+1 h ∈ Vh be the solution of (9) with the previous iterate ψn,i h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Recall Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, under the Assumptions 1–2, one has ��� ��� ��� ˜ψn,i+1 h − ψn,i h ��� ��� ��� N,ψn,i h ≤ ηi L→N, where, ηi L→N := 2 2−Ci N �� ηi,poten L→N �2 + τ � ηi,flux L→N,2 �2� 1 2 8 with ηi,poten L→N := ���θ′(ψn,i h )− 1 2 � L � ψn,i h − ψn,i−1 h � − � θ(ψn,i h ) − θ(ψn,i−1 h ) ����� Th\\T i,ϵ deg , ηi,flux L→N := ���K(θ(ψn,i h ))− 1 2 �� K(θ(ψn,i h )) − K(θ(ψn,i−1 h )) � ∇ � ψn,i h + z � + σi L ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Observe from (9) that δψi+1 h := ˜ψn,i+1 h − ψn,i h ∈ Vh satisfies (θ′(ψn,i h )δψi+1 h , vh) + τ(K(θ(ψn,i h ))∇δψi+1 h , ∇vh) + τ � (K ◦ θ)′(ψn,i h )∇(ψn,i h + z) δψi+1 h , ∇vh � = τ(f n, vh) − (θ(ψn,i h ) − θ(ψn−1 h ), vh) − τ(K(θ(ψn,i h ))∇ψi h, ∇vh), (15) for all vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Inserting the test function vh = δψi+1 h in (15), one has ������δψi+1 h ������2 N,ψn,i h (12) = � Ω � θ′(ψn,i h )|δψi+1 h |2 + τ|K(θ(ψn,i h )) 1 2∇δψi+1 h |2� (15) = −τ � (K ◦ θ)′(ψn,i h )∇(ψn,i h + z) δψi+1 h , ∇δψi+1 h � � �� � =:T1 + τ(f n, δψi+1 h ) − (θ(ψn,i h ) − θ(ψn−1 h ), δψi+1 h ) − τ(K(θ(ψn,i h ))∇(ψn,i h + z), ∇δψi+1 h ) � �� � =:T2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (16a) Calling σi = (K ◦ θ)′(ψn,i h )∇(ψn,i h + z) for brevity, we estimate that T1 := −τ(σiδψi+1 h , ∇δψi+1 h ) ≤ � τ � Ω |K(θ(ψn,i h ))− 1 2σi|2(δψi+1 h )2 � 1 2 � τ � Ω |K(θ(ψn,i h )) 1 2∇δψi+1 h |2 � 1 2 (13) ≤ Ci N �� Ω θ′(ψn,i h )(δψi+1 h )2 � 1 2 � τ � Ω |K(θ(ψn,i h )) 1 2∇δψi+1 h |2 � 1 2 ≤ Ci N 2 � Ω � θ′(ψn,i h )|δψi+1 h |2 + τ|K(θ(ψn,i h )) 1 2∇δψi+1 h |2� = Ci N 2 ������δψi+1 h ������2 N,ψn,i h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (16b) For estimating the last term, we observe from the divergence theorem that − (σi L, ∇δψi+1 h ) = (∇ · σi L, δψi+1 h ) (17) = 1 τ (Πh(L(ψn,i h − ψn,i−1 h ) − (θ(ψn,i h ) − θ(ψn,i−1 h ))), δψi+1 h )T i,ϵ deg = 1 τ (L(ψn,i h − ψn,i−1 h ) − (θ(ψn,i h ) − θ(ψn,i−1 h )), δψi+1 h )T i,ϵ deg The last equality follows from the definition of the projection operator Πh and δψi+1 h ∈ 9 Vh ⊂ Pp(Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Using this result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' along with (8) and δψi+1 h ∈ Vh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' one has T2 := τ(f n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' δψi+1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn−1 h ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' δψi+1 h ) − τ(K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ))∇ψi h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ∇δψi+1 h ) (8) = (L(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' δψi+1 h ) − τ((K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )) − K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )))∇(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h + z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ∇δψi+1 h ) = (L(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' δψi+1 h ) + τ(σi L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ∇δψi+1 h ) − τ((K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )) − K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )))∇(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h + z) + σi L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ∇δψi+1 h ) = (L(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' δψi+1 h )Th\\T i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='ϵ deg − τ((K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )) − K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )))∇(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h + z) + σi L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ∇δψi+1 h ) (14) ≤ (θ′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )− 1 2(L(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' θ′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) 1 2δψi+1 h )Th\\T i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='ϵ deg + τ[ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='flux L→N ] ∥K(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) 1 2∇δψi+1 h ∥ ≤ [ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='poten L→N ] · ∥θ′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) 1 2δψi+1 h ∥ + √τ [ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='flux L→N ] · √τ∥K(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) 1 2∇δψi+1 h ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (16c) Combining (16), using the Cauchy-Schwarz inequality along with the definition of ηi L→N, one has the estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3 Newton to L-scheme switching estimate Assuming that the L-scheme converges unconditionally, after switching to Newton we would want to switch back to the L-scheme only if linearization error of the Newton scheme increases with iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Similar to before, we can estimate if this is going to happen in the (i + 1)th-step, purely from the iterates up to the ith-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For this purpose, we introduce another equilibrated flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 (Equilibrated flux σi L for degenerate regions (Newton scheme)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Recalling Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1, we define σi N ∈ RTp(Th) ∩ H(div, Ω) as ∇ · σi N = � 1 τ Πh(θ′(ψn,i h )(ψn,i h − ψn,i−1 h ) − (θ(ψn,i h ) − θ(ψn,i−1 h ))) in T i,ϵ deg, 0 otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (17) The corresponding result mirroring Proposition 1 is Proposition 2 (Error control of Newton to Newton step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a given ψn,0 h , ψn−1 h ∈ Vh, let {ψn,j h }i+1 j=1 ⊂ Vh solve (9) for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' under Assumptions 1–2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' one has ������ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i+1 h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ������ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ≤ ηi N→L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' where ηi N→L := 2 (2−Ci N) � [ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='poten N→L ]2 + τ[ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='flux N→L ]2� 1 2 with ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='poten N→L := ∥θ′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )− 1 2(θ′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h ) − (θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ) − θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )))∥Th\\T i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='ϵ deg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='flux N→L := ������ � (K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h )) − K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )))∇(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h + z) −(K ◦ θ)′(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h − ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h )∇(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i−1 h + z)) � K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ))− 1 2 +K(θ(ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='i h ))− 1 2σi N ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 10 The proof is identical to the proof of Proposition 1 and hence is left for the avid reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 3 (Effectivity of the estimators ηi L→N and ηi N→L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The estimators ηi L→N and ηi N→L predict the linearization error ηi+1 lin of the (i + 1)th iteration if done using the Newton scheme (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In the cases where the iteration is done indeed using the Newton scheme, the sharpness of the estimate can be measured using the effectivity index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', if (i + 1)th iteration is Newton then (Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' )i := � ηi L→N/ηi+1 lin if ith iteration is L-scheme, ηi N→L/ηi+1 lin if ith iteration is Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (18) Observe that it is always greater than 1 due to Propositions 1 and 2 and an effectivity index close to 1 implies a sharp estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The estimators are expected to be quite accurate since mainly the Cauchy-Schwarz inequality is used to derive them, except for estimate (16b) where the term T1 is bounded above using the global approximation in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This expected sharpness is shown to be the case through the numerical experiments of Section 4, see in particular Figures 5 and 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 A-posteriori estimate based adaptive linearization algorithm With the above estimates in mind, we propose a switching algorithm between the L- scheme and the Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The linearization scheme used at iteration j = i + 1 should be Newton if the linearization error, predicted by the estimators ηi L→N and ηi N→L, is smaller than the linearization error ηi lin of the ith step, see (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, to optimize the algorithm we take a few numerical considerations into account first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Computational considerations To speed up the computations of this switching criteria, we make a few more reductions [Equilibrated flux] If the saturated domain is much smaller than the unsaturated domain, then we take σi L = σi N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' [Switching condition] The condition ηi L→N ≤ ηi lin might be difficult to satisfy if the estimators are not sharp (see Remark 3), and even when it is satisfied it might require large values of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Hence, to expedite the switching between L-scheme and Newton, we will use the criteria ηi L→N < Ctol ηi lin for a constant Ctol > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Adaptive linearization algorithm Under these considerations we propose the following adaptive algorithm: 11 Algorithm 1 L-scheme/Newton a-posteriori switching Require: ψn,0 ∈ L2(Ω) as initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Ensure: Scheme= L-scheme , Ctol = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 for i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='. do if Scheme= L-scheme then Compute iterate using L-scheme , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', (8) if Ci N ≥ 2 then continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' else if ηi L→N ≤ Ctolηi lin then Set Scheme= Newton else Compute iterate using Newton , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', (9) if ηi N→L > ηi lin then Set Scheme= L-scheme Remark 4 (Combining L-scheme adaptivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Appendix A, we further propose an algorithm to adaptively select L in order to expedite the convergence of the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This can directly be implemented in conjunction to Algorithm 1 to improve the convergence speed of the composite scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nevertheless, we have refrained from combining these schemes for the ease of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 5 (Computational cost of the estimators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In the non-degenerate case, the quan- tities Ci N, ηi L→N and ηi N→L, can be directly computed from the iterates ψn,i h and ψn,i−1 h by inserting σi L = σi N = 0, see Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Hence, the cost of computing the estimators is small in comparison to the cost of the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Since the L-scheme iter- ations are less expensive than the Newton iterations, the L/N scheme generally performs better or similarly to the Newton scheme time-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This is evident from the numerical experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' see Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In the degenerate case, global computation are required for computing σi L and σi N if they are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We discuss the computation of these equili- brated fluxes in Appendix B and their computation can be made relatively inexpensive by precomputing the associated stiffness matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The computational cost for the estimators can be reduced even further by evaluating them only for selected iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Nevertheless, we do not pursue this option for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4 Numerical results In this section, we perform several numerical examples that demonstrate the robustness and efficiency of the proposed algorithm for switching between Newton’s method and the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This is done through careful comparison between the switching algorithm, hereafter called the L/N-scheme, the standard Newton method and the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It is important to note that the L-scheme includes a tuning parameter that significantly affects the performance of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As a remedy, we choose two different values, L1 and L2 in the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, L1 is a quasi-optimal choice of tuning parameter and will be defined for each specific subproblem, see Table 2, and L2 = sup {θ′ (ψ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the L/N-scheme, L1 is always chosen for the L-scheme iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' To measure the performance of each separate method, we examine both the number of iterations and computational time that they require to satisfy the stopping criterion ������ψn,j h − ψn,j−1 h ������ L,ψn,j−1 h < 10−7, 12 where |||·|||L,ψn,j−1 h is the iteration and linearization-dependent energy norm for the pressure head, with L ∈ {L, N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, the computational time covers the entire simulations and all experiments were performed on an Acer Swift 3, with an Intel core i7-1165G7-processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In total, three different test cases for the numerical experiments are considered: Test case 1: The first test case is taken from [35], although it is modified in the sense that we disregard surfactant transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, the flow is always partially saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Test case 2: The second test case can be found in [1], and it considers extrac- tion/injection above the water table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Test case 3: The final test case is a known benchmark problem that is studied in [1, 36, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, a time-dependent Dirichlet boundary condition is used to describe the recharge of a groundwater reservoir from a drainage trench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For all test cases, the van Genuchten-Mualem parametrization [33] is used to describe the relation between the saturation, the pressure head and the permeability, θ(ψ) = � � � θR + (θS − θR) � 1 1+(−αψ)n � n−1 n , ψ ≤ 0, θS, ψ > 0, K(Θ(ψ)) = � � � � � Ks (Θ(ψ)) 1 2 � 1 − � 1 − Θ(ψ) n n−1 � n−1 n �2 , ψ ≤ 0, Ks, ψ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (19) Here, Θ(ψ) = θ(ψ) − θR θS − θR , with θS and θR being the water volume and the residual water content respectively, Ks the hydraulic conductivity of the fully saturated porous medium, and α and n soil related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In all of the test-cases, triangular linear conforming finite elements with mesh diameter h are applied together with the implicit Euler time-discretization with time step size τ, as described in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The mesh diameter h and time step size τ vary between the different experiments and will be specified for each individual experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We note that the numerical experiments are expected to perform equivalently for other spatial discretization methods such as the Raviart-Thomas mixed finite elements or discontinuous Galerkin finite elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The finite element implementation is Python based and uses the simulation toolbox PorePy [39] for grid management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It is available for download at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='com/ MrShuffle/RichardsEquation/releases/tag/v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 13 Parameters Test case 1 Test case 2 Test case 3 van Genuchten-Mualem θR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='131 θS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='396 KS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='96 · 10−2 α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='423 n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='06 L-scheme L1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='501·10−3 L2 = Lθ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='501·10−3 Table 2: Parameter values for all test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The parameters are presented in column format, where each column corresponds to the parameters for the specified test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Test case 1: Strictly unsaturated medium In this test case, we consider a strictly unsaturated porous medium, and use the van Genuchten-Mualem parametrization that is described by parameters from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The test case is heavily inspired by [35], and the domain is given by Ω = Ω1 ∪ Ω2, where Ω1 = [0, 1] × [0, 1/4] and Ω2 = [0, 1] × (1/4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We consider the time interval [0, T], where T = τ varies with choice of time step size τ, as we only take one time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As initial condition, we choose the pressure head ψ0(x, z) = � −z − 1/4 (x, z) ∈ Ω1 −4 (x, z) ∈ Ω2, where x represents the positional variable in the horizontal direction and z in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A Dirichlet boundary condition is imposed at the top boundary that complies with the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the rest of the boundary no-flow boundary conditions are used, and the following source term is applied f(x, z) = � 0 (x, z) ∈ Ω1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='06 cos � 4 3π(z) � sin (x) (x, z) ∈ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The solution after one time step with time step size τ = 1, is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 2: Test case 1: Strictly unsaturated medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Pressure head ψ at final time T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Comparison of convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, we discuss the performance and convergence properties of the newly proposed L/N- scheme and compare it to the Newton method and the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Figure 3a, the number of iterations for different choices of the mesh size parameters, with time step size τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As expected the L-scheme is robust and converges in each scenario, for both L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Newton’s method, however, only converges for sufficiently coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Yet, when converging, it converges in fewer iterations than the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Finally, the hybrid L/N method converges in as few if not fewer iterations as the Newton method (when it converges) and converges robustly, and in far fewer iterations than the L-scheme for the other mesh sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Furthermore, a similar experiment is performed for a fixed mesh size h = √ 2/40, and varying time step sizes, see Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For larger time step sizes the Newton method diverges, while the other methods converge robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Again the L/N-scheme converges with the performance expected of Newton’s method, in addition to being as robust as the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We highlight the enormous difference in the number of iterations for the largest time step size τ = 1 in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 0 20 40 60 80 0 10 20 30 40 24 24 25 25 25 26 26 26 33 34 35 35 35 35 35 36 (3/8) (3/7) (2/7) (2/6) (1/7) (1/5) (1/4) (1/4) 8 7 5 5 √ 2/h Number of Iterations (a) Total number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The numbers in the red parentheses correspond to (number of L-scheme iterations/number of Newton it- erations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 20 30 40 50 60 70 80 0 500 1,000 1,500 332 442 577 772 991 1230 469 599 832 1032 1373 1666 518 373 275 193 153 91 104 151 √ 2/h CPU time [s] L1 L2 Newton L/N (b) Computational time in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 3: Test case 1: Strictly unsaturated medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Performance metrics for all lineariza- tion schemes for fixed τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 and varying mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then, the performance of the linearization schemes is compared in terms of computa- tional time, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 3b and Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' One can observe virtually the same performance for the hybrid method as for Newton’s method when the latter converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The former in fact is sometimes slightly faster, due to each L-scheme iteration being slightly less expen- sive than a Newton iteration, see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In addition, the hybrid method continues to show the same performance for the cases in which Newton’s method does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Finally, Figure 3b shows that, for all meshes, the computational time of the L-schemes is consistent with the reported numbers of iterations in Figure 3a with L1 being the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Although it uses more than double the computational time of the L/N-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Overall, the newly proposed L/N-scheme shows the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It is as fast as Newton’s method when it converges, and is significantly more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 1 0 10 20 30 40 55 84 114 (1/7) (1/7) (1/7) (3/8) (3/8) (3/8) (1/7) (1/7) (1/7) (1/4) (1/4) (1/4) τ Number of iterations L/N Newton L1 L2 (a) Number of iterations for different time step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 1 0 500 1,000 1,500 2,000 τ CPU time [s] L/N Newton L1 L2 (b) Total computational time in seconds for dif- ferent time step sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 4: Test case 1: Strictly unsaturated medium: Performance comparison for all of the linearization schemes for different time step sizes and fixed mesh size h = √ 2/40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 6 (Computational time per iteration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' It is known that condition numbers for matrices coming from systems linearized by Newton’s method are higher than for those linearized by the L-scheme [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Therefore, each iteration of Newton’s method, when im- plemented without preconditioning, takes more time than each L-scheme iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Remark 7 (Computational time for the coarsest mesh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The computational times of the coarsest meshes are omitted due to the use of multiprocessing in the implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' This causes the most time consuming part to be the spawn process of the local assembly on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As a result, the computational times for the coarsest meshes are very similar for all the linearization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Switching characteristics Finally, the dynamic switch between the L-scheme and Newton’s method is inspected in further detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Figure 5, the evolution of the indicators for the switch is displayed for a fixed mesh and time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The example particularly demonstrates the ability of the hybrid method to switch back and forth between both linearizations before switching fully to Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In addition, the final number of L-scheme iterations is kept at its minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The plot also shows the effectivity indices introduced in (18) and discussed in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The effectivity index is greater than 1 in all cases, which validates Propositions 1 and 2 and it stays between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='27 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3, implying that the estimators ηi L→N and ηi N→L are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Test case 2: Variably saturated medium The example parameters are as in Table 2, Test case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We consider a variably saturated medium, Ω = Ωgw ∪ Ωvad, where the groundwater zone is Ωgw = [0, 1] × [0, 1/4) and a vadoze zone is Ωvad = [0, 1] × [1/4, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, we consider the time interval [0, T], where T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 and we only take one time step with τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' As initial condition, we choose 16 0 2 4 6 8 10 12 10−5 10−4 10−3 10−2 10−1 100 101 Iteration number ηi N→L/ηi lin ηi L→N/ηi lin (a) Evolution of switching indicators for L/N- scheme where the dashed line is Ctol = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The L/N-scheme oscillates between the lin- earization strategies, but eventually recovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 0 2 4 6 8 10 12 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 Iteration number (Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' )i (b) Efficency index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Notice that the iterations correspond to the ones in Figure 5(a), and that only the ones where the Newton method is performed are counted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', iteration 1,3 and 5 are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 5: Test case 1: Strictly unsaturated medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Evolution of switching indicators for the L/N-scheme and efficiency indices (18) for the Newton iterations (see Remark 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Here, the mesh size is h = √ 2/80 and time step size τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' the pressure head ψ0(x, z) = � −z + 1/4 (x, z) ∈ Ωgw −3 (x, z) ∈ Ωvad, where x represents the positional variable in the horizontal direction and z in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' On the surface a constant Dirichlet boundary condition is imposed, being equal to the initial condition at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For the rest of the boundary no-flow boundary conditions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' We apply the following source term f(x, z) = � 0 (x, z) ∈ Ωgw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='006 cos � 4 3π(z − 1) � sin (2πx) (x, z) ∈ Ωvad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' After one time step the pressure head profile is given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 6: Test case 2: Variably saturated medium: Pressure head profile at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content="0 t0'E- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8 100 20 40 60 80 0 20 40 60 (4/6) (5/5) (5/5) (8/3) (6/3) (8/4) (5/4) (43) (39) (38) (36) (36) (35) (35) (43) (43) (58) (57) (55) (56) (54) (51) (66) (67) √ 2/h Number of iterations (a) Total number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The numbers in the red parentheses correspond to (number of L-scheme iterations/number of Newton itera- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 20 30 40 50 60 70 80 0 500 1,000 1,500 2,000 2,500 (484) (395) (304) (273) (173) (169) (1808) (1442) (1058) (828) (616) (477) (2676) (2141) (1614) (1269) (939) (700) √ 2/h CPU time [s] L1 L2 L/N (b) Computational time in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 7: Test case 2: Variably saturated medium: Performance metrics for all lineariza- tion schemes for fixed τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01 and varying mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Comparison of convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The iteration count for the second test case for different mesh sizes and fixed time step for all linearization schemes is illustrated in Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Again the L-scheme converges in every case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, Newton’s method does not converge for any mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The hybrid method needs the fewest number of iterations, which shows that the dynamic switch is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The CPU time performance of the linearization schemes is compared in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Both versions of the L-scheme takes computational times consistent with the number of iterations, with the simulations with the parameter L1 being less expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, the L-scheme (using L1) requires approximately 373% of the computational time of the hybrid method including the computation of the switching indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In addition, the benefit of a few additional L-scheme iterations further decreases the computational time of the hybrid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Switching characteristics We also give a more in-depth look to the dynamic switch between the Newton’s method and the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Figure 8, the evolution of the switching indicators is shown for a fixed time step and a fixed mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' After 8 L-scheme iterations the switching indicator ηL→N becomes lower than Ctol and then Newton’s method converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' From Figure 7a the number of L-scheme iterations required before the switching indicator becomes small enough to switch to Newton’s method varies with the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Note that for the coarsest mesh no switch to Newton’s method happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 18 0 2 4 6 8 10 12 10−8 10−6 10−4 10−2 100 102 Iteration number ηi N→L/ηi lin ηi L→N/ηi lin (Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' )i Figure 8: Test case 2: Variably saturated medium: Evolution of switching indicators for L/N-scheme for fixed h = √ 2/50 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The dashed line is Ctol = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5, the switching criterion from L-scheme to Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The effectivity indices (18) corresponding to the Newton iterations are also plotted and they remain below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3 Test case 3: Benchmark problem Here, we consider a known benchmark problem [38], also used e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' in [1], which models the recharge of a groundwater reservoir from a drainage trench in two spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The domain Ω ⊂ R2 represents a vertical segment of the subsurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' One portion of the right side of the domain is fixed by a constant Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A time-dependent Dirichlet boundary condition on parts of the upper boundary is used to mimic the drainage trench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' No-flow conditions are utilized on the remaining parts of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The used parameters are given in Table 2 Test case 3, corresponding to silt loam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The geometry is given by Ω = [0, 2] × [0, 3], ΓD1 = [0, 1] × (3), ΓD2 = (2) × [0, 1], ΓN = Ω\\ {ΓD1 ∪ ΓD2} , and the initial pressure head distribution and boundary conditions are ψ(0, x, z) = 1 − z ψ(t, x, z) = � � � � � −2 + 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2t, if t ≤ 1 16, on ΓD1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2, if t > 1 16, on ΓD1, 1 − z, on ΓD2, − K(θ(ψ(t, x, z)))∇(ψ(t, x, z) + z) · ν = 0, on ΓN, where ν is the outward normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The solution is computed over 9 timesteps, where the time unit is in days, with time step size τ = 1/48 and with a regular mesh consisting 19 of 2501 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The pressure head profile at the final time for the L/N-scheme is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Figure 9: Test case 3: Benchmark problem: Pressure head profile at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Itr CPU time [s] L1 274 6136 L2 330 7356 Newton 39 980 L/N (10/30) 1021 Table 3: Test case 3: Benchmark problem: Performance metrics for 2501 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 Comparison of convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The performance of all schemes for test case 3 is displayed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' All schemes converge for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The Newton method requires the least amount of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, the hybrid method only needs one more iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Both uses significantly less iterations than the L-schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For all time steps except one, only one L-scheme iteration is needed per time step, which indicates a successful dynamic switch for almost all time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The computational time for the L-schemes is much higher than both Newton’s method and the hybrid method, which is consistent with the expense per iteration discussed in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' More significantly, the L/N-scheme performs almost the same as Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 5 Conclusions In this paper, we considered solving Richards’ equation, which models the flow of water through saturated/unsaturated porous media (soil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' After applying backward Euler time- discretization and continuous Galerkin finite element space-discretization to Richards’ equation, to solve the resulting nonlinear finite-dimensional problem we developed a hy- brid iterative linearization strategy that combines the L-scheme with the Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='64 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='0The idea behind this is to use the robust, but only first-order convergent L-scheme to stabilize the quadratically convergent Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The switching between the two schemes is done in an adaptive manner using a posteriori indicators which predict the linearization error of the next iteration using a concept of iteration-dependent energy norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' After each iteration, it is checked whether the Newton method is predicted to decrease the linearization error of the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' If so, then the Newton method is used, otherwise, the iteration is done using the L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The hybrid scheme is now robust, but still quadratically convergent after switching to the Newton scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The performance of the hybrid scheme is tested on illustrative, realistic numerical examples which reveal that the scheme is as robust as the L-scheme and it converges in cases where Newton fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Moreover, in cases when Newton converges, the hybrid scheme takes roughly the same amount of iterations and computational time and is considerably faster than even the optimized L-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Lastly, we comment that the scheme is quite general as it can, in principle, be extended to other spatial discretization and linearization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Appendix A An adaptive L-scheme As discussed in Sections 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1, the L-scheme converges unconditionally provided that L ≥ 1 2 supξ∈R θ′(ξ) and the time step size τ is smaller than a constant independent of the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, numerical results in [1] suggest that the optimal rate of convergence of the L-scheme is obtained for a considerably smaller L although convergence cannot always be guaranteed for such values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Hence, to speed up the computations, it is possible to start the iterations with a smaller value of L and then use the a posteriori estimates to decide if L is to be increased or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Analogous to Propositions 1 and 2 we state a result that allows us to do this rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Proposition 3 (Error control of L-scheme).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For a given ψn,0 h , ψn−1 h ∈ Vh, let {ψn,j h }i+1 j=1 ⊂ Vh solve (8) for some i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Then under Assumption 1, ������ψn,i+1 h − ψn,i h ������ L,ψn,i h ≤ ηi L→L, where ηi L→L := � [ηi,poten L→L ]2 + τ[ηi,flux L→L ]2� 1 2 with ηi,poten L→L := ∥L− 1 2(L(ψn,i h − ψn,i−1 h ) − (θ(ψn,i h ) − θ(ψn,i−1 h )))∥, ηi,flux L→L := ���(K(θ(ψn,i h )) − K(θ(ψn,i−1 h )))K(θ(ψn,i h ))− 1 2∇(ψn,i h + z) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The detailed proof is again omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Observe that for the estimate above, neither Assumption 2 nor any separate treatment of the degenerate domains is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 L-adaptive algorithm Based on Proposition 3, we propose an algorithm that selects optimal L-values adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' 21 Algorithm 2 The L-adaptive scheme Require: ψn,0 ∈ L2(Ω) as initial guess, LM := supψ∈R θ′(ψ), and Lm := LM/8 Ensure: CL→L = √ 2, L = Lm for i=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='. do Compute iterate using L-scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=', (8) if ηi L→L > ηi lin then Replace Lm = L, L = min(CL→LL, LM), and continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' else if ηj L→L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='8 ηj lin for j ∈ {i, i − 1, i − 2} then Replace L = max(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='9L, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1Lm) and continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2 Numerical result 0 10 20 30 40 50 60 10−1 100 101 Iteration number ηi L→L/ηi lin Figure 10: Test case 1: Strictly unsaturated medium: L-scheme with L-adaptivity and initial stabilization parameter L0 = L2/8, h = √ 2/40 and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In Figure 10 we show a result where the L-adaptive scheme is superior to a fixed L- approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' In this case, Lθ/2 is too small for convergence due to a large time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Compared with fixed L1 with the same mesh size and time step size, see Figure 4, the number of iterations is improved by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For smaller time steps, the numerical results reveal that Algorithm 2 results in roughly the same number of iterations compared to a fixed and optimized L = L1 lesser than Lθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' But in all examples considered, it uses fewer iterations than simply choosing L = L2 = Lθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' The advantage of such an adaptive technique is that an optimization study of L does not need to be conducted prior to the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' However, since the L-adaptive strategy does not significantly improve the behavior of the L-scheme over the optimized L = L1, we refrained from including it in Algorithm 1 for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Appendix B Computation of equilibrated flux Recalling Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content='2, let us propose a simple algorithm to compute an equilibrated flux σh ∈ RTp(Th) ∩ H(div, Ω) satisfying ∇ · σh = Πhf in T i,ϵ deg, and ∇ · σh = 0 otherwise, where f ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Defining Qh := RTp(Th) ∩ H(div, Ω) and 22 ˜Vh := {vh ∈ Pp(Th)| Tr∂Ω(vh) = 0}, we seek a pair (σh, rh) ∈ Qh × ˜Vh that satisfies the mixed finite element problem, (K(1)−1σh, qh) = (rh, ∇ · qh), ∀ qh ∈ Qh, (20a) (∇ · σh, vh) = (f, vh), ∀ vh ∈ ˜Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' (20b) The advantage of this flux is that it minimizes ∥K(1)− 1 2σh∥ which appears in the estimates in Propositions 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' For practical purposes, a much coarser mesh can be used outside of T i,ϵ deg to compute it, and the stiffness matrix can be precomputed to accelerate the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Acknowledgements The work of JWB is funded in part through the Center of Sustainable Subsurface Re- sources (Norwegian Research Council project 331841) and the ‘FracFlow’ project funded by Equinor, Norway through Akademiaavtalen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' KM acknowledges the support of FWO (Fonds Wetenschappelijk Onderzoek) for funding him through the ‘Junior Postdoctoral Fellowship’ and to Akademiaavtalen for funding his visit to the University of Bergen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' List, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddA0T4oBgHgl3EQfG_-E/content/2301.02055v1.pdf'} +page_content=' Radu, A study on 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0000000000000000000000000000000000000000..7a9ac67787349f914d1999a71beb80dfa453def2 --- /dev/null +++ b/e9A0T4oBgHgl3EQfHf82/content/tmp_files/2301.02061v1.pdf.txt @@ -0,0 +1,1722 @@ +arXiv:2301.02061v1 [math.OC] 5 Jan 2023 +Distributed Control Strategy for Layered Barrier Coverage of +Multi-Agent Systems in Uncertain Environments +Pengyang Fan, Chao Zhai ∗ +August 2022 +Abstract +This paper presents a distributed multi-layer ring barrier coverage algorithm. In order to +achieve single-layer ring barrier coverage, a distributed single-layer ring barrier coverage al- +gorithm that maximises the probability of monitoring is proposed. Considering the security +risks of single-layer barrier coverage, a distributed adjustment mechanism between multiple +layers of barriers is designed and combined with the single-layer ring barrier coverage algo- +rithm to propose a distributed multi-layer ring barrier coverage algorithm. Furthermore, we +present a theoretical analysis of the proposed algorithm to demonstrate its effectiveness and +necessity. Finally, our algorithm is verified by numerical simulation and experiment. +1 +Introduction +Multi-agent systems(MASs) are composed of agents that interact with each other in an environ- +ment. Each agent is a system, MASs are systems in which a large number of agents are grouped +together and realise an overall behaviour or activity. Agents can be natural creatures[1], ar- +tificial robots or mobile sensors[2]. The MASs aims to take a distributed approach to solve +some large and complex problems. Each agent is an independent individual that can perceive +the environment, process information, communicate, learn, and make decisions independently. +Since each agent adopts an independent strategy, coordinated control of multi-agent systems is +∗Pengyang Fan and Chao Zhai are with School of Automation, China University of Geosciences, Wuhan 430074 +China, and with Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems +and Engineering Research Center of Intelligent Technology for Geo-exploration, Ministry of Education, Wuhan +430074 China. Corresponding author: Chao Zhai (email: zhaichao@amss.ac.cn). +1 + +essential in order for the whole system to accomplish a common goal, and this area has attracted +many scholars to conduct research. +Multi-agent coverage control is a hot research topic in multi-agent coordination control. +Multi-agent coverage control refers to a group of agent bodies with mobile, communication, +computing, and learning capabilities to sense the environment and perform a given task in a +distributed manner in a given or indefinite area, such as: search and rescue, missile interception, +monitoring, sweeping, etc[5]. Multi-agent coverage control can be classified into area coverage, +sweeping coverage and barrier coverage according to the area covered by the agent. Area coverage +is a series of operations in which each agent body determines its optimal state in the area +through communication, computation, and coordination and achieves this state through some +control science methods. +The most classic one is the multi-agent coverage algorithm based +on Voronoi partition[3], in which each agent divides a convex region into sub-regions through +communication, and each agent uses a strategy of moving to the center of mass of the sub-region +to maximize the coverage quality. This method can cover a convex region to the maximum +extent. On this basis, many scholars have found many problems and proposed some solutions. +For example, to solve the non-convex region, some scholars proposed the Voronoi center-of-mass +coverage algorithm for non-convex region based on the geodesic Voronoi partition algorithm[4]; +to solve the time-varying density function problem, some scholars proposed the Voronoi center- +of-mass coverage in dynamic environment based on the control barrier function[6]; to solve +the coverage problem in uncertain environment, some scholars proposed the Voronoi center-of- +mass coverage in uncertain environment based on the Bayesian estimation[7]. Sweep coverage +not only requires the agent to reach the designated area but also requires agent to be able to +traverse the entire area to achieve cleaning of the environment. For example, some scholars have +achieved equal-task sweep coverage of a class of regions based on equal-task partitioning methods, +which can improve the overall efficiency[8]. Some scholars have also proposed a multi-agent +sweeping coverage algorithm based on the temperature field approach[9]. Moreover, literature +[10] combines Voronoi segmentation with a temperature field approach to design a distributed +overlay method that enables each agent to have the same workload. +Multi-agent barrier coverage refers to the coverage of a group of agents on a line, which is +usually used to monitor whether a creature or object crosses the line or to intercept objects that +attempt to cross on the line. The literature [11] proposes the definition of barrier coverage and +k-barrier coverage. The k-barrier coverage is further divided into weak k-barrier coverage and +strong k-barrier coverage[11][12][13]. The strong k-barrier coverage means that an intruder is +2 + +detected by at least k agents regardless of any path into or through the target area. Besides, +Chen et al. proposed the concept of local barrier coverage, which can reduce the number of agents +compared to global fence overlays and can also be used for general cases[14]. However, local +barrier coverage is a security risk, as intruders can potentially traverse the area without being +detected. +There are currently many algorithms for barrier coverage and k-barrier coverage. +For example, the coverage-based approach proposed in literature [19] can equip the task of +assigning intrusion probability to intercept the intruded items thus achieving protection of the +target. In addition to this, scholars have designed a distributed algorithm that can achieve a +uniform barrier coverage between two landmarks[15]. Ban et al. investigate the strong k-barrier +coverage problem of mobile sensor networks over open belt using a grid-based approach in [16]. +However, the algorithm can only achieve coverage on a straight line between two points, and +cannot achieve coverage on a curve, nor can it achieve coverage on a closed curve. In practical +applications, if the targets in the area need to be protected or monitored in an all-round way, +the whole boundary of the area needs to be covered. For an enclosed target area, the agents also +needs to be covered within the closed belt. For this reason, Binay et al. designed the algorithm +to move the smart body to the boundary of a simple polygon and thus protect the area inside +the polygon[17]. Moreover, a barrier coverage algorithm has been designed on a circle, and the +agent can be uniformly covered on the circle with limited communication[18]. But a circle is +a kind of convex region, and how to perform barrier coverage on the boundary of non-convex +regions is the inspiration of our research. Moreover, covering only the boundary means that as +soon as the intruder breaks through this layer, the intruder enters the area we need to protect +and the system loses the means to monitor the intruder, which means an increase in security +risks. From a security point of view, a k-barrier coverage is more secure than a single layer +barrier coverage. +To this end, we first designed algorithms that can perform barrier coverage on the boundary +of a class of non-convex regions. The algorithm is applied in the context of monitoring intruders, +and uses a region partition to assign a region to each agent for monitoring, which can eventually +lead to a local maximum monitoring probability. Therefore, we want to design a multi-agent +control algorithm with multi-layer barrier coverage to solve this problem. When an intruder +breaks through a layer, there are still several internal monitoring layers that can continue to +monitor the intruder. +The goal of this paper is to design a distributed multi-agent barrier +coverage algorithm that can implement a multi-layer barrier coverage and can autonomously +adjust the number of agents on each layer to optimize the monitoring quality of the whole +3 + +system. The contributions of this paper are as follows. +1. Design a multi-agent barrier coverage algorithm for a class of non-convex areas which can +maximize intruder monitoring. +2. Develop a distributed adjustment mechanism for the number of agents per layer, which +can optimize the monitoring probability of multi-agent systems. +3. Combining the single-layer fence coverage algorithm and the distributed adjustment mech- +anism of the number of multi-layer agents, we propose the multi-layer barrier coverage +algorithm. +The remainder of this paper is structured as follows: Section 2 presents a single-layer barrier +coverage algorithm for non-convex region boundaries at first and then provides a distributed +adjustment mechanism for the number of agents in a multi-layer coverage region and a multi- +layer barrier coverage algorithm. +Section 3 presents a theoretical verification of the single- +layer barrier coverage algorithm and the multi-layer barrier coverage algorithm proposed in +Section 2 and gives the case when our algorithm is applied to a circle. Section 4 simulates our +algorithm and performs experimental validation on Robotarium. Finally, we conclude the paper +in Section 5. +2 +Problem Formulation +In this section, we will introduce the distributed multi-agent barrier coverage algorithm. Con- +sider a closed curve region D, which can be represented by polar coordinates, the center of the +circle D is denoted by O. Without loss of generality, we can set the center of the circle as the +origin, i.e. O = (0, 0). The boundary of the circle area D is denoted by ∂D. The radius of the +closed curve region is denoted by R(θ), θ ∈ [0, 2π). +2.1 +Single layer barrier coverage +In the application of monitoring intruder, multiple layer barrier coverage is more effective than +single layer barrier coverage. Therefore, we propose a multiple layer barrier coverage algorithm +in this paper. Since this algorithm is based on single layer barrier coverage algorithm, we firstly +introduce the single layer barrier coverage algorithm in this subsection. +In layer k, there are Nk mobile agents that can communicate and monitor. The layer k +can be denoted by Rk(θ), θ ∈ [0, 2π). We assume that agents can all communicate with each +4 + +other if they are on the same layer. We use INk to denote the number of agents on this layer, +INk = {1, 2, ..., Nk}. +We use ρ(θ) to denote the probability that each point on the layer is +invaded by an intruder. We denote the position of agents by P = {p1, p2, ..., pNk}. We specify +an angle for agent ik with respect to the center of the circle O, denoted by ϕik, ik = 1, 2, ..., Nk. +We denote the probabilistic model that the agent ik detects an intruder by f(d(ϕik, θ)), where +d(ϕik, θ) is a distance function about ϕik and θ, and the distance function is Lipschitz continuous, +and d (ϕik, θ) ∝ δ(ϕik, θ). The function δ(ϕik, θ) is denoted by +δ(ϕik, θ) = + + + + + + + +|ϕik − θ − 2π| +if +ϕik − θ > π +|ϕik − θ + 2π| +if +ϕik − θ ≤ −π +|ϕik − θ| +else +(1) +Moreover, f(d(ϕik, θ)) need to meet the following conditions: +1. f(d(ϕik, θ)) is differentiable. +2. f(d(ϕik, θ)) is monotonically decreasing. +Now we give the calculation way of ϕik as follows +ϕik(t) = Ψ(pT +ik(t)). +(2) +where Ψ(x, y) is an operation specified by us, which is calculated as follows +Ψ(x, y) = + + + + + + + + + + + + + + + + + + + + + + + + + +arctan( y +x) + π +x < 0 +arctan( y +x) +y ⩾ 0 +x > 0 +arctan( y +x) + 2π +y < 0 +x > 0 +π +2 +y > 0 +x = 0 +− π +2 +y < 0 +x = 0 +0 +y = 0 +x = 0 +(3) +Combining (2) and (3), it is easy to know that ϕik ∈ [0, 2π), for ik = 1, 2, ..., Nk. Moreover, +agents have the following numbering rules +0 ≤ ϕ1 (0) < ϕ2 (0) < ... < ϕNk (0) < 2π +As the distance increases, the detection probability of the agent will decrease. Therefore, we +stipulate that the agent only detects the area in which it is responsible. Therefore, we propose a +5 + +Agent +Division point +Origin point +i +is +1 +is � +js +1 +js � +j +O +i� +j +� +x +D +Figure 1: +Coverage area D, agent communication radius and monitoring radius, and the color +of the pentagram indicates the working state of the agent. +partition method that can partition the layer k into Nk subareas. In order to achieve partition, +we provide Nk division points on the layer k, denoted by S = {s1, s2, ..., sNk}, and sik ∈ [0, 2π) +represent the phase of these division points. +These division points divide the layer k into Nk sub-areas, which is denoted by E = +{E1, E2, ..., ENk}, Eik is represented as follows +Eik = + + + +{(Rk(θ), θ)|sik ≤ θ < sik+1} +if +sik < sik+1 +{(Rk(θ), θ)|sik ≤ θ < 2π, 0 ≤ θ < sik+1} +otherwise +(4) +and sN+1 = s1. The probability of an intruder invading from Eik is denoted by mik, and mik is +calculated as follows +mik = + + + +� sik+1 +sik +ρ (θ) dθ +if +sik < sik+1 +� 2π +sik ρ (θ) dθ + +� sik+1 +0 +ρ (θ) dθ +otherwise +(5) +In the same way, we use Tik to indicate where the division point exists as follows +Tik = + + + +{(Rk(θ), θ)|ϕik ≤ θ < ϕik+1} +if +ϕik < ϕik+1 +{(Rk(θ), θ)|ϕik ≤ θ < 2π, 0 ≤ θ < ϕik+1} +otherwise +(6) +where ϕNk+1 = ϕ1. +According to the Law of Total Probability, we can give the monitoring probability of the +6 + +multi-agent systems as follows +H(ϕ, S) = +N +� +i=1 +� +Eik +f(d(ϕik, θ))ρ(θ)dθ +(7) +It is not difficult to find that the meaning represented by the quality function (ϕ, S) is the +probability that an intruder intrusion is detected. For applications that detect intruders, the +larger the value of the equation (7) the better the system. Thus the problem of detect intruders +can be transformed into the following optimization problem +max (H (ϕ, S)) +(8) +In order to fit the actual situation, We express the agent dynamics equations with the following +nonholonomic constraint motion equations + + + + + + + + + + + + + +xik = rik cos (ϕik) +yik = rik sin (ϕik) +˙ϕik = ωik +˙rik = ur +ik +(9) +where rik represent the distance between the agent and the center of the circle, i.e. rik = ∥pik∥. +xik and yik are the horizontal and vertical coordinates of pik, and pik(t) = (xik(t), yik(t))T +represent the agent position at the time step t ∈ R+. ωik represents the angular velocity of the +agent, which will be introduced later. ur +ik is denoted as follows +ur +ik = κr (R(ϕik) − rik) , +(10) +where κr is an adjustable parameter. From (9) and (10), we can get the dynamic equation of +the agents is + + + +˙xik = +∂xik +∂rik ˙rik + +∂xik +∂ϕik ˙ +ϕik = ur +ik cos (ϕik) − rikωik sin (ϕik) +˙yik = +∂yik +∂rik ˙rik + +∂yik +∂ϕik ˙ +ϕik = ur +ik sin (ϕik) + rikωik cos (ϕik) +(11) +Using the gradient method for (7), we can get +˙H = +N +� +i=1 +� +Eik +∂f (d(ϕik, θ)) +∂ϕik +ρ (θ) dθ · ωik + +N +� +i=1 +∂H +∂sik +˙sik +(12) +To maximize H, we can set ωik as follows +ωik = κω +� +Eik +∂f (d(ϕik, θ)) +∂ϕik +ρ (θ) dθ +(13) +7 + +Algorithm 1 Single Layer Barrier Coverage Algorithm +Initializate: κr, κω, κs, T ∗ +For ik ∈ INk, ik-th agent performs as follow +1: for t = 1 : T ∗ do +2: +Calculate ϕik by (2) and (3); +3: +while d(ϕik, sik) − d(ϕα, sik) < ε do +4: +Update sik with (15); +5: +end while +6: +Update pik with (15), (10) and (9); +7: end for +where κω is a adjustable constant ,and (13) can guarantee that (12) is not less than zero, which +means that H does not decrease. +We construct the following control input of division points +˙sik = κs(d(ϕik, sik) − d(ϕik−1, sik)) +(14) +where κs is positive constant. In order to apply the algorithm to the multi-layer barrier coverage +algorithm, we need to make some adjustments to the control input. We can find that for agent +ik, which performs barrier coverage, only the states of agent ik − 1 and agent ik + 1 are needed +to complete the algorithm. Moreover, the relative position of agent ik −1 and agent ik +1 is the +closest agent in the clockwise and counterclockwise direction of agent ik, respectively. Therefore, +in multi-layer coverage problem, we use α and β to denote agent ik − 1 and agent ik + 1. We +can rewrite the control input of the agent as follows +˙sik = κs(d(ϕik, sik) − d(ϕα, sik)) +ωik = + + + +κω +� sβ +sik +∂f(d(ϕik ,θ)) +∂ϕik +ρ (θ) dθ, +if sik < sβ +κω +� 2π +sik +∂f(d(ϕik ,θ)) +∂ϕik +ρ (θ) dθ + +� sβ +0 +∂f(d(ϕik ,θ)) +∂ϕik +ρ (θ) dθ. +otherwise +(15) +Finally, we give the single layer barrier distributed coverage algorithm as in Table.1. In this +we ensure that the split point must lie at the midpoint of the curve between the intelligences. +Since the splitting point is virtual, this step is quite fast in practical execution. The multi-layer +barrier coverage algorithm is described next. +8 + +Algorithm 2 Nk Calculate Algorithm +1: for j = 1 : N do +2: +if aj = 1 then +3: +k = kj; +4: +Nk = Nk + 1; +5: +INk = INk ∪ {j}; +6: +end if +7: end for +2.2 +Multi-layer barrier coverage +In this subsection, we will introduce a distributed barrier coverage control algorithm based on +subsection 2.1. +Consider K∗ layers of area to be covered. We use R1(θ), R2(θ), ..., RK∗(θ) to denote the +polar coordinate equation of these layers, and 0 < R1(θ) < R2(θ) < ... < RK∗(θ) < Rmax, for +θ ∈ (0, 2π]. Where Rmax is a positive constant. From subsection 2.1, the number of agents on +layer k is denoted by Nk. In the same way, the number of all agents on the layer is denoted by +NL, and NL = �K∗ +k=1 Nk. +Rather than the number of agents in each layer being fixed, we prefer to find a distributed +method that can automatically allocate the number of agents in each layer. We call this function +as layer swapping. Agent will get a target layer when it is going to do layer swapping. It is easy +to find when an agent moves to its target layer, the agent does not belong to any layer. We +call this class of agents as free agent. On the other hand, agents belong to a layer are called as +layer agent. Moreover, we think that there should be no difference between the states of agents +at the initial moment except their distinct positions. Therefore, all the agents are free agent +at the initial moment in our work. We can think of free agent as stem cell and layer agent as +differentiated cell. The transformation of free agent into layer agent is like the differentiation of +cell. The number of free agent is denoted by NF . The number of all agents is represented by +N, and N = NL + NF. In the initial moment, N = NF . Here we numbered all the agents as +IN = {1, 2, ..., N}. We use ai to denote what type of agent is the agent i, when ai = 0, the agent +is a free agent and when ai = 1, it is a layer agent. And we use the Algorithm 2 to calculate Nk +for each agent, which is the basis of our work. +It is similar to that shown in subsection 2.1, ϕi and pi = (xi, yi)T denote the phase angle +9 + +and position of agent i, respectively. And we use ki to denote the target layer of agent i, and +ki ∈ {0, 1, 2, ..., K}. ki = 0 means agent i has no target layer. In this case, in order to find target +layer agent i will move as follows + + + +˙xi = −riω0 sin (ϕi) , +˙yi = riω0 cos (ϕi) , +(16) +where ω0 is a constant. And when ki ̸= 0, similar to control input (16) , agent will move as +follows + + + +˙xi = κr(R(ϕi) − ri)cos(ϕi), +˙yi = κr(R(ϕi) − ri)sin(ϕi). +(17) +In subsection 2.1, we have numbered the agent. However, there are some difference about +agent number in this subsection. +In layer k, INk = {j|aj × kj = 1}. +As we calculated in +Algorithm 2, we use INk to denote the number set of layer k. If all agents work on layers, there +is such a relationship that IN = �K∗ +k=1 INk. +When the agent is close to a certain layer, the agent needs to consider whether it c an +join the covering task of this layer. +We use rk to denote the range of layer k, and rk := +{(rcos(θ), rsin(θ))|Rk(θ) − ∆ ≤ r ≤ Rk(θ) + ∆}. And ∆ is a is a small enough constant. When +an agent enters rk, we consider that the agent is close to layer k. Moreover ,we consider that +whether agent i can enter the k-th layer depends on agent j already working in the k-th layer, +rather than agent i itself. And only when agent j approves this entry, agent i can enter layer k +to perform the detection task, otherwise agent i should try to move to other layers. If there is +no agent in layer k, the agent will enter the layer k without any problem. +Now, we will introduce the detect state of agent i, when agent i is performing the detect +task. The detect state of agent i is the key variable to judge whether the agent outside the layer +can enter the layer to execute the task. It is easy to know that when an agent is carrying too +much work, its detection capability will decrease. On the other hand, when there are enough +agents in a certain layer, the contribution of agents entering this layer is not as large as that +entering other layers. Therefore, we use ci to denote the detect state of agent i as follows +ci = + + + +1 +ηi > h +0 +otherwise +(18) +where h ∈ (0, 1) is an adjustable parameter, and +ηi = +� +Ei f(d(ϕi, θ))ρ(θ)dθ +� +Ei ρ(θ)dθ +, +(19) +10 + +� � +1 +R � +� � +2 +R � +� � +3 +R +� +O +x +� +Figure 2: Coverage area D, and agent communication radius and monitoring radius. +ηi can be interpreted as the task completion rate. It can also be interpreted as the probability +of being detected by agent i under the condition that intruder invades region Ei. +As shown in Fig.2, there are three layers of area to cover. The color of the star represents +the detection state of the agent, and the color of the circle represents whether the agent is a +free agent. Blue circle means this agent is performing detect task, and yellow circle means this +agent is a free agent and moving to its target layer. Red star means that this agent does not +allow other agents to enter this layer. Green star means that this agent allows other agent enter +this layer. And the star will turn red if ci = 1. +Every free agent has a target layer, we use ki to denote the target layer of agent i, and how +to help the agent find the target layer is the basis of the algorithm. We present Algorithm 3 to +help the agent achieve this function. Our idea is that all agents target the first layer first. If you +cannot enter to the first layer, consider the second, and so on, all the way to the K∗-th layer. +When considering whether to take the k-th layer as the target layer, if there is no agent at the +k-th layer, agent i will choose to target at the k-th layer. If there are agents in k-th layer, agent +i needs to predict whether the agent at layer k can allow entry. +Now, we need to consider how to let an agent enter a layer. As mentioned above, whether +an agent can enter the layer depends on the agent in the layer. Therefore, we use the Algorithm +4 to realize this function. In Algorithm 4, in order to prevent the phase of the agent from being +the same, resulting in the difficulty of setting subsequent division points, the agent will change +its phase when it finds the phase is the same. To avoid agents being preempted by other agents +when they change phase, we need to set ai = 1 first. +It is easy to find that the Algorithm 1 requires agent α and agent β to implement. Therefore, +11 + +Algorithm 3 Target Layer Identification Algorithm +Agent i performs as follows +1: for k = 1 : K∗ do +2: +if Nk = 0 then +3: +ki = k; +4: +else +5: +for j ∈ INk do +6: +if (ϕi, Rk(ϕi)) ∈ Ej then +7: +if cj = 0 then +8: +ki = kj; +9: +end if +10: +end if +11: +end for +12: +end if +13: end for +14: return ki +we design the Algorithm 5 to find the agent α and agent β for agent i. In Algorithm 5, we design +a operation similar to (1) as follows +δ∗(θ1, θ2) = + + + + + + + +θ1 − θ2 − 2π +if +θ1 − θ2 > π +θ1 − θ2 + 2π +if +θ1 − θ2 ≤ −π +θ1 − θ2 +else +(20) +Actually, δ(θ1, θ2) = |δ∗(θ1, θ2)|. The operation can calculate the phase difference from θ1 to +θ2 in the counterclockwise direction. We can find that δ∗(θ1, θ2) ∈ (−π, π], which is difficult +to to compare in algorithm. Therefore, we use Ψ(cos(δ∗(θ1, θ2)), sin(δ∗(θ1, θ2))) to let all the +phase differences be positive. +Then, by finding the minimum of these, the agent α in the +counterclockwise direction can be determined. In the same way, we can also get the agent β in +the other direction. +When the neighbor agent α in clockwise direction changes, the division point si should also +change accordingly. The same is true for counterclockwise which does not change the division +point si. When agent α does not change, agent i can execute Algorithm 1. This ensures that +Algorithm 1 works efficiently. +In addition, the importance of each layer should be different from one another. In general, +12 + +Algorithm 4 Entry Request Algorithm +1: if Nk = 0 then +2: +ai = 1; +3: +α = β = i; +4: +si = ϕi + π; +5: +if si > 2π then +6: +si = si − 2π; +7: +end if +8: else +9: +for j ∈ INk do +10: +if (ϕi, Rk(ϕi)) ∈ Ej then +11: +if cj = 1 then +12: +ki = 0; +13: +else +14: +ai = 1; +15: +while ϕi = ϕj do +16: +Move with (16); +17: +Calculate ϕi by (2) and (3); +18: +end while +19: +end if +20: +end if +21: +end for +22: end if +the more important the inner layer is. Therefore, when the number of agents is insufficient, the +inner layer should be covered first. As shown in Fig.3, when the outer agent finds that the inner +agent needs help, even if it is working well, it will leave the outer layer and head to the inner +layer. We use Algorithm 7 to realize this function. When agent i discovers ϕi ∈ Ej, aj = 1 and +cj = 0 of the inner agent, the agent i transforms itself into a free agent, and the inner layer is +the target layer. +Finally, we present the multi-layer barrier coverage algorithm in Algorithm 8. When agent +i does not have a target layer, the agent will first find a target layer. If agent i cannot find the +target layer with the phase unchanged, the agent will change its phase. After the agent finds +13 + +Algorithm 5 Neighbor Seeking Algorithm +Input: INk, agent i, j state +Output: α, β +1: for j ∈ INk&& j ̸= i do +2: +if Nk = 2 then +3: +α = β = j; +4: +else +5: +if Ψ(cos(δ∗(ϕi, ϕj)), sin(δ∗(ϕi, ϕj))) < Ψ(cos(δ∗(ϕi, ϕα)), sin(δ∗(ϕi, ϕα))) then +6: +α = j; +7: +end if +8: +if Ψ(cos(δ∗(ϕj, ϕi)), sin(δ∗(ϕj, ϕi))) < Ψ(cos(δ∗(ϕβ, ϕi)), sin(δ∗(ϕβ, ϕi))) then +9: +β = j; +10: +end if +11: +end if +12: end for +13: Return α and β; +(a) +(b) +Figure 3: Diagram of agent moving to other layer +the target layer, the agent moves to the target layer. When the agent reaches the target layer, +it will request to enter the target layer. If the request is rejected, the agent looks for another +target layer. When the request is granted, the agent enters the layer to perform the coverage +task. In order to ensure the smooth progress of the algorithm, the intelligent experience obtains +the neighbor information at all times. Finally, when the agent finds that the inner layer needs +help, it stops coverage and helps the inner layer instead. We use P to denote the detection +14 + +Algorithm 6 Division Point Set Algorithm +Initializate: z = α +1: Run Algorithm 5; +2: if z ̸= α then +3: +The division point is set as si = ϕi − 1 +2δ(ϕi, ϕα); +4: +if si < 0 then +5: +si = si + 2π; +6: +end if +7: else +8: +Run Algorithm 1; +9: end if +Algorithm 7 Weight-based Layer Change Algorithm +1: if ki > 1 then +2: +Run Algorithm 2; +3: +for j ∈ INki−1 do +4: +if ϕi ∈ Ej && aj = 1 && cj = 0 then +5: +Set ki = kj and ai = 0; +6: +end if +7: +end for +8: end if +probability of the algorithm. P is calculated as follows +P = 1 − +Nk +� +k=1 +(1 − Pk), +(21) +where Pk is the detected probability of layer k. +In the next section, we will theoretically demonstrate the effectiveness of the proposed algo- +rithm. +3 +Main Results +Lemma 3.1. For fixed agents position, the set of midpoints of T guarantees the maximum of +joint monitoring probability H. +15 + +Algorithm 8 Multi-layer Barrier Coverage Algorithm +Initializate: k = 1, ai = 0, ci = 0 +For i ∈ IN, i-th agent performs as follow +1: while ai = 0 do +2: +while ki = 0 do +3: +Run Algorithm 2; +4: +Run Algorithm 3; +5: +Move with (16); +6: +end while +7: +while pi /∈ rki do +8: +Move to the target layer with (17); +9: +end while +10: +Run Algorithm 2; +11: +Run Algorithm 4; +12: +while ai = 1 do +13: +Run Algorithm 2; +14: +Run Algorithm 6; +15: +Run Algorithm 7; +16: +Update ci with (18) and (19); +17: +end while +18: end while +Proof. By taking the partial derivative of H with respect to si, one gets +∂H +∂si += [f (d(ϕi−1, si)) − f (d(ϕi, si))] ρ (si) . +It is observed that if f(d(ϕi−1, si)) = f(d(ϕi, si)) for i = 1, 2, ..., N, we can get ∂H +∂S = 0. Ac- +cording to the description of the properties of f(·) in Section II, this means that ∂H +∂S = 0 can be +achieved with only d(ϕi−1, si) = d(ϕi, si) for i = 1, 2, ..., N. Since the distinct agents position, +d(ϕi−1, si) = d(ϕi, si) means division point is the midpoint of Ti. Moreover, the Hessian matrix +of the function of coverage quality (7) satisfies +∇2H = [ ∂2U +∂si∂sj +] ∈ RN×N += diag(α1, α2, ..., αN), +16 + +where αi = (∂f(d(ϕi−1,si)) +∂si +− ∂f(d(ϕi,si)) +∂si +)ρ(si). Since f(·) is monotonically decreasing and d (ϕi, si) ∝ +δ(ϕi, si), combining with equation (1), we can get ∂f(d(ϕi−1,si)) +∂si +< 0 and ∂f(d(ϕi,si)) +∂si +ρ(si) > 0. This +means αi < 0 for i = 1, 2, ..., N. Therefore, we can get ∇2H < 0, which implies this lemma. +Theorem 3.1. Dynamic system (9) and (14) ensure that the function (7) reach the local max- +imum value. +Proof. Construct the following Lyapunov function +V (ϕ, S) = 1 +H . +Since si ∈ [0, 2π), f(d(ϕi, θ)) > 0 and ρ(θ) ≥ 0, we can find that V > 0. Moreover, f(d(ϕi, θ)) +and ρ(θ) are bounded, which implies V1 is bounded. Taking the derivative of the Lyapunov +function, we find that +˙V (ϕ, S) = − 1 +H2 · ˙H, +From (7), we can find that V (ϕ, S) > 0.The time derivative of (7) with respect to the compound +dynamics (9) and (14) is given by +˙H = +N +� +i=1 +∂H +∂ϕi +ωi + +N +� +i=1 +∂H +∂si +˙si += +N +� +i=1 +�� +Ei +∂f (d(ϕi, θ)) +∂ϕi +ρ (θ) dθ +�2 ++ κs +N +� +i=1 +[f (d(ϕi−1, si)) − f (d(ϕi, si))] (d(ϕi, si) − d(ϕi−1, si)) ρ (si) +Since [f (d(ϕi−1, si)) − f (d(ϕi, si))] (d(ϕi, si) − d(ϕi−1, si)) ≥ 0, we can find that dH +dt ≥ 0. On +the other hand, the derivative of Lyapuonv function satisfies ˙V1 ≤ 0. According to local invariant +set theorem, the state of the system will converge to the set of {(ϕ, s)| ˙V1 = 0}. From the equation +(7), we can know that H is bounded. Therefore, if and only if dH +dt = 0, ˙V1 = 0. And in this +case, the division points are located in the middle of the arc lengths between agents. In the +meantime, agents are located in the set that {ϕi| +� +Ei +∂f(d(ϕi,θ)) +ϕ +ρ(θ)dθ = 0}. Therefore, V reach +the local minimum value. On the other hand, H reach the local maximum value. +Lemma 3.2. For a working agent i, the agent i will never collide with the division point. +Proof. We assume that the agent i enters the layer at time t∗, we can get the following relation- +ship +δ∗(sβ, ϕi) > 0, δ∗(ϕi, sα) > 0 +17 + +si +s +s +0 +Figure 4: Diagram of the phase location of agent i +Let us first consider that agent i will not collide with division point sβ. As shown in Fig.4, we +use E1 +i , E2 +i , E3 +i , E4 +i to denote the area between two phases. We use Lβ to denote the distance +between agent i and division point sβ, and Lβ is represented as follows +Lβ = K(δ∗(sβ, ϕi)), +where K(·) is a class K function. Next, in combination with Equation (20), we take the derivative +of Lβ to get +˙Lβ = K1 · ( ˙sβ − ˙ϕi) += K1 · (κs(δ∗(ϕβ, sβ) − δ∗(sβ, ϕi)) − +� +Ei +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ) += K1 · (κs(δ∗(ϕβ, sβ) − δ∗(sβ, ϕi)) − +� +E3 +i +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ − +� +E2 +i +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ) +where K1 = +∂K(δ∗(sβ,ϕi)) +∂δ∗(sβ,ϕi) +> 0. +As ϕi −→ sβ, we can find that +� +E3 +i +∂f(d(ϕi,θ)) +∂ϕi +ρ(θ)dθ −→ 0, +∂f(d(ϕi,θ)) +∂ϕi +< 0 in the range of E2 +i , and δ∗(ϕβ, sβ) > 0. Therefore, we can get that +˙Lβ > −K1 · κs(δ∗(sβ, ϕi)), +which means that Lβ > 0 holds within the interval of t ≥ t∗. In the same way, we use the Lα +to denote the distance between agent i and division point si, i.e.Lα = K(δ∗(ϕi, si)), and we can +18 + +also get the following +˙Li = K2 · ( ˙ϕi − ˙si) += K2 · ( +� +Ei +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ − κs(δ∗(ϕi, si) − δ∗(si, ϕα))) += K2 · ( +� +Ei +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ − κs(δ∗(ϕi, si) − δ∗(si, ϕα))) += K2 · ( +� +E2 +i +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ + +� +E3 +i +∂f(d(ϕi, θ)) +∂ϕi +ρ(θ)dθ + κsδ∗(si, ϕα) − κsδ∗(ϕi, si)) +where K2 = +∂K(δ∗(si,ϕi)) +∂δ∗(si,ϕi) +> 0. +As ϕi −→ si, we can find that +� +E2 +i +∂f(d(ϕi,θ)) +∂ϕi +ρ(θ)dθ −→ 0, +∂f(d(ϕi,θ)) +∂ϕi +> 0 in the range of E3 +i , and δ∗(si, ϕα) > 0. Therefore, we can get that +˙Lα > −K2 · κs(δ∗(ϕi, si)), +which means that Lα > 0 holds within the interval of t ≥ t∗. Because of Lα > 0 and Lβ > 0, +the agent will not collide with the division point. +Lemma 3.3. The division points never collide with each other. +Proof. From lemma 3.2, we can know that the distance between division point si and sβ can be +denoted as Li = Lα + Lβ. Obviously, Li is the length of Ei. Therefore, we can get that Li > 0 +holds within the interval of t ≥ t∗, which implies this lemma. +We use Lk to denote the length of layer k. To demonstrate our conclusion, we discover the +following lemmas. +Lemma 3.4. For agent i working at layer k, if Li = min{j ∈ INk|Lj}, we have Li ≤ Lk +Nk . +Proof. From Table 6, The region of the k-th layer will be divided without remainder by the +agents working in the k-th layer. Therefore, we can get the following relation +Lk = +� +j∈INk +Lj. +From Lemma 3.3, we have Li > 0, for i ∈ INk. Since Li = min{j ∈ INk|Lj}, we have Lj ≥ Li, +for j ∈ INK. Therefore, the above formula can be rewritten as +Lk = +� +j∈INk +Lj ≥ Nk × Li +which means that Li ≤ Lk +Nk . +19 + +Lemma 3.5. For agent i working at layer k, as t −→ ∞, if Li = max{j ∈ INk|Lj} and Nk ≥ 2, +we have Lk +Nk ≤ Li ≤ Lk +2 . +Proof. Similar to Lemma 3.4, since Li = max{j ∈ INk|Lj}, we have Lj ≤ Li for j ∈ INK, which +means that Li ≥ Lk +Nk . +As shown in Figure 4, we use li to denote the length of E3 +i ∪ E4 +i . We can get the following +relation +Lk = +� +j∈INk +lj. +From Lemma 3.1 and control input (14), as t −→ ∞, the agent i have following relation +Li = 0.5(li + lα), +Since li + lα ≤ Lk, we get Li ≤ Lk +2 . +In our algorithm, the number of agents on the k-th layer is not fixed. Obviously, we can get +a relation as follows Pk(t) ≥ 0, for t ≥ 0. +Lemma 3.6. For the layer k, if the agent i leaves this layer and Nk ≥ 2, the maximal reduction +of the detect probability of the k-th layer can be calculated as follows +Pk′ = +� +E2 +i +(f (d (ϕi, θ)) − f (d (si, ϕi) + d (si, θ))) ρ (θ) dθ ++ +� +E3 +i +(f (d (ϕi, θ)) − f (d (sβ, ϕi) + d (sβ, θ))) ρ (θ) dθ +Proof. In our algorithm, when the agent i enters or leaves, there is only a change in the moni- +toring probability of E2 +i and E3 +i for all regions in layer k. We assume that agent i leaves layer +k at time ti, and the new division point is at the position of ϕi. From the Lemma 3.1, then we +can get the following formula +Pk(ti + ε) ≥ Pk(ti) − +� +E2 +i +(f (d (ϕi, θ)) − f (d (ϕα, θ))) ρ (θ) dθ ++ +� +E3 +i +(f (d (ϕi, θ)) − f (d (ϕβ, θ))) ρ (θ) dθ, +where ε is an infinitesimal. From Table 1, we can know that si and sβ will converge to the +midpoint of lα and li. As shown in Fig.4, the length of E1 +i is the same as that of E2 +i , and the +20 + +length of E3 +i is the same as that of E4 +i . Therefore, the variation of the detect probability of the +k-th layer can rewritten as follows +Pk′ ≥ +� +E2 +i +(f (d (ϕi, θ)) − f (d (si, ϕi) + d (si, θ))) ρ (θ) dθ ++ +� +E3 +i +(f (d (ϕi, θ)) − f (d (sβ, ϕi) + d (sβ, θ))) ρ (θ) dθ, +which implies this lemma. +Lemma 3.7. For the layer k, if the agent i enters this layer, the minimal increase of the detect +probability of the k-th layer can be calculated as follows +P ′ +k = +� +E2 +i ∪E3 +i +f (d (ϕi, θ)) ρ (θ) dθ − +� sβ +s∗ +β +f (d (ϕβ, θ)) ρ (θ) dθ − +� s∗ +β +si +f (d (ϕα, θ)) ρ (θ) dθ +where s∗ +β is the division point in lα before agent i enters layer k, and if Nk = 0, then d(ϕα, si) = 0, +d(ϕα, θ) = 0. +Proof. As Lemma 3.6 said, agent entry will only change the detect probabilities of E2 +i and E3 +i +in the k-th layer. Assuming that the agent enters layer k after time ti, We can know the detect +probability of this area as follows +Pk(ti) = P ∗ +k (ti) + +� s∗ +β +si +f (d (ϕα, θ)) ρ (θ) dθ + +� sβ +s∗ +β +f (d (ϕβ, θ)) ρ (θ) dθ +where P ∗ +k indicates that the k-th layer does not consider the monitoring probability of E2 +i and +E3 +i . After the agent i enters the k layer, from Theorem 3.1 the above formula is rewritten as +Pk(ti + ε) ≥ P ∗ +k (ti + ε) + +� +E2 +i ∪E3 +i +f (d (ϕi, θ)) ρ (θ) dθ +where P ∗ +k (ti + ε) = P ∗ +k (ti). Therefore, we can get P ′ +k as follows +P ′ +k ≥ +� +E2 +i ∪E3 +i +f (d (ϕi, θ)) ρ (θ) dθ − +� sβ +s∗ +β +f (d (ϕβ, θ)) ρ (θ) dθ − +� s∗ +β +si +f (d (ϕα, θ)) ρ (θ) dθ +which implies this lemma. +According to the above conclusions, we can get the following theorem +Theorem 3.2. For a multi-agent multi-layer barrier coverage system with Nk layers, if the +agent i working on the k-th layer satisfies the following inequality, +P ′ +k < (1 − Pk)P ′ +v +1 − Pv − P ′v +the detection probability (21) of the system will increase if the agent i enters the v-th layer. +21 + +Proof. Without loss of generality, we can assume that when the multi-agent coverage system is +at time t1, agent i works at layer k; when the system is at time t2, agent i works at layer v. +And, at the two moments, except that the working place of agent i is different, other agents are +still working in the same layer. Therefore, we can get the following equation by (21) +P(t1) = 1 − (1 − P1)(1 − P2)...(1 − Pk)...(1 − Pv)...(1 − PNk). +From Lemma 3.6 and Lemma 3.7, we can get the following equation +P(t2) ≥ 1 − (1 − P1)(1 − P2)...(1 − Pk + Pk′)...(1 − Pv − Pv′)...(1 − PNk) +let 1 − (1 − P1)(1 − P2)...(1 − Pk + Pk′)...(1 − Pv − Pv′)...(1 − PNk)) > P(t1), we can get +P(t1) > P(t2), which implies this theorem. Simplify the above formula to get +P ′ +k < (1 − Pk)P ′ +v +1 − Pv − P ′v +This completes the proof. +Corollary 3.1. For a single-layer barrier coverage system with fixed division points, when the +layer is a circle with a radius R0, d adopts the geodesic distance obtained on the layer and the +detection model of the agent is a Gaussian probability model, i.e. f(d) = e−d2/γ2 if the radius +R0 satisfies R0 ≤ +√ +2γ +2π , dynamic system (9) ensure that the function (7) reaches the maximum +value. +Proof. By taking the partial derivative of (7) with respect to ϕi, we get +∂H +∂ϕi += +� +Ei +∂f (d(ϕi, θ)) +∂ϕi +ρ (θ) dθ +Substituting f(d) = e−d2/γ2 into the above formula yields +∂H +∂ϕi += +� +E1 +i +e− d2 +γ2 +� +−2 d +γ2 +� ∂d +∂ϕi +ρ (θ) + +� +E2 +i +e− d2 +γ2 +� +−2 d +γ2 +� ∂d +∂ϕi +ρ (θ) +The integral is segmented because the geodesic distance d is not derivable when θ = ϕi. And +∂d +∂ϕi = Ro as θ ∈ E2 +i , +∂d +∂ϕi = −Ro as θ ∈ E3 +i . +We take the partial derivative of the above formula with respect to ϕi to get +∂2H +∂ϕi2 = +� +E2 +i +e +− d2 +γ2 +� +4d2 +γ4 − 2 +γ2 +� � ∂d +∂ϕi +�2 +ρ (θ) dθ + +� +E3 +i +e +− d2 +γ2 +� +4d2 +γ4 − 2 +γ2 +� � ∂d +∂ϕi +�2 +ρ (θ) dθ +22 + +From Lemma 3.5, we can get d ≤ πR0. If R0 ≤ +√ +2γ +2π , we have ∂2H +∂ϕi2 < 0. Moreover, the Hessian +matrix of the function of coverage quality (7) satisfies +∇2H = [ +∂2U +∂ϕi∂ϕj +] = diag(∂2H +∂ϕ2 +1 +, ∂2H +∂ϕ2 +2 +, ..., ∂2H +∂ϕ2 +N +) ∈ RN×N. +This means H has a unique maximum. Combining with Theorem 3.1, the dynamic system (9) +will ensure the function (7) reaches the maximum value. +4 +Case Studies +In this section, we will give some simulation and experiment results to verify our coverage +algorithm. We implemented our algorithm on MATLAB 2022a. Now, we give the multi-agent +barrier coverage algorithm in Table 8. +4.1 +Numerical simulation +We designed 3 layers of area. There are 50 agents needs to cover on these three layers to monitor +the invasion of intruders. These three layers are designed as follows + + + + + + + + + +R1(θ) = 1 + 0.15 sin(4θ), +R2(θ) = 2 + 0.15 sin(10θ), +R3(θ) = 3 + 0.15 sin(40θ). +(22) +The probabilistic model is given by f(d(ϕi, θ)) = exp(−d(ϕi, θ)2), where the distance function +d is calculated as follows +d(ϕi, θ) = + + + + + + + +���� +� θ +ϕi +� +R(θ)2 + R′(θ)2dθ +���� , +if +���� +� θ +ϕi +� +R(θ)2 + R′(θ)2dθ +���� ≤ +Lki +2 +Lki − +���� +� θ +ϕi +� +R(θ)2 + R′(θ)2dθ +���� . +otherwise +where d is Lipschitz continuous. The density function is ρ(θ) = +θ +2π2 . We set the adjustable +parameters as follows + + + + + + + +κr = 0.1, +κω = 0.01, +κs = 0.05. +As shown in Fig.5, we place the agent inside the innermost layer. +All agents gradually +expand outwards, and finally cover all three layers. And we intercept the position results of the +algorithm at 4 time points, which are 0s, 8s, 16s and 24s respectively. +23 + +-4 +-2 +0 +2 +4 +(a) +-4 +-2 +0 +2 +4 +t=0s +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +-4 +-2 +0 +2 +4 +(b) +-4 +-2 +0 +2 +4 +t=8s +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +-4 +-2 +0 +2 +4 +(c) +-4 +-2 +0 +2 +4 +t=16s +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +-4 +-2 +0 +2 +4 +(d) +-4 +-2 +0 +2 +4 +t=24s +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Figure 5: Snapshots of simulation results. Circles denote the mobile agents, and the stars refer +to the division points. +As shown in Fig.5, when the algorithm first starts running, all agents are in the innermost +inner region. After the algorithm runs for 8 seconds, 6 agents have been covered on the first +layer, and some agents have moved to the second layer. Combined with Figure 7, after the +algorithm runs for about 13 seconds, the detect probability of the third layer decreases. When +the algorithm runs to 16 seconds, we find that some agents are moving from the third layer to +the second layer. This is because the Algorithm 7, when the inner agent is not well qualified for +its detection task, the outer agent will leave the outer layer and go to the inner layer to help +the inner agent. When the algorithm runs for 24 seconds, the multi-agent systems is basically +stable, and most of the agents are already working on the layer. Around the circle with a radius +of 4, some agents are patrolling, looking for any agents that need help, and when found, these +24 + +-4 +-2 +0 +2 +4 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Figure 6: The final result of the system state. +patrolling agents will take action. Finally, we give the results of the algorithm running to the +last moment of the system in Figure 6. We can find that on each layer, the agents are denser +where the invasion probability is high. Moreover, there are still free agents patrolling the circle +of radius 4. +In Fig. 7, we show how the detection probability of the system and each layer changes over +time. We can see that when the second and third layers have no agents, the total detection +probability is the same as that of the first layer. When the second layer and the third layer +have agents working one after another, the monitoring probability of the agents has a significant +increase. Finally, it can be found that the detection probability of the multi-layer fence coverage +algorithm exceeds 99.99%. +We also did controlled experiments with multi-layer barrier coverage and single layer barrier +coverage. As shown in Fig.8, the detection probability of the multi-layer barrier coverage was +inferior to that of the single-layer fence cover for the initial period, but once agents moved to +the second layer, the detection probability of the multi-layer barrier coverage reversed to that +of the single-layer fence cover, and was higher than that of the single-layer for the rest of the +time. +We counted the final detection probability of single-layer barrier coverage and multi-layer +barrier coverage with different number of smart bodies, as shown in Fig.9. It can be found that +25 + +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Time(s) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Detection probability +Layer1 +Layer2 +Layer3 +Total +10 +12 +14 +16 +18 +20 +Time(s) +0.998 +0.9985 +0.999 +0.9995 +1 +Detection probability +Layer1 +Layer2 +Layer3 +Total +Figure 7: Detection probability of each layer and total system. +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Time(s) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Detection probability +Single-layer +Multi-layer +5 +10 +15 +20 +25 +Time(s) +0.96 +0.97 +0.98 +0.99 +1 +Detection probability +Figure 8: Difference between single layer barrier coverage and multi-layer barrier coverage for +the same number of agents. +26 + +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Agent Number +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Detection probability +20 +30 +40 +50 +Agent Number +0.994 +0.996 +0.998 +1 +Detection probability +Figure 9: Difference in detection probability between single and multi-layer barrier coverage for +the same number of agents. +there is no difference in the detect probability between single and multi-layer barrier coverage +when the number of agents is small. However, the detection probability of the multi-layer barrier +coverage is significantly higher than that of the single-layer barrier coverage when the number of +agents gradually increases. When the number of smart bodies is large enough, the increase in the +number of smart bodies is of little help to the single-layer barrier coverage. When the number +of agents is 50, the detection probability of single-layer barrier coverage reaches 99.8 percent, +while the detection probability of multi-layer barrier coverage is very close to 100 percent. +5 +Conclusions +This paper presented a distributed multi-agent barrier coverage algorithm. First, a single-layer +barrier coverage quality function was designed based on the probabilistic model of intrusion and +a single-layer barrier coverage algorithm was designed based on the gradient method. Then a +layer-to-layer adjustment mechanism was proposed based on the single-layer algorithm, which +adjusts the number of agents on each layer so that the coverage quality of the whole system was +improved. Then some theoretical analyses were given to theoretically verify the stability and +27 + +effectiveness of the single-layer algorithm and the necessity of the multi-layer algorithm, and the +theoretical results were given in some special cases. Finally, the effectiveness of our algorithm +was verified by simulation and the practicality of the algorithm was verified by experiment. +6 +Appendix +Acknowledgment +The Project was supported by the Fundamental Research Funds for the Central Universities, +China University of Geosciences (Wuhan). +References +[1] Vicsek, Tam´as, et al. ”Novel type of phase transition in a system of self-driven particles.” +Physical review letters 75.6 (1995): 1226. +[2] Wilson S, Glotfelter P, Wang L, et al. 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AIAA Journal of Guidance, +Control, and Dynamics, pp.1-7, 2016. +29 + diff --git a/e9A0T4oBgHgl3EQfHf82/content/tmp_files/load_file.txt b/e9A0T4oBgHgl3EQfHf82/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e420cf3046d4aaed787ac52396cdf31b711b1f48 --- /dev/null +++ b/e9A0T4oBgHgl3EQfHf82/content/tmp_files/load_file.txt @@ -0,0 +1,638 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf,len=637 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='02061v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='OC] 5 Jan 2023 Distributed Control Strategy for Layered Barrier Coverage of Multi-Agent Systems in Uncertain Environments Pengyang Fan, Chao Zhai ∗ August 2022 Abstract This paper presents a distributed multi-layer ring barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In order to achieve single-layer ring barrier coverage, a distributed single-layer ring barrier coverage al- gorithm that maximises the probability of monitoring is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Considering the security risks of single-layer barrier coverage, a distributed adjustment mechanism between multiple layers of barriers is designed and combined with the single-layer ring barrier coverage algo- rithm to propose a distributed multi-layer ring barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Furthermore, we present a theoretical analysis of the proposed algorithm to demonstrate its effectiveness and necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, our algorithm is verified by numerical simulation and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 1 Introduction Multi-agent systems(MASs) are composed of agents that interact with each other in an environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Each agent is a system, MASs are systems in which a large number of agents are grouped together and realise an overall behaviour or activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Agents can be natural creatures[1], ar- tificial robots or mobile sensors[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The MASs aims to take a distributed approach to solve some large and complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Each agent is an independent individual that can perceive the environment, process information, communicate, learn, and make decisions independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since each agent adopts an independent strategy, coordinated control of multi-agent systems is ∗Pengyang Fan and Chao Zhai are with School of Automation, China University of Geosciences, Wuhan 430074 China, and with Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems and Engineering Research Center of Intelligent Technology for Geo-exploration, Ministry of Education, Wuhan 430074 China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Corresponding author: Chao Zhai (email: zhaichao@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 1 essential in order for the whole system to accomplish a common goal, and this area has attracted many scholars to conduct research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Multi-agent coverage control is a hot research topic in multi-agent coordination control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Multi-agent coverage control refers to a group of agent bodies with mobile, communication, computing, and learning capabilities to sense the environment and perform a given task in a distributed manner in a given or indefinite area, such as: search and rescue, missile interception, monitoring, sweeping, etc[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Multi-agent coverage control can be classified into area coverage, sweeping coverage and barrier coverage according to the area covered by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Area coverage is a series of operations in which each agent body determines its optimal state in the area through communication, computation, and coordination and achieves this state through some control science methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The most classic one is the multi-agent coverage algorithm based on Voronoi partition[3], in which each agent divides a convex region into sub-regions through communication, and each agent uses a strategy of moving to the center of mass of the sub-region to maximize the coverage quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' This method can cover a convex region to the maximum extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' On this basis, many scholars have found many problems and proposed some solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For example, to solve the non-convex region, some scholars proposed the Voronoi center-of-mass coverage algorithm for non-convex region based on the geodesic Voronoi partition algorithm[4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' to solve the time-varying density function problem, some scholars proposed the Voronoi center- of-mass coverage in dynamic environment based on the control barrier function[6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' to solve the coverage problem in uncertain environment, some scholars proposed the Voronoi center-of- mass coverage in uncertain environment based on the Bayesian estimation[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Sweep coverage not only requires the agent to reach the designated area but also requires agent to be able to traverse the entire area to achieve cleaning of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For example, some scholars have achieved equal-task sweep coverage of a class of regions based on equal-task partitioning methods, which can improve the overall efficiency[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Some scholars have also proposed a multi-agent sweeping coverage algorithm based on the temperature field approach[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, literature [10] combines Voronoi segmentation with a temperature field approach to design a distributed overlay method that enables each agent to have the same workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Multi-agent barrier coverage refers to the coverage of a group of agents on a line, which is usually used to monitor whether a creature or object crosses the line or to intercept objects that attempt to cross on the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The literature [11] proposes the definition of barrier coverage and k-barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The k-barrier coverage is further divided into weak k-barrier coverage and strong k-barrier coverage[11][12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The strong k-barrier coverage means that an intruder is 2 detected by at least k agents regardless of any path into or through the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Besides, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' proposed the concept of local barrier coverage, which can reduce the number of agents compared to global fence overlays and can also be used for general cases[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' However, local barrier coverage is a security risk, as intruders can potentially traverse the area without being detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' There are currently many algorithms for barrier coverage and k-barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For example, the coverage-based approach proposed in literature [19] can equip the task of assigning intrusion probability to intercept the intruded items thus achieving protection of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In addition to this, scholars have designed a distributed algorithm that can achieve a uniform barrier coverage between two landmarks[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Ban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' investigate the strong k-barrier coverage problem of mobile sensor networks over open belt using a grid-based approach in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' However, the algorithm can only achieve coverage on a straight line between two points, and cannot achieve coverage on a curve, nor can it achieve coverage on a closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In practical applications, if the targets in the area need to be protected or monitored in an all-round way, the whole boundary of the area needs to be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For an enclosed target area, the agents also needs to be covered within the closed belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For this reason, Binay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' designed the algorithm to move the smart body to the boundary of a simple polygon and thus protect the area inside the polygon[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, a barrier coverage algorithm has been designed on a circle, and the agent can be uniformly covered on the circle with limited communication[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' But a circle is a kind of convex region, and how to perform barrier coverage on the boundary of non-convex regions is the inspiration of our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, covering only the boundary means that as soon as the intruder breaks through this layer, the intruder enters the area we need to protect and the system loses the means to monitor the intruder, which means an increase in security risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From a security point of view, a k-barrier coverage is more secure than a single layer barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' To this end, we first designed algorithms that can perform barrier coverage on the boundary of a class of non-convex regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The algorithm is applied in the context of monitoring intruders, and uses a region partition to assign a region to each agent for monitoring, which can eventually lead to a local maximum monitoring probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we want to design a multi-agent control algorithm with multi-layer barrier coverage to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When an intruder breaks through a layer, there are still several internal monitoring layers that can continue to monitor the intruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The goal of this paper is to design a distributed multi-agent barrier coverage algorithm that can implement a multi-layer barrier coverage and can autonomously adjust the number of agents on each layer to optimize the monitoring quality of the whole 3 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Design a multi-agent barrier coverage algorithm for a class of non-convex areas which can maximize intruder monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Develop a distributed adjustment mechanism for the number of agents per layer, which can optimize the monitoring probability of multi-agent systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Combining the single-layer fence coverage algorithm and the distributed adjustment mech- anism of the number of multi-layer agents, we propose the multi-layer barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Section 2 presents a single-layer barrier coverage algorithm for non-convex region boundaries at first and then provides a distributed adjustment mechanism for the number of agents in a multi-layer coverage region and a multi- layer barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Section 3 presents a theoretical verification of the single- layer barrier coverage algorithm and the multi-layer barrier coverage algorithm proposed in Section 2 and gives the case when our algorithm is applied to a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Section 4 simulates our algorithm and performs experimental validation on Robotarium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, we conclude the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 2 Problem Formulation In this section, we will introduce the distributed multi-agent barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Con- sider a closed curve region D, which can be represented by polar coordinates, the center of the circle D is denoted by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Without loss of generality, we can set the center of the circle as the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' O = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The boundary of the circle area D is denoted by ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The radius of the closed curve region is denoted by R(θ), θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 Single layer barrier coverage In the application of monitoring intruder, multiple layer barrier coverage is more effective than single layer barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we propose a multiple layer barrier coverage algorithm in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since this algorithm is based on single layer barrier coverage algorithm, we firstly introduce the single layer barrier coverage algorithm in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In layer k, there are Nk mobile agents that can communicate and monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The layer k can be denoted by Rk(θ), θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We assume that agents can all communicate with each 4 other if they are on the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use INk to denote the number of agents on this layer, INk = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', Nk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use ρ(θ) to denote the probability that each point on the layer is invaded by an intruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We denote the position of agents by P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', pNk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We specify an angle for agent ik with respect to the center of the circle O, denoted by ϕik, ik = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We denote the probabilistic model that the agent ik detects an intruder by f(d(ϕik, θ)), where d(ϕik, θ) is a distance function about ϕik and θ, and the distance function is Lipschitz continuous, and d (ϕik, θ) ∝ δ(ϕik, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The function δ(ϕik, θ) is denoted by δ(ϕik, θ) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 |ϕik − θ − 2π| if ϕik − θ > π |ϕik − θ + 2π| if ϕik − θ ≤ −π |ϕik − θ| else (1) Moreover, f(d(ϕik, θ)) need to meet the following conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' f(d(ϕik, θ)) is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' f(d(ϕik, θ)) is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Now we give the calculation way of ϕik as follows ϕik(t) = Ψ(pT ik(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' (2) where Ψ(x, y) is an operation specified by us, which is calculated as follows Ψ(x, y) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 arctan( y x) + π x < 0 arctan( y x) y ⩾ 0 x > 0 arctan( y x) + 2π y < 0 x > 0 π 2 y > 0 x = 0 − π 2 y < 0 x = 0 0 y = 0 x = 0 (3) Combining (2) and (3), it is easy to know that ϕik ∈ [0, 2π), for ik = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, agents have the following numbering rules 0 ≤ ϕ1 (0) < ϕ2 (0) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' < ϕNk (0) < 2π As the distance increases, the detection probability of the agent will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we stipulate that the agent only detects the area in which it is responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we propose a 5 Agent Division point Origin point i is 1 is � js 1 js � j O i� j � x D Figure 1: Coverage area D, agent communication radius and monitoring radius, and the color of the pentagram indicates the working state of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' partition method that can partition the layer k into Nk subareas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In order to achieve partition, we provide Nk division points on the layer k, denoted by S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', sNk}, and sik ∈ [0, 2π) represent the phase of these division points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' These division points divide the layer k into Nk sub-areas, which is denoted by E = {E1, E2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', ENk}, Eik is represented as follows Eik = \uf8f1 \uf8f2 \uf8f3 {(Rk(θ), θ)|sik ≤ θ < sik+1} if sik < sik+1 {(Rk(θ), θ)|sik ≤ θ < 2π, 0 ≤ θ < sik+1} otherwise (4) and sN+1 = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The probability of an intruder invading from Eik is denoted by mik, and mik is calculated as follows mik = \uf8f1 \uf8f2 \uf8f3 � sik+1 sik ρ (θ) dθ if sik < sik+1 � 2π sik ρ (θ) dθ + � sik+1 0 ρ (θ) dθ otherwise (5) In the same way, we use Tik to indicate where the division point exists as follows Tik = \uf8f1 \uf8f2 \uf8f3 {(Rk(θ), θ)|ϕik ≤ θ < ϕik+1} if ϕik < ϕik+1 {(Rk(θ), θ)|ϕik ≤ θ < 2π, 0 ≤ θ < ϕik+1} otherwise (6) where ϕNk+1 = ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' According to the Law of Total Probability, we can give the monitoring probability of the 6 multi-agent systems as follows H(ϕ, S) = N � i=1 � Eik f(d(ϕik, θ))ρ(θ)dθ (7) It is not difficult to find that the meaning represented by the quality function (ϕ, S) is the probability that an intruder intrusion is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For applications that detect intruders, the larger the value of the equation (7) the better the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Thus the problem of detect intruders can be transformed into the following optimization problem max (H (ϕ, S)) (8) In order to fit the actual situation, We express the agent dynamics equations with the following nonholonomic constraint motion equations \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xik = rik cos (ϕik) yik = rik sin (ϕik) ˙ϕik = ωik ˙rik = ur ik (9) where rik represent the distance between the agent and the center of the circle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' rik = ∥pik∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' xik and yik are the horizontal and vertical coordinates of pik, and pik(t) = (xik(t), yik(t))T represent the agent position at the time step t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' ωik represents the angular velocity of the agent, which will be introduced later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' ur ik is denoted as follows ur ik = κr (R(ϕik) − rik) , (10) where κr is an adjustable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From (9) and (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' we can get the dynamic equation of the agents is \uf8f1 \uf8f2 \uf8f3 ˙xik = ∂xik ∂rik ˙rik + ∂xik ∂ϕik ˙ ϕik = ur ik cos (ϕik) − rikωik sin (ϕik) ˙yik = ∂yik ∂rik ˙rik + ∂yik ∂ϕik ˙ ϕik = ur ik sin (ϕik) + rikωik cos (ϕik) (11) Using the gradient method for (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' we can get ˙H = N � i=1 � Eik ∂f (d(ϕik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' θ)) ∂ϕik ρ (θ) dθ · ωik + N � i=1 ∂H ∂sik ˙sik (12) To maximize H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' we can set ωik as follows ωik = κω � Eik ∂f (d(ϕik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' θ)) ∂ϕik ρ (θ) dθ (13) 7 Algorithm 1 Single Layer Barrier Coverage Algorithm Initializate: κr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' κω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' κs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' T ∗ For ik ∈ INk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' ik-th agent performs as follow 1: for t = 1 : T ∗ do 2: Calculate ϕik by (2) and (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 3: while d(ϕik, sik) − d(ϕα, sik) < ε do 4: Update sik with (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 5: end while 6: Update pik with (15), (10) and (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 7: end for where κω is a adjustable constant ,and (13) can guarantee that (12) is not less than zero, which means that H does not decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We construct the following control input of division points ˙sik = κs(d(ϕik, sik) − d(ϕik−1, sik)) (14) where κs is positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In order to apply the algorithm to the multi-layer barrier coverage algorithm, we need to make some adjustments to the control input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can find that for agent ik, which performs barrier coverage, only the states of agent ik − 1 and agent ik + 1 are needed to complete the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, the relative position of agent ik −1 and agent ik +1 is the closest agent in the clockwise and counterclockwise direction of agent ik, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, in multi-layer coverage problem, we use α and β to denote agent ik − 1 and agent ik + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can rewrite the control input of the agent as follows ˙sik = κs(d(ϕik, sik) − d(ϕα, sik)) ωik = \uf8f1 \uf8f2 \uf8f3 κω � sβ sik ∂f(d(ϕik ,θ)) ∂ϕik ρ (θ) dθ, if sik < sβ κω � 2π sik ∂f(d(ϕik ,θ)) ∂ϕik ρ (θ) dθ + � sβ 0 ∂f(d(ϕik ,θ)) ∂ϕik ρ (θ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' otherwise (15) Finally, we give the single layer barrier distributed coverage algorithm as in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In this we ensure that the split point must lie at the midpoint of the curve between the intelligences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since the splitting point is virtual, this step is quite fast in practical execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The multi-layer barrier coverage algorithm is described next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 8 Algorithm 2 Nk Calculate Algorithm 1: for j = 1 : N do 2: if aj = 1 then 3: k = kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: Nk = Nk + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 5: INk = INk ∪ {j};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 6: end if 7: end for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 Multi-layer barrier coverage In this subsection, we will introduce a distributed barrier coverage control algorithm based on subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Consider K∗ layers of area to be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use R1(θ), R2(θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', RK∗(θ) to denote the polar coordinate equation of these layers, and 0 < R1(θ) < R2(θ) < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' < RK∗(θ) < Rmax, for θ ∈ (0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Where Rmax is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, the number of agents on layer k is denoted by Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the same way, the number of all agents on the layer is denoted by NL, and NL = �K∗ k=1 Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Rather than the number of agents in each layer being fixed, we prefer to find a distributed method that can automatically allocate the number of agents in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We call this function as layer swapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Agent will get a target layer when it is going to do layer swapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It is easy to find when an agent moves to its target layer, the agent does not belong to any layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We call this class of agents as free agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' On the other hand, agents belong to a layer are called as layer agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, we think that there should be no difference between the states of agents at the initial moment except their distinct positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, all the agents are free agent at the initial moment in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can think of free agent as stem cell and layer agent as differentiated cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The transformation of free agent into layer agent is like the differentiation of cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The number of free agent is denoted by NF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The number of all agents is represented by N, and N = NL + NF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the initial moment, N = NF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Here we numbered all the agents as IN = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use ai to denote what type of agent is the agent i, when ai = 0, the agent is a free agent and when ai = 1, it is a layer agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And we use the Algorithm 2 to calculate Nk for each agent, which is the basis of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It is similar to that shown in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, ϕi and pi = (xi, yi)T denote the phase angle 9 and position of agent i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And we use ki to denote the target layer of agent i, and ki ∈ {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' ki = 0 means agent i has no target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In this case, in order to find target layer agent i will move as follows \uf8f1 \uf8f2 \uf8f3 ˙xi = −riω0 sin (ϕi) , ˙yi = riω0 cos (ϕi) , (16) where ω0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And when ki ̸= 0, similar to control input (16) , agent will move as follows \uf8f1 \uf8f2 \uf8f3 ˙xi = κr(R(ϕi) − ri)cos(ϕi), ˙yi = κr(R(ϕi) − ri)sin(ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' (17) In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, we have numbered the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' However, there are some difference about agent number in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In layer k, INk = {j|aj × kj = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As we calculated in Algorithm 2, we use INk to denote the number set of layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If all agents work on layers, there is such a relationship that IN = �K∗ k=1 INk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the agent is close to a certain layer, the agent needs to consider whether it c an join the covering task of this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use rk to denote the range of layer k, and rk := {(rcos(θ), rsin(θ))|Rk(θ) − ∆ ≤ r ≤ Rk(θ) + ∆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And ∆ is a is a small enough constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When an agent enters rk, we consider that the agent is close to layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover ,we consider that whether agent i can enter the k-th layer depends on agent j already working in the k-th layer, rather than agent i itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And only when agent j approves this entry, agent i can enter layer k to perform the detection task, otherwise agent i should try to move to other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If there is no agent in layer k, the agent will enter the layer k without any problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Now, we will introduce the detect state of agent i, when agent i is performing the detect task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The detect state of agent i is the key variable to judge whether the agent outside the layer can enter the layer to execute the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It is easy to know that when an agent is carrying too much work, its detection capability will decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' On the other hand, when there are enough agents in a certain layer, the contribution of agents entering this layer is not as large as that entering other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we use ci to denote the detect state of agent i as follows ci = \uf8f1 \uf8f2 \uf8f3 1 ηi > h 0 otherwise (18) where h ∈ (0, 1) is an adjustable parameter, and ηi = � Ei f(d(ϕi, θ))ρ(θ)dθ � Ei ρ(θ)dθ , (19) 10 � � 1 R � � � 2 R � � � 3 R � O x � Figure 2: Coverage area D, and agent communication radius and monitoring radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' ηi can be interpreted as the task completion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It can also be interpreted as the probability of being detected by agent i under the condition that intruder invades region Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2, there are three layers of area to cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The color of the star represents the detection state of the agent, and the color of the circle represents whether the agent is a free agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Blue circle means this agent is performing detect task, and yellow circle means this agent is a free agent and moving to its target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Red star means that this agent does not allow other agents to enter this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Green star means that this agent allows other agent enter this layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And the star will turn red if ci = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Every free agent has a target layer, we use ki to denote the target layer of agent i, and how to help the agent find the target layer is the basis of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We present Algorithm 3 to help the agent achieve this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Our idea is that all agents target the first layer first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If you cannot enter to the first layer, consider the second, and so on, all the way to the K∗-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When considering whether to take the k-th layer as the target layer, if there is no agent at the k-th layer, agent i will choose to target at the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If there are agents in k-th layer, agent i needs to predict whether the agent at layer k can allow entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Now, we need to consider how to let an agent enter a layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As mentioned above, whether an agent can enter the layer depends on the agent in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we use the Algorithm 4 to realize this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In Algorithm 4, in order to prevent the phase of the agent from being the same, resulting in the difficulty of setting subsequent division points, the agent will change its phase when it finds the phase is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' To avoid agents being preempted by other agents when they change phase, we need to set ai = 1 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It is easy to find that the Algorithm 1 requires agent α and agent β to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, 11 Algorithm 3 Target Layer Identification Algorithm Agent i performs as follows 1: for k = 1 : K∗ do 2: if Nk = 0 then 3: ki = k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: else 5: for j ∈ INk do 6: if (ϕi, Rk(ϕi)) ∈ Ej then 7: if cj = 0 then 8: ki = kj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 9: end if 10: end if 11: end for 12: end if 13: end for 14: return ki we design the Algorithm 5 to find the agent α and agent β for agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In Algorithm 5, we design a operation similar to (1) as follows δ∗(θ1, θ2) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 θ1 − θ2 − 2π if θ1 − θ2 > π θ1 − θ2 + 2π if θ1 − θ2 ≤ −π θ1 − θ2 else (20) Actually, δ(θ1, θ2) = |δ∗(θ1, θ2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The operation can calculate the phase difference from θ1 to θ2 in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can find that δ∗(θ1, θ2) ∈ (−π, π], which is difficult to to compare in algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we use Ψ(cos(δ∗(θ1, θ2)), sin(δ∗(θ1, θ2))) to let all the phase differences be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Then, by finding the minimum of these, the agent α in the counterclockwise direction can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the same way, we can also get the agent β in the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the neighbor agent α in clockwise direction changes, the division point si should also change accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The same is true for counterclockwise which does not change the division point si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When agent α does not change, agent i can execute Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' This ensures that Algorithm 1 works efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In addition, the importance of each layer should be different from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In general, 12 Algorithm 4 Entry Request Algorithm 1: if Nk = 0 then 2: ai = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 3: α = β = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: si = ϕi + π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 5: if si > 2π then 6: si = si − 2π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 7: end if 8: else 9: for j ∈ INk do 10: if (ϕi, Rk(ϕi)) ∈ Ej then 11: if cj = 1 then 12: ki = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 13: else 14: ai = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 15: while ϕi = ϕj do 16: Move with (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 17: Calculate ϕi by (2) and (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 18: end while 19: end if 20: end if 21: end for 22: end if the more important the inner layer is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, when the number of agents is insufficient, the inner layer should be covered first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3, when the outer agent finds that the inner agent needs help, even if it is working well, it will leave the outer layer and head to the inner layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use Algorithm 7 to realize this function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When agent i discovers ϕi ∈ Ej, aj = 1 and cj = 0 of the inner agent, the agent i transforms itself into a free agent, and the inner layer is the target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, we present the multi-layer barrier coverage algorithm in Algorithm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When agent i does not have a target layer, the agent will first find a target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If agent i cannot find the target layer with the phase unchanged, the agent will change its phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' After the agent finds 13 Algorithm 5 Neighbor Seeking Algorithm Input: INk, agent i, j state Output: α, β 1: for j ∈ INk&& j ̸= i do 2: if Nk = 2 then 3: α = β = j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: else 5: if Ψ(cos(δ∗(ϕi, ϕj)), sin(δ∗(ϕi, ϕj))) < Ψ(cos(δ∗(ϕi, ϕα)), sin(δ∗(ϕi, ϕα))) then 6: α = j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 7: end if 8: if Ψ(cos(δ∗(ϕj, ϕi)), sin(δ∗(ϕj, ϕi))) < Ψ(cos(δ∗(ϕβ, ϕi)), sin(δ∗(ϕβ, ϕi))) then 9: β = j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 10: end if 11: end if 12: end for 13: Return α and β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' (a) (b) Figure 3: Diagram of agent moving to other layer the target layer, the agent moves to the target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the agent reaches the target layer, it will request to enter the target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If the request is rejected, the agent looks for another target layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the request is granted, the agent enters the layer to perform the coverage task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In order to ensure the smooth progress of the algorithm, the intelligent experience obtains the neighbor information at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, when the agent finds that the inner layer needs help, it stops coverage and helps the inner layer instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use P to denote the detection 14 Algorithm 6 Division Point Set Algorithm Initializate: z = α 1: Run Algorithm 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 2: if z ̸= α then 3: The division point is set as si = ϕi − 1 2δ(ϕi, ϕα);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: if si < 0 then 5: si = si + 2π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 6: end if 7: else 8: Run Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 9: end if Algorithm 7 Weight-based Layer Change Algorithm 1: if ki > 1 then 2: Run Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 3: for j ∈ INki−1 do 4: if ϕi ∈ Ej && aj = 1 && cj = 0 then 5: Set ki = kj and ai = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 6: end if 7: end for 8: end if probability of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' P is calculated as follows P = 1 − Nk � k=1 (1 − Pk), (21) where Pk is the detected probability of layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the next section, we will theoretically demonstrate the effectiveness of the proposed algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 3 Main Results Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For fixed agents position, the set of midpoints of T guarantees the maximum of joint monitoring probability H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 15 Algorithm 8 Multi-layer Barrier Coverage Algorithm Initializate: k = 1, ai = 0, ci = 0 For i ∈ IN, i-th agent performs as follow 1: while ai = 0 do 2: while ki = 0 do 3: Run Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4: Run Algorithm 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 5: Move with (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 6: end while 7: while pi /∈ rki do 8: Move to the target layer with (17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 9: end while 10: Run Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 11: Run Algorithm 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 12: while ai = 1 do 13: Run Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 14: Run Algorithm 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 15: Run Algorithm 7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 16: Update ci with (18) and (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 17: end while 18: end while Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' By taking the partial derivative of H with respect to si, one gets ∂H ∂si = [f (d(ϕi−1, si)) − f (d(ϕi, si))] ρ (si) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It is observed that if f(d(ϕi−1, si)) = f(d(ϕi, si)) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', N, we can get ∂H ∂S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Ac- cording to the description of the properties of f(·) in Section II, this means that ∂H ∂S = 0 can be achieved with only d(ϕi−1, si) = d(ϕi, si) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since the distinct agents position, d(ϕi−1, si) = d(ϕi, si) means division point is the midpoint of Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, the Hessian matrix of the function of coverage quality (7) satisfies ∇2H = [ ∂2U ∂si∂sj ] ∈ RN×N = diag(α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', αN), 16 where αi = (∂f(d(ϕi−1,si)) ∂si − ∂f(d(ϕi,si)) ∂si )ρ(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since f(·) is monotonically decreasing and d (ϕi, si) ∝ δ(ϕi, si), combining with equation (1), we can get ∂f(d(ϕi−1,si)) ∂si < 0 and ∂f(d(ϕi,si)) ∂si ρ(si) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' This means αi < 0 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get ∇2H < 0, which implies this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Dynamic system (9) and (14) ensure that the function (7) reach the local max- imum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Construct the following Lyapunov function V (ϕ, S) = 1 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since si ∈ [0, 2π), f(d(ϕi, θ)) > 0 and ρ(θ) ≥ 0, we can find that V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, f(d(ϕi, θ)) and ρ(θ) are bounded, which implies V1 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Taking the derivative of the Lyapunov function, we find that ˙V (ϕ, S) = − 1 H2 · ˙H, From (7), we can find that V (ϕ, S) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='The time derivative of (7) with respect to the compound dynamics (9) and (14) is given by ˙H = N � i=1 ∂H ∂ϕi ωi + N � i=1 ∂H ∂si ˙si = N � i=1 �� Ei ∂f (d(ϕi, θ)) ∂ϕi ρ (θ) dθ �2 + κs N � i=1 [f (d(ϕi−1, si)) − f (d(ϕi, si))] (d(ϕi, si) − d(ϕi−1, si)) ρ (si) Since [f (d(ϕi−1, si)) − f (d(ϕi, si))] (d(ϕi, si) − d(ϕi−1, si)) ≥ 0, we can find that dH dt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' On the other hand, the derivative of Lyapuonv function satisfies ˙V1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' According to local invariant set theorem, the state of the system will converge to the set of {(ϕ, s)| ˙V1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From the equation (7), we can know that H is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, if and only if dH dt = 0, ˙V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And in this case, the division points are located in the middle of the arc lengths between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the meantime, agents are located in the set that {ϕi| � Ei ∂f(d(ϕi,θ)) ϕ ρ(θ)dθ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, V reach the local minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' On the other hand, H reach the local maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For a working agent i, the agent i will never collide with the division point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We assume that the agent i enters the layer at time t∗, we can get the following relation- ship δ∗(sβ, ϕi) > 0, δ∗(ϕi, sα) > 0 17 si s s 0 Figure 4: Diagram of the phase location of agent i Let us first consider that agent i will not collide with division point sβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4, we use E1 i , E2 i , E3 i , E4 i to denote the area between two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use Lβ to denote the distance between agent i and division point sβ, and Lβ is represented as follows Lβ = K(δ∗(sβ, ϕi)), where K(·) is a class K function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Next, in combination with Equation (20), we take the derivative of Lβ to get ˙Lβ = K1 · ( ˙sβ − ˙ϕi) = K1 · (κs(δ∗(ϕβ, sβ) − δ∗(sβ, ϕi)) − � Ei ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ) = K1 · (κs(δ∗(ϕβ, sβ) − δ∗(sβ, ϕi)) − � E3 i ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ − � E2 i ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ) where K1 = ∂K(δ∗(sβ,ϕi)) ∂δ∗(sβ,ϕi) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As ϕi −→ sβ, we can find that � E3 i ∂f(d(ϕi,θ)) ∂ϕi ρ(θ)dθ −→ 0, ∂f(d(ϕi,θ)) ∂ϕi < 0 in the range of E2 i , and δ∗(ϕβ, sβ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get that ˙Lβ > −K1 · κs(δ∗(sβ, ϕi)), which means that Lβ > 0 holds within the interval of t ≥ t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In the same way, we use the Lα to denote the distance between agent i and division point si, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='Lα = K(δ∗(ϕi, si)), and we can 18 also get the following ˙Li = K2 · ( ˙ϕi − ˙si) = K2 · ( � Ei ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ − κs(δ∗(ϕi, si) − δ∗(si, ϕα))) = K2 · ( � Ei ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ − κs(δ∗(ϕi, si) − δ∗(si, ϕα))) = K2 · ( � E2 i ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ + � E3 i ∂f(d(ϕi, θ)) ∂ϕi ρ(θ)dθ + κsδ∗(si, ϕα) − κsδ∗(ϕi, si)) where K2 = ∂K(δ∗(si,ϕi)) ∂δ∗(si,ϕi) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As ϕi −→ si, we can find that � E2 i ∂f(d(ϕi,θ)) ∂ϕi ρ(θ)dθ −→ 0, ∂f(d(ϕi,θ)) ∂ϕi > 0 in the range of E3 i , and δ∗(si, ϕα) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get that ˙Lα > −K2 · κs(δ∗(ϕi, si)), which means that Lα > 0 holds within the interval of t ≥ t∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Because of Lα > 0 and Lβ > 0, the agent will not collide with the division point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The division points never collide with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2, we can know that the distance between division point si and sβ can be denoted as Li = Lα + Lβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Obviously, Li is the length of Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get that Li > 0 holds within the interval of t ≥ t∗, which implies this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We use Lk to denote the length of layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' To demonstrate our conclusion, we discover the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For agent i working at layer k, if Li = min{j ∈ INk|Lj}, we have Li ≤ Lk Nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From Table 6, The region of the k-th layer will be divided without remainder by the agents working in the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get the following relation Lk = � j∈INk Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3, we have Li > 0, for i ∈ INk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Since Li = min{j ∈ INk|Lj}, we have Lj ≥ Li, for j ∈ INK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, the above formula can be rewritten as Lk = � j∈INk Lj ≥ Nk × Li which means that Li ≤ Lk Nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 19 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For agent i working at layer k, as t −→ ∞, if Li = max{j ∈ INk|Lj} and Nk ≥ 2, we have Lk Nk ≤ Li ≤ Lk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Similar to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4, since Li = max{j ∈ INk|Lj}, we have Lj ≤ Li for j ∈ INK, which means that Li ≥ Lk Nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Figure 4, we use li to denote the length of E3 i ∪ E4 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can get the following relation Lk = � j∈INk lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 and control input (14), as t −→ ∞, the agent i have following relation Li = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5(li + lα), Since li + lα ≤ Lk, we get Li ≤ Lk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In our algorithm, the number of agents on the k-th layer is not fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Obviously, we can get a relation as follows Pk(t) ≥ 0, for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For the layer k, if the agent i leaves this layer and Nk ≥ 2, the maximal reduction of the detect probability of the k-th layer can be calculated as follows Pk′ = � E2 i (f (d (ϕi, θ)) − f (d (si, ϕi) + d (si, θ))) ρ (θ) dθ + � E3 i (f (d (ϕi, θ)) − f (d (sβ, ϕi) + d (sβ, θ))) ρ (θ) dθ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In our algorithm, when the agent i enters or leaves, there is only a change in the moni- toring probability of E2 i and E3 i for all regions in layer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We assume that agent i leaves layer k at time ti, and the new division point is at the position of ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, then we can get the following formula Pk(ti + ε) ≥ Pk(ti) − � E2 i (f (d (ϕi, θ)) − f (d (ϕα, θ))) ρ (θ) dθ + � E3 i (f (d (ϕi, θ)) − f (d (ϕβ, θ))) ρ (θ) dθ, where ε is an infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From Table 1, we can know that si and sβ will converge to the midpoint of lα and li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4, the length of E1 i is the same as that of E2 i , and the 20 length of E3 i is the same as that of E4 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, the variation of the detect probability of the k-th layer can rewritten as follows Pk′ ≥ � E2 i (f (d (ϕi, θ)) − f (d (si, ϕi) + d (si, θ))) ρ (θ) dθ + � E3 i (f (d (ϕi, θ)) − f (d (sβ, ϕi) + d (sβ, θ))) ρ (θ) dθ, which implies this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For the layer k, if the agent i enters this layer, the minimal increase of the detect probability of the k-th layer can be calculated as follows P ′ k = � E2 i ∪E3 i f (d (ϕi, θ)) ρ (θ) dθ − � sβ s∗ β f (d (ϕβ, θ)) ρ (θ) dθ − � s∗ β si f (d (ϕα, θ)) ρ (θ) dθ where s∗ β is the division point in lα before agent i enters layer k, and if Nk = 0, then d(ϕα, si) = 0, d(ϕα, θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6 said, agent entry will only change the detect probabilities of E2 i and E3 i in the k-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Assuming that the agent enters layer k after time ti, We can know the detect probability of this area as follows Pk(ti) = P ∗ k (ti) + � s∗ β si f (d (ϕα, θ)) ρ (θ) dθ + � sβ s∗ β f (d (ϕβ, θ)) ρ (θ) dθ where P ∗ k indicates that the k-th layer does not consider the monitoring probability of E2 i and E3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' After the agent i enters the k layer, from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 the above formula is rewritten as Pk(ti + ε) ≥ P ∗ k (ti + ε) + � E2 i ∪E3 i f (d (ϕi, θ)) ρ (θ) dθ where P ∗ k (ti + ε) = P ∗ k (ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get P ′ k as follows P ′ k ≥ � E2 i ∪E3 i f (d (ϕi, θ)) ρ (θ) dθ − � sβ s∗ β f (d (ϕβ, θ)) ρ (θ) dθ − � s∗ β si f (d (ϕα, θ)) ρ (θ) dθ which implies this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' According to the above conclusions, we can get the following theorem Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For a multi-agent multi-layer barrier coverage system with Nk layers, if the agent i working on the k-th layer satisfies the following inequality, P ′ k < (1 − Pk)P ′ v 1 − Pv − P ′v the detection probability (21) of the system will increase if the agent i enters the v-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Without loss of generality, we can assume that when the multi-agent coverage system is at time t1, agent i works at layer k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' when the system is at time t2, agent i works at layer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And, at the two moments, except that the working place of agent i is different, other agents are still working in the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Therefore, we can get the following equation by (21) P(t1) = 1 − (1 − P1)(1 − P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − PNk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='7, we can get the following equation P(t2) ≥ 1 − (1 − P1)(1 − P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pk + Pk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pv − Pv′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − PNk) let 1 − (1 − P1)(1 − P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pk + Pk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − Pv − Pv′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='(1 − PNk)) > P(t1), we can get P(t1) > P(t2), which implies this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Simplify the above formula to get P ′ k < (1 − Pk)P ′ v 1 − Pv − P ′v This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' For a single-layer barrier coverage system with fixed division points, when the layer is a circle with a radius R0, d adopts the geodesic distance obtained on the layer and the detection model of the agent is a Gaussian probability model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' f(d) = e−d2/γ2 if the radius R0 satisfies R0 ≤ √ 2γ 2π , dynamic system (9) ensure that the function (7) reaches the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' By taking the partial derivative of (7) with respect to ϕi, we get ∂H ∂ϕi = � Ei ∂f (d(ϕi, θ)) ∂ϕi ρ (θ) dθ Substituting f(d) = e−d2/γ2 into the above formula yields ∂H ∂ϕi = � E1 i e− d2 γ2 � −2 d γ2 � ∂d ∂ϕi ρ (θ) + � E2 i e− d2 γ2 � −2 d γ2 � ∂d ∂ϕi ρ (θ) The integral is segmented because the geodesic distance d is not derivable when θ = ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And ∂d ∂ϕi = Ro as θ ∈ E2 i , ∂d ∂ϕi = −Ro as θ ∈ E3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We take the partial derivative of the above formula with respect to ϕi to get ∂2H ∂ϕi2 = � E2 i e − d2 γ2 � 4d2 γ4 − 2 γ2 � � ∂d ∂ϕi �2 ρ (θ) dθ + � E3 i e − d2 γ2 � 4d2 γ4 − 2 γ2 � � ∂d ∂ϕi �2 ρ (θ) dθ 22 From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5, we can get d ≤ πR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' If R0 ≤ √ 2γ 2π , we have ∂2H ∂ϕi2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, the Hessian matrix of the function of coverage quality (7) satisfies ∇2H = [ ∂2U ∂ϕi∂ϕj ] = diag(∂2H ∂ϕ2 1 , ∂2H ∂ϕ2 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=', ∂2H ∂ϕ2 N ) ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' This means H has a unique maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Combining with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, the dynamic system (9) will ensure the function (7) reaches the maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4 Case Studies In this section, we will give some simulation and experiment results to verify our coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We implemented our algorithm on MATLAB 2022a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Now, we give the multi-agent barrier coverage algorithm in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 Numerical simulation We designed 3 layers of area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' There are 50 agents needs to cover on these three layers to monitor the invasion of intruders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' These three layers are designed as follows \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 R1(θ) = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 sin(4θ), R2(θ) = 2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 sin(10θ), R3(θ) = 3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 sin(40θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' (22) The probabilistic model is given by f(d(ϕi, θ)) = exp(−d(ϕi, θ)2), where the distance function d is calculated as follows d(ϕi, θ) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ���� � θ ϕi � R(θ)2 + R′(θ)2dθ ���� , if ���� � θ ϕi � R(θ)2 + R′(θ)2dθ ���� ≤ Lki 2 Lki − ���� � θ ϕi � R(θ)2 + R′(θ)2dθ ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' otherwise where d is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' The density function is ρ(θ) = θ 2π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We set the adjustable parameters as follows \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 κr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1, κω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='01, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5, we place the agent inside the innermost layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' All agents gradually expand outwards, and finally cover all three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' And we intercept the position results of the algorithm at 4 time points, which are 0s, 8s, 16s and 24s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 23 4 2 0 2 4 (a) 4 2 0 2 4 t=0s 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 4 2 0 2 4 (b) 4 2 0 2 4 t=8s 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 4 2 0 2 4 (c) 4 2 0 2 4 t=16s 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 4 2 0 2 4 (d) 4 2 0 2 4 t=24s 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 Figure 5: Snapshots of simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Circles denote the mobile agents, and the stars refer to the division points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5, when the algorithm first starts running, all agents are in the innermost inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' After the algorithm runs for 8 seconds, 6 agents have been covered on the first layer, and some agents have moved to the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Combined with Figure 7, after the algorithm runs for about 13 seconds, the detect probability of the third layer decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the algorithm runs to 16 seconds, we find that some agents are moving from the third layer to the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' This is because the Algorithm 7, when the inner agent is not well qualified for its detection task, the outer agent will leave the outer layer and go to the inner layer to help the inner agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the algorithm runs for 24 seconds, the multi-agent systems is basically stable, and most of the agents are already working on the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Around the circle with a radius of 4, some agents are patrolling, looking for any agents that need help, and when found, these 24 4 2 0 2 4 4 3 2 1 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 Figure 6: The final result of the system state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' patrolling agents will take action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, we give the results of the algorithm running to the last moment of the system in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can find that on each layer, the agents are denser where the invasion probability is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Moreover, there are still free agents patrolling the circle of radius 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 7, we show how the detection probability of the system and each layer changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We can see that when the second and third layers have no agents, the total detection probability is the same as that of the first layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the second layer and the third layer have agents working one after another, the monitoring probability of the agents has a significant increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, it can be found that the detection probability of the multi-layer fence coverage algorithm exceeds 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We also did controlled experiments with multi-layer barrier coverage and single layer barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='8, the detection probability of the multi-layer barrier coverage was inferior to that of the single-layer fence cover for the initial period, but once agents moved to the second layer, the detection probability of the multi-layer barrier coverage reversed to that of the single-layer fence cover, and was higher than that of the single-layer for the rest of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' We counted the final detection probability of single-layer barrier coverage and multi-layer barrier coverage with different number of smart bodies, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' It can be found that 25 0 10 20 30 40 50 60 70 80 90 100 Time(s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9 1 Detection probability Layer1 Layer2 Layer3 Total 10 12 14 16 18 20 Time(s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9995 1 Detection probability Layer1 Layer2 Layer3 Total Figure 7: Detection probability of each layer and total system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 90 100 Time(s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9 1 Detection probability Single-layer Multi-layer 5 10 15 20 25 Time(s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='99 1 Detection probability Figure 8: Difference between single layer barrier coverage and multi-layer barrier coverage for the same number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 26 0 5 10 15 20 25 30 35 40 45 50 Agent Number 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='9 1 Detection probability 20 30 40 50 Agent Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='998 1 Detection probability Figure 9: Difference in detection probability between single and multi-layer barrier coverage for the same number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' there is no difference in the detect probability between single and multi-layer barrier coverage when the number of agents is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' However, the detection probability of the multi-layer barrier coverage is significantly higher than that of the single-layer barrier coverage when the number of agents gradually increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the number of smart bodies is large enough, the increase in the number of smart bodies is of little help to the single-layer barrier coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' When the number of agents is 50, the detection probability of single-layer barrier coverage reaches 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content='8 percent, while the detection probability of multi-layer barrier coverage is very close to 100 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 5 Conclusions This paper presented a distributed multi-agent barrier coverage algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' First, a single-layer barrier coverage quality function was designed based on the probabilistic model of intrusion and a single-layer barrier coverage algorithm was designed based on the gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Then a layer-to-layer adjustment mechanism was proposed based on the single-layer algorithm, which adjusts the number of agents on each layer so that the coverage quality of the whole system was improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Then some theoretical analyses were given to theoretically verify the stability and 27 effectiveness of the single-layer algorithm and the necessity of the multi-layer algorithm, and the theoretical results were given in some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' Finally, the effectiveness of our algorithm was verified by simulation and the practicality of the algorithm was verified by 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9A0T4oBgHgl3EQfHf82/content/2301.02061v1.pdf'} diff --git a/edFJT4oBgHgl3EQfTCwr/content/tmp_files/2301.11502v1.pdf.txt b/edFJT4oBgHgl3EQfTCwr/content/tmp_files/2301.11502v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d4f3edaf521d24a9b3c2be977794aeb90283c2c --- /dev/null +++ b/edFJT4oBgHgl3EQfTCwr/content/tmp_files/2301.11502v1.pdf.txt @@ -0,0 +1,1694 @@ +A Mixed-integer Linear Formulation for Dynamic Modified Stochastic +p-Median Problem in a Competitive Supply Chain Network Design +Amir Hossein Sadeghia,∗, Ziyuan Sunb, Amirreza Sahebi Fakhrabada, Hamid Arzanic, Robert B. +Handfieldd +aDepartment of Industrial and Systems Engineering, North Carolina State University, NC, USA +bDepartment of Mechanical, Industrial and Aerospace Engineering, Concordia University, Qu´ebec, Canada +cRotman School of Management, University of Toronto, Ontario, Canada +dDepartment of Business Management Poole College of Management, North Carolina State University, NC, USA +Abstract +The Dynamic Modified Stochastic p-Median Problem (DMS-p-MP) is an important problem in +supply chain network design, as it deals with the optimal location of facilities and the allocation +of demand in a dynamic and uncertain environment. In this research paper, we propose a mixed- +integer linear formulation for the DMS-p-MP, which captures the key features of the problem +and allows for efficient solution methods. The DMS-p-MP adds two key features to the classical +problem: (1) it considers the dynamic nature of the problem, where the demand is uncertain +and changes over time, and (2) it allows for the modification of the facility locations over time, +subject to a fixed number of modifications. The proposed model is using robust optimization in +order to address the uncertainty of demand by allowing for the optimization of solutions that are +not overly sensitive to small changes in the data or parameters. To manage the computational +challenges presented by large-scale DMS-p-MP networks, a Lagrangian relaxation (LR) algorithm +is employed. Our computational study in a real-life case study demonstrates the effectiveness of +the proposed formulation in solving the DMS p-Median Problem. According to the results, the +number of opened and closed buildings remains unchanged as the time horizon increases. This is +due to the periodic nature of our demand. This formulation can be applied to real-world problems, +providing decision-makers with an effective tool to optimize their supply chain network design in +a dynamic and uncertain environment. +Keywords: +p-Median Problem; Supply Chain Network Design; Dynamic Allocation; Robust +Optimization; Lagrangian Relaxation +1. Introduction +Mobile grocery stores, also known as grocery trucks (GT) or pop-up grocery stores, have +emerged as a new trend in the retail industry. GTs typically stock a wide variety of fresh fruits +and vegetables, as well as other grocery items. They are usually equipped with refrigeration units +to keep the food fresh and have a payment system in place for customers to buy the products. +These mobile stores bring convenience and accessibility to customers in under-served or rural +areas, or in areas where traditional brick-and-mortar grocery stores are not present Fahlevi et al. +(2019); Nader (2018). The trend of grocery trucks is expected to continue to grow in the coming +∗Corresponding author +Email address: asadegh3@ncsu.edu (Amir Hossein Sadeghi) +Preprint submitted to Logistics, MDPI +January 30, 2023 +arXiv:2301.11502v1 [math.OC] 27 Jan 2023 + +years as more and more people look for ways to improve access to healthy food in their communities. +In the U.S., the mobile food vendors market is valued at 1.16 billion dollars in 2021 and is predicted +to grow at a rate of 6.4% annually from 2022 to 2030, driven by the increasing trend of culinary +arts and the preference of young people for different dining experiences over the traditional dining +in restaurants Research (2022). +GTs can also be used to provide food to people in emergency situations, like natural disasters or +power outages Hecht et al. (2019); Pelling (2001), as well as address food insecurity by increasing +access to healthy food options. GTs are valuable resources for alleviating food insecurity, and these +trucks are often operated by non-profit organizations, local governments, and community groups +that want to address food insecurity and promote healthy eating Mohan et al. (2013). They can +help to address food insecurity in several ways: 1. Accessibility: Grocery trucks can bring fresh +produce and other grocery items to low-income or rural areas where there is limited access to +traditional grocery stores and supermarkets. This makes it easier for residents in these areas to +access healthy food options; 2. Affordability: Many grocery trucks accept SNAP (Supplemental +Nutrition Assistance Program) benefits and other forms of government assistance, making it more +affordable for low-income families to purchase healthy food; 3. Education: Some grocery trucks +provide educational programs and cooking demonstrations that teach customers how to prepare +healthy meals. This can help to improve the overall health and nutrition of the community; 4. +Variety: Grocery trucks can also offer a wider variety of fresh produce and other grocery items +than traditional corner stores or convenience stores, which may only carry a limited selection of +products. +Overall, grocery trucks are a creative and innovative way to address food insecurity and improve +access to healthy food in communities that need it the most. However, the success of a mobile +grocery store is heavily dependent on the location where it is parked Esparza et al. (2014); Wessel +(2012). Various factors should be considered when selecting a location for a grocery/food truck, +such as dynamic demand, visibility, accessibility, and competition. +Logistics for mobile grocery stores refer to the process of planning, coordinating, and controlling +the movement and storage of goods, services and information from the point of origin to the point of +consumption. This includes transportation, inventory management, warehousing, and distribution +of products. +For mobile grocery stores, logistics also includes the planning and coordination +of the routes and schedule of the mobile store, as well as the management of the supply chain +and inventory. This includes sourcing products from suppliers, managing inventory levels and +restocking the mobile store as needed, and coordinating delivery schedules. +Effective logistics management is crucial for the success of mobile grocery stores, as it can impact +delivery times, cost, and overall customer satisfaction. By optimizing logistics, mobile grocery +store owners and operators can improve their business by reducing costs and increasing efficiency, +ultimately resulting in better customer service Restuputri et al. (2022); Bourlakis and Weightman +(2008). In this regard, we first provide a comprehensive literature review of the proposed location- +allocation models in the literature in section 2, then provide a mathematical formulation for our +model considering the uncertainty of demand in section 3. Sections 4 and 5 show the application +of our proposed model in the real-world mobile grocery location problem and provide insights, +respectively. +2. Literature Review +Location science acts on a significant role in modern development in different disciplines, in- +cluding business, economics, computer science, geography, military, transportation, etc. +With +2 + +historical records, the first optimal location problem was proposed by Pierre de Fermat to find +the geometric median among three points. The problem was solved by Evangelista Torricelli soon +after. This is undoubtedly logical that the single facility minimum Euclidean distance problem +is named Fermat-Torricelli problem (Known as the most famous Weber problem, the distances +are looked upon as the weights of nodes). Laporte et al. (2019) is a book that comprehensively +introduces the development and forecast of the subject of location science. +The beginning of the new era of location science was around the 1960s. Berge (1957) investigates +the problem of finding the minimum coverage on a graph originally. Miehle (1958); Cooper (1963) +raises the best known p-median problem, which is to pick the exact p of facilities to open that +minimizes the total transportation cost. Followed by, Hakimi (1964, 1965) releases Hakimi’s node +optimally theorem to demonstrate that the optimal solution on a continuous graph for absolute +median problems is always on the graph’s vertices, which means many network problems can simply +transfer to discrete version problems. Hakimi (1964) introduces the concept of absolute center to +find the decision of satisfying the mini-max. As for solving more complex realistic optimization +tasks in the next decade, mixed-integer linear programming (MILP) was widely used to deal with +location problems. Balinski (1965); ReVelle and Swain (1970); Toregas et al. (1971) are early works +to use MILP to solve the uncapacitated facility location problem, discrete p-median problem, and +covering-location problem, respectively. +The deterministic model means all model parameters are known with certainty; nevertheless, +unpredictability or randomness always occurs in real life. Stochastic programming or robust opti- +mization is the solution to reduce the impact of uncertainties in predicting. Researchers have been +quite interested in topics to combine location models with stochastic programming methods in the +last few decades. Louveaux (1986); Louveaux and Peeters (1992) modify the uncapacitated facil- +ity location and p-median problems with stochastic variables, including the customer’s demand, +the unit cost of satisfying the customer from a specific location, and the fixed cost of locating at +the candidate location. Laporte et al. (1994) assumes the customer’s demand is stochastic for the +capacitated facility location problem. Current et al. (1998) studies the number of opening facilities +is uncertain. For improving customer service, based on p initial opening facilities, Berman and +Drezner (2008) allows increasing r new opening facilities depending on the customer’s demand, +where 0 ≤ r ≤ q. Sonmez and Lim (2012) is the opposite of the previous article, r existing facilities +have to be closed, where 0 ≤ r ≤ q ≤ p. Liu and Song (2022) considers the covering-type location +problems under demand uncertainties. Snyder (2006); Correia and Saldanha-da Gama (2019) are +overviews of facility location under uncertainty. +In reality, the vast majority of practical location problems can take into consideration time. +The setting of the problem does not rely on time, called ‘single-period’ or ‘static’; conversely, it is +called ‘multi-period’ or ‘dynamic’ if the decision varies with different parameters at each period. +Ballou (1968); Sweeney and Tatham (1976) are early works about locating and re-locating a single +facility within a time span. Scott (1971) continuous the previous works for locating multiple facili- +ties. Wesolowsky (1973) merges the conception of the Weber problem with the multi-period facility +location problem to find the best opening facility in each time period. Warszawski (1973); Cavalier +and Sherali (1985); Drezner (1995); Hakimi et al. (1999) studies the dynamic uncapacitated fa- +cility location problem, location problems on networks, network p-median problems, and network +center problems, respectively. Wesolowsky (1973); Wesolowsky and Truscott (1975); Galv˜ao and +Santiba˜nez-Gonzalez (1992); Dias et al. (2007) involve facilities’ opening and closing costs in mod- +els. Moreover, Ahmed and Garcia (2003); Romauch and Hartl (2005); Marques and Dias (2013) +present dynamic location problems under uncertain environments. Nickel and Saldanha-da Gama +(2019) is a survey of dynamic facility location problems. +3 + +In the past century, selecting an optimal retail location in a competitive environment has +been much discussed. The concept of stability in the competition was first proposed in Hotbllino +(1929). Reilly (1931); Converse (1949) are early works that mention the retailing models about +the relationship between customer consumption, the size of the facility, distance, et cetera. Huff +(1964, 1966) presents further progress: Huff’s formulation of the competitive function of allocating +to customers; depends on the facility’s attraction and distance to the customer (Jiang et al. (2019); +Liang et al. (2020); Jiang et al. (2021) collect social media data to evaluate the facility’s attraction +to coincide with the current trend). On account of Huff’s formulation, Nakanishi and Cooper (1974) +adds more possible factors to extend the original formulation with the least squares approach. +Drezner and Zerom (2023) is a recent paper discusses circumstances in the competitive environment +to avoid cut-throat peer competition that eliminates existing facilities if a new facility is allocated. +To sum up, Ghosh and MacLafferty (1987); Eiselt et al. (1993); Ashtiani (2016); Eiselt et al. (2019) +are surveys of the competitive location models. +Emerging as the times require, applications of relocatable facilities have the benefits of adapting +to the times to optimize the cost and meet different needs. Many multifarious applications have +been studied for relocatable facilities. Cho and Lee (2021); Bhandawat (2018); Kalra et al. (2022) +study location optimization problems for mobile vendors or food trucks in interurban environments +to serve more customers. Adler et al. (2014) focuses on allocating police routine patrol vehicles on +the city streets to improve public security. Contardo et al. (2012) investigates practical problems +for the bike-sharing system: there is no available bike for renting and capacitated stations without +an empty spot for returning at rush hour. +The article provides a solution to the dynamical +model to balance the inventory of unused bikes among stations with adequate inventory and short +inventory. R˘aboac˘a et al. (2020) lies their research on how to temporarily locate transportable +charging stations for the electric vehicle under potential constraints with optimal charging demand. +Glaeser et al. (2019) interrelates to the distribution problem of E-commerce. Due to the high charge +of door-to-door delivery, there is an innovative solution that customers can order online first, then +pick up offline at a location where delivery trucks allow parking and maximize the profit of the +logistics company. An analogous application is given in Cao and Qi (2022) about the unmanned +market on wheels for stall economy; dis-similarly, the emphasis is on planning routes based on the +reformative vehicle problem model. +3. Modelling Process and Methods +This section presents the development of a mixed-integer linear programming (MILP) model +to address the research problem. +3.1. Notation list +The following are the notations used in the formulation of the model, including sets, parameters, +and decision variables: +4 + +Table 1: Notations used for mathematical modeling of DMS-p-MP +Sets +T +Set of time periods in the planning horizon; t ∈ {1, 2, . . . , |T|} +K +Set of categories (groups) in the area +B +Set of candidate locations; i, j ∈ {1, 2, . . . , |B|} +where based on our problem, the candidate locations (j) are the same as demand nodes (i). +Parameters +cij +Unit cost of satisfying demand of location i from facility j +γo +Mobile store’s opening cost in each location +γc +Mobile store’s closing cost in each location +dit +Demand of location i at day t +p +Available number of mobile stores in each day +mk +Maximum number of stores allowed in group k +nk +Minimum number of stores allowed in group k +Decision Variables +xijt +Fraction of demand of i that is supplied from j at day t +yjt +Binary variables that is 1 if a mobile store is located at j at day t, and is 0 otherwise +ajt +Auxiliary binary variables which 1 if a store is located in j +at day t and will not be located in j at day t + 1 (,i.e., closing variable), and is 0 otherwise +bjt +Auxiliary binary variable which is 1 if a store is not located in j +at day t and will be located in j at day t + 1 (,i.e., opening variable), and is 0 otherwise +The set of all candidate locations (buildings) in group k is denoted by Fk. We know B = +∪(Fk)k∈[K], that it means each building should be at least in one group. Note that ∩(Fk)k∈[K] can +be nonempty, meaning that one building can be a member of more than one group. Moreover, we +do not need to define opening and closing costs (ajt, bjt) as binary variables since our minimization +model and constraints force them to be 0 or 1. In order to have a feasible solution, we assume +that p must satisfy the following inequalities: +� +k∈K +nk ≤ p ≤ +� +k∈K +mk, +meaning that there is a solution that satisfies the model’s assumptions. +3.2. Mathematical Formulation +We defined all parameters and variables of our problem. The goal is to minimize the total cost +of the system. We can model our problem as follows: +5 + +Optimal Cost = min +� +t∈T +� +j∈B +� +i∈B +ditcijxijt + +� +t∈T\{|T|} +� +j∈B +(γcajt + γobjt) +(1a) +s.t. +� +j∈B +xijt = 1, +∀i ∈ B, t ∈ T, +(1b) +� +j∈B +yjt = p, +∀t ∈ T, +(1c) +xijt − yjt ≤ 0, +∀i, j ∈ B, t ∈ T, +(1d) +yjt − yj(t+1) − ajt ≤ 0, +∀j ∈ B, t ∈ T \ {|T|}, +(1e) +yj(t+1) − yjt − bjt ≤ 0, +∀j ∈ B, t ∈ T \ {|T|}, +(1f) +� +j∈Fk +yjt ≤ mk, +∀k ∈ K, t ∈ T, +(1g) +� +j∈Fk +yjt ≥ nk, +∀k ∈ K, t ∈ T, +(1h) +yjt ∈ {0, 1}, +∀j ∈ B, t ∈ T, +(1i) +ajt, bjt ≥ 0, +∀j ∈ B, t ∈ T, +(1j) +xijt ≥ 0, +∀i, j ∈ B, t ∈ T. +(1k) +where constraint (1b) ensures the demand of all locations over the time horizon must be satisfied, +constraint (1c) ensures the number of mobile stores allocated in each day should be p, constraint +(1d) shows the relation between continuous and binary variables, i.e., if xijt > 0, then yjt should +be 1. Constraints (1e) and (1f) ensure that we consider the closing and opening costs, respectively. +Constraints (1g) and (1h) ensure that we are following the policies regarding the limits on the +number of stores in each group. Finally, constraint (1i) is the binary constraint, and (1j), and (1k) +are the non-negativity constraints. We assume that at the beginning of day 1, dit is stochastic for +all i ∈ B and all t ∈ T. +3.3. Solution Approach +The problem at hand is a mixed-integer mathematical programming model that involves un- +certainty. We will outline a state-of-the-art methodology to tackle these characteristics in the +following manner: +3.3.1. Robust Optimization +In this section, an overview of the robust optimization approach proposed by Bertsimas and +Sim (2004) is presented. To do so, the following linear programming model is considered: +Min +� +j +cjxj +s.t. +� +j +˜aijxj ≤ bi; +∀i +xj ≥ 0; +∀j +(2) +6 + +where the technological coefficients ˜aij are assumed to be uncertain. +In other words, each +coefficient ˜aij is regarded as an independent, symmetric, and bounded parameter, which can take +values in [aij − ˆaij, aij + ˆaij], i.e. ˜aij ∈ [aij − ˆaij, aij + ˆaij]. In this definition, aij and ˆaij denote +the nominal value and the maximum deviation from the nominal value, respectively. Associated +with each row i in problem (1) is Ji, which is defined as the set of all coefficients in row i that are +subject to uncertainty. Furthermore, a scaled deviation ηij ∈ [−1, 1] is defined for each uncertain +coefficient ˜aij as ηij = ˜aij − aij +ˆaij +that represents the scaled perturbation of ˜aij from its nominal +value aij. +Bertsimas and Sim (2004) also introduced a parameter Γi ∈ [0, |Ji|] as the budget of uncertainty +for each constraint i, where |Ji| denotes the number of elements of set Ji Bertsimas and Sim (2004). +In fact, Γi is the maximum number of parameters that can really deviate from their nominal +values for each constraint i. The parameter Γi that bounds the total scaled deviation of uncertain +parameters as � +j∈Ji |ηij| ≤ Γi adjusts the robustness of the proposed method against the level of +solution conservatism. In particular, Γi = 0 represents the nominal or deterministic formulation, +whereas Γi = |ηij| relates to the worst-case formulation in which all uncertain parameters are +fixed at their worst-case values from the uncertainty set. However, the decision maker can make a +trade-off between the protection level of constraint i and the degree of conservatism of the solution +if Γi ∈ (0, |Ji|). Therefore, the budget of uncertainty Γi that is an input to the robust optimization +model can specify how risk-averse the decision-maker is. +Bertsimas and Sim (2004) proposed a nonlinear programming model as follows, which is equiv- +alent to the uncertain model (1): +Min +� +j +cjxj +s.t. +� +j +aijxj + max +Ω { +� +j∈Si +ˆaijxj + (Γi − ⌊Γi⌋)ˆaitixj} ≤ bi; +∀i +xj ≥ 0; +∀j +(3) +where Ω = {Si ∪ {ti}|Si ⊆ Ji, Si = ⌊Γi⌋ , ti ∈ Ji \ Si} is defined as the uncertainty set. For a +given optimal solution x∗ of the problem (2), Bertsimas and Sim (2004) demonstrated that the +protection function for constraint i against uncertainty, which is βi(x∗, Γi) = max +Ω {� +j∈Si ˆaijxj + +(Γi − ⌊Γi⌋)ˆaitixj} can be formulated as the following linear programming problem: +βi(x∗, Γi) = Max +� +j∈Ji +ˆaij|x∗ +j|ηij +s.t. +� +j∈Ji +ηij ≤ Γi; +∀i +0 ≤ ηij ≤ 1; +∀i, j +(4) +According to the theory of strong duality, since problem (3) is always feasible and bounded for +all Γi ∈ [0, |Ji|], its dual problem is feasible and bounded as well. Therefore, replacing the dual +problem of the problem (3) into (2), Bertsimas and Sim (2004) derived the robust formulation of +7 + +the uncertain linear programming problem (1) as follows: +Min +� +j +cjxj +s.t. +� +j +aijxj + βiΓi + +� +j∈Ji +µij ≤ bi; +∀i +βi + µij ≥ ˆaijxj; +∀i, j ∈ Ji +µij ≥ 0; +∀i, j ∈ Ji +βi ≥ 0; +∀i +xj ≥ 0; +∀j +(5) +where βi and µij are dual variables associated with the first and second constraints in programming +problem (3), respectively. +if the number of uncertain coefficients in constraint i that perturb from their respective nominal +values is less than or equal to Γi, then the optimal solution from the robust problem (4) will remain +always feasible. However, if more than Γi coefficients deviate from their nominal values, then the +probability of violating constraint i for an optimal solution x∗ +j is calculated as follows: +Pr( +� +j +˜aijx∗ +j < bi) ≤ 1 − ϕ(Γi − 1 +� +|Ji| +) +(6) +where ϕ(.) is the cumulative distribution function of a standard normal random variable. +The robust optimization equivalent of the proposed problem (1) is: +min +x,y,a,b max +d∈U dc1x + c2(a + b) +(7a) +s.t. (1b) − (1k) +(7b) +or equivalently: +min +x,y,a,bθ + c2(a + b) +(8a) +s.t. (1b) − (1k) +(8b) +θ ≥ max +d∈U dc1x +(8c) +where U is the uncertainty set for demand, and a, b are decision vectors for ajt, bjt respectively. +Depending on the problem setting, one can use the box, budget, or conic uncertainty set. Among +these, the budget uncertainty set is widely used, both due to its intuitive interpretation and +tractability. The budget uncertainty set for the demand is: +U = {d ∈ R|D|×|D| | d = d + γξ, ξ||∞ ≤ 1, +� +i,j∈D +ξit ≤ Γ} +where Γ is the uncertainty budget. Using this definition we have: +min +x,y,a,bθ + c2(a + b) +(9a) +s.t. (1b) − (1k) +(9b) +max +ξ∞≤1,ξ1≤Γ ξc1x ≤ θ − ¯dc1x +(9c) +8 + +Let’s focus on the left-hand-side of equation (3c): +max +ξ∞≤1,ξ||1≤Γ γξc1x → max +ξ +γξc1x, +s.t. : +� +i,t +|ξit| ≤ Γ, −1 ≤ ξit ≤ 1 +(10a) +Let ait = |ξit|, then we have: +max γξc1x +(11a) +s.t. +� +i,t +ait ≤ Γ +[π1] +(11b) +ξit ≤ 1 +[π2 +it] +(11c) +− ξit ≤ 1 +[π3 +it] +(11d) +ξit − ait ≤ 0 +[π4 +it] +(11e) +− ξit − ait ≤ 0 +[π5 +it] +(11f) +ξit ∈ R, ait ∈ R+ +(11g) +Since the model is a minimization problem, we need to obtain the dual problem whose objective +is also minimization: The dual problem is: +min +π +Γπ1 + +� +i,t∈D +(π2 +it + π3 +it) +(12a) +s.t. +π1 ≥ π4 +it + π5 +it +∀i, t ∈ D +(12b) +π ∈ R+ +(12c) +Finally the DMS-p-MP reformulation of the problem becomes as: +min +x,y,a,bθ + c2(a + b) +(13a) +s.t. (1b) − (1k) +(13b) +Γπ1 + π2 + π3 + ¯dc1x ≤ θ +(13c) +π1 ≥ π4 +it + π5 +it +∀i, t ∈ D +(13d) +π2 +it − π3 +it + π4 +it − π5 +it = γitc1 +itxijt +∀i, t ∈ D, ∀j ∈ B +(13e) +πκ +it ≥ 0 +∀i ∈ B, t ∈ T, κ ∈ {1, 2, 3, 4, 5} +(13f) +where πκ, x, and d is the vector of πκ +it’s, dit, and xijt’s respectively. +3.3.2. Lagrangian relaxation +In discrete location theory, one of the basic models is the p-median problem and it is an NP-hard +problem, as with most location problems Mladenovi´c et al. (2007). The MILP model presented in +section 3 is typically solved in practice with numerous residence areas (demand points), potential +facility locations, and time horizons. As a result of the large-scale property of the developed model, +it presents substantial computational difficulty when applied to real problems, which cannot be +addressed by commercial optimization software, such as Cplex, Xpress or Gurobi. For supply chain +optimization problems involving such computational complexity, Lagrangian relaxation (LR) is +widely used (Diabat et al., 2013; Duong and Bui, 2018; Rafie-Majd et al., 2018; Kheirabadi et al., +2019; Hamdan and Diabat, 2020) in the literature. +9 + +Therefore, this section develops an LR approach for solving the presented MILP problem. The +LR method is an iterative algorithm that provides the upper and lower bounds of the optimal +objective value as well as the estimation of the optimality gap of the feasible established solution +in each iteration (Daskin, 1997). +The LR method used in this paper includes the general steps as follows: 1) Relax one of +the constraints by multiplying it by a Lagrange multiplier and bringing the constraint into the +objective function, 2) Solve the model to find the optimal values of the relaxed problem, 3) Find +the feasible solution to the original problem by using the resulting decision variables found in step +2, 4) Compute the lower bound using the solution obtained from the relaxed problem in step 2, +5) Use the subgradient optimization method to modify the Lagrange multiplier assigned to the +violated constraint, return to step 2 after finding the new multiplier(s) for the Lagrange variable. +The algorithm terminates whenever the lower bound is close enough to the upper bound. +Step 1. +Solving the Relaxed Problem. In order to make the problem easier to solve, the con- +straints in this study are relaxed, even if this relaxation may lead to infeasibility (Daskin, 1997). +Specifically, constraints (1b), (1c) are relaxed, resulting in the following Lagrangian dual problem: +min +x,y,a,bθ + c2(a + b)+ +� +j∈B +� +i∈B +� +t∈T +λ1 +bt(xijt − 1)+ +� +j∈B +� +t∈T +λ2 +t(yjt − p) +(14a) +s.t. (1d) - (1k), (13c) - (13f) +(14b) +The optimal value of the objective function in the Lagrangian dual problem above, which is +defined using non-negative Lagrange multipliers (λ1 +bt, λ2 +t), serves as a lower bound for the mixed- +integer linear programming problem with constraints numbered (14a) to (14g). +Step 2. Finding a Feasible Solution and an Upper Bound. In most situations, the solution to +the Lagrangian dual problem is not feasible due to the relaxation of constraints (1b) and (1c). +However, it is possible to obtain a feasible solution that provides an upper bound on the single +objective MILP model by solving this model and setting the decision variables xijt, yjt, ajt and bjt +to the optimal values obtained from solving the Lagrangian dual problem. +Step 3. Finding a Lower Bound, and Updating the Lagrange Multipliers. During each iteration +of the Lagrangian procedure, the Lagrangian multipliers λ1 +bt, λ2 +t are updated and new lower and +upper bounds are subsequently derived. There are various methods in the literature, such as cutting +planes (Kelley, 1960), sub-gradients (Daskin, 1997), and bundling (Borghetti et al., 2003), that +can be used for this purpose. In this paper, the sub-gradient approach is employed to update the +Lagrangian multipliers because it is a widely recognized and commonly used method. According +to the sub-gradient procedure (Daskin, 1997), the Lagrange multipliers at the (n + 1) iteration are +calculated as follows: +(λ1 +bt)n+1 = max +� +0, (λ1 +bt)n − τ n +1 (xijt − 1) +� +(15) +(λ2 +t)n+1 = max +� +0, (λ2 +t)n − τ n +2 (yjt − p) +� +(16) +10 + +The step sizes τ n +1 , and τ n +2 in the algorithm are defined as follows: +τ n +1 = +αn(UB − LBn) +� +j∈B +� +i∈B +� +t∈T λ1 +bt(xijt − 1)2 +(17) +τ n +2 = +αn(UB − LBn) +� +j∈B +� +t∈T λ2 +t(yjt − p)2 +(18) +The term α is simply a constant that will be changed during each iteration of the algorithm as +described above. α is initialized to 2. If the lower bound, LB, has not increased in an iteration, +then its value will be halved. Additionally, let UB be the best upper bound (the one with the +smallest value) that has been discovered thus far, and LBn is the lower bound obtained at iteration +n. +Step 4. Termination Criteria. The algorithm will stop when one of the following conditions is met +(Daskin, 1997): +• A predetermined number of iterations have been completed. +• The lower bound is equal to the upper bound (UB = LBn) or is close enough to the upper +bound (UB − LBn < 0.1). +• The value of α becomes small +4. Numerical Experiment +4.1. Case description +In this section, we discuss a case in which a company wants to place some mobile grocery +stores on the campus of the University of Waterloo. These self-service mobile stores can serve all +students and staff. Figure 1 shows a sample picture of these stores. Specifically, the company +wants to propose a plan to specify the optimal locations of stores needed on campus dynamically +(every day) based on the demand variation on campus. They update their plan every month, i.e., +they decide on all days at the beginning of each month. In this research, we aim to find the optimal +locations of stores over the time horizon (one month). Based on Section 3, |T| = 28, |K| = 6, and +|B| = 91. +Figure 2 shows the University of Waterloo campus map. All buildings on campus are categorized +into five groups: Service and Administrative Buildings, Academic Buildings, Residence Buildings, +Research Park Buildings, and others. The University of Waterloo asked the company to follow +some rules regarding each group. Specifically, there are some limitations on the number of stores +needed for each group for each day. More details and the list of buildings included in each group +will be provided in the following sections. +Each building has a specific demand on each day. Obviously, it is not (financially) feasible to +place one store in each building. The company has only p stores that in each period (day) they +want to place all of them on campus. We consider a demand cost, for students and staff in a +building that does not have a store and they need to go to other buildings. The goal is to place +the stores to satisfy the demand of all buildings while minimizing the costs. There are two more +costs in our problem: opening cost and closing cost. Although the stores are mobile, we need to +consider an opening (closing) cost if we want to open (close) a store at the beginning of each day. +In fact, we are capturing all transportation costs needed to change a store’s location. +11 + +Figure 1: Sample Picture of Mobile Grocery Store from PYMNTS.com (2018) +The demand of each building may change day-to-day because of different reasons such as +weekends, events, etc. Then, it is an excellent motivation for day-to-day decision-making following +demand variation. In this research, we assume that our demand prediction is accurate. Then, we +can finalize all decisions for each day of the month, once at the beginning of the month, i.e., our +demand is deterministic. +We can model this problem as a modified multi-period p-median problem. +Generally, our +problem has two modifications compared to the simple multi-period p-median problem. First, we +are considering opening and closing costs over the time horizon. Second, there are some extra +constraints raised from University of Waterloo policies. +12 + +mobde +Sab +MOBILE +CONVENIENCE +STORE +Welcome!Figure 2: University of Waterloo - Campus Map +4.2. Data Collection +We divide the University of Waterloo’s (UW’s) all facilities according to their functionality into +six different segments: Academic Buildings, Parking Spots, Student Residence Buildings, Research +Park Buildings, Athletic Buildings, and University Plaza respectively. The workspace is based on +the UW’s official map includes 91 facilities, and we also make an assumption that the population +in each facility within a segment is all equivalent. Segments indicate in the Table 2 are as follows, +13 + +(1) +(2) +WES CRASAMW +WES GRAHAN WR +375 +RESEARCH+TECHNOLOGY PARK +DAVIDJOHNSTON +340 +S +FRANK TOMPA DR +NEATHE +TOMEA +X +LAKE +COLUMBIA +EL +(5) +LEGEND +PARKING +ECZ +AccessibleParking +Meter Parking +Motorcycle Parking +SYMBOLS +LLAGEGREDN +SLC +C +PermitParking +Accesslble Entrances +Short-termParking +BulldingCodes +Visitor Parking +Construction Site and +Future Site of Bullding +COLOURCODES +Grand RiverCarShare +Academic/AdministrativeBulldings +GRT +Grand RiverTransit +RoadsandParkingLots +Greyhound +CityRoadsand Parking Lots +GO Transit +Pathways +Help Line Telephone +Residence Bulldings +Information +Water +PubllcTelephone +ResearchParkBulldings +300 metres +Service VehicleTable 2: The Functional Classification for All Facilities in the Workspace +Segment +Buildings Included in the Segment (Building +Id) +Academic Buildings +COG (1), COM (2), CPH (3), RA2 (4), M3 +(5), ML (6), RAC (7), GSC (8), GH (9), +BRH (10), SLC (11), OWE (12), HMN (13), +MC (14), TC (15), KOC (16), C2 (17), EV3 +(18), OPT (19), DC (20), SCH (21), FED +(22), QNC (23), AL (24), HS (25), B2 (26), +EV2 (27), REN (28), ESC (29), EV1 (30), +STJ (31), EIT (32), HH (33), STP (34), Bl +(35), PAS (36), CGR (37), E3 (38), EC3 (39), +BMH (40), PHY (41), EC1 (42), LHI (43), +NHI (44), EC2 (45), UC (46), LIB (47), ECH +(48), ERC (49), E2 (50), ES (51), CSB (52), +RCH (53), E6 (54). +Parking +Parking CL (55), Parking A (56), Parking X +(57), Parking C (58), Parking W (59), Park- +ing OV (60), Parking V (61), Parking S (62), +Parking K (63), Parking J (64), Parking R +(65), Parking P (66), Parking T (67), Park- +ing M (68), Parking L (69), Parking D (70), +Parking EC (71), Parking HV (72), Parking +N (73), Parking UWP (74). +Residence Buildings +CLN (75), CLV (76), MKV (77), V1 (78), +REV (79), TH (80), MHR (81), UWP (82). +Research Park Buildings +445 (83), 375 (84), 340 (85), 275 (86), ACW +(87), 300 (88). +Athletic Buildings +CLF (89), PAC (90). +University Plaza +Plaza (91). +We use Python’s OpenCV toolbox by Bradski and Kaehler (2000) to determine facilities’ specific +locations by fixing pixel’s coordinates in the workspace. Students and staff allows going through +buildings, it is likely for us to use Euclidean distance measurement +l2 = +� +(x − a)2 + (y − b)2 +to find the distance between the two pairs in the data set is as follows on the Figure 3. +14 + +Figure 3: Coordinates for All Facilities on the Workspace +From the UW’s website, we obtained the rough number of students and staff for each faculty, +the rough number of students and staff for each university college, and the capacities of each +residence. A general assumption is that 10% of university students will use one of two gyms (CIF +and PAC). Tables 3, 4, 5, 6, 7, 8 below are the population of all classifications for each segment. +Table 3: Statistical Results for Academic Buildings +Segment (1) +Population +Engineering Faculty +11000 +Mathematics Faculty +9260 +Science Faculty +6000 +Health Faculty +3643 +Arts Faculty +3000 +Environment Faculty +3000 +Others +1200 +Total population +37103 +Total facilities +54 +Average per facility +687 +Table 4: Statistical Results for Parking Spots +Segment (2) +Population +Campus parking +4000 +Total population +4000 +Total facilities +20 +Average per facility +200 +15 + +Segment Academic Athletic Parking XResearch Park →Residence +University Plaza +800 +Parking T +MHR +Parking P +Parking C +CGR +Parking HV +EV +HH +Parking A +Parking UWP +EV2 EVI +AL_ +UWP +600 +SCH +STP +DWE +ParkingD +ML +GH +RCH +CPH +NH +LIB +E2 ++ +PHY + STJ +Plaza +REN +BI +E3 +B2 +EIT +ESC +Longitude +QNC +ES +HS +C2 +ECH +SLC +MC +DC +400 +ParkingM +PAC +M3 + Scaled L +Gsc +REV +TH +ERC +CsB +COM +i EC2 +VI +uc +LHI +MKV +BMH +Parking L +■ +ECI +Parking V +Parking S +Parking K +Parking J +FED +Parking R +Parking N +EC3 +Parking w +Parking OV +CLV +OPT +Parking CL +200 +Parking X +XNWH +BRH +X +CLN +COG +X +ACW +B No.300 +B No.275 +X +X +B No. 340 +B No. 375 +RA2 +RAC +XB No. 445 +200 +400 +600 +800 +1000 +Scaled LatitudeTable 5: Statistical Results for Student Residence Buildings +Segment (3) +Population +Columbia Lake Village - North +404 +Columbia Lake Village - South +400 +William Lyon Mackenzie King Village +320 +Student Village 1 +1381 +Ron Eydt Village +960 +Tutors’ Houses +100 +Minota Hagey Residence +70 +University of Waterloo Place +1650 +Others +200 +Total population +5285 +Total facilities +8 +Average per facility +660 +Table 6: Statistical Results for Research Parking Buildings +Segment (4) +Population +David Johnson Research Park +4000 +Others +100 +Total population +4100 +Total facilities +6 +Average per facility +683 +Table 7: Statistical Results for Athletic Buildings +Segment (5) +Population +Columbia Icefield +2100 +Physical Activities Complex +2100 +Others +100 +Total population +4300 +Total facilities +2 +Average per facility +2150 +Table 8: Statistical Results for University Plaza +Segment (6) +Population +University Shops Plaza +3200 +Total population +3200 +Total facilities +1 +Average per facility +3200 +Given in the previous section that the time horizon T = 28 (approximately equivalent to a +month), it can also separate into four periods (weeks). Based on the functionality of each segment, +16 + +with time-varying from Monday to Sunday, we may empirically define the facility in the different +segments and the different days in utilization rate Ukt. For instance, the utilization rates for the +academic buildings are 100 out of 100, but there are 30 out of 100 during the weekend. The +complete estimated utilization rates for different functional buildings for each day over a week are +as the following Table 9: +Table 9: The Estimated Utilization Rate for Different Functional Buildings for Each Day over a Week +Functionality +Monday +Tuesday +Wednesday +Thursday +Friday +Saturday +Sunday +Academic Buildings +100 +90 +90 +80 +90 +30 +30 +Parking Spots +100 +100 +100 +100 +100 +20 +20 +Residence Buildings +50 +50 +50 +50 +60 +100 +100 +Research Park Buildings +100 +90 +90 +80 +90 +10 +10 +Athletic Buildings +50 +50 +50 +60 +50 +100 +100 +University Plaza +100 +100 +70 +80 +90 +50 +50 +After that, the demand dkt of each building in segment k on day t, where t ∈ {1, . . . , 7}, in a +week is able to define as, +dkt = Ukt × Total Population(k) +Total Facilities(k) . +As for having a unified standard, we calculate the lower bound of opening facilities in constraint +(1h) by taking a floor of 70% of the number of facilities in segment k over the total facilities on +the workspace, +nk = +� +p × the Number of Facilities in Segment (k) +91 +× 70% +� +. +For example, n1 = +� +18 × 54 +91 × 70% +� += ⌊7.47⌋ = 7. Similarly, the upper bound of opening facilities +in constraint (1g) by taking a ceiling of 130% of the number of facilities in segment k over the +total facilities on the workspace, +mk = +� +p × the Number of Facilities in Segment (k) +91 +× 130% +� +. +The following Table 10 is the maximum (minimum) number of Robomarts are allowed in +segment k, +Table 10: Bounds on the number of Robomarts +Functionality +Min (nk) +Max (mk) +Academic Buildings +7 +14 +Parking Lots +2 +6 +Residence Buildings +1 +3 +Research Park Buildings +0 +2 +Athletic Buildings +0 +1 +University Plaza +0 +1 +17 + +4.3. Computational Experiments +In this section, we use all parameters discussed in the Data Collection section, to solve our +problem. All numerical experiments have been run on an Apple M1 processor, limited to 16 GB +of RAM. Gurobi has access to 8 physical cores, 8 logical processors, using up to 8 threads. This +model is MILP and has M(= 239512) variables and N(= 239694) constraints (excluding sign +constraints). We use Gurobi version 9.5.1 by Bixby (2007) to solve this optimization model. Since +default settings in Gurobi generally work well, we are keeping all settings as default. Specifically, +we use Gurobi’s API embedded in Python. +MILP models are generally solved using a linear-programming based branch-and-bound algo- +rithm. The Gurobi provides advanced implementations of the latest MILP algorithms including +deterministic parallel, nontraditional search, heuristics, solution improvement, cutting planes, and +symmetry breaking. +Based on section 5, the demand is weekly periodic. Then, we expect the model to make the +same decision over the weeks. Table 11 shows what buildings are open during the planning horizon. +We just show the days that we change our decisions. For instance, the buildings that are open +between day 1 and day 6 are the same. +Table 11 shows that we only open new buildings and close the current buildings over the +weekend. We again change our decision on weekdays. It is expected because the utilization rates +of some of our segments are significantly different during the weekend. +Specifically, we close buildings 12, 55, and 83 (that base on Table 3 are Conrad Grebel university +college, Parking lots, Research building 375, respectively) and open buildings 63, 77, and 81 (that +are Parking P, Student Village, University of Waterloo Place respectively). The interesting point +is that all buildings 63, 77, and 81 are around students’ residences. Students are mostly in the +residence area instead of the academic campus during the weekends. Then, it is worth closing +some stores and opening new ones in students’ residences. +Table 11: Results - Optimal Objective Value = 43175.7, CPU Time = 12.99 Seconds +t +Open Buildings +1 (Monday) +0, 2, 3, 5, 10, 12, 17, 27, 34, 40, +44, 48, 54, 55, 76, 83, 89, 90 +6 (Saturday) +0, 2, 3, 5, 10, 17, 27, 34, 40, 44, +48, 54, 63, 76, 77, 81, 89, 90 +8 (Monday) +0, 2, 3, 5, 10, 12, 17, 27, 34, 40, +44, 48, 54, 55, 76, 83, 89, 90 +13 (Saturday) +0, 2, 3, 5, 10, 17, 27, 34, 40, 44, +48, 54, 63, 76, 77, 81, 89, 90 +15 (Monday) +0, 2, 3, 5, 10, 12, 17, 27, 34, 40, +44, 48, 54, 55, 76, 83, 89, 90 +20 (Saturday) +0, 2, 3, 5, 10, 17, 27, 34, 40, 44, +48, 54, 63, 76, 77, 81, 89, 90 +22 (Monday) +0, 2, 3, 5, 10, 12, 17, 27, 34, 40, +44, 48, 54, 55, 76, 83, 89, 90 +27 (Saturday) +0, 2, 3, 5, 10, 17, 27, 34, 40, 44, +48, 54, 63, 76, 77, 81, 89, 90 +The detail solution is attached to this report in an excel file, containing 28 sheets, each sheet +18 + +for each day. +4.4. Sensitivity Analysis +In this section, we will change the parameters to see how variability can affect on our results. +4.4.1. Time Horizon +First, we analyze the impact of the length of time horizon on our results. Table 12 shows +the results when we increase (decrease) the time horizon. For instance, as we increase the time +horizon, it will take more time to find the optimal solution. Besides, our optimal objective value +would increase since we have are adding more positive terms to our cost. +Table 12: Sensitivity Analysis on Time Horizon +T +Id of Opened Build- +ings +Id of Closed Build- +ings +Objective Value +CPU Time (S) +14 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +21570.1 +12.43 +21 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +32372.9 +6.08 +28 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +43175.7 +12.99 +35 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +53978.5 +17.02 +42 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +64781.3 +20.40 +19 + +Figure 4: The impact of the time horizon on the objective functions and CPU process time. +Moreover, as we see the opened and closed buildings remain the same as we increase the time +horizon. The reason is that our demand is periodic over the weeks. By increasing the number of +weeks, the optimal decision to open and close some stores would be the same. Figure 5 summarize +the results. +4.4.2. Number of Facilities to Be Located (P) +Now we change P and see how it affects our results. Variability of P is important since it would +help the company to decide the number of stores they want to buy and invent on the campus. +Note that we could also consider a fixed cost of buying each store in our model (i.e., we could +add Price.P to the objective function). However, since in our optimization model P is fixed and +given, we don’t need to consider it. Figure ?? shows the results when we change P. First, it seems +that the running time is not directly dependent on P. However, P is determining the complexity +of the problem. Besides, the objective value is obviously decreasing as we increase P since we +didn’t consider the price of each store. +20 + +Sensitivity Analysis on Time Horizon +Objective Value +CPU Time +20 +60000 +50000 +15 +40000 +10 +30000 +20000 +14 +21 +28 +35 +42 +Time horizonFigure 5: The impact of the number of facilities on the objective functions and CPU process time. +Also, Table 13 shows how our decision changes in different P. There is one interesting point in +our decisions. Compare P = 12 with P = 15. In P = 15, we don’t use the same opened building +we used in P = 12. It shows as we want to add one building, we may no longer need another +building. +21 + +Sensitivity Analysis on P +jective Value +(S) +CPU +Cqo +55000 +90 +50000 +60 +45000 +30 +40000 +35000 +12 +15 +18 +21 +24 +PTable 13: Sensitivity Analysis on P +P +Id of Opened Build- +ings +Id of Closed Build- +ings +Objective Value +CPU Time (S) +12 +2, 5, 10, 17, 27, 35, +44, 54, 65, 76, 83, +90, (63, 89) +63, 89, (65, 83) +58616.4 +58.07 +15 +2, 3, 5, 10, 27, 35, +45, 48, 63, 66, 74, +76, 83, 89 , 90 (81, +17, 61, 79) +81, 17, 61, 79, (83, +3, 66, 76) +49402.9 +7.79 +18 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +43175.7 +12.99 +21 +0, 2, 5, 7, 10, 12, 17, +23, 26, 34, 40, 44, +48, 53, 54, 55, 76, +83, 85, 89, 90, (64, +77, 81, 3, 79) +64, 77, 81, 3, 79, (7, +55, 83, 76, 85) +38266.5 +7.89 +24 +0, 2, 3, 5, 6, 10, 12, +17, 23, 34, 40, 44, +48, 53, 54, 61, 67, +76, 80, 81, 83, 87, +89, 90, (64, 78) +64, 78, (80, 87) +34874.4 +107.60 +4.4.3. Cost Coefficients +Followed by two previous sections, we want to discuss the effects of changing opening and +closing costs together on our results. Table 14 shows the results when we increase (decrease) the +cost coefficients. Considering the case that costs are both zero, the model tries to open (and close) +any building in each period. In other words, it does not care how many building wants to open +or close each day. Another noteworthy point is that running time is increasing as we increase the +costs. It shows that the trade-off between keeping current buildings and opening (closing) other +buildings is becoming important and challenging to the model. +22 + +Table 14: Sensitivity Analysis on Cost Coefficients (together) +γo(c) +Id of Opened Build- +ings +Id of Closed Build- +ings +Objective Value +CPU Time (S) +0 +0, 2, 3, 10, 12, 17, +26, 35, 45, 48, 54, +65, 76, 83, 86, 88, +90, (1, 4, 5, 23, 26, +27, 34, 40, 44, 56, +63, 81, 89) +4, 5, 27, 34, 40, 44, +56, 63, 81, 89, (23, +26, 35, 45, 65, 86, +88) +42900 +3.90 +2.5 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 79, 81) +63, 77, 79, 81, (12, +55, 76, 83) +43049.3 +9.34 +5 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (63, 77, 81) +63, 77, 81, (12, 55, +83) +43175.7 +12.99 +10 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (80, 81) +80, 81, (12, 83) +43382.3 +58.45 +20 +0, 2, 3, 5, 10, 12, 17, +27, 34, 40, 44, 48, +54, 55, 76, 83, 89, +90, (81) +81, (83) +43658.2 +82.58 +Figure 6 shows that the initially opened buildings remain the same and starts to dynamically +be changed as we increase the opening/closing cost. Moreover, the objective value increase as we +increase the opening cost. Also, for the large opening costs, we prefer not to open new buildings +(and close the current open buildings) since it would be costly. +23 + +Figure 6: The impact of opening/closing cost of facilities on the objective functions. +5. Discussion +This study has presented a mixed-integer linear formulation for the Dynamic Modified Stochas- +tic p-Median Problem in a Competitive Supply Chain Network Design. The proposed model takes +into account the robust optimization and time horizon as a novel approach, which enables the +decision-maker to consider uncertainty and short-term changes in the supply chain network de- +sign. The robust optimization approach used in this study allows for the consideration of different +scenarios and uncertainty in demand and supply, which is crucial in real-world applications. Addi- +tionally, the time horizon approach allows for the dynamic nature of the problem to be captured, +which is important in today’s fast-paced business environment. +The proposed model was tested using computational experiments, and the results demonstrate +the effectiveness of the proposed approach in handling the dynamic and stochastic nature of the +problem. The results also provide valuable insights for practitioners and researchers in the field +of supply chain network design. The proposed model can be extended and applied to other sim- +ilar problems in the field, such as facility location, transportation and logistics, and inventory +management. +One of the main contributions of this study is the integration of robust optimization and time +horizon in the mixed-integer linear formulation for the Dynamic Modified Stochastic p-Median +Problem. The robust optimization approach allows for the consideration of different scenarios and +uncertainty, while the time horizon approach allows for the dynamic nature of the problem to be +captured. This integration provides a more realistic and practical solution to the problem, which +can be useful for practitioners and researchers in the field. +Another important contribution of this study is the application of the proposed model to a +competitive supply chain network design problem. This application is relevant and valuable as it +provides insights into how the proposed model can be used in a real-world context. The results of +24 + +Sensitivity Analysis on Cost Coefficients (together) +Count +Value +Facility +Objective +44000 +40 +43750 +30 +43500 +20 +43250 +10 +43000 +0 +0.0 +2.5 +5.0 +10.0 +20.0 +Cost coefficient +Facility status +Closed +Openedthe computational experiments demonstrate the effectiveness of the proposed model in handling +the dynamic and stochastic nature of the problem and provide valuable insights for practitioners +and researchers in the field. For instance, we can change over the problem definition to solve any +other location problems. Covering location problems are valuable to be investigated, such as trying +to find the optimal number of Robomarts PYMNTS.com (2018) that can serve all facilities on the +workspace if the single Robomart can only serve facilities within limited miles. +This study has the potential usefulness of the proposed model in the context of a pandemic +and quarantine. The ability to handle uncertainty and short-term changes in the supply chain +network design. The robust optimization approach used in this study allows for the consideration +of different scenarios and uncertainty in the demand and supply, which is crucial in a pandemic +context. The proposed model can be used to design and optimize supply chain networks that are +more resilient and adaptable to the changing conditions caused by a pandemic. This can help +businesses and organizations to minimize disruptions and maintain the continuity of operations +during challenging times. +One limitation of this study is that the Robomart can only be located at specific facilities +(nodes) in the proposed model. In reality, however, the Robomart can also be located somewhere +on the route between two nodes (edges). This limitation may affect the validity and applicability +of the proposed model in real-world scenarios. To address this limitation, it is possible to consider +analogous absolute p-median problems to simulate reality. This approach would involve the inclu- +sion of edge-based locations for the Robomart in the model, which would provide a more realistic +representation of the problem. However, this would require additional mathematical development +and computational resources, and would be a subject for future research. +Another potential area of future research is to make the problem an Adaptive Robust Opti- +mization (ARO) problem Sun et al. (2021). In this approach, the number of trucks p would be +determined in the first stage and other variables in the second stage. This would enable a more +flexible and dynamic approach to supply chain network design, as the number of trucks can be +adjusted in response to changes in demand and supply. +In summary, the proposed DMS-p-MP model takes into account the robust optimization and +time horizon as a novel approach, which enables the decision-maker to consider uncertainty and +short-term changes in the supply chain network design. The results of the computational experi- +ments demonstrate the effectiveness of the proposed model in handling the dynamic and stochastic +nature of the problem and provide valuable insights for practitioners and researchers in the field. +The proposed model can be extended and applied to other similar problems in the field of supply +chain network design. 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Journal of Urban Design 17, 511–531. +30 + diff --git a/edFJT4oBgHgl3EQfTCwr/content/tmp_files/load_file.txt b/edFJT4oBgHgl3EQfTCwr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4cd59b847f9f544de3cadf1d16aee81d5e4cfc7f --- /dev/null +++ b/edFJT4oBgHgl3EQfTCwr/content/tmp_files/load_file.txt @@ -0,0 +1,1561 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf,len=1560 +page_content='A Mixed-integer Linear Formulation for Dynamic Modified Stochastic p-Median Problem in a Competitive Supply Chain Network Design Amir Hossein Sadeghia,∗, Ziyuan Sunb, Amirreza Sahebi Fakhrabada, Hamid Arzanic, Robert B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Handfieldd aDepartment of Industrial and Systems Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' USA bDepartment of Mechanical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Industrial and Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Concordia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Qu´ebec,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Canada cRotman School of Management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' University of Toronto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Canada dDepartment of Business Management Poole College of Management,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' North Carolina State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' NC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' USA Abstract The Dynamic Modified Stochastic p-Median Problem (DMS-p-MP) is an important problem in supply chain network design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' as it deals with the optimal location of facilities and the allocation of demand in a dynamic and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this research paper, we propose a mixed- integer linear formulation for the DMS-p-MP, which captures the key features of the problem and allows for efficient solution methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The DMS-p-MP adds two key features to the classical problem: (1) it considers the dynamic nature of the problem, where the demand is uncertain and changes over time, and (2) it allows for the modification of the facility locations over time, subject to a fixed number of modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The proposed model is using robust optimization in order to address the uncertainty of demand by allowing for the optimization of solutions that are not overly sensitive to small changes in the data or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' To manage the computational challenges presented by large-scale DMS-p-MP networks, a Lagrangian relaxation (LR) algorithm is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Our computational study in a real-life case study demonstrates the effectiveness of the proposed formulation in solving the DMS p-Median Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' According to the results, the number of opened and closed buildings remains unchanged as the time horizon increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This is due to the periodic nature of our demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This formulation can be applied to real-world problems, providing decision-makers with an effective tool to optimize their supply chain network design in a dynamic and uncertain environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Keywords: p-Median Problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Supply Chain Network Design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Dynamic Allocation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Robust Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Lagrangian Relaxation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Introduction Mobile grocery stores, also known as grocery trucks (GT) or pop-up grocery stores, have emerged as a new trend in the retail industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' GTs typically stock a wide variety of fresh fruits and vegetables, as well as other grocery items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' They are usually equipped with refrigeration units to keep the food fresh and have a payment system in place for customers to buy the products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' These mobile stores bring convenience and accessibility to customers in under-served or rural areas, or in areas where traditional brick-and-mortar grocery stores are not present Fahlevi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Nader (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The trend of grocery trucks is expected to continue to grow in the coming ∗Corresponding author Email address: asadegh3@ncsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='edu (Amir Hossein Sadeghi) Preprint submitted to Logistics, MDPI January 30, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='11502v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='OC] 27 Jan 2023 years as more and more people look for ways to improve access to healthy food in their communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', the mobile food vendors market is valued at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='16 billion dollars in 2021 and is predicted to grow at a rate of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4% annually from 2022 to 2030, driven by the increasing trend of culinary arts and the preference of young people for different dining experiences over the traditional dining in restaurants Research (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' GTs can also be used to provide food to people in emergency situations, like natural disasters or power outages Hecht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Pelling (2001), as well as address food insecurity by increasing access to healthy food options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' GTs are valuable resources for alleviating food insecurity, and these trucks are often operated by non-profit organizations, local governments, and community groups that want to address food insecurity and promote healthy eating Mohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' They can help to address food insecurity in several ways: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Accessibility: Grocery trucks can bring fresh produce and other grocery items to low-income or rural areas where there is limited access to traditional grocery stores and supermarkets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This makes it easier for residents in these areas to access healthy food options;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Affordability: Many grocery trucks accept SNAP (Supplemental Nutrition Assistance Program) benefits and other forms of government assistance, making it more affordable for low-income families to purchase healthy food;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Education: Some grocery trucks provide educational programs and cooking demonstrations that teach customers how to prepare healthy meals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This can help to improve the overall health and nutrition of the community;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Variety: Grocery trucks can also offer a wider variety of fresh produce and other grocery items than traditional corner stores or convenience stores, which may only carry a limited selection of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Overall, grocery trucks are a creative and innovative way to address food insecurity and improve access to healthy food in communities that need it the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, the success of a mobile grocery store is heavily dependent on the location where it is parked Esparza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Wessel (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Various factors should be considered when selecting a location for a grocery/food truck, such as dynamic demand, visibility, accessibility, and competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Logistics for mobile grocery stores refer to the process of planning, coordinating, and controlling the movement and storage of goods, services and information from the point of origin to the point of consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This includes transportation, inventory management, warehousing, and distribution of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For mobile grocery stores, logistics also includes the planning and coordination of the routes and schedule of the mobile store, as well as the management of the supply chain and inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This includes sourcing products from suppliers, managing inventory levels and restocking the mobile store as needed, and coordinating delivery schedules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Effective logistics management is crucial for the success of mobile grocery stores, as it can impact delivery times, cost, and overall customer satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' By optimizing logistics, mobile grocery store owners and operators can improve their business by reducing costs and increasing efficiency, ultimately resulting in better customer service Restuputri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Bourlakis and Weightman (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this regard, we first provide a comprehensive literature review of the proposed location- allocation models in the literature in section 2, then provide a mathematical formulation for our model considering the uncertainty of demand in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Sections 4 and 5 show the application of our proposed model in the real-world mobile grocery location problem and provide insights, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Literature Review Location science acts on a significant role in modern development in different disciplines, in- cluding business, economics, computer science, geography, military, transportation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' With 2 historical records, the first optimal location problem was proposed by Pierre de Fermat to find the geometric median among three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The problem was solved by Evangelista Torricelli soon after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This is undoubtedly logical that the single facility minimum Euclidean distance problem is named Fermat-Torricelli problem (Known as the most famous Weber problem, the distances are looked upon as the weights of nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019) is a book that comprehensively introduces the development and forecast of the subject of location science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The beginning of the new era of location science was around the 1960s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Berge (1957) investigates the problem of finding the minimum coverage on a graph originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Miehle (1958);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Cooper (1963) raises the best known p-median problem, which is to pick the exact p of facilities to open that minimizes the total transportation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Followed by, Hakimi (1964, 1965) releases Hakimi’s node optimally theorem to demonstrate that the optimal solution on a continuous graph for absolute median problems is always on the graph’s vertices, which means many network problems can simply transfer to discrete version problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Hakimi (1964) introduces the concept of absolute center to find the decision of satisfying the mini-max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' As for solving more complex realistic optimization tasks in the next decade, mixed-integer linear programming (MILP) was widely used to deal with location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Balinski (1965);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ReVelle and Swain (1970);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Toregas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1971) are early works to use MILP to solve the uncapacitated facility location problem, discrete p-median problem, and covering-location problem, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The deterministic model means all model parameters are known with certainty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' nevertheless, unpredictability or randomness always occurs in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Stochastic programming or robust opti- mization is the solution to reduce the impact of uncertainties in predicting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Researchers have been quite interested in topics to combine location models with stochastic programming methods in the last few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Louveaux (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Louveaux and Peeters (1992) modify the uncapacitated facil- ity location and p-median problems with stochastic variables, including the customer’s demand, the unit cost of satisfying the customer from a specific location, and the fixed cost of locating at the candidate location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1994) assumes the customer’s demand is stochastic for the capacitated facility location problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Current et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1998) studies the number of opening facilities is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For improving customer service, based on p initial opening facilities, Berman and Drezner (2008) allows increasing r new opening facilities depending on the customer’s demand, where 0 ≤ r ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Sonmez and Lim (2012) is the opposite of the previous article, r existing facilities have to be closed, where 0 ≤ r ≤ q ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Liu and Song (2022) considers the covering-type location problems under demand uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Snyder (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Correia and Saldanha-da Gama (2019) are overviews of facility location under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In reality, the vast majority of practical location problems can take into consideration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The setting of the problem does not rely on time, called ‘single-period’ or ‘static’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' conversely, it is called ‘multi-period’ or ‘dynamic’ if the decision varies with different parameters at each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Ballou (1968);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Sweeney and Tatham (1976) are early works about locating and re-locating a single facility within a time span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Scott (1971) continuous the previous works for locating multiple facili- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Wesolowsky (1973) merges the conception of the Weber problem with the multi-period facility location problem to find the best opening facility in each time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Warszawski (1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Cavalier and Sherali (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Drezner (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Hakimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1999) studies the dynamic uncapacitated fa- cility location problem, location problems on networks, network p-median problems, and network center problems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Wesolowsky (1973);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Wesolowsky and Truscott (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Galv˜ao and Santiba˜nez-Gonzalez (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Dias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2007) involve facilities’ opening and closing costs in mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Moreover, Ahmed and Garcia (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Romauch and Hartl (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Marques and Dias (2013) present dynamic location problems under uncertain environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Nickel and Saldanha-da Gama (2019) is a survey of dynamic facility location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3 In the past century, selecting an optimal retail location in a competitive environment has been much discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The concept of stability in the competition was first proposed in Hotbllino (1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Reilly (1931);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Converse (1949) are early works that mention the retailing models about the relationship between customer consumption, the size of the facility, distance, et cetera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Huff (1964, 1966) presents further progress: Huff’s formulation of the competitive function of allocating to customers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' depends on the facility’s attraction and distance to the customer (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2021) collect social media data to evaluate the facility’s attraction to coincide with the current trend).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' On account of Huff’s formulation, Nakanishi and Cooper (1974) adds more possible factors to extend the original formulation with the least squares approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Drezner and Zerom (2023) is a recent paper discusses circumstances in the competitive environment to avoid cut-throat peer competition that eliminates existing facilities if a new facility is allocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' To sum up, Ghosh and MacLafferty (1987);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Eiselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Ashtiani (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Eiselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019) are surveys of the competitive location models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Emerging as the times require, applications of relocatable facilities have the benefits of adapting to the times to optimize the cost and meet different needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Many multifarious applications have been studied for relocatable facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Cho and Lee (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Bhandawat (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Kalra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2022) study location optimization problems for mobile vendors or food trucks in interurban environments to serve more customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Adler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2014) focuses on allocating police routine patrol vehicles on the city streets to improve public security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Contardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2012) investigates practical problems for the bike-sharing system: there is no available bike for renting and capacitated stations without an empty spot for returning at rush hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The article provides a solution to the dynamical model to balance the inventory of unused bikes among stations with adequate inventory and short inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' R˘aboac˘a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2020) lies their research on how to temporarily locate transportable charging stations for the electric vehicle under potential constraints with optimal charging demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Glaeser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2019) interrelates to the distribution problem of E-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Due to the high charge of door-to-door delivery, there is an innovative solution that customers can order online first, then pick up offline at a location where delivery trucks allow parking and maximize the profit of the logistics company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' An analogous application is given in Cao and Qi (2022) about the unmanned market on wheels for stall economy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' dis-similarly, the emphasis is on planning routes based on the reformative vehicle problem model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Modelling Process and Methods This section presents the development of a mixed-integer linear programming (MILP) model to address the research problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Notation list The following are the notations used in the formulation of the model, including sets, parameters, and decision variables: 4 Table 1: Notations used for mathematical modeling of DMS-p-MP Sets T Set of time periods in the planning horizon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' , |T|} K Set of categories (groups) in the area B Set of candidate locations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' i, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' , |B|} where based on our problem, the candidate locations (j) are the same as demand nodes (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='cij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Unit cost of satisfying demand of location i from facility j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='γo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Mobile store’s opening cost in each location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='γc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Mobile store’s closing cost in each location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='dit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Demand of location i at day t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Available number of mobile stores in each day ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='mk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Maximum number of stores allowed in group k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='nk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Minimum number of stores allowed in group k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Decision Variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='xijt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Fraction of demand of i that is supplied from j at day t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='yjt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Binary variables that is 1 if a mobile store is located at j at day t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' and is 0 otherwise ajt Auxiliary binary variables which 1 if a store is located in j at day t and will not be located in j at day t + 1 (,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', closing variable), and is 0 otherwise bjt Auxiliary binary variable which is 1 if a store is not located in j at day t and will be located in j at day t + 1 (,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', opening variable), and is 0 otherwise The set of all candidate locations (buildings) in group k is denoted by Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We know B = ∪(Fk)k∈[K], that it means each building should be at least in one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Note that ∩(Fk)k∈[K] can be nonempty, meaning that one building can be a member of more than one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Moreover, we do not need to define opening and closing costs (ajt, bjt) as binary variables since our minimization model and constraints force them to be 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In order to have a feasible solution, we assume that p must satisfy the following inequalities: � k∈K nk ≤ p ≤ � k∈K mk, meaning that there is a solution that satisfies the model’s assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Mathematical Formulation We defined all parameters and variables of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The goal is to minimize the total cost of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We can model our problem as follows: 5 Optimal Cost = min � t∈T � j∈B � i∈B ditcijxijt + � t∈T\\{|T|} � j∈B (γcajt + γobjt) (1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � j∈B xijt = 1, ∀i ∈ B, t ∈ T, (1b) � j∈B yjt = p, ∀t ∈ T, (1c) xijt − yjt ≤ 0, ∀i, j ∈ B, t ∈ T, (1d) yjt − yj(t+1) − ajt ≤ 0, ∀j ∈ B, t ∈ T \\ {|T|}, (1e) yj(t+1) − yjt − bjt ≤ 0, ∀j ∈ B, t ∈ T \\ {|T|}, (1f) � j∈Fk yjt ≤ mk, ∀k ∈ K, t ∈ T, (1g) � j∈Fk yjt ≥ nk, ∀k ∈ K, t ∈ T, (1h) yjt ∈ {0, 1}, ∀j ∈ B, t ∈ T, (1i) ajt, bjt ≥ 0, ∀j ∈ B, t ∈ T, (1j) xijt ≥ 0, ∀i, j ∈ B, t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1k) where constraint (1b) ensures the demand of all locations over the time horizon must be satisfied, constraint (1c) ensures the number of mobile stores allocated in each day should be p, constraint (1d) shows the relation between continuous and binary variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', if xijt > 0, then yjt should be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Constraints (1e) and (1f) ensure that we consider the closing and opening costs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Constraints (1g) and (1h) ensure that we are following the policies regarding the limits on the number of stores in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Finally, constraint (1i) is the binary constraint, and (1j), and (1k) are the non-negativity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We assume that at the beginning of day 1, dit is stochastic for all i ∈ B and all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Solution Approach The problem at hand is a mixed-integer mathematical programming model that involves un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We will outline a state-of-the-art methodology to tackle these characteristics in the following manner: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Robust Optimization In this section, an overview of the robust optimization approach proposed by Bertsimas and Sim (2004) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' To do so, the following linear programming model is considered: Min � j cjxj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � j ˜aijxj ≤ bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i xj ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀j (2) 6 where the technological coefficients ˜aij are assumed to be uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In other words, each coefficient ˜aij is regarded as an independent, symmetric, and bounded parameter, which can take values in [aij − ˆaij, aij + ˆaij], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ˜aij ∈ [aij − ˆaij, aij + ˆaij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this definition, aij and ˆaij denote the nominal value and the maximum deviation from the nominal value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Associated with each row i in problem (1) is Ji, which is defined as the set of all coefficients in row i that are subject to uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Furthermore, a scaled deviation ηij ∈ [−1, 1] is defined for each uncertain coefficient ˜aij as ηij = ˜aij − aij ˆaij that represents the scaled perturbation of ˜aij from its nominal value aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Bertsimas and Sim (2004) also introduced a parameter Γi ∈ [0, |Ji|] as the budget of uncertainty for each constraint i, where |Ji| denotes the number of elements of set Ji Bertsimas and Sim (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In fact, Γi is the maximum number of parameters that can really deviate from their nominal values for each constraint i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The parameter Γi that bounds the total scaled deviation of uncertain parameters as � j∈Ji |ηij| ≤ Γi adjusts the robustness of the proposed method against the level of solution conservatism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In particular, Γi = 0 represents the nominal or deterministic formulation, whereas Γi = |ηij| relates to the worst-case formulation in which all uncertain parameters are fixed at their worst-case values from the uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, the decision maker can make a trade-off between the protection level of constraint i and the degree of conservatism of the solution if Γi ∈ (0, |Ji|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Therefore, the budget of uncertainty Γi that is an input to the robust optimization model can specify how risk-averse the decision-maker is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Bertsimas and Sim (2004) proposed a nonlinear programming model as follows, which is equiv- alent to the uncertain model (1): Min � j cjxj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � j aijxj + max Ω { � j∈Si ˆaijxj + (Γi − ⌊Γi⌋)ˆaitixj} ≤ bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i xj ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀j (3) where Ω = {Si ∪ {ti}|Si ⊆ Ji, Si = ⌊Γi⌋ , ti ∈ Ji \\ Si} is defined as the uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For a given optimal solution x∗ of the problem (2), Bertsimas and Sim (2004) demonstrated that the protection function for constraint i against uncertainty, which is βi(x∗, Γi) = max Ω {� j∈Si ˆaijxj + (Γi − ⌊Γi⌋)ˆaitixj} can be formulated as the following linear programming problem: βi(x∗, Γi) = Max � j∈Ji ˆaij|x∗ j|ηij s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � j∈Ji ηij ≤ Γi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i 0 ≤ ηij ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i, j (4) According to the theory of strong duality, since problem (3) is always feasible and bounded for all Γi ∈ [0, |Ji|], its dual problem is feasible and bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Therefore, replacing the dual problem of the problem (3) into (2), Bertsimas and Sim (2004) derived the robust formulation of 7 the uncertain linear programming problem (1) as follows: Min � j cjxj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � j aijxj + βiΓi + � j∈Ji µij ≤ bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i βi + µij ≥ ˆaijxj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i, j ∈ Ji µij ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i, j ∈ Ji βi ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀i xj ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ∀j (5) where βi and µij are dual variables associated with the first and second constraints in programming problem (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' if the number of uncertain coefficients in constraint i that perturb from their respective nominal values is less than or equal to Γi, then the optimal solution from the robust problem (4) will remain always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, if more than Γi coefficients deviate from their nominal values, then the probability of violating constraint i for an optimal solution x∗ j is calculated as follows: Pr( � j ˜aijx∗ j < bi) ≤ 1 − ϕ(Γi − 1 � |Ji| ) (6) where ϕ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=') is the cumulative distribution function of a standard normal random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The robust optimization equivalent of the proposed problem (1) is: min x,y,a,b max d∈U dc1x + c2(a + b) (7a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1b) − (1k) (7b) or equivalently: min x,y,a,bθ + c2(a + b) (8a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1b) − (1k) (8b) θ ≥ max d∈U dc1x (8c) where U is the uncertainty set for demand, and a, b are decision vectors for ajt, bjt respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Depending on the problem setting, one can use the box, budget, or conic uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Among these, the budget uncertainty set is widely used, both due to its intuitive interpretation and tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The budget uncertainty set for the demand is: U = {d ∈ R|D|×|D| | d = d + γξ, ξ||∞ ≤ 1, � i,j∈D ξit ≤ Γ} where Γ is the uncertainty budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Using this definition we have: min x,y,a,bθ + c2(a + b) (9a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1b) − (1k) (9b) max ξ∞≤1,ξ1≤Γ ξc1x ≤ θ − ¯dc1x (9c) 8 Let’s focus on the left-hand-side of equation (3c): max ξ∞≤1,ξ||1≤Γ γξc1x → max ξ γξc1x, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' : � i,t |ξit| ≤ Γ, −1 ≤ ξit ≤ 1 (10a) Let ait = |ξit|, then we have: max γξc1x (11a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' � i,t ait ≤ Γ [π1] (11b) ξit ≤ 1 [π2 it] (11c) − ξit ≤ 1 [π3 it] (11d) ξit − ait ≤ 0 [π4 it] (11e) − ξit − ait ≤ 0 [π5 it] (11f) ξit ∈ R, ait ∈ R+ (11g) Since the model is a minimization problem, we need to obtain the dual problem whose objective is also minimization: The dual problem is: min π Γπ1 + � i,t∈D (π2 it + π3 it) (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' π1 ≥ π4 it + π5 it ∀i, t ∈ D (12b) π ∈ R+ (12c) Finally the DMS-p-MP reformulation of the problem becomes as: min x,y,a,bθ + c2(a + b) (13a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1b) − (1k) (13b) Γπ1 + π2 + π3 + ¯dc1x ≤ θ (13c) π1 ≥ π4 it + π5 it ∀i, t ∈ D (13d) π2 it − π3 it + π4 it − π5 it = γitc1 itxijt ∀i, t ∈ D, ∀j ∈ B (13e) πκ it ≥ 0 ∀i ∈ B, t ∈ T, κ ∈ {1, 2, 3, 4, 5} (13f) where πκ, x, and d is the vector of πκ it’s, dit, and xijt’s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Lagrangian relaxation In discrete location theory, one of the basic models is the p-median problem and it is an NP-hard problem, as with most location problems Mladenovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The MILP model presented in section 3 is typically solved in practice with numerous residence areas (demand points), potential facility locations, and time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' As a result of the large-scale property of the developed model, it presents substantial computational difficulty when applied to real problems, which cannot be addressed by commercial optimization software, such as Cplex, Xpress or Gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For supply chain optimization problems involving such computational complexity, Lagrangian relaxation (LR) is widely used (Diabat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Duong and Bui, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Rafie-Majd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Kheirabadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Hamdan and Diabat, 2020) in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 9 Therefore, this section develops an LR approach for solving the presented MILP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The LR method is an iterative algorithm that provides the upper and lower bounds of the optimal objective value as well as the estimation of the optimality gap of the feasible established solution in each iteration (Daskin, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The LR method used in this paper includes the general steps as follows: 1) Relax one of the constraints by multiplying it by a Lagrange multiplier and bringing the constraint into the objective function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 2) Solve the model to find the optimal values of the relaxed problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3) Find the feasible solution to the original problem by using the resulting decision variables found in step 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 4) Compute the lower bound using the solution obtained from the relaxed problem in step 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 5) Use the subgradient optimization method to modify the Lagrange multiplier assigned to the violated constraint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' return to step 2 after finding the new multiplier(s) for the Lagrange variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The algorithm terminates whenever the lower bound is close enough to the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Solving the Relaxed Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In order to make the problem easier to solve, the con- straints in this study are relaxed, even if this relaxation may lead to infeasibility (Daskin, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Specifically, constraints (1b), (1c) are relaxed, resulting in the following Lagrangian dual problem: min x,y,a,bθ + c2(a + b)+ � j∈B � i∈B � t∈T λ1 bt(xijt − 1)+ � j∈B � t∈T λ2 t(yjt − p) (14a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (1d) - (1k), (13c) - (13f) (14b) The optimal value of the objective function in the Lagrangian dual problem above, which is defined using non-negative Lagrange multipliers (λ1 bt, λ2 t), serves as a lower bound for the mixed- integer linear programming problem with constraints numbered (14a) to (14g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Finding a Feasible Solution and an Upper Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In most situations, the solution to the Lagrangian dual problem is not feasible due to the relaxation of constraints (1b) and (1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, it is possible to obtain a feasible solution that provides an upper bound on the single objective MILP model by solving this model and setting the decision variables xijt, yjt, ajt and bjt to the optimal values obtained from solving the Lagrangian dual problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Finding a Lower Bound, and Updating the Lagrange Multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' During each iteration of the Lagrangian procedure, the Lagrangian multipliers λ1 bt, λ2 t are updated and new lower and upper bounds are subsequently derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' There are various methods in the literature, such as cutting planes (Kelley, 1960), sub-gradients (Daskin, 1997), and bundling (Borghetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', 2003), that can be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this paper, the sub-gradient approach is employed to update the Lagrangian multipliers because it is a widely recognized and commonly used method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' According to the sub-gradient procedure (Daskin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 1997),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' the Lagrange multipliers at the (n + 1) iteration are calculated as follows: (λ1 bt)n+1 = max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (λ1 bt)n − τ n 1 (xijt − 1) � (15) (λ2 t)n+1 = max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (λ2 t)n − τ n 2 (yjt − p) � (16) 10 The step sizes τ n 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' and τ n 2 in the algorithm are defined as follows: τ n 1 = αn(UB − LBn) � j∈B � i∈B � t∈T λ1 bt(xijt − 1)2 (17) τ n 2 = αn(UB − LBn) � j∈B � t∈T λ2 t(yjt − p)2 (18) The term α is simply a constant that will be changed during each iteration of the algorithm as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' α is initialized to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' If the lower bound, LB, has not increased in an iteration, then its value will be halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Additionally, let UB be the best upper bound (the one with the smallest value) that has been discovered thus far, and LBn is the lower bound obtained at iteration n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Termination Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The algorithm will stop when one of the following conditions is met (Daskin, 1997): A predetermined number of iterations have been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The lower bound is equal to the upper bound (UB = LBn) or is close enough to the upper bound (UB − LBn < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The value of α becomes small 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Numerical Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Case description In this section, we discuss a case in which a company wants to place some mobile grocery stores on the campus of the University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' These self-service mobile stores can serve all students and staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Figure 1 shows a sample picture of these stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Specifically, the company wants to propose a plan to specify the optimal locations of stores needed on campus dynamically (every day) based on the demand variation on campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' They update their plan every month, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', they decide on all days at the beginning of each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this research, we aim to find the optimal locations of stores over the time horizon (one month).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Based on Section 3, |T| = 28, |K| = 6, and |B| = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Figure 2 shows the University of Waterloo campus map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' All buildings on campus are categorized into five groups: Service and Administrative Buildings, Academic Buildings, Residence Buildings, Research Park Buildings, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The University of Waterloo asked the company to follow some rules regarding each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Specifically, there are some limitations on the number of stores needed for each group for each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' More details and the list of buildings included in each group will be provided in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Each building has a specific demand on each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Obviously, it is not (financially) feasible to place one store in each building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The company has only p stores that in each period (day) they want to place all of them on campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We consider a demand cost, for students and staff in a building that does not have a store and they need to go to other buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The goal is to place the stores to satisfy the demand of all buildings while minimizing the costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' There are two more costs in our problem: opening cost and closing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Although the stores are mobile, we need to consider an opening (closing) cost if we want to open (close) a store at the beginning of each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In fact, we are capturing all transportation costs needed to change a store’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 11 Figure 1: Sample Picture of Mobile Grocery Store from PYMNTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='com (2018) The demand of each building may change day-to-day because of different reasons such as weekends, events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Then, it is an excellent motivation for day-to-day decision-making following demand variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this research, we assume that our demand prediction is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Then, we can finalize all decisions for each day of the month, once at the beginning of the month, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', our demand is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We can model this problem as a modified multi-period p-median problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Generally, our problem has two modifications compared to the simple multi-period p-median problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' First, we are considering opening and closing costs over the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Second, there are some extra constraints raised from University of Waterloo policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 12 mobde Sab MOBILE CONVENIENCE STORE Welcome!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Figure 2: University of Waterloo - Campus Map 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Data Collection We divide the University of Waterloo’s (UW’s) all facilities according to their functionality into six different segments: Academic Buildings, Parking Spots, Student Residence Buildings, Research Park Buildings, Athletic Buildings, and University Plaza respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The workspace is based on the UW’s official map includes 91 facilities, and we also make an assumption that the population in each facility within a segment is all equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Segments indicate in the Table 2 are as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='WES CRASAMW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='WES GRAHAN WR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='375 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='RESEARCH+TECHNOLOGY PARK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='DAVIDJOHNSTON ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='FRANK TOMPA DR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='NEATHE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='TOMEA ' 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+page_content='Help Line Telephone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Residence Bulldings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='PubllcTelephone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='ResearchParkBulldings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='300 metres ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Service VehicleTable 2: The Functional Classification for All Facilities in the Workspace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Buildings Included in the Segment (Building ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Id) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Academic Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='COG (1),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' GH (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' BRH (10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' SLC (11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' OWE (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' HMN (13),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' MC (14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' TC (15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' KOC (16),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' C2 (17),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EV3 (18),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' OPT (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' DC (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' SCH (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' FED (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' QNC (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' AL (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' HS (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' B2 (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EV2 (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' REN (28),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ESC (29),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EV1 (30),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' STJ (31),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EIT (32),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' HH (33),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' STP (34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Bl (35),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' PAS (36),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' CGR (37),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' E3 (38),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EC3 (39),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' BMH (40),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' PHY (41),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EC1 (42),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' LHI (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' NHI (44),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' EC2 (45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' UC (46),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' LIB (47),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ECH (48),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ERC (49),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' E2 (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ES (51),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' CSB (52),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' RCH (53),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' E6 (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Parking Parking CL (55), Parking A (56), Parking X (57), Parking C (58), Parking W (59), Park- ing OV (60), Parking V (61), Parking S (62), Parking K (63), Parking J (64), Parking R (65), Parking P (66), Parking T (67), Park- ing M (68), Parking L (69), Parking D (70), Parking EC (71), Parking HV (72), Parking N (73), Parking UWP (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Residence Buildings CLN (75), CLV (76), MKV (77), V1 (78), REV (79), TH (80), MHR (81), UWP (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Research Park Buildings 445 (83), 375 (84), 340 (85), 275 (86), ACW (87), 300 (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Athletic Buildings CLF (89), PAC (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' University Plaza Plaza (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We use Python’s OpenCV toolbox by Bradski and Kaehler (2000) to determine facilities’ specific locations by fixing pixel’s coordinates in the workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Students and staff allows going through buildings, it is likely for us to use Euclidean distance measurement l2 = � (x − a)2 + (y − b)2 to find the distance between the two pairs in the data set is as follows on the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 14 Figure 3: Coordinates for All Facilities on the Workspace From the UW’s website, we obtained the rough number of students and staff for each faculty, the rough number of students and staff for each university college, and the capacities of each residence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' A general assumption is that 10% of university students will use one of two gyms (CIF and PAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Tables 3, 4, 5, 6, 7, 8 below are the population of all classifications for each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 3: Statistical Results for Academic Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Engineering Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='11000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Mathematics Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='9260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Science Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Health Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3643 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Arts Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Environment Faculty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='37103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='687 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 4: Statistical Results for Parking Spots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Campus parking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='15 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='275 X X B No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 340 B No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 375 RA2 RAC XB No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 445 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Scaled LatitudeTable 5: Statistical Results for Student Residence Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Columbia Lake Village - North ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='404 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Columbia Lake Village - South ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='William Lyon Mackenzie King Village ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='320 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Student Village 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1381 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Ron Eydt Village ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='960 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Tutors’ Houses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Minota Hagey Residence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='University of Waterloo Place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1650 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5285 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='660 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 6: Statistical Results for Research Parking Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='David Johnson Research Park ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='683 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 7: Statistical Results for Athletic Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Columbia Icefield ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Physical Activities Complex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 8: Statistical Results for University Plaza ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Segment (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='University Shops Plaza ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total population ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Total facilities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Average per facility ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Given in the previous section that the time horizon T = 28 (approximately equivalent to a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='month),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' it can also separate into four periods (weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Based on the functionality of each segment, 16 with time-varying from Monday to Sunday, we may empirically define the facility in the different segments and the different days in utilization rate Ukt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For instance, the utilization rates for the academic buildings are 100 out of 100, but there are 30 out of 100 during the weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='complete estimated utilization rates for different functional buildings for each day over a week are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='as the following Table 9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Table 9: The Estimated Utilization Rate for Different Functional Buildings for Each Day over a Week ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Functionality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Monday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Tuesday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Wednesday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Thursday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Friday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Saturday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Sunday ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Academic Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Parking Spots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Residence Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Research Park Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='Athletic Buildings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='University Plaza ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='After that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' the demand dkt of each building in segment k on day t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' where t ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' , 7}, in a week is able to define as, dkt = Ukt × Total Population(k) Total Facilities(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' As for having a unified standard, we calculate the lower bound of opening facilities in constraint (1h) by taking a floor of 70% of the number of facilities in segment k over the total facilities on the workspace, nk = � p × the Number of Facilities in Segment (k) 91 × 70% � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For example, n1 = � 18 × 54 91 × 70% � = ⌊7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='47⌋ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Similarly, the upper bound of opening facilities in constraint (1g) by taking a ceiling of 130% of the number of facilities in segment k over the total facilities on the workspace, mk = � p × the Number of Facilities in Segment (k) 91 × 130% � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The following Table 10 is the maximum (minimum) number of Robomarts are allowed in segment k, Table 10: Bounds on the number of Robomarts Functionality Min (nk) Max (mk) Academic Buildings 7 14 Parking Lots 2 6 Residence Buildings 1 3 Research Park Buildings 0 2 Athletic Buildings 0 1 University Plaza 0 1 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Computational Experiments In this section, we use all parameters discussed in the Data Collection section, to solve our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' All numerical experiments have been run on an Apple M1 processor, limited to 16 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Gurobi has access to 8 physical cores, 8 logical processors, using up to 8 threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This model is MILP and has M(= 239512) variables and N(= 239694) constraints (excluding sign constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We use Gurobi version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1 by Bixby (2007) to solve this optimization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Since default settings in Gurobi generally work well, we are keeping all settings as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Specifically, we use Gurobi’s API embedded in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' MILP models are generally solved using a linear-programming based branch-and-bound algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The Gurobi provides advanced implementations of the latest MILP algorithms including deterministic parallel, nontraditional search, heuristics, solution improvement, cutting planes, and symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Based on section 5, the demand is weekly periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Then, we expect the model to make the same decision over the weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 11 shows what buildings are open during the planning horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We just show the days that we change our decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For instance, the buildings that are open between day 1 and day 6 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 11 shows that we only open new buildings and close the current buildings over the weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' We again change our decision on weekdays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' It is expected because the utilization rates of some of our segments are significantly different during the weekend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Specifically, we close buildings 12, 55, and 83 (that base on Table 3 are Conrad Grebel university college, Parking lots, Research building 375, respectively) and open buildings 63, 77, and 81 (that are Parking P, Student Village, University of Waterloo Place respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The interesting point is that all buildings 63, 77, and 81 are around students’ residences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Students are mostly in the residence area instead of the academic campus during the weekends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Then, it is worth closing some stores and opening new ones in students’ residences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 11: Results - Optimal Objective Value = 43175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='7, CPU Time = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='99 Seconds t Open Buildings 1 (Monday) 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 54,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 55,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 76,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 76,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 77,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 90 8 (Monday) 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 54,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 76,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 77,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 81,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 90 The detail solution is attached to this report in an excel file,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' containing 28 sheets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' each sheet 18 for each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Sensitivity Analysis In this section, we will change the parameters to see how variability can affect on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Time Horizon First, we analyze the impact of the length of time horizon on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 12 shows the results when we increase (decrease) the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For instance, as we increase the time horizon, it will take more time to find the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Besides, our optimal objective value would increase since we have are adding more positive terms to our cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 12: Sensitivity Analysis on Time Horizon T Id of Opened Build- ings Id of Closed Build- ings Objective Value CPU Time (S) 14 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 21570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='43 21 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 32372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='08 28 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 43175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='99 35 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 53978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='02 42 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 64781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='40 19 Figure 4: The impact of the time horizon on the objective functions and CPU process time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Moreover, as we see the opened and closed buildings remain the same as we increase the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The reason is that our demand is periodic over the weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' By increasing the number of weeks, the optimal decision to open and close some stores would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Figure 5 summarize the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Number of Facilities to Be Located (P) Now we change P and see how it affects our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Variability of P is important since it would help the company to decide the number of stores they want to buy and invent on the campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Note that we could also consider a fixed cost of buying each store in our model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=', we could add Price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='P to the objective function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, since in our optimization model P is fixed and given, we don’t need to consider it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' shows the results when we change P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' First, it seems that the running time is not directly dependent on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, P is determining the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Besides, the objective value is obviously decreasing as we increase P since we didn’t consider the price of each store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 20 Sensitivity Analysis on Time Horizon Objective Value CPU Time 20 60000 50000 15 40000 10 30000 20000 14 21 28 35 42 Time horizonFigure 5: The impact of the number of facilities on the objective functions and CPU process time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Also, Table 13 shows how our decision changes in different P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' There is one interesting point in our decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Compare P = 12 with P = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In P = 15, we don’t use the same opened building we used in P = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' It shows as we want to add one building, we may no longer need another building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 21 Sensitivity Analysis on P jective Value (S) CPU Cqo 55000 90 50000 60 45000 30 40000 35000 12 15 18 21 24 PTable 13: Sensitivity Analysis on P P Id of Opened Build- ings Id of Closed Build- ings Objective Value CPU Time (S) 12 2, 5, 10, 17, 27, 35, 44, 54, 65, 76, 83, 90, (63, 89) 63, 89, (65, 83) 58616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='07 15 2, 3, 5, 10, 27, 35, 45, 48, 63, 66, 74, 76, 83, 89 , 90 (81, 17, 61, 79) 81, 17, 61, 79, (83, 3, 66, 76) 49402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='79 18 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 43175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='99 21 0, 2, 5, 7, 10, 12, 17, 23, 26, 34, 40, 44, 48, 53, 54, 55, 76, 83, 85, 89, 90, (64, 77, 81, 3, 79) 64, 77, 81, 3, 79, (7, 55, 83, 76, 85) 38266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='89 24 0, 2, 3, 5, 6, 10, 12, 17, 23, 34, 40, 44, 48, 53, 54, 61, 67, 76, 80, 81, 83, 87, 89, 90, (64, 78) 64, 78, (80, 87) 34874.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Cost Coefficients Followed by two previous sections, we want to discuss the effects of changing opening and closing costs together on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Table 14 shows the results when we increase (decrease) the cost coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Considering the case that costs are both zero, the model tries to open (and close) any building in each period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In other words, it does not care how many building wants to open or close each day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Another noteworthy point is that running time is increasing as we increase the costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' It shows that the trade-off between keeping current buildings and opening (closing) other buildings is becoming important and challenging to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 22 Table 14: Sensitivity Analysis on Cost Coefficients (together) γo(c) Id of Opened Build- ings Id of Closed Build- ings Objective Value CPU Time (S) 0 0, 2, 3, 10, 12, 17, 26, 35, 45, 48, 54, 65, 76, 83, 86, 88, 90, (1, 4, 5, 23, 26, 27, 34, 40, 44, 56, 63, 81, 89) 4, 5, 27, 34, 40, 44, 56, 63, 81, 89, (23, 26, 35, 45, 65, 86, 88) 42900 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 79, 81) 63, 77, 79, 81, (12, 55, 76, 83) 43049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='34 5 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (63, 77, 81) 63, 77, 81, (12, 55, 83) 43175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='99 10 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (80, 81) 80, 81, (12, 83) 43382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='45 20 0, 2, 3, 5, 10, 12, 17, 27, 34, 40, 44, 48, 54, 55, 76, 83, 89, 90, (81) 81, (83) 43658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='58 Figure 6 shows that the initially opened buildings remain the same and starts to dynamically be changed as we increase the opening/closing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Moreover, the objective value increase as we increase the opening cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Also, for the large opening costs, we prefer not to open new buildings (and close the current open buildings) since it would be costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 23 Figure 6: The impact of opening/closing cost of facilities on the objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Discussion This study has presented a mixed-integer linear formulation for the Dynamic Modified Stochas- tic p-Median Problem in a Competitive Supply Chain Network Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The proposed model takes into account the robust optimization and time horizon as a novel approach, which enables the decision-maker to consider uncertainty and short-term changes in the supply chain network de- sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The robust optimization approach used in this study allows for the consideration of different scenarios and uncertainty in demand and supply, which is crucial in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Addi- tionally, the time horizon approach allows for the dynamic nature of the problem to be captured, which is important in today’s fast-paced business environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The proposed model was tested using computational experiments, and the results demonstrate the effectiveness of the proposed approach in handling the dynamic and stochastic nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The results also provide valuable insights for practitioners and researchers in the field of supply chain network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The proposed model can be extended and applied to other sim- ilar problems in the field, such as facility location, transportation and logistics, and inventory management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' One of the main contributions of this study is the integration of robust optimization and time horizon in the mixed-integer linear formulation for the Dynamic Modified Stochastic p-Median Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The robust optimization approach allows for the consideration of different scenarios and uncertainty, while the time horizon approach allows for the dynamic nature of the problem to be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This integration provides a more realistic and practical solution to the problem, which can be useful for practitioners and researchers in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Another important contribution of this study is the application of the proposed model to a competitive supply chain network design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This application is relevant and valuable as it provides insights into how the proposed model can be used in a real-world context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The results of 24 Sensitivity Analysis on Cost Coefficients (together) Count Value Facility Objective 44000 40 43750 30 43500 20 43250 10 43000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='0 Cost coefficient Facility status Closed Openedthe computational experiments demonstrate the effectiveness of the proposed model in handling the dynamic and stochastic nature of the problem and provide valuable insights for practitioners and researchers in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' For instance, we can change over the problem definition to solve any other location problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Covering location problems are valuable to be investigated, such as trying to find the optimal number of Robomarts PYMNTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content='com (2018) that can serve all facilities on the workspace if the single Robomart can only serve facilities within limited miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This study has the potential usefulness of the proposed model in the context of a pandemic and quarantine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The ability to handle uncertainty and short-term changes in the supply chain network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The robust optimization approach used in this study allows for the consideration of different scenarios and uncertainty in the demand and supply, which is crucial in a pandemic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The proposed model can be used to design and optimize supply chain networks that are more resilient and adaptable to the changing conditions caused by a pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This can help businesses and organizations to minimize disruptions and maintain the continuity of operations during challenging times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' One limitation of this study is that the Robomart can only be located at specific facilities (nodes) in the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In reality, however, the Robomart can also be located somewhere on the route between two nodes (edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This limitation may affect the validity and applicability of the proposed model in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' To address this limitation, it is possible to consider analogous absolute p-median problems to simulate reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This approach would involve the inclu- sion of edge-based locations for the Robomart in the model, which would provide a more realistic representation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' However, this would require additional mathematical development and computational resources, and would be a subject for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' Another potential area of future research is to make the problem an Adaptive Robust Opti- mization (ARO) problem Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In this approach, the number of trucks p would be determined in the first stage and other variables in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' This would enable a more flexible and dynamic approach to supply chain network design, as the number of trucks can be adjusted in response to changes in demand and supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' In summary, the proposed DMS-p-MP model takes into account the robust optimization and time horizon as a novel approach, which enables the decision-maker to consider uncertainty and short-term changes in the supply chain network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFJT4oBgHgl3EQfTCwr/content/2301.11502v1.pdf'} +page_content=' The results of the computational experi- ments demonstrate the effectiveness of the proposed model in 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photovoltaic effect in antiferromagnetic +MnPSe3 monolayer +Qianqian Xue, Xingchi Mu, Yan Sun, Jian Zhou* +Center for Alloy Innovation and Design, State Key Laboratory for Mechanical Behavior of +Materials, Xi'an Jiaotong University, Xi'an, 710049 China +*Email: jianzhou@xjtu.edu.cn +Abstract +Valleytronics that uses the inequivalent electronic states at the band extrema in +semiconductors have been considered to play a vital role in the future information read/write +technology. In the current work, we theoretically show that sizable valley contrasting bulk +photovoltaic (BPV) effect could emerge, even when the total photocurrent is symmetrically +forbidden. We illustrate our theory by using a prototypical two-dimensional antiferromagnetic +semiconductor, MnPSe3 monolayer, that is 𝒫𝒯-symmetric (𝒫 and 𝒯 refer to spatially inversion +and time-reversal operators, respectively). We show that the Néel vector well controls the magnetic +point group at the Γ point, and the BPV current direction. In addition, k-dependent photocurrent +generally arises due to the reduction of little group constraints at the valley. This leads to hidden +valley-polarized photoconductivity which could reach a magnitude of 1350 μA/V2, observable +experimentally. We further predict that the MnPSe3 monolayer could be two-dimensional +ferrotoroidic, again depending on its Néel vector direction, which can be determined via +magnetoelectric response measurements. Our work provides an exemplary platform for paving the +route to future opto-spintronic and opto-valleytronic devices in a single antiferromagnetic +nanomaterial. + + + + +2 +I. Introduction +Manipulation and control of the electronic state at band extrema, known as band valleys, have +received tremendous attention during the past decade [1-4]. Different valleys in semiconductors +usually locate in the momentum space with a large separation, guaranteeing their robustness +against smooth geometric deformation and low energy phonon excitations. Valleytronics that uses +valley degree of freedom to constitute the binary logic states (similar as the charge and spin in +electronics and spintronics, respectively), holds the potential for ultrafast and efficient information +and data read/write applications [5-7]. Even though early studies on electronic valleys were mainly +focusing on silicon (dated back to 1970s) [8,9], recently discovered two-dimensional (2D) lattices +have significantly promoted its advances [10-13]. Currently, in order to detect the valleytronic +feature [14,15], one usually uses optical absorption spectrum such as circular or linear dichroism +spectroscopy [16-18], and electrical approaches such as (quantum) valley Hall effect [13,19-21]. +Note that electric signal is more realistic and facile for nanoelectronic devices, yet the Hall effect +measurement requires depositing electrodes onto samples, which may introduce unwanted +impurities or disorders. +In this work, we propose another valley contrasting feature that is stimulated by noncontacting +optical illumination and can be measured and probed electrically. We discuss such optoelectronic +responses via bulk photovoltaic (BPV) effect [22] in 2D antiferromagnetic (AFM) honeycomb +materials [23]. The AFM systems that composes compensated spin polarization are found to be +advantageous due to the absence of stray field and ultrafast spin dynamics [24]. Hence, they give +rise to large information storage density and high switching efficiency in real operations. We apply +group theory analysis and conduct first-principles density functional theory (DFT) calculations to +show that valley contrasting BPV photocurrent could emerge in a prototypical AFM MnPSe3 +monolayer. The MnPSe3 belongs to 2D transition metal phosphorus chalcogenides family, usually +denoted as TMPX3 (TM = Cr, Mn, Fe, Co, Ni, and X = S and Se). Depending on the transition metal +species, this series of materials exhibit different magnetic patterns. Among them, the MnPSe3 +monolayer shows Néel-type AFM configuration [25], which is 𝒫𝒯-symmetric (𝒫 refers to +inversion symmetry and 𝒯 denotes time-reversal symmetry). As both 𝒫 and 𝒯 are broken, it holds +non-degenerate valley polarizations [26]. Previous nonlinear optics theory has demonstrated that +the BPV current under linearly polarized light (LPL) irradiation shows magnetic injection current +(MIC) feature [27]. Our calculation suggests a sizable MIC density (1D current density on the + + +3 +order of 0.1–1 A/cm) could emerge under an intermediate light intensity (electric field component +on the order of 0.1 MV/cm). In addition, we show that the AFM Néel vector L (= MMn1 – MMn2, +Mn1 and Mn2 denote the two Mn sites in the unit cell) could effectively tune the symmetry +constraints for the MIC generation. At a generic k point in the momentum space, the symmetry +reduction could lead to hidden photocurrent generation, as revealed by a simple group theory +approach. This suggests that ubiquitous valley contrasting MIC exists in the AFM MnPSe3 +monolayer. The Néel vector is strongly coupled with in-plane mechanical deformation, i.e., the in- +plane magnetocrystalline anisotropy energy (MAE) can be manipulated via small uniaxial strains. +Finally, we suggest that the MnPSe3 monolayer also hosts a L-dependent toroidal moment that can +be measured by magnetoelectric responses, showing magnetically harnessed ferrotoroidicity. + +II. Computational Details +We perform DFT calculations in the Vienna ab initio simulation package (VASP) [28,29] that +uses the generalized gradient approximation (GGA) method in the solid state Perdew-Burke- +Enzerhof (PBEsol) [30] form to treat the exchange-correlation interaction. Projector augmented- +wave (PAW) [31] method is used to describe the core electrons, while the valence electrons are +expanded by a planewave basis set with its kinetic cutoff energy setting to be 400 eV. The first +Brillouin zone (BZ) is represented by (12×12×1) Monkhorst-Pack k-mesh grids [32], and the +strong correlation on the Mn-d orbital is treated by adding a Hubbard U correction [33,34] with +effective value of 5 eV. This has been widely adopted in previous works [25], and we note that the +exact U value does not affect our main conclusion. If not indicated explicitly, spin-orbit coupling +(SOC) is included self-consistently in all our calculations. In order to simulate 2D materials in +periodic boundary condition, we add a vacuum space of over 20 Å in the out-of-plane z direction, +which could eliminate the nearest neighbor image layer interactions. The convergence criteria of +total energy and Hellman-Feynman force components are set as 1 × 10–7 eV and 1 × 10–3 eV/Å, +respectively. We fit the DFT calculated electronic states by using maximally localized Wannier +functions based on Mn-s and d, P-p, and Se-p orbitals, as implemented in the Wannier90 code +[35,36], which are used to evaluate the BPV photoconductivity and magnetoelectric coupling +components. + + + +4 +III. Results +Geometric, electronic structure and symmetry arguments. The atomic structure of MnPSe3 +monolayer is plotted in Fig. 1(a). Geometrically, each P dimer is vertically sandwiched by six Se +atoms. These P2Se6 moieties are embedded in the hollow sites of Mn honeycomb sublattice +framework. Without considering spin polarization, it belongs to 𝑃3%1𝑚 layer group (3%𝑚 point +group), which contains 𝒞3z rotation, 𝒞2y rotation, and a mirror reflection ℳ!. Hence, the inversion +symmetry 𝒫 is also preserved. The Néel-type AFM configuration guarantees the unit cell being +hexagonal lattice [the black rhombus in Fig. 1(a)], containing two Mn sites that carry antiparallel +spin polarization. Before including SOC, the electronic states in the two spin channels (majority +and minority) are degenerate [Fig. 1(b)], with the valence and conduction band valleys locating at +the corner of the first BZ (K and K' points). The direct bandgap value is calculated to be 1.697 eV, +consistent with previous reports [23]. + +FIG. 1. (a) Top and side views of the MnPSe3 monolayer. The crystalline mirror reflection +ℳ! is indicated by the horizontal dashed line, and the black rhombus represents the unit cell. (b) +Band dispersion along the high symmetric k-path without including SOC effect. (c) The first BZ, +with high symmetric points of Γ = (0, 0, 0), K = (1/3, 1/3, 0), M = (0, 1/2, 0), and K' = (–1/3, 2/3, +0) in direct coordinates. (d) Schematic plots of band edge positions with bandgap values when the +Néel vector L is along the x, y, and z axes with SOC included. Note that each band is actually +doubly degenerate due to antiunitary 𝒫𝒯-symmetry. + +(b) +3 +(eV) +2 +M +spin up +1 +y + spin down +Mn +0 +x +P +Z +.1 +Se +. +M +K +K' +M +b2 +(c) +(d) +K +M +K +b1 +1.6927 eV +.6928 eV +1.6926 eV +1.6926 eV +1.7202 eV +1.6557 eV +24.5 mel +K +K +K +K +Lx +Lly +L12 +5 + +The inclusion of SOC breaks the spin rotational symmetry. In this case, the system becomes +𝒫𝒯-preserved, so that each band is still doubly degenerated due to its antiunitary symmetry. Since +the spin angular momentum transforms as a pseudovector, its direction would determine the +magnetic point group and the valley splitting. We list the basic symmetric arguments for 𝐋 ∥ 𝐱-, +𝐋 ∥ 𝐲-, and 𝐋 ∥ 𝐳- in Table I, (⋅̂ denotes the Cartesian unit vector). Here, x and y refer to the zigzag +and armchair directions of the Mn honeycomb sublattice, respectively. Our MAE calculations +reveal that in-plane spin polarization (𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-) is energetically more favorable than the +out-of-plane spin polarization (𝐋 ∥ 𝐳-) by ~0.5 meV per unit cell (or ~0.023 μJ/cm2), also tabulated +in Table I. Note that the K and K' valleys are connected via 𝒯𝐾 = 𝐾′ or ℳ"𝐾 = 𝐾′. Hence, one +can deduce that when 𝐋 ∥ 𝐲-, the two valleys are degenerate. On the other hand, the valley +degeneracy lifts for 𝐋 ∥ 𝐱- (marginal) and 𝐋 ∥ 𝐳- (bandgap differ by 64 meV at the two valleys). Our +calculations can be seen in Fig. 1(d) for schematic plots and Fig. S1 in Supplemental Material (SM) +for details. + +Table I. Magnetic point group, mirror reflection, and relative energies for L along three +Cartesian axes. +Neel vector +Magnetic point +group at Γ +Mirror +reflection +Relative energy +(μeV per unit cell) +Magnetic group +at K valley +𝐋 ∥ 𝐱- +2#/𝑚 +ℳ! +0 +𝑚 +𝐋 ∥ 𝐲- +2/𝑚′ +ℳ!𝒯 +1.63 + 𝑚′ +𝐋 ∥ 𝐳- +3%′𝑚 +ℳ! +494 +3𝑚′ + +Valley contrasting bulk photovoltaic effect. Next, we show that the mirror reflection difference +with respect to Néel vector L could lead to contrasting BPV photocurrent generation. We will +focus on LPL irradiation, which, according to nonlinear optics theory, generates MIC for 𝒫𝒯- +symmetric systems [37,38]. The current density is evaluated via +𝒥$ = 𝜂%% +$ (0; 𝜔, −𝜔)𝐸%(𝜔)𝐸%(−𝜔), (1) + + +6 +where i and j refer to in-plane Cartesian axes (x or y), and 𝐄(𝜔) is the optical alternating electric +field with the angular frequency 𝜔. The MIC in 𝒫𝒯-systems can be viewed as a cousin effect to +the normal injection current [27,37] that appears in nonmagnetic materials (𝒯-preserved, 𝒫-broken) +under circularly polarized light. It arises from the velocity difference between the valence and +conduction bands, which increases linearly with time and saturates at the carrier lifetime. +According to band theory, its length-gauge form formula is +𝜂%% +$ (0; 𝜔, −𝜔) = − +&'(! +)ℏ" ∫ ++"𝐤 +()')" ∑ +𝑓/0Δ/0 +$ +𝑔/0 +%% 𝛿(𝜔/0 − 𝜔) +/0 +. (2) +Here, 𝑓/0 = 𝑓/ − 𝑓0 and Δ/0 +$ += 𝑣// +$ +− 𝑣00 +$ measure the Fermi-Dirac occupation and group +velocity differences between the band 𝑚 and 𝑛, respectively. The Kronecker delta function +𝛿(𝜔/0 − 𝜔), represented by the Lorentz function with a broadening factor of 0.02 eV, guarantees +the energy conservation law, with ℏ𝜔/0 = ℏ𝜔/ − ℏ𝜔0 referring to the eigenenergy difference. +The MIC generation is scaled by quantum metric tensor 𝑔/0 +%% += 2 ∑ +Re N𝑟/#0$ +% +𝑟0$/# +% +P +1,3 +, where 𝜇 +and 𝜈 represent the degenerate band indices, and the position matrix is 𝑟0/ +% += S𝑛T𝑟̂%T𝑚U = +40567%5/8 +%9&' . +All these quantities are k-dependent which are omitted for clarity reason. According to previous +works [38], we take the carrier lifetime 𝜏 to be a universal value of 0.2 ps, comparable to most +experimental observations in 2D materials. This carrier lifetime can be effectively controlled by +the sample quality, disorder level, temperature, etc., and can be characterized by the electrical +conductance according to the Drude model. The integral is performed in the whole 2D first BZ. +We take the effective thickness as d = 0.6 nm, measured from its bulk counterpart. Then, we divide +the 2D photoconductivity [μA⋅nm/V2, according to Eq. (2)] by d, so that it adopts the conventional +3D photoconductivity unit (μA/V2). +Before performing DFT calculations, we briefly analyze the magnetic point group for each +case and their implication for BPV photocurrents. The highest symmetry exists when the Néel +vector is parallel to z, 𝐋 ∥ 𝐳-, and the system belongs to magnetic point group 3%′𝑚 = 𝐶:6⨂𝒫𝒯. +Since we are focus on the MIC (injection current under LPL) that is invariant under 𝒫𝒯, we focus +on the 𝐶:6 point group (character table can be found in Tables S1). For the electric field and current +in the 2D (xy) plane, the irreducible representation for current and second order symmetric field +are Γ𝒥 = 𝐸 and Γ(𝐄𝐄)( = 𝐴=⨁𝐸. Hence, one has Γ𝒥⨂Γ(𝐄𝐄)( = 𝐴=⨁𝐴)⨁2𝐸, allowing only one + + +7 +nonzero independent MIC component, which will be shown to be 𝜂"" +" = −𝜂!! +" = −𝜂"! +! and 𝜂"" +! = +𝜂!! +! = 𝜂"! +" = 0. If we focus on the valley K (or K'), the spatial inversion is no longer preserved, +giving its magnetic little group of 3𝑚′ = 𝐶:⨁𝐶>𝒯. Thus, the symmetry argument at each valley +for MIC generation follows 𝐶: (since MIC is 𝒫𝒯-symmetric which is inconsistent with 𝐶>𝒯), +yielding Γ𝒥⨂Γ(𝐄𝐄)( = 𝐸⨂(𝐴⨁𝐸) = 2𝐴⨁2𝐸 with two allowed and independent MIC components +(Table S2). This indicates that momentum-dependent hidden MIC exists. This is akin to the hidden +spin polarization (or spin Hall effect) as discovered locally in centrosymmetric ionic compounds +and antiferroelectric materials [39-43], which arises in the real space due to the reduced symmetry +constraints on each sector. Similar confinements can be found for in-plane Néel vector, which +breaks the three-fold rotation 𝒞3z. The 𝐋 ∥ 𝐱- belongs to 2′/𝑚 = 𝐶>⨂𝒫𝒯, and the allowed MIC +generation can be deduced from the 𝐶> point group (see its character table in Table S3). We then +have x-flowing MIC satisfying Γ𝒥⨂Γ(𝐄𝐄∗)( = 𝐴#⨂(2𝐴#⨁𝐴##) = 2𝐴#⨁𝐴′′. The two independent +MIC would be 𝜂"" +" and 𝜂!! +" . The 𝐋 ∥ 𝐲- is 2/𝑚′ = 𝐶)⨂𝒫𝒯, and one can perform similar arguments +to yield same results as in 𝐋 ∥ 𝐲-. At the valleys, their 180°-rotation symmetry (2# or 2) is no longer +preserved. Hence, they both exhibit valley-dependent finite MIC components that are forbidden in +the whole system. +The switching of L strongly affects the velocity texture distribution in k-space, so that the +MIC direction depends on the Néel vector [see Eq. (2)]. In detail, when 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳-, the ℳ! +reflection assigns ℳ!Δ!\𝑘", 𝑘!^ = −Δ!(𝑘", −𝑘!). On the contrary, for the 𝐋 ∥ 𝐲- case, we have +ℳ!𝒯Δ"\𝑘", 𝑘!^ = −Δ"(−𝑘", 𝑘!). Since the quantum metric tensor is almost unchanged in these +cases, one easily deduces that the x-flowing MIC is forbidden when 𝐋 ∥ 𝐲-, while the y-flowing +MIC is zero when 𝐋 ∥ 𝐱- or 𝐋 ∥ 𝐳-. Note that here we assume that the x- (or y-) polarized LPL is +illuminated. For a general polarization angle, such symmetry arguments may be changed, and the +final MIC will be a linear combination from the x-LPL and y-LPL. + + +8 + +FIG. 2. Calculated MIC photoconductivity under x and y-polarized LPL for (a) 𝐋 ∥ 𝐱-, (b) 𝐋 ∥ +𝐲-, and (c) 𝐋 ∥ 𝐳-. In (c), we note that 𝜂"" +" (𝜔) = −𝜂!! +" (𝜔), arising from the 𝒞3z rotation. Such +symmetry is broken when L lies in-plane. Quantum metric distribution between the top valence +and bottom conduction bands 𝑔6? +""(𝐤) over the first BZ for (d) 𝐋 ∥ 𝐱-, (e) 𝐋 ∥ 𝐲-, and (f) 𝐋 ∥ 𝐳-, and +𝑔6? +!!(𝐤) for (g) 𝐋 ∥ 𝐱-, (h) 𝐋 ∥ 𝐲-, and (i) 𝐋 ∥ 𝐳-. + +Our first-principles calculations confirm the above qualitative results. As shown in Figs. 2(a)– +2(c), we plot the calculated MIC generation photoconductivity. One clearly observes that 𝜂%% +! = 0 +for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳- (i = x or y). When L is switched to along y, the 𝜂%% +" becomes zero. For the +symmetrically allowed current, the magnitude of photoconductivity reaches ~360 μA/V2 (𝐋 ∥ 𝐳-). + +(a) +(b) +(c) +LIx +Lly +LZ +(μA/V2) +400 +(μA/V2) +300 +na +300 +nr = 0 +naa +nua +0 +200 +200 +200 +nyy +=0 +lyy +Photoconductivity +Photoconductivity +Photoconductivity +0 +100 +100 +nyy +yy +0 +0 +0 +-200 +-100 +-100 +-200 +-200 +-400 +2 +4 +6 +0 +0 +2 +6 +0 +2 +4 +6 +w(eV) +w(eV) +w(eV) +(d) +(e) +() +g (k) +g(k) +gx (k) +8 +8 +0.8 +0.8 +0.8 +7 +7 +6 +6 +0.4 +0.4 +6 +0.4 +5 +5 +5 +(t-y)n +ky(A +y)ny +0 +4 +0 +4 +0 +4 +3 +3 +3 +-0.4 +-0.4 +-0.4 +2 +2 +2 +1 +1 +1 +-0.8 +-0.8 +-0.8 +0 +0 +-0.4 +0 +0.4 +-0.4 +0 +0.4 +-0.4 +0.4 +0 +ka(A-1) +ka(A-1) +ka(A-1) +(g) +(h) +(i) +gu (k) +yy +gc (k) +8 +0.8 +0.8 +8 +0.8 +7 +7 +7 +6 +6 +6 +0.4 +0.4 +0.4 +5 +5 +5 +y)n +V +0 +ky(A +0 +4 +0 +4 +3 +3 +3 +-0.4 +-0.4 +-0.4 +2 +2 +1 +1 +1 +-0.8 +-0.8 +-0.8 +0 +0 +-0.4 +0 +0.4 +-0.4 +0 +0.4 +-0.4 +0 +0.4 +ka(A-1) +ka(A-1) +ka(A-1) +9 +It indicates that if we take the electric field magnitude of 0.1 V/nm (at the photon energy of 3.2 +eV, or wavelength of 387 nm), corresponding to 1.33 × 10@ W/cm2 light intensity, one could +generate ~3.6 μA/nm2 current density. Across the lateral size of 1 nm (note that the effective +thickness is d = 6 Å), the current reaches 2.16 μA. Normal to the MIC, no net photocurrent can be +detected. This vividly suggests that switching magnetic moment direction could drastically rotate +the MIC generation direction. Such a large contrast can be directly measured via closed-circuit +current or open-circuit voltage. The quantum metric between the top valence band and bottom +conduction band (both doubly degenerate) 𝑔6? +""(𝐤) and 𝑔6? +!!(𝐤) for 𝐋 ∥ 𝐱-, 𝐋 ∥ 𝐲-, 𝐋 ∥ 𝐳- are shown +in Figs. 2(d)–2(i). We can see symmetry argument of 𝑔6? +%% \𝑘", 𝑘!^ = 𝑔6? +%% \𝑘", −𝑘!^, (𝑖 = 𝑥 or 𝑦) +for ℳ! and 𝑔6? +%% \𝑘", 𝑘!^ = 𝑔6? +%% \−𝑘", 𝑘!^ for ℳ!𝒯. In addition, we note that the time-reversal +symmetry 𝒯 that flips the Néel vector L (e.g., between 𝐋 ∥ 𝐱- and 𝐋 ∥ −𝐱-, or from 𝐋 ∥ 𝐳- to 𝐋 ∥ −𝐳-) +also reverses the MIC generation while keeping the magnitude, as the 𝜂%% +$ is scaled by velocity +operator and being 𝒯-odd (see Fig. S2). +We then show that sizable valley contrasting MIC emerges even when the net current is zero. +In order to explicitly see this, we plot the k-resolved MIC contributions, namely, the integrand of +Eq. (2), for the symmetrically forbidden currents [Figs. 3(a)–3(c)]. The incident photon frequency +is selected to be ℏω = 1.8 eV, slightly above the bandgap. In all these cases, the mirror symmetry +constraints (Table I) are clearly observed. Main contributions appear in the vicinity of K (and K') +valleys, in agreement with the joint density of states at this incident energy. Even though the +contributions around one valley show both positive and negative ridges, their summation is +nonzero. In Figs. 3(d)–(f), we plot the valley-dependent MIC, which is integrated in the triangular +k-space around each valley, whose area equals to half of BZ. We see that both valleys contribute +significant MIC generation, reaching a photoconductivity of ~1500 μA/V2 (for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳-) or +~600 μA/V2 (for 𝐋 ∥ 𝐲-). In each case, the MICs from two valleys flow oppositely with the same +magnitude, giving zero net MIC generation. This hidden valley-dependent BPV current has been +largely overlooked previously, and may find its potential applications in 2D valleytronic devices. +We propose that this result provides another scheme to use the valley current, in addition to the +electrically triggered (quantum) valley Hall effect. Experimentally, one could design a magnetic +heterostructure [Fig. 3(g)] to separate and measure such valley-dependent MIC. Valley contrasting +MIC could accumulate at a domain wall between two AFM configurations that are time-reversal + + +10 +with each other (e.g., 𝐋 ∥ 𝐳- and 𝐋 ∥ −𝐳-), so that the current contributed from a specific valley K +(or K') in both domains flow to their boundary. Similar valley-dependent MIC also exists for the +symmetrically allowed components, as plotted in Fig. S3. The two valleys also hold oppositely +flowing MIC, but they do not completely cancel each other. + +FIG. 3. BZ contribution of MIC integrand 𝜚"" +$ (𝐤, 𝜔) = ∑ +𝑓/0Δ/0 +$ +𝑔/0 +"" 𝛿(𝜔/0 − 𝜔) +/0 + under +x-PLP irradiation for (a) 𝐋 ∥ 𝐱- (𝑗 = 𝑦), (b) 𝐋 ∥ 𝐲- (𝑗 = 𝑥), and (c) 𝐋 ∥ 𝐳- (𝑗 = 𝑦). Here the incident +photon energy is ℏ𝜔 = 1.8 eV. (d)–(f) plot their corresponding valley contrasting MIC +contributions. (g) Schematic plot of magnetic heterostructure with two AFM configurations that +are time-reversal with each other (e.g., between 𝐋 ∥ 𝐳- and 𝐋 ∥ −𝐳-). Valley-dependent BPV current +would accumulate at their domain boundary. + +MAE modulation under strains. One may wonder how to harness the in-plane MAE (𝐸ABC = +𝐸𝐋∥𝐱7 − 𝐸𝐋∥𝐲7), so that the Néel vector L can be pinned along x or y. We show that a uniaxial strain +could further split the energy difference between 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-. Our numerical results are shown +in Fig. 4. At the equilibrium (strain-free) state, the 𝐋 ∥ 𝐱- is almost degenerate with 𝐋 ∥ 𝐲- (EMAE = + +Qxx +(a) +(b) +(c) +0.8 +115 +0.8 +0.8 +6 +6 +10 +4 +4 +0.4 +0.4 +0.4 +5 +2 +2 +C +0 +A +0 +0 +0 +0 +0 +-2 +-5 +-2 +-0.4 +-0.4 +-0.4 +-4 +-4 +-10 +-0.8 +-6 +-0.8 +-0.8 +-6 +-15 +-0.4 +0 +0.4 +-0.4 +0 +0.4 +-0.4 +0.4 +0 +ka(A-1) +ka(A-1) +ka(A-1) +(d) +(f) +(e) +1500 +600 +1500 +n(K) +n(K) +n(K) +1000 +400 +1000 +n(K) +n(K') +n(K') +(μA/V2) +500 +200 +500 +(μA) +0 +0 +0 +3 +-500 +-500 +-1000 +-400 +-1000 +-1500 +-600 +-1500 +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +w(eV) +w(eV) +w(eV) +(g) +K +LⅡ2 +LⅡ-2 +11 +−1.63 μeV per unit cell). Under tensile strain along x (εxx), the EMAE is negative, so that the Néel +vector L prefers the x direction. On the other hand, the y-tensile strain increases the EMAE, aligning +the L along y. In both cases, a small strain of 3% (about elastic energy of 49 meV in one unit cell) +could enhance the MAE magnitude to be ~35 μeV per unit cell, which is large enough to be +distinguished in low temperature experiments. Furthermore, we find that such a small strain will +not significantly change the band structure in these cases, and the MIC photoconductivity +marginally change their values. + +FIG. 4. Total energy difference (per unit cell) between 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲- under uniaxial strain ε +along x and y. The out-of-plane AFM configuration 𝐋 ∥ 𝐳- is always much higher in energy under +such small strains. + +L-dependent ferrotoroidicity and magnetoelectric responses. In addition to the optically +induced nonlinear current, we now show that the ferrotoroidicity is also sensitive to the Néel vector +direction. Ferrotoroidicity has been discovered to be another primary ferroic order, compensating +the ferroelasticity (𝒫-even, 𝒯-even), ferroelectricity (𝒫-odd, 𝒯-even), and ferromagnetism (𝒫- +even, 𝒯-odd) [44,45]. It reverses its sign under either 𝒫 or 𝒯, and is defined by toroidal moment +𝐭 = ∑ 𝐫% × 𝐬% +% +, where 𝐫% and 𝐬% represent the position and spin vectors of ion-i, and the summation +runs over all sites in the supercell. Previous theoretical and experimental works have disclosed a +few ferrotoroidic bulk materials, such as LiCoPO4 [46], LiFeSi2O6 [47], and defective SrTiO3 [48]. +Here, we suggest that the 2D MnPSe3 monolayer also holds ferrotoroidic order with sizable out- +of-plane toroidal moment, if the Néel vector is pointing away from y. As shown in Fig. 5(a), we + + +12 +schematically depict the Mn sublattice distributions in a rectangular supercell, which, without loss +of generality, is uniaxially strained. The supercell contains four Mn sites, which locate on two co- +centered circles with radius of 𝑟= and 𝑟). At the equilibrium state, these two green circles are +identical, 𝑟= = 𝑟), and the angle θ = 30°. Geometrically, if x-tensile strain is applied, θ reduces and +𝑟= < 𝑟); y-tension will increase θ and makes 𝑟= > 𝑟). + +FIG. 5. (a) Schematic plot of ferrotoroidicity in a rectangular supercell. Blue and red circles +represent the antiparallel spin polarized Mn sites, and the two green circles (radius 𝑟= and 𝑟)) are +co-centered. Detailed explanation can be found in the main text. (b) Calculated magnetoelectric +coefficient for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-. The abscissa axis denotes the chemical potential relative to the +Fermi energy. + +We can directly estimate the toroidal moment 𝐭 in this supercell setup. Note that similar as the +electric polarization 𝐏, here t is also multi-valued with respect to a quanta, depending on the choice +of origin. Nonetheless, we can use this position-spin cross product definition to determine its +existence. When the Mn spin polarization is along x, namely, 𝐬 = (±𝑠", 0,0), it can be directly +deduced that 𝐭 = (0,0, 𝑡H) and 𝑡H = − ∑ +𝑦%𝑠%," +I +%J= += −2𝑠"(𝑟= − 𝑟) sin 𝜃) ≠ 0. Hence, it shows a +nonzero z component toroidal moment. If 𝐬 = (0, ±𝑠!, 0), one could easily find 𝐭 = (0,0,0), +constrained by the ℳ!𝒯 operation, even though the supercell is uniaxially strained. This can also +be understood by analyzing the 𝐶)K point group (Table S4). The 𝐋 ∥ 𝐱- belongs to the magnetic +point group 2′/𝑚, which gives negative character for both 𝐶) rotation and inversion 𝑖. Hence, one +has Γ/ = 𝐵L in which all other elements are represented by +1. The vertically aligned electric field + +(b) +d +0.01 +nm/V) +0 +X +r1 +0.01 +(μB +-0.02 +文 +-0.03 +Lly +a +-0.04 +0 +1 +2 +13 +and spin polarization are presented by Γ𝐄 = 𝐴L and Γ𝐌 = 𝐵N (or Γ𝐄 = 𝐵L and Γ𝐌 = 𝐴N ), +respectively. Then we have Γ/ ⊗ Γ𝐄 ⊗ Γ𝐌 = 𝐴N, which is symmetrically allowed. On the other +hand, the magnetic point group of 2/𝑚′ for 𝐋 ∥ 𝐲- gives Γ/ = 𝐵N. Thus, Γ/ ⊗ Γ𝐄 ⊗ Γ𝐌 = 𝐴L, +being forbidden. Note that flipping Neel vector L corresponds to time-reversal operation, thus the +ferrotoroidic vector t is also reversed between L and –L. +The ferrotoroidicity can be reflected by the nondiagonal elements of magnetoelectric response +coefficient tensor 𝛼CA, defined as 𝑀$ = 𝛼%$ +CA𝐸%. According to Kubo perturbation theory, 𝛼CA can +be calculated by +𝛼%$ +CA = +(O +ℏ ∫ ++"𝐤 +()')" Im ∑ +P*&6&* +% /*& ++ +(9*&Q%/&)" +0,S ++ +(O +ℏ 𝜏 ∫ ++"𝐤 +()')" ∑ 𝑣00 +% 𝑚00 +$ 𝛿(𝜔0 − 𝜇) +0 +. (3) +Here, 𝑚S0 +$ = S𝑙𝐤T𝑚•$T𝑛𝐤U ≃ −2⟨𝑙𝐤|𝑠̂$|𝑛𝐤⟩ is the magnetic moment matrix element, and only spin +contribution is taken account in this work. 𝐴 refers to the area of unit cell, hence 𝛼%$ +CA measures +the total magnetic moments in the unit cell induced by an in-plane static electric field 𝐸% (also +called as Rashba-Edelstein coefficient [49,50]). The first term arises from the interband +contribution, while the second term evaluates the intrinsic contributions from the Fermi surface, +being τ1-dependent (similar as the MIC generation). According to previous discussions [51], 𝑇T ∼ +𝜖%$T𝛼%$ +CA where 𝜖%$T is the Levi-Civita symbol (with Einstein summation convention). Thus, we +plot the nondiagonal difference \𝛼"! +CA − 𝛼!" +CA^ as a function of chemical potential 𝜇 in Fig. 5(b). It +is clear that when 𝐋 ∥ 𝐲-, the \𝛼"! +CA − 𝛼!" +CA^ = 0, consistent with previous discussions. The 𝐋 ∥ 𝐱- +pattern gives finite magnetoelectric responses. When the chemical potential lies inside the bandgap, +only extrinsic interband contribution exists, which is found to be 0.03 μB×nm/V (μB is Bohr +magneton). Upon n- or p-type doping, the intrinsic Fermi surface term comes in, which +significantly increases \𝛼"! +CA − 𝛼!" +CA^ to the order of 0.01 μB×nm/V. Thus, an intermediate electric +field with 0.1 V/nm strength yields about 10–3 μB magnetic moment variation, being sufficiently +large for experimental observation. Considering the magnetic exchange parameter Jex of MnPSe3 +is 0.12 meV/(μB)2 [25], we can estimate the effective magnetic field of the magnetoelectric +coupling to be ~20 mT per (V/nm). This magnetoelectric responses serve as an indirect and +complementary demonstration for magnetic moment direction dependent ferrotoroidicity. + + + +14 +IV. Discussion +Before conclusion, we would like to remark a few points. In addition to charge current, recent +advances have been extending the BPV effect into spin degree of freedom, namely, bulk spin +photovoltaic (BSPV) generation [38,52,53]. Here, we follow this route and compute the BSPV +photoconductivity. Previous works [27] have demonstrated that the LPL-induced spin +photocurrent belongs to the shift current nature for 𝒫𝒯-symmetric systems, rather than MIC +mechanism for the electric charge current. The spin current operator is defined as 𝕁Š%$ = += +) \𝑣-%𝑠̂$ + 𝑠̂$𝑣-%^, where we adopt spin parallel to Néel vector, namely, j is along L. Our calculation +results are plotted in Fig. S4. We find that no matter L is parallel to x, y, or z, the BSPV current +always flows along y, making the x-propagating spin current symmetrically forbidden. This is +because the spin current operator contains a surplus spin operator that transforms as pseudovector, +compared with electric charge current operator. Nevertheless, valley-dependent spin photocurrents +still exist, even though in the forbidden net spin current situation. +The SOC effect plays an essential role in not only breaking the spin rotational symmetry, but +significantly affecting the MIC generation. In order to show this, we adjust the SOC strength by +multiplying a tuning factor 𝜆 ∈ [0,1]. Here 𝜆 = 0 turns off the SOC effect, and 𝜆 = 1 refers full +SOC. As shown in Fig. S5, zero MIC is generated when the SOC is absent. We find that as SOC +is gradually increased, the symmetrically allowed MIC photoconductivity almost linearly +enhances. Note that when SOC is turned on, the spin magnetic quantum number is not conserved +and one cannot calculate the MIC from the two spin channels separately. Such SOC variation +effects do not largely affect the BSPV current, which remains to be finite regardless with 𝜆. We +note that such SOC tunable BPV effect in AFM 𝒫𝒯-symmetric systems is different from the +nonmagnetic (𝒯-symmetric, 𝒫-broken) materials [38], where spin photocurrent linearly enhances +with 𝜆 but the electric charge current remains almost unchanged. +The AFM pattern also determines the symmetry constraints. In this work, we focus on the +Néel pattern in the MnPSe3 monolayer, as determined by recent experiments [54,55]. If other +transition metals are used, e.g., FePX3 and CrPX3 (X = S or Se), stripe or zigzag AFM pattern could +become energetically optimal [23,25,56,57]. In these cases, the system is 𝒫-symmetric, rather than +𝒫𝒯. According to previous discussions, the second order nonlinear BPV current vanishes for 𝒫- +symmetric systems, regardless the local spin polarization directions. In addition, the band extrema + + +15 +in these cases do not locate at the K (or K') point, hence we cannot determine valley-polarized BPV +effect here, even though k-resolved BPV photocurrents do not completely vanish. +Inversion symmetry could also preserve in Néel AFM patterns when we stack two monolayers +together and form a MnPSe3 bilayer. In order to illustrate this, we calculate the BPV +photoconductivity, and the results are shown in Fig. S6. One could see that depending on the spin +polarization patterns between the two monolayers, the whole system can be either 𝒫 or 𝒫𝒯. Hence, +zero or finite photocurrent emerges in these cases. This is akin to the recently proposed sliding +ferroelectricity [58-60] that arises from atomic interfacial mismatch between the two layers, while +here it is the magnetic order that is mismatched at the interface. Such interlayer spin order adjusted +symmetry in bilayer AFM materials is beyond the scope in the current work and will be discussed +in detail elsewhere. + +V. Conclusion +In summary, we conduct group theory and first-principles calculations on MnPSe3 monolayer +to show that photoinduced MIC generation in AFM 𝒫𝒯-symmetric materials sensitively depends +on the spin polarization Néel vector L. The symmetry arguments, especially mirror reflection, vary +by switching L. In addition, we show that sizable valley contrasting photocurrents could exist in +AFM 𝒫𝒯-symmetric materials, even though the net MIC component is symmetrically constrained +to be zero. The Néel vector direction can be well-tuned by applying external uniaxial stress, which +also harnesses the toroidal moment and the magnetoelectric coupling. This work provides an in- +depth examination on the AFM magnetic order implications on various electrical and optical +responses, and paves the route to realizing nanoscale optoelectronic, optospintronic, and +optovalleytronic devices with ultrafast kinetics. + +Acknowledgments. This work was supported by the National Natural Science Foundation of +China (NSFC) under Grant No. 11974270. The computational resources from HPC platform of +Xi’an Jiaotong University are also acknowledged. + + + + +16 +References: +[1] +A. Rycerz, J. Tworzydło, and C. W. J. Beenakker, Valley filter and valley valve in graphene, +Nature Phys. 3, 172 (2007). +[2] +D. 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Vizner Stern et al., Interfacial ferroelectricity by van der Waals sliding, Science 372, +1462 (2021). + + +S1 + +Supplemental Material for Valley contrasting bulk photovoltaic effect in +antiferromagnetic MnPSe3 monolayer +Qianqian Xue1, Xingchi Mu1, Yan Sun1, Jian Zhou1,* +1Center for Alloy Innovation and Design, State Key Laboratory for Mechanical +Behavior of Materials, Xi'an Jiaotong University, Xi'an, 710049 China +*Email: jianzhou@xjtu.edu.cn +1. Supplemental Tables. +Table S1. Character table for ���, the symmetric (Γ(��)�) and antisymmetric +(Γ(��)�) electric field representations are generated in 2D xy plane. + +E +2��(�) +3σ� +basic function +�� +1 +1 +1 +� +�� +1 +1 +−1 +�� +� +2 +−1 +0 +(�, �) (��, ��) +Γ(��)� +3 +0 +1 + +Γ(��)� +1 +1 +−1 + + +Table S2. Character table for �� (� = ���� +� ). + +E +��(�) +��(�)� +basic function +� +1 +1 +1 +�, �� +� +1 +� +�∗ +� + ��; �� + ��� +1 +�∗ +� +� − ��; �� − ��� +Γ(��)� +3 +0 +0 + +Γ(��)� +1 +1 +1 + + +Table S3. Character table for �� (assuming mirror vertical to �). + +E +σ� +basic function +�� +1 +1 +�, �, �� +��� +1 +−1 +�, ��, �� +Γ(��)� +3 +1 + +Γ(��)� +1 +−1 + + + +S2 + +Table S4. Character table for ��� (assuming axis along �). + +E +��(�) +� +�� +basic +function +�� +1 +1 +1 +1 +�� +�� +1 +−1 +1 +−1 +��, �� +�� +1 +1 +−1 +−1 +� +�� +1 +−1 +−1 +1 +�, � + +2. Supplemental Figures + +FIG. S1. Band dispersion along the high symmetric k-path including SOC effect. +(a) � ∥ ��, (b) � ∥ ��, and (c) � ∥ ��. + + +FIG. S2. MIC photoconductivity under x and y-polarized LPL for (a) � ∥ −��, (b) +� ∥ −��, and (c) � ∥ −��. One sees that time-reversal operation reverses current. + + +FIG. S3. The MIC photoconductivity under x-polarized LPL for (a) � ∥ �� (� = �), +(b) � ∥ �� (� = �), and (c) � ∥ �� (� = �). + +e +D +rgym +E +0 +0. +E +0-1 +-1 +FL- +M +K +r +K' +M +M +K +r +K' +M +M +K +K' +MP -400 +P +P -400 +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +w(eV) +w(eV) +w(eV)(b) +(c) +(a) +L I- +LIl-y +LI-2(b) +(a) +(c) +LIIX +LIly +L2 +600 +1500 +600200 +200 +400 +V2 +uA/V2) +100 +100 +0 +μA +200n(K) +n(K) +n(K) +400 +1000 +400 +n(K') +n(K)0 +Tyy +luctivity +luctivity +0 +uctivity +y +-100 +0 +-100001 +0-200 +二0 +hotocond +hotocond +tocon +-200 +-200 +212 +-300 +=0 +-300 +yy-200 +-400 +-1000 +400-600 +-1500 +0 +2 +4 +6 +0 +2 +4 +6 +0 +2 +4 +6 +w(eV) +w(eV) +w(eV)(b) +(c) +(a) +LIX +LIly +LI23S3 + + +FIG. S4. Calculated spin current photoconductivity under x and y-polarized LPL +(a) � ∥ �� , (b) � ∥ ��, and (c) � ∥ �� . + + +FIG. S5. (a) MIC photoconductivity ��� +� and (b) spin current photoconductivity +��� +��� under � ∥ �� when SOC coefficient � increases from 0 (SOC totally turned off) +to 1 (full SOC is included). One sees that the charge current almost linearly increases +with SOC effect, while the spin current is not largely affected. + + +FIG. S6. (a) Bilayer MnPSe3 structure in � and �� magnetic stacking patterns. +They only differ in magnetic configurations, while the atomic coordinates are the same. +(b) MIC of the � ∥ �� in the �� magnetic pattern. The � magnetic pattern gives zero +BPV photoconductivity. + +0P-0 +P +P-o +2 +4 +2 +4 +0 +0 +0 +2 +4 +6 +6 +6 +w(eV) +w(eV) +w(eV)-200 +cono-600(b) +(a) +102 +4 +９Z00 +100 +e5入= 0.25 += 0.25 +200 +入= 0.5 +0.50.75 +-300 +入= 0.75 +入=1 +入=10 +2 +4 +6 +2 +4 +6 +0 +w(eV)(a) +(b) +(c) +Ly +LIX +LI2(a) +p +PT +(b)CS +CS +2 +5600 +00400 +Tlyy +200Tyy +hhl +uctivity +ysy +ysy +uctivity +uctivity +hh +0 +0 +0lotocond +notocond +notocond +5 +5 +-5 \ No newline at end of file diff --git a/htE0T4oBgHgl3EQf6gL5/content/tmp_files/load_file.txt b/htE0T4oBgHgl3EQf6gL5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7cf56deda1aefb1bee344838e315cede412113cb --- /dev/null +++ b/htE0T4oBgHgl3EQf6gL5/content/tmp_files/load_file.txt @@ -0,0 +1,1048 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf,len=1047 +page_content="1 Valley contrasting bulk photovoltaic effect in antiferromagnetic MnPSe3 monolayer Qianqian Xue, Xingchi Mu, Yan Sun, Jian Zhou* Center for Alloy Innovation and Design, State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an, 710049 China Email: jianzhou@xjtu." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='cn Abstract Valleytronics that uses the inequivalent electronic states at the band extrema in semiconductors have been considered to play a vital role in the future information read/write technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In the current work, we theoretically show that sizable valley contrasting bulk photovoltaic (BPV) effect could emerge, even when the total photocurrent is symmetrically forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We illustrate our theory by using a prototypical two-dimensional antiferromagnetic semiconductor, MnPSe3 monolayer, that is 𝒫𝒯-symmetric (𝒫 and 𝒯 refer to spatially inversion and time-reversal operators, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We show that the Néel vector well controls the magnetic point group at the Γ point, and the BPV current direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition, k-dependent photocurrent generally arises due to the reduction of little group constraints at the valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This leads to hidden valley-polarized photoconductivity which could reach a magnitude of 1350 μA/V2, observable experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We further predict that the MnPSe3 monolayer could be two-dimensional ferrotoroidic, again depending on its Néel vector direction, which can be determined via magnetoelectric response measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our work provides an exemplary platform for paving the route to future opto-spintronic and opto-valleytronic devices in a single antiferromagnetic nanomaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Introduction Manipulation and control of the electronic state at band extrema, known as band valleys, have received tremendous attention during the past decade [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Different valleys in semiconductors usually locate in the momentum space with a large separation, guaranteeing their robustness against smooth geometric deformation and low energy phonon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Valleytronics that uses valley degree of freedom to constitute the binary logic states (similar as the charge and spin in electronics and spintronics, respectively), holds the potential for ultrafast and efficient information and data read/write applications [5-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Even though early studies on electronic valleys were mainly focusing on silicon (dated back to 1970s) [8,9], recently discovered two-dimensional (2D) lattices have significantly promoted its advances [10-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Currently, in order to detect the valleytronic feature [14,15], one usually uses optical absorption spectrum such as circular or linear dichroism spectroscopy [16-18], and electrical approaches such as (quantum) valley Hall effect [13,19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that electric signal is more realistic and facile for nanoelectronic devices, yet the Hall effect measurement requires depositing electrodes onto samples, which may introduce unwanted impurities or disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In this work, we propose another valley contrasting feature that is stimulated by noncontacting optical illumination and can be measured and probed electrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We discuss such optoelectronic responses via bulk photovoltaic (BPV) effect [22] in 2D antiferromagnetic (AFM) honeycomb materials [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The AFM systems that composes compensated spin polarization are found to be advantageous due to the absence of stray field and ultrafast spin dynamics [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, they give rise to large information storage density and high switching efficiency in real operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We apply group theory analysis and conduct first-principles density functional theory (DFT) calculations to show that valley contrasting BPV photocurrent could emerge in a prototypical AFM MnPSe3 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The MnPSe3 belongs to 2D transition metal phosphorus chalcogenides family, usually denoted as TMPX3 (TM = Cr, Mn, Fe, Co, Ni, and X = S and Se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Depending on the transition metal species, this series of materials exhibit different magnetic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Among them, the MnPSe3 monolayer shows Néel-type AFM configuration [25], which is 𝒫𝒯-symmetric (𝒫 refers to inversion symmetry and 𝒯 denotes time-reversal symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' As both 𝒫 and 𝒯 are broken, it holds non-degenerate valley polarizations [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Previous nonlinear optics theory has demonstrated that the BPV current under linearly polarized light (LPL) irradiation shows magnetic injection current (MIC) feature [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our calculation suggests a sizable MIC density (1D current density on the 3 order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='1–1 A/cm) could emerge under an intermediate light intensity (electric field component on the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='1 MV/cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition, we show that the AFM Néel vector L (= MMn1 – MMn2, Mn1 and Mn2 denote the two Mn sites in the unit cell) could effectively tune the symmetry constraints for the MIC generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' At a generic k point in the momentum space, the symmetry reduction could lead to hidden photocurrent generation, as revealed by a simple group theory approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This suggests that ubiquitous valley contrasting MIC exists in the AFM MnPSe3 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The Néel vector is strongly coupled with in-plane mechanical deformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', the in- plane magnetocrystalline anisotropy energy (MAE) can be manipulated via small uniaxial strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Finally, we suggest that the MnPSe3 monolayer also hosts a L-dependent toroidal moment that can be measured by magnetoelectric responses, showing magnetically harnessed ferrotoroidicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Computational Details We perform DFT calculations in the Vienna ab initio simulation package (VASP) [28,29] that uses the generalized gradient approximation (GGA) method in the solid state Perdew-Burke- Enzerhof (PBEsol) [30] form to treat the exchange-correlation interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Projector augmented- wave (PAW) [31] method is used to describe the core electrons, while the valence electrons are expanded by a planewave basis set with its kinetic cutoff energy setting to be 400 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The first Brillouin zone (BZ) is represented by (12×12×1) Monkhorst-Pack k-mesh grids [32], and the strong correlation on the Mn-d orbital is treated by adding a Hubbard U correction [33,34] with effective value of 5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This has been widely adopted in previous works [25], and we note that the exact U value does not affect our main conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' If not indicated explicitly, spin-orbit coupling (SOC) is included self-consistently in all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In order to simulate 2D materials in periodic boundary condition, we add a vacuum space of over 20 Å in the out-of-plane z direction, which could eliminate the nearest neighbor image layer interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The convergence criteria of total energy and Hellman-Feynman force components are set as 1 × 10–7 eV and 1 × 10–3 eV/Å, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We fit the DFT calculated electronic states by using maximally localized Wannier functions based on Mn-s and d, P-p, and Se-p orbitals, as implemented in the Wannier90 code [35,36], which are used to evaluate the BPV photoconductivity and magnetoelectric coupling components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Results Geometric, electronic structure and symmetry arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The atomic structure of MnPSe3 monolayer is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Geometrically, each P dimer is vertically sandwiched by six Se atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' These P2Se6 moieties are embedded in the hollow sites of Mn honeycomb sublattice framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Without considering spin polarization, it belongs to 𝑃3%1𝑚 layer group (3%𝑚 point group), which contains 𝒞3z rotation, 𝒞2y rotation, and a mirror reflection ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='. Hence, the inversion symmetry 𝒫 is also preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The Néel-type AFM configuration guarantees the unit cell being hexagonal lattice [the black rhombus in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 1(a)], containing two Mn sites that carry antiparallel spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Before including SOC, the electronic states in the two spin channels (majority and minority) are degenerate [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" 1(b)], with the valence and conduction band valleys locating at the corner of the first BZ (K and K' points)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The direct bandgap value is calculated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='697 eV, consistent with previous reports [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) Top and side views of the MnPSe3 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The crystalline mirror reflection ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' is indicated by the horizontal dashed line, and the black rhombus represents the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (b) Band dispersion along the high symmetric k-path without including SOC effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" (c) The first BZ, with high symmetric points of Γ = (0, 0, 0), K = (1/3, 1/3, 0), M = (0, 1/2, 0), and K' = (–1/3, 2/3, 0) in direct coordinates." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (d) Schematic plots of band edge positions with bandgap values when the Néel vector L is along the x, y, and z axes with SOC included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that each band is actually doubly degenerate due to antiunitary 𝒫𝒯-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (b) 3 (eV) 2 M spin up 1 y spin down Mn 0 x P Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='1 Se .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" M K K' M b2 (c) (d) K M K b1 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6927 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6928 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6926 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6926 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='7202 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6557 eV 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='5 mel K K K K Lx Lly L12 5 The inclusion of SOC breaks the spin rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In this case, the system becomes 𝒫𝒯-preserved, so that each band is still doubly degenerated due to its antiunitary symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Since the spin angular momentum transforms as a pseudovector, its direction would determine the magnetic point group and the valley splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We list the basic symmetric arguments for 𝐋 ∥ 𝐱-, 𝐋 ∥ 𝐲-, and 𝐋 ∥ 𝐳- in Table I, (⋅̂ denotes the Cartesian unit vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Here, x and y refer to the zigzag and armchair directions of the Mn honeycomb sublattice, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our MAE calculations reveal that in-plane spin polarization (𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-) is energetically more favorable than the out-of-plane spin polarization (𝐋 ∥ 𝐳-) by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='5 meV per unit cell (or ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='023 μJ/cm2), also tabulated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that the K and K\' valleys are connected via 𝒯𝐾 = 𝐾′ or ℳ"𝐾 = 𝐾′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, one can deduce that when 𝐋 ∥ 𝐲-, the two valleys are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' On the other hand, the valley degeneracy lifts for 𝐋 ∥ 𝐱- (marginal) and 𝐋 ∥ 𝐳- (bandgap differ by 64 meV at the two valleys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our calculations can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 1(d) for schematic plots and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S1 in Supplemental Material (SM) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Magnetic point group, mirror reflection, and relative energies for L along three Cartesian axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Neel vector Magnetic point group at Γ Mirror reflection Relative energy (μeV per unit cell) Magnetic group at K valley 𝐋 ∥ 𝐱- 2#/𝑚 ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 0 𝑚 𝐋 ∥ 𝐲- 2/𝑚′ ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='𝒯 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='63 𝑚′ 𝐋 ∥ 𝐳- 3%′𝑚 ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 494 3𝑚′ Valley contrasting bulk photovoltaic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Next, we show that the mirror reflection difference with respect to Néel vector L could lead to contrasting BPV photocurrent generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We will focus on LPL irradiation, which, according to nonlinear optics theory, generates MIC for 𝒫𝒯- symmetric systems [37,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The current density is evaluated via 𝒥$ = 𝜂%% $ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 𝜔, −𝜔)𝐸%(𝜔)𝐸%(−𝜔), (1) 6 where i and j refer to in-plane Cartesian axes (x or y), and 𝐄(𝜔) is the optical alternating electric field with the angular frequency 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The MIC in 𝒫𝒯-systems can be viewed as a cousin effect to the normal injection current [27,37] that appears in nonmagnetic materials (𝒯-preserved, 𝒫-broken) under circularly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' It arises from the velocity difference between the valence and conduction bands, which increases linearly with time and saturates at the carrier lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' According to band theory, its length-gauge form formula is 𝜂%% $ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" 𝜔, −𝜔) = − &'(!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' )ℏ" ∫ +"𝐤 ()\')" ∑ 𝑓/0Δ/0 $ 𝑔/0 %% 𝛿(𝜔/0 − 𝜔) /0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (2) Here, 𝑓/0 = 𝑓/ − 𝑓0 and Δ/0 $ = 𝑣// $ − 𝑣00 $ measure the Fermi-Dirac occupation and group velocity differences between the band 𝑚 and 𝑛, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The Kronecker delta function 𝛿(𝜔/0 − 𝜔), represented by the Lorentz function with a broadening factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='02 eV, guarantees the energy conservation law, with ℏ𝜔/0 = ℏ𝜔/ − ℏ𝜔0 referring to the eigenenergy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" The MIC generation is scaled by quantum metric tensor 𝑔/0 %% = 2 ∑ Re N𝑟/#0$ % 𝑟0$/# % P 1,3 , where 𝜇 and 𝜈 represent the degenerate band indices, and the position matrix is 𝑟0/ % = S𝑛T𝑟̂%T𝑚U = 40567%5/8 %9&' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' All these quantities are k-dependent which are omitted for clarity reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' According to previous works [38], we take the carrier lifetime 𝜏 to be a universal value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='2 ps, comparable to most experimental observations in 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This carrier lifetime can be effectively controlled by the sample quality, disorder level, temperature, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', and can be characterized by the electrical conductance according to the Drude model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The integral is performed in the whole 2D first BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We take the effective thickness as d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6 nm, measured from its bulk counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Then, we divide the 2D photoconductivity [μA⋅nm/V2, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (2)] by d, so that it adopts the conventional 3D photoconductivity unit (μA/V2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Before performing DFT calculations, we briefly analyze the magnetic point group for each case and their implication for BPV photocurrents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The highest symmetry exists when the Néel vector is parallel to z, 𝐋 ∥ 𝐳-, and the system belongs to magnetic point group 3%′𝑚 = 𝐶:6⨂𝒫𝒯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Since we are focus on the MIC (injection current under LPL) that is invariant under 𝒫𝒯, we focus on the 𝐶:6 point group (character table can be found in Tables S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' For the electric field and current in the 2D (xy) plane, the irreducible representation for current and second order symmetric field are Γ𝒥 = 𝐸 and Γ(𝐄𝐄)( = 𝐴=⨁𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, one has Γ𝒥⨂Γ(𝐄𝐄)( = 𝐴=⨁𝐴)⨁2𝐸, allowing only one 7 nonzero independent MIC component, which will be shown to be 𝜂"" " = −𝜂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' " = −𝜂"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' and 𝜂"" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' = 𝜂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' = 𝜂"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' " = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" If we focus on the valley K (or K'), the spatial inversion is no longer preserved, giving its magnetic little group of 3𝑚′ = 𝐶:⨁𝐶>𝒯." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Thus, the symmetry argument at each valley for MIC generation follows 𝐶: (since MIC is 𝒫𝒯-symmetric which is inconsistent with 𝐶>𝒯), yielding Γ𝒥⨂Γ(𝐄𝐄)( = 𝐸⨂(𝐴⨁𝐸) = 2𝐴⨁2𝐸 with two allowed and independent MIC components (Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This indicates that momentum-dependent hidden MIC exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This is akin to the hidden spin polarization (or spin Hall effect) as discovered locally in centrosymmetric ionic compounds and antiferroelectric materials [39-43], which arises in the real space due to the reduced symmetry constraints on each sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Similar confinements can be found for in-plane Néel vector, which breaks the three-fold rotation 𝒞3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The 𝐋 ∥ 𝐱- belongs to 2′/𝑚 = 𝐶>⨂𝒫𝒯, and the allowed MIC generation can be deduced from the 𝐶> point group (see its character table in Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We then have x-flowing MIC satisfying Γ𝒥⨂Γ(𝐄𝐄∗)( = 𝐴#⨂(2𝐴#⨁𝐴##) = 2𝐴#⨁𝐴′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The two independent MIC would be 𝜂"" " and 𝜂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' " .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The 𝐋 ∥ 𝐲- is 2/𝑚′ = 𝐶)⨂𝒫𝒯, and one can perform similar arguments to yield same results as in 𝐋 ∥ 𝐲-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' At the valleys, their 180°-rotation symmetry (2# or 2) is no longer preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, they both exhibit valley-dependent finite MIC components that are forbidden in the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The switching of L strongly affects the velocity texture distribution in k-space, so that the MIC direction depends on the Néel vector [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In detail, when 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳-, the ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' reflection assigns ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='Δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='\\𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^ = −Δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (𝑘", −𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' On the contrary, for the 𝐋 ∥ 𝐲- case, we have ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='𝒯Δ"\\𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^ = −Δ"(−𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Since the quantum metric tensor is almost unchanged in these cases, one easily deduces that the x-flowing MIC is forbidden when 𝐋 ∥ 𝐲-, while the y-flowing MIC is zero when 𝐋 ∥ 𝐱- or 𝐋 ∥ 𝐳-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that here we assume that the x- (or y-) polarized LPL is illuminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' For a general polarization angle, such symmetry arguments may be changed, and the final MIC will be a linear combination from the x-LPL and y-LPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Calculated MIC photoconductivity under x and y-polarized LPL for (a) 𝐋 ∥ 𝐱-, (b) 𝐋 ∥ 𝐲-, and (c) 𝐋 ∥ 𝐳-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In (c), we note that 𝜂"" " (𝜔) = −𝜂!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' " (𝜔), arising from the 𝒞3z rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Such symmetry is broken when L lies in-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Quantum metric distribution between the top valence and bottom conduction bands 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' ""(𝐤) over the first BZ for (d) 𝐋 ∥ 𝐱-, (e) 𝐋 ∥ 𝐲-, and (f) 𝐋 ∥ 𝐳-, and 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (𝐤) for (g) 𝐋 ∥ 𝐱-, (h) 𝐋 ∥ 𝐲-, and (i) 𝐋 ∥ 𝐳-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our first-principles calculations confirm the above qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 2(a)– 2(c), we plot the calculated MIC generation photoconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' One clearly observes that 𝜂%% !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' = 0 for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳- (i = x or y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' When L is switched to along y, the 𝜂%% " becomes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' For the symmetrically allowed current, the magnitude of photoconductivity reaches ~360 μA/V2 (𝐋 ∥ 𝐳-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) (b) (c) LIx Lly LZ (μA/V2) 400 (μA/V2) 300 na 300 nr = 0 naa nua 0 200 200 200 nyy =0 lyy Photoconductivity Photoconductivity Photoconductivity 0 100 100 nyy yy 0 0 0 200 100 100 200 200 400 2 4 6 0 0 2 6 0 2 4 6 w(eV) w(eV) w(eV) (d) (e) () g (k) g(k) gx (k) 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 7 7 6 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 5 5 5 (t-y)n ky(A y)ny 0 4 0 4 0 4 3 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 2 2 2 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 ka(A-1) ka(A-1) ka(A-1) (g) (h) (i) gu (k) yy gc (k) 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 7 7 7 6 6 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 5 5 5 y)n V 0 ky(A 0 4 0 4 3 3 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 2 2 1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 ka(A-1) ka(A-1) ka(A-1) 9 It indicates that if we take the electric field magnitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='1 V/nm (at the photon energy of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='2 eV, or wavelength of 387 nm), corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='33 × 10@ W/cm2 light intensity, one could generate ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='6 μA/nm2 current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Across the lateral size of 1 nm (note that the effective thickness is d = 6 Å), the current reaches 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='16 μA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Normal to the MIC, no net photocurrent can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This vividly suggests that switching magnetic moment direction could drastically rotate the MIC generation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Such a large contrast can be directly measured via closed-circuit current or open-circuit voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The quantum metric between the top valence band and bottom conduction band (both doubly degenerate) 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' ""(𝐤) and 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (𝐤) for 𝐋 ∥ 𝐱-, 𝐋 ∥ 𝐲-, 𝐋 ∥ 𝐳- are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 2(d)–2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We can see symmetry argument of 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' %% \\𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^ = 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' %% \\𝑘", −𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^, (𝑖 = 𝑥 or 𝑦) for ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' and 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' %% \\𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^ = 𝑔6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' %% \\−𝑘", 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='^ for ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='𝒯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition, we note that the time-reversal symmetry 𝒯 that flips the Néel vector L (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', between 𝐋 ∥ 𝐱- and 𝐋 ∥ −𝐱-, or from 𝐋 ∥ 𝐳- to 𝐋 ∥ −𝐳-) also reverses the MIC generation while keeping the magnitude, as the 𝜂%% $ is scaled by velocity operator and being 𝒯-odd (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We then show that sizable valley contrasting MIC emerges even when the net current is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In order to explicitly see this, we plot the k-resolved MIC contributions, namely, the integrand of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (2), for the symmetrically forbidden currents [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 3(a)–3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The incident photon frequency is selected to be ℏω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 eV, slightly above the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In all these cases, the mirror symmetry constraints (Table I) are clearly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" Main contributions appear in the vicinity of K (and K') valleys, in agreement with the joint density of states at this incident energy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Even though the contributions around one valley show both positive and negative ridges, their summation is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 3(d)–(f), we plot the valley-dependent MIC, which is integrated in the triangular k-space around each valley, whose area equals to half of BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We see that both valleys contribute significant MIC generation, reaching a photoconductivity of ~1500 μA/V2 (for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐳-) or ~600 μA/V2 (for 𝐋 ∥ 𝐲-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In each case, the MICs from two valleys flow oppositely with the same magnitude, giving zero net MIC generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This hidden valley-dependent BPV current has been largely overlooked previously, and may find its potential applications in 2D valleytronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We propose that this result provides another scheme to use the valley current, in addition to the electrically triggered (quantum) valley Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Experimentally, one could design a magnetic heterostructure [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 3(g)] to separate and measure such valley-dependent MIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Valley contrasting MIC could accumulate at a domain wall between two AFM configurations that are time-reversal 10 with each other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=", 𝐋 ∥ 𝐳- and 𝐋 ∥ −𝐳-), so that the current contributed from a specific valley K (or K') in both domains flow to their boundary." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Similar valley-dependent MIC also exists for the symmetrically allowed components, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The two valleys also hold oppositely flowing MIC, but they do not completely cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' BZ contribution of MIC integrand 𝜚"" $ (𝐤, 𝜔) = ∑ 𝑓/0Δ/0 $ 𝑔/0 "" 𝛿(𝜔/0 − 𝜔) /0 under x-PLP irradiation for (a) 𝐋 ∥ 𝐱- (𝑗 = 𝑦), (b) 𝐋 ∥ 𝐲- (𝑗 = 𝑥), and (c) 𝐋 ∥ 𝐳- (𝑗 = 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Here the incident photon energy is ℏ𝜔 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (d)–(f) plot their corresponding valley contrasting MIC contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (g) Schematic plot of magnetic heterostructure with two AFM configurations that are time-reversal with each other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', between 𝐋 ∥ 𝐳- and 𝐋 ∥ −𝐳-).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Valley-dependent BPV current would accumulate at their domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' MAE modulation under strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' One may wonder how to harness the in-plane MAE (𝐸ABC = 𝐸𝐋∥𝐱7 − 𝐸𝐋∥𝐲7), so that the Néel vector L can be pinned along x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We show that a uniaxial strain could further split the energy difference between 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our numerical results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' At the equilibrium (strain-free) state, the 𝐋 ∥ 𝐱- is almost degenerate with 𝐋 ∥ 𝐲- (EMAE = Qxx (a) (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 6 6 10 4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 5 2 2 C 0 A 0 0 0 0 0 2 5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 4 4 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='8 6 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content="4 0 ka(A-1) ka(A-1) ka(A-1) (d) (f) (e) 1500 600 1500 n(K) n(K) n(K) 1000 400 1000 n(K) n(K') n(K') (μA/V2) 500 200 500 (μA) 0 0 0 3 500 500 1000 400 1000 1500 600 1500 0 2 4 6 0 2 4 6 0 2 4 6 w(eV) w(eV) w(eV) (g) K LⅡ2 LⅡ-2 11 −1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='63 μeV per unit cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Under tensile strain along x (εxx), the EMAE is negative, so that the Néel vector L prefers the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' On the other hand, the y-tensile strain increases the EMAE, aligning the L along y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In both cases, a small strain of 3% (about elastic energy of 49 meV in one unit cell) could enhance the MAE magnitude to be ~35 μeV per unit cell, which is large enough to be distinguished in low temperature experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Furthermore, we find that such a small strain will not significantly change the band structure in these cases, and the MIC photoconductivity marginally change their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Total energy difference (per unit cell) between 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲- under uniaxial strain ε along x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The out-of-plane AFM configuration 𝐋 ∥ 𝐳- is always much higher in energy under such small strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' L-dependent ferrotoroidicity and magnetoelectric responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition to the optically induced nonlinear current, we now show that the ferrotoroidicity is also sensitive to the Néel vector direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Ferrotoroidicity has been discovered to be another primary ferroic order, compensating the ferroelasticity (𝒫-even, 𝒯-even), ferroelectricity (𝒫-odd, 𝒯-even), and ferromagnetism (𝒫- even, 𝒯-odd) [44,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' It reverses its sign under either 𝒫 or 𝒯, and is defined by toroidal moment 𝐭 = ∑ 𝐫% × 𝐬% % , where 𝐫% and 𝐬% represent the position and spin vectors of ion-i, and the summation runs over all sites in the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Previous theoretical and experimental works have disclosed a few ferrotoroidic bulk materials, such as LiCoPO4 [46], LiFeSi2O6 [47], and defective SrTiO3 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Here, we suggest that the 2D MnPSe3 monolayer also holds ferrotoroidic order with sizable out- of-plane toroidal moment, if the Néel vector is pointing away from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 5(a), we 12 schematically depict the Mn sublattice distributions in a rectangular supercell, which, without loss of generality, is uniaxially strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The supercell contains four Mn sites, which locate on two co- centered circles with radius of 𝑟= and 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' At the equilibrium state, these two green circles are identical, 𝑟= = 𝑟), and the angle θ = 30°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Geometrically, if x-tensile strain is applied, θ reduces and 𝑟= < 𝑟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' y-tension will increase θ and makes 𝑟= > 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) Schematic plot of ferrotoroidicity in a rectangular supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Blue and red circles represent the antiparallel spin polarized Mn sites, and the two green circles (radius 𝑟= and 𝑟)) are co-centered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Detailed explanation can be found in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (b) Calculated magnetoelectric coefficient for 𝐋 ∥ 𝐱- and 𝐋 ∥ 𝐲-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The abscissa axis denotes the chemical potential relative to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We can directly estimate the toroidal moment 𝐭 in this supercell setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that similar as the electric polarization 𝐏, here t is also multi-valued with respect to a quanta, depending on the choice of origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Nonetheless, we can use this position-spin cross product definition to determine its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' When the Mn spin polarization is along x, namely, 𝐬 = (±𝑠", 0,0), it can be directly deduced that 𝐭 = (0,0, 𝑡H) and 𝑡H = − ∑ 𝑦%𝑠%," I %J= = −2𝑠"(𝑟= − 𝑟) sin 𝜃) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, it shows a nonzero z component toroidal moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' If 𝐬 = (0, ±𝑠!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', 0), one could easily find 𝐭 = (0,0,0), constrained by the ℳ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='𝒯 operation, even though the supercell is uniaxially strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This can also be understood by analyzing the 𝐶)K point group (Table S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The 𝐋 ∥ 𝐱- belongs to the magnetic point group 2′/𝑚, which gives negative character for both 𝐶) rotation and inversion 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, one has Γ/ = 𝐵L in which all other elements are represented by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The vertically aligned electric field (b) d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='01 nm/V) 0 X r1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='01 (μB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='02 文 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='03 Lly a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='04 0 1 2 13 and spin polarization are presented by Γ𝐄 = 𝐴L and Γ𝐌 = 𝐵N (or Γ𝐄 = 𝐵L and Γ𝐌 = 𝐴N ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Then we have Γ/ ⊗ Γ𝐄 ⊗ Γ𝐌 = 𝐴N, which is symmetrically allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' On the other hand, the magnetic point group of 2/𝑚′ for 𝐋 ∥ 𝐲- gives Γ/ = 𝐵N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Thus, Γ/ ⊗ Γ𝐄 ⊗ Γ𝐌 = 𝐴L, being forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that flipping Neel vector L corresponds to time-reversal operation, thus the ferrotoroidic vector t is also reversed between L and –L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The ferrotoroidicity can be reflected by the nondiagonal elements of magnetoelectric response coefficient tensor 𝛼CA, defined as 𝑀$ = 𝛼%$ CA𝐸%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' According to Kubo perturbation theory, 𝛼CA can be calculated by 𝛼%$ CA = (O ℏ ∫ +"𝐤 ()\')" Im ∑ P*&6&* % /*& + (9*&Q%/&)" 0,S + (O ℏ 𝜏 ∫ +"𝐤 ()\')" ∑ 𝑣00 % 𝑚00 $ 𝛿(𝜔0 − 𝜇) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (3) Here, 𝑚S0 $ = S𝑙𝐤T𝑚•$T𝑛𝐤U ≃ −2⟨𝑙𝐤|𝑠̂$|𝑛𝐤⟩ is the magnetic moment matrix element, and only spin contribution is taken account in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 𝐴 refers to the area of unit cell, hence 𝛼%$ CA measures the total magnetic moments in the unit cell induced by an in-plane static electric field 𝐸% (also called as Rashba-Edelstein coefficient [49,50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The first term arises from the interband contribution, while the second term evaluates the intrinsic contributions from the Fermi surface, being τ1-dependent (similar as the MIC generation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' According to previous discussions [51], 𝑇T ∼ 𝜖%$T𝛼%$ CA where 𝜖%$T is the Levi-Civita symbol (with Einstein summation convention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Thus, we plot the nondiagonal difference \\𝛼"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' CA − 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='" CA^ as a function of chemical potential 𝜇 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' It is clear that when 𝐋 ∥ 𝐲-, the \\𝛼"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' CA − 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='" CA^ = 0, consistent with previous discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The 𝐋 ∥ 𝐱- pattern gives finite magnetoelectric responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' When the chemical potential lies inside the bandgap, only extrinsic interband contribution exists, which is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='03 μB×nm/V (μB is Bohr magneton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Upon n- or p-type doping, the intrinsic Fermi surface term comes in, which significantly increases \\𝛼"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' CA − 𝛼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='" CA^ to the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='01 μB×nm/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Thus, an intermediate electric field with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='1 V/nm strength yields about 10–3 μB magnetic moment variation, being sufficiently large for experimental observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Considering the magnetic exchange parameter Jex of MnPSe3 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='12 meV/(μB)2 [25], we can estimate the effective magnetic field of the magnetoelectric coupling to be ~20 mT per (V/nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This magnetoelectric responses serve as an indirect and complementary demonstration for magnetic moment direction dependent ferrotoroidicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 14 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Discussion Before conclusion, we would like to remark a few points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition to charge current, recent advances have been extending the BPV effect into spin degree of freedom, namely, bulk spin photovoltaic (BSPV) generation [38,52,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Here, we follow this route and compute the BSPV photoconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Previous works [27] have demonstrated that the LPL-induced spin photocurrent belongs to the shift current nature for 𝒫𝒯-symmetric systems, rather than MIC mechanism for the electric charge current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The spin current operator is defined as 𝕁Š%$ = = ) \\𝑣-%𝑠̂$ + 𝑠̂$𝑣-%^, where we adopt spin parallel to Néel vector, namely, j is along L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Our calculation results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We find that no matter L is parallel to x, y, or z, the BSPV current always flows along y, making the x-propagating spin current symmetrically forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This is because the spin current operator contains a surplus spin operator that transforms as pseudovector, compared with electric charge current operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Nevertheless, valley-dependent spin photocurrents still exist, even though in the forbidden net spin current situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The SOC effect plays an essential role in not only breaking the spin rotational symmetry, but significantly affecting the MIC generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In order to show this, we adjust the SOC strength by multiplying a tuning factor 𝜆 ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Here 𝜆 = 0 turns off the SOC effect, and 𝜆 = 1 refers full SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S5, zero MIC is generated when the SOC is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We find that as SOC is gradually increased, the symmetrically allowed MIC photoconductivity almost linearly enhances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Note that when SOC is turned on, the spin magnetic quantum number is not conserved and one cannot calculate the MIC from the two spin channels separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Such SOC variation effects do not largely affect the BSPV current, which remains to be finite regardless with 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' We note that such SOC tunable BPV effect in AFM 𝒫𝒯-symmetric systems is different from the nonmagnetic (𝒯-symmetric, 𝒫-broken) materials [38], where spin photocurrent linearly enhances with 𝜆 but the electric charge current remains almost unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The AFM pattern also determines the symmetry constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In this work, we focus on the Néel pattern in the MnPSe3 monolayer, as determined by recent experiments [54,55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' If other transition metals are used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', FePX3 and CrPX3 (X = S or Se), stripe or zigzag AFM pattern could become energetically optimal [23,25,56,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In these cases, the system is 𝒫-symmetric, rather than 𝒫𝒯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' According to previous discussions, the second order nonlinear BPV current vanishes for 𝒫- symmetric systems, regardless the local spin polarization directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" In addition, the band extrema 15 in these cases do not locate at the K (or K') point, hence we cannot determine valley-polarized BPV effect here, even though k-resolved BPV photocurrents do not completely vanish." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Inversion symmetry could also preserve in Néel AFM patterns when we stack two monolayers together and form a MnPSe3 bilayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In order to illustrate this, we calculate the BPV photoconductivity, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' One could see that depending on the spin polarization patterns between the two monolayers, the whole system can be either 𝒫 or 𝒫𝒯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Hence, zero or finite photocurrent emerges in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This is akin to the recently proposed sliding ferroelectricity [58-60] that arises from atomic interfacial mismatch between the two layers, while here it is the magnetic order that is mismatched at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Such interlayer spin order adjusted symmetry in bilayer AFM materials is beyond the scope in the current work and will be discussed in detail elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Conclusion In summary, we conduct group theory and first-principles calculations on MnPSe3 monolayer to show that photoinduced MIC generation in AFM 𝒫𝒯-symmetric materials sensitively depends on the spin polarization Néel vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The symmetry arguments, especially mirror reflection, vary by switching L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' In addition, we show that sizable valley contrasting photocurrents could exist in AFM 𝒫𝒯-symmetric materials, even though the net MIC component is symmetrically constrained to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The Néel vector direction can be well-tuned by applying external uniaxial stress, which also harnesses the toroidal moment and the magnetoelectric coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This work provides an in- depth examination on the AFM magnetic order implications on various electrical and optical responses, and paves the route to realizing nanoscale optoelectronic, optospintronic, and optovalleytronic devices with ultrafast kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' This work was supported 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' 118, e2115703118 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' [59] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Wu, Two-Dimensional van der Waals Ferroelectrics: Scientific and Technological Opportunities, ACS Nano 15, 9229 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' [60] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Vizner Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=', Interfacial ferroelectricity by van der Waals sliding, Science 372, 1462 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=" S1 Supplemental Material for Valley contrasting bulk photovoltaic effect in antiferromagnetic MnPSe3 monolayer Qianqian Xue1, Xingchi Mu1, Yan Sun1, Jian Zhou1,* 1Center for Alloy Innovation and Design, State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an, 710049 China Email: jianzhou@xjtu." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content='cn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Supplemental Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Character table for ���, the symmetric (Γ(��)�) and antisymmetric (Γ(��)�) electric field representations are generated in 2D xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' E 2��(�) 3σ� basic function �� 1 1 1 � �� 1 1 −1 �� � 2 −1 0 (�, �) (��, ��) Γ(��)� 3 0 1 Γ(��)� 1 1 −1 Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Character table for �� (� = ���� � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' E ��(�) ��(�)� basic function � 1 1 1 �, �� � 1 � �∗ � + ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' �� + ��� 1 �∗ � � − ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' �� − ��� Γ(��)� 3 0 0 Γ(��)� 1 1 1 Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Character table for �� (assuming mirror vertical to �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' E σ� basic function �� 1 1 �, �, �� ��� 1 −1 �, ��, �� Γ(��)� 3 1 Γ(��)� 1 −1 S2 Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Character table for ��� (assuming axis along �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' E ��(�) � �� basic function �� 1 1 1 1 �� �� 1 −1 1 −1 ��, �� �� 1 1 −1 −1 � �� 1 −1 −1 1 �, � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Supplemental Figures FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' Band dispersion along the high symmetric k-path including SOC effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) � ∥ ��, (b) � ∥ ��, and (c) � ∥ ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' MIC photoconductivity under x and y-polarized LPL for (a) � ∥ −��, (b) � ∥ −��, and (c) � ∥ −��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' One sees that time-reversal operation reverses current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The MIC photoconductivity under x-polarized LPL for (a) � ∥ �� (� = �), (b) � ∥ �� (� = �), and (c) � ∥ �� (� = �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' e D rgym E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) MIC photoconductivity ��� � and (b) spin current photoconductivity ��� ��� under � ∥ �� when SOC coefficient � increases from 0 (SOC totally turned off) to 1 (full SOC is included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' One sees that the charge current almost linearly increases with SOC effect, while the spin current is not largely affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (a) Bilayer MnPSe3 structure in � and �� magnetic stacking patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' They only differ in magnetic configurations, while the atomic coordinates are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' (b) MIC of the � ∥ �� in the �� magnetic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/htE0T4oBgHgl3EQf6gL5/content/2301.02766v1.pdf'} +page_content=' The � magnetic pattern gives zero BPV photoconductivity.' metadata={'source': 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Rocha1, P. Kerfriden2, and F.P. van der Meer1 +1Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.O. Box 5048, 2600GA Delft, The Netherlands +2Mines Paris, PSL University, Centre des mat´eriaux, 63-65 Rue Henri-Auguste Desbrueres BP87, F-91003 ´Evry, France +Abstract +In this work we present a hybrid physics-based and data-driven learning approach to construct surrogate models +for concurrent multiscale simulations of complex material behavior. We start from robust but inflexible physics-based +constitutive models and increase their expressivity by allowing a subset of their material parameters to change in time +according to an evolution operator learned from data. This leads to a flexible hybrid model combining a data-driven +encoder and a physics-based decoder. Apart from introducing physics-motivated bias to the resulting surrogate, the +internal variables of the decoder act as a memory mechanism that allows path dependency to arise naturally. We +demonstrate the capabilities of the approach by combining an FNN encoder with several plasticity decoders and +training the model to reproduce the macroscopic behavior of fiber-reinforced composites. The hybrid models are able +to provide reasonable predictions of unloading/reloading behavior while being trained exclusively on monotonic data. +Furthermore, in contrast to traditional surrogates mapping strains to stresses, the specific architecture of the hybrid +model allows for lossless dimensionality reduction and straightforward enforcement of frame invariance by using +strain invariants as the feature space of the encoder. +Keywords: Concurrent multiscale (FE2) modeling, Surrogate modeling, Hybrid learning +1 +Introduction +Recent advances in materials science and manufacturing techniques are paving the way for the design of materials +with highly-tailored microstructures, including metamaterials [1, 2], novel composite material systems [3, 4], printed +cementitious materials [5] and multifunctional living materials [6]. The common thread in these new developments +is a shift from traditional design focused on tailoring structures to material constraints towards tailoring material +microstructures to macroscopic constraints. This shift in turn requires the development of highly-detailed models of +material behavior across spatial scales and a shift to virtual structural certification, as trial-and-error design becomes +infeasible [7–9]. +Scale bridging has been traditionally performed through a bottom-up approach: physics-based constitutive models +at smaller scales are calibrated using experiments and used to perform numerical simulations (using e.g. the Finite +Element (FE) method) on representative lower-scale domains from which higher-scale physics-based models can be +calibrated [10, 11]. However, physics-based constitutive models come with a priori assumptions that often fail to +reproduce complex lower-scale behavior [10]. The alternative is to opt for an FE2 (or Computational Homogenization) +approach: lower-scale FE models are embedded at every Gauss point of a higher-scale model and material behavior is +directly upscaled with no constitutive assumptions at the higher scale [12–14]. Yet, the computational cost associated +with repeatedly solving a large number of micromodels quickly becomes a bottleneck, in particular for many-query +procedures such as design exploration and optimization that require several higher-scale simulations to be performed. +Since the bottleneck of FE2 lies in computing lower-scale models, a popular approach to reduce computational +effort is to substitute the original FE micromodels with either structure-preserving reduced-order models [15–21] or +purely data-driven surrogates [22–27] trained offline. More recently, Recurrent Neural Networks (RNN) have become +the model of choice especially for strain path-dependent materials, with a large body of literature dedicated to their use +and tuning to different applications [28–34]. RNNs can reproduce complex long-term time dependencies in material +behavior by learning latent representations of the material state, making them fast and flexible surrogates. However, +1 +arXiv:2301.13547v1 [math.NA] 31 Jan 2023 + +these learned representations are not a priori related to actual thermodynamic internal state variables and the model +is therefore poorly interpretable (see [35] for an interesting discussion on the subject). Furthermore, training for +path dependency requires sampling from a potentially infinite-dimensional space of arbitrarily-long strain paths. This +means training RNNs to reproduce complex material behavior often requires an inordinate amount of data (curse of +dimensionality) and their purely data-driven nature limits their ability to extrapolate away from paths seen during +training. +In order to address these drawbacks, a growing number of recent works are shifting focus to models with a fusion of +data-driven and physics-based components. Inspired by physics-informed neural networks ([36]), the authors in [37] +opt for data-driven models with physics-inspired bias by enforcing thermodynamic principles in a weak sense through +an augmented loss function. In a similar vein, the model in [38] learns hyperelasticity by linking together several +carefully crafted neural nets to represent quantities with clear physical meaning, improving the interpretability of the +resulting model. In [39] the authors extend a similar hyperelastic surrogate with a network that learns plastic flow +direction and the evolution of a yield surface parametrized by a level set function, resulting in a hyperelastic-plastic +model with superior extrapolation capabilities. A common thread in the aforementioned approaches, however, is that +their learning architectures are heavily dependent on the type of model being learned (e.g. hyperelasticity, plasticity), +making extensions to other models a convoluted task. In contrast, the authors in [40, 41] propose a surrogate for +heterogeneous micromodels constructed by directly employing unmodified versions of the constitutive models used +for the micro constituents and using a customized network architecture to infer a homogenization operator from data +that combines their responses. Nevertheless, the method employs a highly-specialized iterative online prediction +routine requiring extra implementation effort and with increased computational overhead when compared to that of +traditional surrogates mapping strains to stresses. Finally, in [42–44] a dictionary of candidate physics-based models +is assumed and the role of machine learning shifts instead to that of performing model selection and/or design of +experiments. +In this work we explore an alternative approach for constructing hybrid surrogate models for path-dependent +multiscale simulations. We start from the premise that existing physics-based models — e.g. the ones used to describe +microscale constituents — are not flexible enough to reproduce macroscale behavior but nonetheless encapsulate +crucial physical features such as frame invariance and loading/unloading conditions. It is our aim to avoid learning +these features directly from data, as that would require either an excessively large dataset or a highly-specialized +learning architecture. We therefore opt for keeping the constitutive model as intact as possible and instead increasing +flexibility by allowing some (or all) of its material parameters to evolve in time. The resulting model can be seen in +Fig. 1: a data-driven encoder that learns the evolution of a set of material properties is linked to a physics-based material +model decoder that maps strains to stresses. In contrast to other strategies in literature, we keep the architecture as +general as possible: a general feature extractor parses macroscopic strains into features for the encoder — which can +be as simple as the strains themselves or other derived quantities (e.g. strain invariants) — and any type of constitutive +model can in principle act as decoder (e.g. hyperelasticity, plasticity, damage). By relegating stress computations to +the decoder, we effectively introduce physics-based bias to the model.1 Furthermore, by letting the material model +handle the evolution of its own internal variables, the model benefits from a recurrent component with interpretable +memory structure that allows path dependency to arise naturally. The strategy we explore here is related to the one we +propose in [46], but in that work we let an encoder learn local strain distributions for several virtual material points +with fixed properties. We see the two approaches as being complementary, and therefore with potential for being used +in combination to form a flexible range of hybrid surrogates. +The remainder of the work is organized as follows. Section 2 contains a primer on concurrent multiscale (FE2) +modeling and discusses the difficulties of training purely data-driven surrogates. In Section 3, we particularize the +model of Fig. 1 to the case of a feedforward neural network encoder and discuss aspects related to offline training and +online numerical stabilization. In Section 4 we assess the performance of the hybrid model in reproducing the behavior +of fiber-reinforced composites using different encoder features and decoder models. Finally, some concluding remarks +and future research directions are discussed in Section 5. +1In purely data-driven surrogates, we accept some bias in exchange for reduced variance — e.g. by employing regularization or adopting prior +distributions for model parameters [45] — in order to counter overfitting and improve generalization. But in that case the bias is merely a way to +reduce complexity, with no physical interpretation and no a priori impact on the extrapolation capabilities of the model. +2 + +F +φt +D +θt +M +εΩ +t +Gauss point +σΩ +t +Gauss point +εΩ +t +Gauss point +material model +feature extractor +data-driven model +macroscale FE model +Gauss point +features +evolving material +properties +encoder +decoder +αΩ +t−1 +internal variables +αΩ +t +Figure 1: The hybrid surrogate combining a data-driven encoder for material parameters and a physics-based material +model decoder. +2 +Concurrent multiscale (FE2) modeling +In this section we present a short discussion on FE2 modeling. The goal is not to be comprehensive — the inter- +ested reader is referred to [13, 14] for detailed discussions on the subject — but rather to expose the computational +bottleneck associated with the method and pinpoint where surrogate models can be used to alleviate the issue. We +then demonstrate how a Recurrent Neural Network (RNN) can be used as surrogate model and showcase some of the +difficulties associated with their training and their extrapolation capabilities. +2.1 +Scale separation and coupling +In FE2 we assume the problem being solved can be split into a homogeneous macroscopic domain Ω and a heteroge- +neous microscopic domain ω ≪ Ω where small-scale geometric features are resolved. Here we opt for a first-order +homogenization approach assuming the displacements on both scales can be related by: +uω = εΩxω + �u +(1) +where microscopic displacements uω are split into a linear contribution proportional to the macroscopic strains εΩ +and a fluctuation term �u that accounts for microscopic heterogeneities. +Since εΩ varies throughout the macroscopic domain, a micromodel for ω is embedded at each Gauss point in Ω +and a microscopic boundary-value equilibrium problem assuming small displacements and strains is solved: +∇ · σω = 0 +εω = 1 +2 +� +∇uω + (∇uω)T� +(2) +microscopic stress σω is related to microscopic strain εω with traditional physics-based constitutive models for each +phase in the heterogeneous domain. In the general case where the material models feature internal variables α, we can +write the constitutive update for the microscale domain as: +Mω +� +αω +t = A +� +εω +t , αω +t−1, θω� +σω +t = S (εω +t , αω +t , θω) +(3) +where θω are the material parameters of the microscopic constituents, the operators A and S can be split into an +arbitrary number of blocks with different models (e.g. elasticity, elastoplasticity, damage) for the different material +phases, and αω is a concatenation of the internal variables of every microscopic Gauss point and therefore fully +describes the path-dependent state of the microscopic problem. +In order to determine the strains εΩ that serve as boundary conditions for the micromodels, a macroscopic small- +strain equilibrium problem is solved: +∇ · σΩ = 0 +εΩ = 1 +2 +� +∇uΩ + +� +∇uΩ�T� +(4) +3 + +but this time no constitutive assumptions are adopted. Macroscale stresses are instead directly homogenized from the +microscopic response: +σΩ = 1 +|ω| +� +ω +σωdω +(5) +which couples the macroscopic strain εΩ with the microscopic solution. Since Eq. (1) also couples the solutions in the +opposite direction, a bidirectional coupling is formed which requires the two-scale equilibrium problem to be solved +iteratively. +2.2 +Data-driven surrogate modeling +The coupled problem of Section 2.1 is extremely computationally demanding. The lower-scale domain ω usually +features complicated geometric features and must therefore be modeled with dense FE meshes in order to ensure +accuracy. Worse yet, an independent microscopic problem must be solved at every integration point in Ω for every +iteration of every time step of the simulation. This nested nature quickly forms a computational bottleneck. +Since the bulk of the computational effort lies in solving the micromodels, a popular approach to make multiscale +analysis viable for practical applications is to substitute the microscopic FE models by data-driven surrogates. The +idea is to perform a number of micromodel simulations under representative boundary conditions and use the resulting +stress-strain pairs to train a machine learning model to be deployed when performing the actual two-scale simulations +of interest. Naturally, the approach tacitly assumes that the number of offline micromodel computations required to +train the model is much smaller than the number of times the microscopic behavior will be computed online. In the +following, we use a simple example to demonstrate a number of difficulties associated with training such a model to +reproduce path-dependent material behavior. +2.3 +Example: A one-dimensional RNN surrogate +For this demonstration, we train a Long Short-term Memory (LSTM) network [47] to reproduce one-dimensional +(single stress/strain component) elastoplasticity. The architecture of the model is shown in Fig. 2a and is implemented +in PyTorch [48]. In order to minimize the risk of overfitting, a pragmatic model selection procedure is performed by +first training the model with several non-monotonic strain paths and gradually increasing cell size until reasonable +accuracy is obtained. This leads to a parsimonious model with a single LSTM cell with 5 latent units. +At this point it is interesting to draw a parallel between the network and the micromodel whose behavior is being re- +produced: the concatenation of the hidden state h and cell state c of the LSTM cell can be seen as a lower-dimensional +surrogate for the set of microscopic internal variables αω of Eq. (3). However, in contrast to the variables in α, the +latent variables h and c have no physical interpretation and evolve purely according to heuristic memory mechanisms +that mimic patterns inferred during training. +First, we train the LSTM using only monotonic data. Since only one strain component is being modeled, this +initial dataset is composed simply of one strain path in tension and one in compression. The trained model is then +used to predict a tension path with one unloading-reloading cycle. Having never seen unloading during training, the +network reverses course and unloads on top of its loading path (Fig. 2b). This result is hardly surprising, but sheds +light on the potentially deceiving nature of the training procedure: even though we are only concerned with a single +strain component, predictions actually take place in an augmented space that describes strain paths in time which can +be arbitrarily high-dimensional (as paths can be arbitrarily long). +We can further demonstrate this manifestation of the curse of dimensionality with the two additional examples of +Fig. 3. In Fig. 3a we train the network with two unloading paths and it fails to predict a third one at an intermediate +strain level. Here it can be deceiving to assume the third path can be interpolated from the other two: in the 48- +dimensional space of strain paths (we use paths with 48 time steps each) the network is actually operating far away +from training data. In Fig. 3b the network tries to reproduce a path seen during training but we first let the material +rest at zero strain for five time steps before loading starts and for another five time steps at the end of the path. With +purely data-driven latent dynamics, the initial rest disturbs the memory structure of the network and causes large +deviations for a large portion of the path. For the rest at the end of the path, we see that the surrogate fails to predict +the characteristic that the stress does not change upon constant deformation. +Training data-driven models to accurately reproduce path dependency is therefore not straightforward: their latent +representations of material state are not interpretable and even phenomena as trivial as resting at zero strain must be +learned from data. At the core of successful applications of RNNs to this task are either extensive datasets obtained +4 + +εΩ +t +LSTM +σΩ +t +ht−1 +ct−1 +ht +ct +(a) Model architecture +0 +10 +20 +30 +40 +0 +20 +40 +60 +80 +100 +120 +Time step [-] +Stress [MPa] +RVE +RNN (unseen) +(b) Failure to predict unseen unloading +Figure 2: An LSTM recurrent neural network as surrogate for 1D path-dependent material behavior trained with only +monotonic data. +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0 +50 +100 +150 +Strain [-] +Stress [MPa] +RVE +RNN (trained) +RNN (unseen) +(a) Unloading at new strain level +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0 +50 +100 +150 +200 +Strain [-] +Stress [MPa] +RVE +RNN (trained) +RNN (unseen) +(b) Stopping for 5 steps at ε = 0 and ε = 0.1 +Figure 3: 1D LSTM surrogate trained with unloading/reloading and used to predict unseen unloading paths. +5 + +εΩ +t−1 +φt−1 +θt−1 +αΩ +t−1 +σΩ +t−1 +εΩ +t−1 +αΩ +t−2 +... +feature +extractor +(fixed) +encoder +(learned) +decoder +(fixed) +εΩ +t +φt +θt +αΩ +t +σΩ +t +εΩ +t +εΩ +t+1 +φt+1 +θt+1 +αΩ +t+1 +αΩ +t+2... +σΩ +t+1 +εΩ +t+1 +Figure 4: Graph representation of the hybrid model architecture combining a data-driven encoder and a physics-based +decoder. Filled circles represent observable variables and hollow circles represent latent variables. +with carefully crafted sampling strategies [33, 49] or highly tailored datasets for specific macroscopic problems [28]. +Alternatively, active learning frameworks may be used to skip offline training altogether [50, 51], but at the cost of +producing slower surrogates. +3 +A hybrid surrogate model +In this work we attempt to avoid the curse of dimensionality by relegating to a physics-based material model some +of the tasks the RNN of Section 2.3 has to explicitly learn from data. In this section, we further formalize the hy- +brid approach of Fig. 1 by looking at the roles of each model component and their dependencies in time. We then +particularize the model for the case of a feedforward neural network (FNN) encoder and discuss feature selection and +numerical stabilization strategies. +3.1 +Evolving material parameters +Physics-based material models are traditionally formulated with a fixed set of parameters θ either directly computed +from a specific set of (numerical) experiments or indirectly from stress-strain measurements in a Maximum Likelihood +Estimation (MLE) approach2. Here we start from the premise that letting (part of) θ evolve in time increases flexibility +and allows the model to capture more complex material behavior. Conversely, keeping the remainder of the model +intact improves interpretability and provides physics-based bias to the data-driven model tasked to learn this evolution. +In Fig. 4, the hybrid model of Fig. 1 is unrolled in time for a number of consecutive time steps and represented as +a graph showing the dependencies between variables. Filled and hollow nodes represent observed and latent variables, +respectively, and are color coded to represent the different model components in Fig. 1. Similar to the microscale +models of Eq. (3), we assume the constitutive behavior at the macroscale is given by a physics-based material model: +MΩ +� +� +� +αΩ +t = A +� +εΩ +t , αΩ +t−1, θΩ +t +� +σΩ +t = S +� +εΩ +t , αΩ +t , θΩ +t +� +(6) +but now with time-dependent parameters θt. Note that the model response at time t depends on the material state at +time t−1 through a set of internal variables αΩ +t−1 (Fig. 4). This gives the model a recurrent nature not unlike that of the +RNN of Fig. 2a with its state variables c and h. The advantage here is that α has clear physical interpretation (plastic +strains, damage variables, etc) and its evolution is handled by the fixed operator A composed of clearly interpretable +algorithmic steps grounded in physics and/or classical material phenomenology (e.g. a return mapping algorithm). +2The parameters θ can also be estimated through Bayesian inference and would therefore be described by a multivariate probability density +instead of a fixed set of values. Regardless, that density would still be stationary in time. +6 + +On the encoder side, we let the material properties θ evolve according to an evolution operator D whose shape is +learned from data: +θt = D (ϕt) +(7) +as a function of a set of features ϕ that are themselves obtained from the macroscopic strains through a feature extractor +F: +ϕt = F +� +εΩ +t +� +(8) +where ϕt could be simply the strains themselves or other quantities derived from it. More importantly, note that θt +depends only on the current features ϕt and we therefore assume the encoder is not recurrent (Fig. 4). This choice +effectively limits the flexibility of D and makes the hybrid surrogate fully rely on the more robust model MΩ to explain +path-dependent phenomena, helping counter the curse of dimensionality associated with sampling strain paths. For +instance, it opens up the possibility to train the surrogate exclusively with monotonic data, as we will demonstrate in +the examples of Section 4. +In the following sections, we particularize the model for the case of D being a fully-connected neural network and +for specific choices of F and M. Nevertheless, the general architecture of Figs. 1 and 4 is meant to be as flexible as +possible: +• The nature and dimensionality of ϕ is not tied to that of εΩ since strains are also given directly to MΩ; +• Other machine learning models for regression can also be used as D, and it could in principle be split into +different models handling the evolution of different subsets of θ. Any number of model parameters may also be +left out of θ and either fixed as constants or optimized to constant values during training; +• No assumption is made on the form of MΩ or the nature or dimensionality of αΩ. Instead of a single model, +it could also for instance be a mixture of physics-based models combined with analytical homogenization tech- +niques. +3.2 +Feature extractors +A pragmatic choice for F is to simply assume ϕ is the macroscopic strain vector εΩ itself. It is also a familiar one, +as we can then relate the resulting model to conventional surrogates mapping strains to stresses. However, since +macroscopic strains are also directly passed on to the decoder, the architecture gives us the freedom to experiment +with different features. +Fig. 5 shows the two model architectures we explore in this work. For the two variants in Fig. 5a we either use εΩ +itself or a set of small-strain invariants of the macroscopic strain tensor of increasing dimensionality: +IΩ +ε = +�Iε +1 +� +or +IΩ +ε = +�Iε +1 +Iε +2 +� +(9) +where the variants are given by the well-known expressions: +Iε +1 = tr (ε) , +Iε +2 = 1 +2 +� +tr (ε)2 − tr +� +ε2�� +(10) +Additionally, since the current study focus on elastoplasticity, it is also interesting to explore feature spaces including +invariants from the deviatoric strain tensor: +IΩ +ε = +�Jε +2 +� +or +IΩ +ε = +�Iε +1 +Jε +2 +� +(11) +where: +Jε +2 = 1 +3 (Iε +1)2 − Iε +2 +(12) +By using features based on invariants and since the decoder material model is itself already frame invariant for small +strains, it follows that the resulting surrogate will naturally inherit this beneficial characteristic. This stands in contrast +with traditional black-box surrogates mapping strains to stresses. Furthermore, opting for invariant-based features can +be seen as a physics-based dimensionality reduction operation that can potentially reduce the amount of data needed +to train the hybrid model. +7 + +· · · +εΩ +or +IΩ +ε +· · · +· · · +· · · +· · · +· · · +M +σΩ, DΩ +α +Feedforward NN +θ +εΩ +(a) Macroscopic strains or invariants directly used as fea- +tures +· · · +α +or +I¯σ +· · · +· · · +· · · +· · · +· · · +M +σΩ, DΩ +α +Feedforward NN +θ +M +α +εΩ +Precalibrated model +εΩ +(b) Features extracted from an informative microscopic +constitutive model +Figure 5: The two types of FNN-based model architectures explored in this work, with different feature extraction +steps. +We also investigate the possibility of extracting features from the outputs of a precalibrated physics-based material +model M subjected to the same strain path seen at the macroscale (Fig. 5b). Note that this specific architecture +introduces an additional recurrent component to the model through the set α of internal variables of M. From a +machine learning perspective, the role of M would be analogous to that of a temporal convolution operator or an RNN +cell appended to the encoder. The key difference, however, is that M is fixed a priori and therefore should not require +extra sampling effort with respect to the more straightforward extractor in Fig. 5a. +Naturally, different choices for M yield models with distinct learning capabilities, and we therefore assume M +encapsulates relevant information about not only the current values of εΩ but also of its history. In the present scenario +where the data is coming from micromodel computations, we opt for the intuitive choice of having M be one of the +known constitutive models used to describe the microscopic material phases. We can therefore conceptually see M as +an imaginary representative material point at the microscale that is always subjected to the average micromodel strain. +We then use either a subset of its internal variables α or a set of invariants I¯σ of its stress outputs as features. +3.3 +Neural network encoder +For simplicity, we opt for modeling the evolution of θ using classical feedforward neural networks with fully- +connected layers. As both architectures in Fig. 5 ultimately compute macroscopic stresses given macroscopic strains, +we can use supervised learning to train the model with a straightforward Maximum Likelihood approach. Gather- +ing the complete set of network weights in a vector w and seeing the complete surrogate as a monolithic model that +computes an approximation �σ for stresses, we adopt the following observation model for the snapshot stresses σ: +σ = �σ (ε, w) + ξ, +ξ ∼ N +� +ξ|0, β−1I +� +(13) +where the superscript Ω is dropped for convenience, I is an identity matrix, and ξ is an additive Gaussian noise3. +Under the assumption of a squared loss, maximizing the likelihood of a training dataset with N observations amounts +to minimizing the loss function [45]: +L = 1 +2 +N +� +n=1 +∥σn − �σ (εn, w)∥2 +(14) +with the variance of the noise that explains data misfit being simply β = N/2L. The resulting loss function is the +same one used for conventional data-driven surrogates and is therefore straightforward to implement. +Nevertheless, it is worth noting that since we cannot directly observe θ, computing the gradients of L with respect +to w involves backpropagating derivatives through the decoder M. Furthermore, since w affects the evolution of the +internal variables α, backpropagation in time becomes necessary. Starting from Eq. (14) and walking back through +3Even though our observations come from a computer model and can be considered noiseless, the surrogate �σ is in general not arbitrarily flexible +and the random variable ξ is therefore still necessary to explain why the model does not exactly fit every single observation in the dataset. +8 + +the graph of Fig. 4, the gradient of the loss at time step t of a given strain path is given by: +∂Lt +∂w = ∂L +∂�σt +� +� +� +∂�σt +∂θt +∂θt +∂w + ∂�σt +∂αt +∂αt +∂θt +∂θt +∂w + ∂�σt +∂αt +1 +� +¯t=t−1 +� +� +� +� +¯t+1 +� +˜t=t +∂α˜t +∂α˜t−1 +� +� ∂α¯t +∂θ¯t +∂θ¯t +∂w +� +� +� +� +� +(15) +where the remaining gradient chain ∂θ/∂w is computed with conventional backpropagation through the network. If +M is implemented in a code base that allows for automatic differentiation (e.g. in PyTorch), these time dependencies +are naturally taken into account as long as a persistent gradient tape is used within each strain path4. In this work we +instead implement network training directly into an existing FE code, and therefore opt for the pragmatic approach of +computing all partial derivatives of quantities derived from M using finite differences. +Finally, in order to enforce upper and lower bounds for θ and avoid unphysical parameter values (e.g. negative +elasticity moduli), we apply sigmoid activation to the final layer of the network and scale the parameters back from a +[0, 1] range using predefined bounds: +θi = θlow +i ++ θσ +i +� +θupp +i +− θlow +i +� +(16) +3.4 +Material decoders +As previously mentioned, any constitutive model can in principle be used as M. For the present study we focus on +reproducing elastoplasticity and therefore narrow our choices down to the following set of potential decoders with +increasing levels of complexity. The simplest one is a linear-elastic isotropic material with no internal variables: +σij = Dijklεkl +with +Dijkl = G (δijδkl + δilδjk) + +� +K − 2 +3G +� +δijδkl +(17) +where index notation is used for convenience. For this model, θ comprises only the bulk and shear moduli K and G, +or equivalently the Young’s modulus E the Poisson’s ratio ν. +The second decoder option is a simple plasticity model with J2 (von Mises) flow. The stress update in this case +becomes: +σij = Dijkl +� +εij − εp +ij +� +(18) +where strain is additively decomposed into elastic and plastic (εp) contributions. The yield criterium and plastic flow +rule are given by: +φ = +� +3Jσ +2 − σy ≤ 0 +and +∆εp +ij = ∆γ +� +3 +2 +Sij +∥Sij∥F +(19) +where S is the deviatoric part of the stresses, γ is a plastic multiplier, σy is a yield stress parameter and we write +the Frobenius norm as ∥·∥F. In order to keep the model as simple as possible, we assume σy is a material constant +and therefore end up with a perfectly-plastic model with associative flow. The internal variables of this model are +components of the plastic strain vector εp and the only new material parameter is the yield stress σy. +Finally, we also consider the more complex pressure-dependent, non-associative plasticity model proposed by +Melro et al. [52]. Stress update is the same as in Eq. (18), but yield surface and plastic flow are given by: +φ = 6Jσ +2 + 2Iσ +1 (σc − σt) − 2σcσt ≤ 0 +and +∆εp +ij = ∆γ +� +3Sij + 1 − 2νp +1 + νp +Iσ +1 δij +� +(20) +where δij is the Kronecker delta, σt and σc are yield stresses in tension and compression, respectively, and νp is a new +parameter controlling plastic contraction and allowing for compressible plastic flow. Hardening can be described by +making the yield stresses general functions of εp, but when used as a decoder we assume σt and σc do not depend on +εp and instead let the decoder D describe their evolution. +The model by Melro et al. [52] is also the one used to describe the microscopic material phase responsible for +the nonlinear behavior observed when homogenizing micromodel response, and can therefore be seen as the natural +choice for M. Nevertheless, the other two decoders can provide interesting insights on the effect of introducing +different levels of bias to the hybrid model. +4This is already the case for RNNs, so switching from RNNs to the present model should require little to no changes to the way training is +performed. +9 + +3.5 +Online predictions and inherited stability +The architecture of Fig. 1 is developed to be minimally intrusive and allow for existing material models to be used +as decoders with minimum effort. We therefore implement the online routine of the model as a wrapper around an +existing implementation of M. The basic structure of the wrapper can be seen in Algorithm 1. The hybrid nature +of the model allows for a robust approach that ensures the numerical stability of the original model M is inherited +by the surrogate. This is achieved by only updating θ at the end of each time step, after the global implicit Newton- +Raphson scheme converges. Material properties are therefore fixed while the global solver is iterating, and that means +the tangent stiffness D comes directly from M and inherits its stability features. +Algorithm 1: Material wrapper implementing the online component of the hybrid surrogate. +Input: strain εΩ +new at macroscopic Gauss point +Output: stress σΩ and stiffness DΩ at macroscopic Gauss point +1 use nested model with converged parameters and internal state: +� +σΩ, DΩ, αnew +� +← M +� +εΩ +new, αold, θ +� +; +2 if global solver has converged : +3 +store latest converged strain: εold ← εnew; +4 +commit material history: αold ← αnew; +5 +compute new features: ϕnew ← F (εnew); +6 +update model parameters for the upcoming time step: θ ← D (ϕnew); +7 if first global iteration of time step and Gauss point is unstable : +8 +stabilize encoder: D ← stabilizeNetwork +� +εΩ +new +� +; +9 +recompute features: ϕold ← F +� +εΩ +old +� +; +10 +recompute model parameters for the current time step: θ ← D (ϕold); +11 return σΩ, DΩ +As an example, the J2 plasticity model of Eq. (19) is unconditionally stable as long as its hardening modulus h ≥ 0 +for any +� +εΩ +t , αΩ +t +� +, which is the case for the perfectly-plastic version we consider here. It then follows that any hybrid +surrogate with J2 decoder is also unconditionally stable. Note that this is only possible because strains are directly +passed on to the decoder and would therefore not be an option for conventional surrogates (e.g. the RNN of Fig. 3). For +those surrogates, the tangent stiffness would come directly from the jacobian of a highly-flexible data-driven model, +often at the cost of numerical stability. +3.6 +Numerical stabilization +Nevertheless, the decoder M may be inherently unstable even with fixed material constants. This is for instance the +case for the model by Melro et al. [52]: the non-associative flow rule of Eq. (20) can cause the tangent stiffness +DΩ to lose positive definiteness under certain strain conditions and for certain combinations of model parameters. To +accommodate such a scenario and open up the possibility for online model adaptivity in other contexts, we propose a +scheme for updating the encoder D on the fly in order to enforce extra constraints locally. +Back to Algorithm 1, at the beginning of a new time step we keep θ fixed to the one obtained with converged strains +from the previous step and let the solver make a first strain prediction. After this first iteration, a stability criterion +is checked and used to define a new loss function that can be used to update network weights in case instability is +detected. Here we employ the determinant of the acoustic tensor Q: +Q = nT +d DΩnd +(21) +where nd is the vector normal to the strain localization direction creating the instability, which we find through an +angle sweep procedure as in [53]. We then use det (Q) as a metric of stability and trigger a retraining procedure in +case a negative value is detected. We then introduce a new loss function: +LQ = −⟨det (Q)⟩− +det (Q0) +(22) +10 + +Linear-elastic fibers +Elastoplastic matrix +Figure 6: The micromodel used in the examples of this work. +where ⟨·⟩− extracts the negative part of its operand and Q0 is a reference value for the acoustic tensor computed at +the start of the simulation. We minimize this new loss at every unstable point for a small number of epochs with +low learning rate, and to discourage significant drifts from the original model we finish the stabilization procedure +by updating the network using the original loss of Eq. (14) for a single minibatch. Finally, θ is updated using the +retrained model and is kept fixed for the remaining iterations5. Note that the local constraint of Eq. (22) is therefore +only enforced in a soft way and remaining instabilities might still cause the global solver to diverge, in which case +we cancel the current increment, go back to the beginning of the time step and allow for the procedure to be triggered +again. +4 +Numerical examples +The proposed model was implemented in an in-house Finite Element code developed using the open-source C++ +numerical analysis library Jem/Jive [54]. In order to allow for seamless online retraining, network training was also +implemented within the same code. We start this section by describing the datasets and model selection strategies used +to build the surrogates. We then investigate the performance of the approach under several choices of encoders and +decoders. Finally, we use the model within an FE2 simulation and demonstrate the online stabilization procedure of +Section 3.5. All simulations are performed on cluster nodes equipped with Xeon E5-2630V4 processors and 128 GB +RAM running CentOS 7. +4.1 +Data sampling and model selection +Models are trained to reproduce the behavior of the fiber-reinforced composite micromodel shown in Fig. 6. Fibers are +modeled as linear-elastic and the matrix is described by the pressure-dependent non-associative elastoplastic model +by Melro et al. [52] (Section 3.4). Microscale material properties are adopted from [10]. The microscopic geometry +shown in Fig. 6 results from an RVE study performed in [10] and is therefore considered representative. Following +the discussion in Section 3, our aim is to investigate up to which extent it is possible to circumvent the curse of dimen- +sionality associated with path dependency by training surrogates exclusively on monotonic strain paths and having +time-dependent behavior arise naturally from a physics-based decoder. We therefore limit ourselves to monotonic +paths for training. For consistency, we also employ exclusively motononic data to perform model selection. +For efficiency, we limit the present investigation to 2D simulations (i.e. three strain components) in the plane +perpendicular to the fibers, but nevertheless expect the discussion and conclusions to generalize to 3D simulations +as long as appropriate orthotropic decoders are employed. Datasets with 2000 monotonic strain paths are generated +under both plane strain and plane stress assumptions. Fig. 7 shows the complete plane strain dataset, with a similar +one also being generated for plane stress. Each path is generated with an FE2 simulation of a single macroscopic +element under displacement control along a fixed direction in strain space sampled from a uniform distribution. To +circumvent convergence issues, we employ an adaptive time stepping technique that progressively reduces time step +size when the simulation does not converge and gradually increases it back for subsequent increments. The simulations +are stopped once a strain norm of 10 % is reached. As the adaptive scheme leads to paths with different numbers of +5Changing D and therefore θ after every iteration would not work in favor of improving stability, but rather have the opposite effect. +11 + +−0.1 +−0.05 +0 +0.05 +0.1 +−0.1 +−0.05 +0 +0.05 +0.1 +εxx [-] +εyy [-] +−0.1 +−0.05 +0 +0.05 +0.1 +−0.1 +−0.05 +0 +0.05 +0.1 +εxx [-] +γxy [-] +−0.1 +−0.05 +0 +0.05 +0.1 +−0.1 +−0.05 +0 +0.05 +0.1 +εyy [-] +γxy [-] +−0.1 +−0.05 +0 +0.05 +0.1 +−600 +−400 +−200 +0 +200 +εxx [-] +σxx [MPa] +−0.1 +−0.05 +0 +0.05 +0.1 +−600 +−400 +−200 +0 +200 +εyy [-] +σyy [MPa] +−0.1 +−0.05 +0 +0.05 +0.1 +−100 +−50 +0 +50 +100 +γxy [-] +τxy [MPa] +Figure 7: The complete plane strain dataset used to train the surrogates, comprising 2000 monotonic strain-stress +paths. A similar dataset is generated under plane stress conditions. +time increments, we balance the dataset by ensuring every path is composed of 30 steps with strain norms as equally +spaced as possible. +To keep model selection straightforward and avoid the need for cumbersome k-fold cross validation or bootstrap- +ping, we train a preliminary model with enough flexibility and an extensive training dataset and gradually increase the +size of the validation set until the validation error converges to a good estimate of the expected prediction error [55]. +This results in validation sets with 500 paths selected at random from the original datasets, leaving 1500 paths to be +used for training. We then perform model selection by gradually increasing the complexity of our FNN encoders until +the validation error stabilizes. From experimenting with different architectures, we find that encoders with 5 hidden +layers of 50 units each with Scaled Exponential Linear Unit (SELU) [56] activation provide enough flexibility for all +the examples treated here. To ensure enough regularization when computing learning curves with small datasets, we +employ Bernoulli dropout layers with a rate of 1 % after every hidden layer. Networks are trained for 20 000 epochs +and the model with lowest historical validation error is kept after training, further reducing the risk of overfitting on +small datasets. +To assess the capabilities of the trained surrogates, we compute an additional test dataset comprising 50 monotonic, +−0.01 +0 +0.01 +0.02 +0.03 +0.04 +0 +50 +100 +150 +Strain [-] +Stress [MPa] +xx +yy +xy +(a) Monotonic +−0.1 +−0.05 +0 +0.05 +−200 +−100 +0 +Strain [-] +Stress [MPa] +xx +yy +xy +(b) Unloading-reloading +−0.02 +0 +0.02 0.04 0.06 0.08 +0.1 +−200 +−100 +0 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +(c) Slow cycling +Figure 8: Examples from a test dataset with 50 paths of each type. They are not used to train any of the networks or +perform model selection. +12 + +0 +20 +40 +60 +80 +100 120 140 +101 +102 +Number of training paths [] +Mean validation error [MPa] +[εxx εyy γxy] → FNN +� +Iε +1/Iε +2 +� +→ FNN +[εxx εyy γxy] → Elastic +� +Iε +1/Iε +2 +� +→ Elastic +(a) Expectations over 50 datasets of each size +5 +10 +15 +20 +25 +30 +0 +50 +100 +150 +200 +250 +Number of training paths [] +Mean validation error [MPa] +[εxx εyy γxy] → FNN +� +Iε +1/Iε +2 +� +→ Elastic +(b) Detailed comparison including standard devia- +tions +Figure 9: Learning curves of models with elastic decoders and conventional FNN models. Mean error over the 500 +validation monotonic paths. +50 unloading-reloading and 50 slow cycling paths, examples of which are shown in Fig. 8. To keep the comparisons +fair, none of these paths are used to perform model selection and are therefore only considered after the surrogates are +trained. We will use example curves like those from Fig. 8 for visual inspection of the model performance, but also +the complete sets of 50 curves each for more rigorous statistical analysis. +4.2 +Elastic decoder +It is interesting to first consider the simple linear-elastic decoder of Eq. (17), as it has no internal variables and therefore +leads to a surrogate model comparable in nature to a conventional FNN trained on stress-strain pairs. As we will +demonstrate, however, the limited physical bias provided by such simple model already proves advantageous. Here +we let both elastic properties be controlled by the learned encoder: +θ = +�E +ν� +(23) +where the bounds 101 < E < 105 and 0 < ν < 0.5 are enforced as described in Eq. (16). +We first perform a feature selection study and investigate how efficiently the model learns as the size of the dataset +is increased. From the original plane strain training dataset of 1500 monotonic strain paths, we draw datasets with +sizes ranging between 1 and 150 paths without replacement and use them to train networks with different encoder +features. To get a reliable estimate of the expected prediction error, we repeat this process 50 times for each dataset +size and encoder type, and for comparison we also do the same for conventional FNNs trained directly on stress targets +(keeping the same architecture but going directly from the final hidden layer to stresses). This amounts to a total of +3400 trained networks from which we can compute an estimate of the prediction error by averaging ∥σ − �σ∥ over the +500 paths left for validation. +Fig. 9a plots averages of the validation error over the 50 training datasets used for each size. Although the hybrid +architecture does not show an advantage over the FNN when the encoder is trained on strain features, there is a clear +gain in learning speed when using only the two first strain invariants as features. Apart from accelerating learning +and resulting in lossless dimensionality reduction, using invariants also results in a surrogate which is frame invariant +under small strains. For comparison, we also train a conventional FNN on the same set of features, but those are +unsurprisingly not enough to describe general strain states and much of the material response is interpreted by the +FNN as observation noise. We zoom into the first part of the learning curves in Fig. 9b, this time also showing single +standard deviation uncertainty bands coming from the variance among the 50 training datasets. The hybrid network +outperforms conventional FNNs in the low data regime and tends to be less sensitive to changes in dataset starting +from about 20 training paths. Nevertheless, the extra flexibility of conventional FNNs allow them to achieve lower +validation errors if significantly more training paths are used. +Training the invariant-based hybrid network with the complete dataset of 1500 curves leads to surrogates with +validation errors of about 4 MPa, accurately representing the monotonic behavior of the original micromodel. Fig. 10 +13 + +0 +0.02 +0.04 +0.06 +0.08 +0 +50 +100 +150 +Strain [-] +Stress [MPa] +xx +yy +xy +Micromodel +I1/I2 →Elastic +(a) Monotonic +−0.1 +−0.05 +0 +0.05 +−300 +−200 +−100 +0 +Strain [-] +Stress [MPa] +xx +yy +xy +Micromodel +I1/I2 →Elastic +(b) Unloading-reloading +−0.04−0.02 +0 +0.02 0.04 0.06 0.08 +−200 +−100 +0 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +Micromodel +I1/I2 →Elastic +(c) Slow cycling +Figure 10: Performance of the elastic decoder model for different test scenarios. +−0.1 −0.08 −0.06 −0.04 −0.02 +0 +−400 +−200 +0 +Strain [-] +Stress [MPa] +xx +yy +xy +Micromodel +I1/I2 → Elastic +−0.05 +0 +0.05 +0.1 +−200 +−100 +0 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +Micromodel +I1/I2 → Elastic +Figure 11: Predicting unloading with a linear-elastic decoder through history-aware feature extraction. +shows representative predictions of this model for paths from the test set. As expected, this surrogate with no internal +variables is not capable of predicting non-monotonic strain paths, and effectively behaves like a hyperelastic material +model just as the conventional FNN would. +Nevertheless, the flexible and interpretable encoder-decoder architecture of Fig. 1 allows for new creative ap- +proaches in feature selection. As a demonstration, we keep the trained network of Fig. 10 intact and only modify its +feature extractor to introduce a simple path-dependent mechanism: +ϕT ≡ +� +I +ε +1 +I +ε +2 +� +T = argmax +0 σt. +We expand upon the feature selection study of Fig. 9 by looking at several feature extractors coming both directly +from strains and from the output of a precalibrated Melro model M with the same properties used at the microscale +(Fig. 5b). Aside from the familiar choice of strain features ([εxx εyy γxy] → Melro), we look into invariants of the +strain tensor ([Iε +1 Iε +2] → Melro), combinations including invariants of the deviatoric strain tensor ([Jε +2] → Melro, +[Iε +1 Jε +2] → Melro), plastic strain internal variables coming from the precalibrated feature extractor ( +� +εp +xx εp +yy γp +xy +� +→ +Melro) and stress invariants coming from the extractor ( +� +Iσ +1 Jσ +2 +� +→ Melro). We also include the precalibrated me- +somodel by Vogler et al. [57] and selected curves from Fig. 9a for comparison purposes. As before, we train 50 +networks of each type for each size of dataset ranging from 1 to 150 paths drawn from the original dataset with 1500 +paths. Each trained network is then used to compute the validation error over the 500 monotonic validation paths and +the 150 test paths (50 extra monotonic paths, 50 paths with unloading-reloading and 50 slow cycle paths). This results +in an extensive study comprising 6800 trained networks and over one million test set simulations. +Results are summarized in Fig. 16, with each point in a curve being the average over 50 networks. Once again using +invariants as features proves beneficial, leading to lossless dimensionality reduction and frame invariant surrogates. +All tested models perform better than the precalibrated mesomodel, with a gap of more than one order of magnitude +for the best performing surrogates. Interestingly, models with Melro-based decoders seem to learn as fast and be as +flexible as models with elastic decoders, already for the monotonic curves in the validation dataset. This suggests +that the new decoder does not impose extra undesirable bias in learning the specific material behavior treated here +other than the assumptions that had already been introduced by elasticity (e.g. symmetries and couplings encoded by +the elastic stiffness tensor). Any benefits reaped when extrapolating to non-monotonic paths, as we will see in the +following, are therefore obtained at a negligible price in terms of monotonic behavior accuracy. This stands in contrast +with the discussion on the J2 decoder of the previous section. +17 + +0 +0.02 +0.04 +0.06 +0.08 +0 +50 +100 +150 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[Jε +2] → Melro +Meso +(a) Model with second deviatoric strain invariant as +feature +0 +0.02 +0.04 +0.06 +0.08 +0 +50 +100 +150 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +� +εp +xx εp +yy γp +xy +� +→ Melro +Meso +(b) Model with plastic strain features +Figure 17: Monotonic test set predictions from feature-deficient Melro models (complete training dataset with 1500 +paths). +0 +20 +40 +60 +80 +100 +120 +140 +101 +102 +103 +Number of training paths [] +Mean test error [MPa] +Precalibrated mesomodel +[Iε +1 Iε +2] → Elastic +[εxx εyy γxy] → Melro +[Iε +1 Iε +2] → Melro +[Iε +1 Jε +2] → Melro +[Jε +2] → Melro +� +εp +xx εp +yy γp +xy +� +→ Melro +� +Iσ +1 Jσ +2 +� +→ Melro +(a) Errors for complete paths +0 +20 +40 +60 +80 +100 +120 +140 +101 +102 +103 +Number of training paths [] +Mean error (non-monotonic) [MPa] +Precalibrated mesomodel +[Iε +1 Iε +2] → Elastic +[εxx εyy γxy] → Melro +[Iε +1 Iε +2] → Melro +[Iε +1 Jε +2] → Melro +[Jε +2] → Melro +� +εp +xx εp +yy γp +xy +� +→ Melro +� +Iσ +1 Jσ +2 +� +→ Melro +(b) Errors including only unloading/reloading steps +Figure 18: Learning curves for unloading-reloading test errors of Melro-decoded surrogates (averages of 50 datasets). +18 + +−0.1 −0.08−0.06−0.04−0.02 +0 +0.02 0.04 +−200 +−100 +0 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[εxx εyy γxy] → Melro +Meso +−0.05 +0 +0.05 +0.1 +−100 +0 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[Iε +1 Iε +2] → Melro +Meso +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +−50 +0 +50 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[Iε +1 Jε +2] → Melro +Meso +−0.02 0 +0.02 0.04 0.06 0.08 0.1 0.12 +−100 +0 +100 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +� +Iσ +1 Jσ +2 +� +→ Melro +Meso +Figure 19: Response of Melro-decoded surrogates with different features for selected unloading/reloading test paths +(1500 monotonic training paths). +Although Fig. 16 is not enough to discern between several of our encoder choices, it is interesting to take a closer +look at the two clearly underperforming options. Fig. 17 shows predictions from [Jε +2] → Melro and +� +εp +xx εp +yy γp +xy +� +→ +Melro for the same monotonic test path. The model with a single feature struggles to predict the entirety of the path, +indicating that further reducing the dimensionality of the feature space is not possible for this dataset. The oscillatory +stress predictions make this model unsuitable for online stress evaluation in a multiscale setting. For the model with +plastic strain features, the feature extractor shows no plastic strains until high stress levels while in the micromodel +plasticity starts much earlier, forcing the surrogate to remain in the elastic regime until a sudden jump brings it back +to the expected path. +Moving to unloading-reloading paths, we compare the performance of different feature sets by plotting the average +test error over the 50 unloading-reloading paths in Fig. 18a. Here an interesting observation can be made: even the +surrogate [Iε +1 Iε +2] → Elastic — which cannot predict unloading at all — attains a lower test error than the precalibrated +mesomodel. This apparent contradiction can be explained by plotting in Fig. 18b the average error computed only at +unloading or reloading time steps: use of an elastic decoder — and therefore of a conventional FNN or an RNN trained +with insufficient data — excels at predicting monotonic response but is consistently inaccurate for non-monotonic +paths and shows little improvement when more monotonic paths are added to the training dataset. +In contrast, the best-performing Melro models are consistently more accurate than the precalibrated mesomodel +even when trained on very little data. We plot in Fig. 19 selected representative unloading paths from the test dataset +for four of the surrogates. Unloading is once again well captured without having been seen during training, and +since it emerges from a purely physical mechanism, it is reasonable to expect unloading at different points along the +path to yield comparable results (c.f. Fig. 3a). Nevertheless, relatively small differences in unloading slope can still +lead to large differences in stress at the end of the unloading branches. Furthermore, the model can struggle with +tension-compression switches and predict spurious hysteresis loops. +Indeed, we observe a consistent inability by the models to properly predict switches between tension and com- +pression within the same path. This becomes clear when looking at slow cycling test paths composed of several of +these switches (Fig. 8c). We plot learning curves for the test error on slow cycling paths in Fig. 20, for complete paths +19 + +0 +20 +40 +60 +80 +100 +120 +140 +101 +102 +103 +Number of training paths [] +Mean test error [MPa] +Precalibrated mesomodel +[Iε +1 Iε +2] → Elastic +[εxx εyy γxy] → Melro +[Iε +1 Iε +2] → Melro +[Iε +1 Jε +2] → Melro +[Jε +2] → Melro +� +εp +xx εp +yy γp +xy +� +→ Melro +� +Iσ +1 Jσ +2 +� +→ Melro +(a) Errors for complete paths +0 +20 +40 +60 +80 +100 +120 +140 +101 +102 +103 +Number of training paths [] +Mean error (non-monotonic) [MPa] +Precalibrated mesomodel +[Iε +1 Iε +2] → Elastic +[εxx εyy γxy] → Melro +[Iε +1 Iε +2] → Melro +[Iε +1 Jε +2] → Melro +[Jε +2] → Melro +� +εp +xx εp +yy γp +xy +� +→ Melro +� +Iσ +1 Jσ +2 +� +→ Melro +(b) Errors including only unloading/reloading steps +Figure 20: Slow cycling test errors for Melro-decoded surrogates (averages of 50 datasets for each size). +as well as exclusively for the non-monotonic branches of the paths. In contrast with results up until now, here we +see larger differences in performance for different feature sets. As expected, elastic decoders are once again shown +to be unsuitable to predict non-monotonic paths, and the difference here is even more pronounced than in for single- +unloading paths (c.f. Fig. 18) as most of the path is composed of unloading/reloading branches. The model encoded +with stress invariants coming from an elastoplastic feature extractor performs best among the models we test. But +crucially, none of the surrogates manages to surpass the precalibrated mesomodel in this case. +As a demonstration, we select a representative path from the test dataset and plot predictions made with four differ- +ent feature sets in Fig. 21. As expected, larger errors are observed for more pronounced tension-compression switches +as models either over- or undershoot the stress levels at compression-tension switch points. Interestingly, most models +manage to converge back to the correct stress path after reloading, since hardening behavior is completely dictated +by their non-recurrent data-driven encoders. The exception is the model with stress invariant features ( +� +Iσ +1 Jσ +2 +� +→ +Melro), performing significantly better than the rest but showing a number of undesired oscillations in stress response +due to the (physically) recurrent nature of its features forcing its neural network encoder to operate in extrapolation. +4.5 +FE2 example +We conclude our discussion with an FE2 demonstration using the proposed hybrid surrogate. We model the tapered +macroscopic bar with geometry and boundary conditions shown in Fig. 22. The model is meshed with 1620 linear +triangles with a single Gauss point each and is loaded in tension until plastic strain localization takes place. The +combination of the tapered geometry with the several circular voids along the model result in a complex range of stress +states throughout the model. In contrast to the cases considered so far, this example also covers non-proportional strain +paths. To facilitate convergence, the substepping approach proposed in [58] is employed and an adaptive stepping +algorithm is used at the macroscale that automatically reduces time step size and recomputes the current increment if +either the micro- or macroscopic Newton-Raphson solver fails to converge. +We use the +� +Iσ +1 Jσ +2 +� +→ Melro model of the previous section as surrogate, trained on the complete set of 1500 +monotonic training strain paths. The global load-displacement curve at the right edge of the model is plotted for the +full-order FE2 solution and using the hybrid surrogate in Fig. 23a. Since we update decoder properties in an explicit +fashion (i.e. once per time step, see Algorithm 1), we use a displacement increment ∆u = 3.5 × 10−3 mm for the +approximate model, 10 times smaller than the one used for the full-order model. +As mentioned in Section 3.5, the model by Melro et al. can suffer from numerical stability issues even with fixed +material properties, and it is reasonable to expect these issues to become worse when letting properties evolve with +time. Indeed, with no additional stabilization the model using the network fails to converge at the point marked in +Fig. 23a. In contrast, the stabilization procedure of Section 3.5 allows for a complete path to be obtained. For this +first result, we stabilize the network for 5 epochs with a learning rate of 1 × 10−5 for the stabilization loss (Eq. (22)) +and 1 × 10−9 for retraining on a single monotonic training path selected at random. We also consider a model with +an unloading/reloading switch after the onset of macroscopic plasticity. Results are shown in Fig. 23b. The surrogate +approximates the full-order behavior fairly accurately and several orders of magnitude faster than the full-order model. +20 + +−0.02 0 +0.02 0.04 0.06 0.08 0.1 0.12 +−400 +−200 +0 +200 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[εxx εyy γxy] → Melro +Meso +−0.02 0 +0.02 0.04 0.06 0.08 0.1 0.12 +−800 +−600 +−400 +−200 +0 +200 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[Iε +1 Iε +2] → Melro +Meso +−0.02 0 +0.02 0.04 0.06 0.08 0.1 0.12 +−400 +−200 +0 +200 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +[Iε +1 Jε +2] → Melro +Meso +−0.02 0 +0.02 0.04 0.06 0.08 0.1 0.12 +−400 +−200 +0 +200 +Strain [-] +Stress [MPa] +xx +yy +xy +Micro +� +Iσ +1 Jσ +2 +� +→ Melro +Meso +Figure 21: Response of Melro-decoded surrogates with different features for selected slow cycling test paths (1500 +monotonic training paths). +full-order micromodel +σΩ +DΩ +εΩ +· · · +I¯σ +· · · +· · · +· · · +· · · +· · · +M +σΩ +DΩ +α +5 × 50 SELU +evolving Melro +θ +M +α +εΩ +microscale Melro +εΩ +Figure 22: FE2 example: geometry, mesh and boundary conditions. Full-order (left) and surrogate-based (right) FE2 +simulations are compared. +21 + +0 +0.5 +1 +1.5 +0 +50 +100 +150 +Displacement [mm] +Load [N] +Full-order FE2 (90 h runtime) +Stabilized surrogate (2 min runtime) +Unstabilized surrogate +(a) Monotonic +0 +0.5 +1 +1.5 +0 +50 +100 +150 +Displacement [mm] +Load [N] +Full-order FE2 (113 h runtime) +Stabilized surrogate (2 min runtime) +(b) Unloading/reloading +Figure 23: FE2example: load-displacement curves with and without online stabilization, compared to the ground-truth +solution. +0 +100 +200 +300 +400 +500 +10 +20 +30 +Time increment [] +Mean validation error [MPa] +2 epochs +5 epochs +10 epochs +50 epochs +100 epochs +a +No retraining +1-path retraining +(a) 500-path validation loss after stabilization +0 +0.5 +1 +1.5 +0 +50 +100 +150 +Displacement [mm] +Load [N] +2 epochs +5 epochs +10 epochs +50 epochs +100 epochs +a +No retraining +1-path retraining +(b) Load-displacement curves +Figure 24: Performance of the surrogate model for stabilization strategies of varying intensities with and without +retraining after stabilization. +22 + +0 +100 +200 +300 +400 +500 +0 +200 +400 +600 +800 +1,000 +Time increment [] +Cumulative execution time [s] +2 epochs +5 epochs +10 epochs +50 epochs +100 epochs +a +No retraining +1-path retraining +(a) Computational overhead due to stabilization +0 +100 +200 +300 +400 +500 +100 +101 +102 +103 +104 +105 +Time increment [] +Unstable points (cumulative) [] +2 epochs +5 epochs +10 epochs +50 epochs +100 epochs +a +No retraining +1-path retraining +(b) Number of detected unstable strain states +Figure 25: Impact of stabilization regime on execution time and number of unstable points throughout the simulation. +We now look closer on the performance of the proposed online stabilization approach. We empirically find that +retraining the network until every violating material point is fully stabilized is not strictly necessary in order to achieve +convergence, and therefore opting for a small number of stabilization epochs proves to be an efficient approach. +It is nevertheless interesting to investigate the impact of the number of stabilization epochs and of the subsequent +retraining minibatch on the original dataset. We solve the monotonic example of Fig. 23a with different numbers of +stabilization/retraining epochs ranging from 2 to 100 and compute the validation loss (on the 500-path validation set +used for model selection) at the end of every macroscopic time increment in order to keep track of how much the +stabilized network deviates from its original pretrained state. +Results are shown together with the corresponding load-displacement curves in Fig. 24. All curves remain stable +at first, as stabilization is only triggered when the first unstable points are detected. From that point, models which +do not undergo retraining after stabilization lose accuracy at a rate proportional to the number of stabilization epochs. +However, this unintuitively does not lead to improved global stability: the loss of accuracy by the surrogate leads to +spurious global softening (c.f. Fig. 24b) which in turn leads to further need for stabilization. Models stabilized for +50 and 100 epochs continuously fail to converge and we opt for terminating the simulation after 100 cancelled time +increments. On the other hand, models retrained with as little as a single strain path (out of the original 1500) after +each stabilization epoch are able to maintain the original model accuracy while offering enough stability gains to allow +the simulation to converge until the final step, with little change in global behavior for different stabilization regimes. +More insight can be obtained on the different stabilization strategies by plotting the cumulative execution time of +the simulation and the cumulative number of detected unstable strain states with time increments for different numbers +of stabilization epochs. Results can be seen in Fig. 25. In general, simulations without retraining tend to run faster +and result in improved stability, although any gains are quickly overshadowed by losses in accuracy (c.f. Fig. 24). +Stabilizing for more epochs results in a reduction in the total number of unstable points detected, but beyond 5 epochs +this does not result in an overall reduction in the computational cost of the simulation given the increased effort spent +on individual stabilization operations. +As one final result, we run the monotonic simulation with the hybrid surrogate for different time step sizes. As +previously mentioned, the hybrid approach allows for explicit update of θ within an implicit simulation by obtaining +the tangent stiffness matrix directly from the decoder. This however introduces a time step size dependency whose +impact merits investigation. We plot in Fig. 26 predictions with step sizes spanning four orders of magnitude, including +the same one used to obtain the full-order response. The combination of the explicit property update with the online +stabilization procedure indeed introduces an upper bound for time step size for this specific problem. It stands to +reason that the sensitivity to time step size also depends on the choice of decoder and on which material properties are +included in θ. Further investigation into the matter in future works is therefore warranted. +23 + +0 +0.5 +1 +1.5 +0 +50 +100 +150 +Displacement [mm] +Load [N] +Full-order (3.5 × 10−2 mm) +∆u = 7.0 × 10−2 mm +∆u = 5.0 × 10−2 mm +∆u = 3.5 × 10−2 mm +∆u = 3.5 × 10−3 mm +∆u = 3.5 × 10−4 mm +∆u = 3.5 × 10−5 mm +Figure 26: FE2 example: Effect of time step size on surrogate predictions. +5 +Conclusions +In this paper, we propose a hybrid surrogate modeling architecture for multiscale modeling of heterogeneous materials. +The model is composed of a data-driven encoder for material properties and a physics-based decoder that computes +stresses. In the resulting architecture, the encoder increases the flexibility of existing material models by letting their +properties evolve in time, while the decoder provides beneficial bias and interpretability to the model. The model is +conceived with flexibility in mind, allowing existing implementations of physics-based material models to be used +with no extra modifications. Furthermore, by letting the decoder directly receive strain inputs, the encoder architecture +is highly flexible and allows for preservation of frame independence. A semi-explicit online prediction algorithm is +also proposed that allows for imposing extra constraints to model behavior in a semi-supervised way. +We demonstrate the architecture by reproducing pressure-dependent elastoplastic behavior coming from homog- +enized fiber-reinforced composite micromodels. The simple model with a linear-elastic decoder learned faster than +conventional data-driven surrogates, allowed for lossless feature space dimensionality reduction through the use of +strain invariants, and was able to approximate path-dependent behavior through a simple history-aware feature ex- +tractor. Models with perfectly-plastic J2 decoders were shown to successfully learn nonlinear hardening and pressure +dependency and predict unloading-reloading while being trained exclusively on monotonic data, outperforming a +state-of-the-art mesomodel for composites in accuracy for arbitrary loading directions. Employing as decoder the +same plasticity model used at the microscale led to highly-accurate monotonic response and fairly accurate extrapo- +lation to unloading/reloading behavior. Finally, the model was used to solve a complex FE2 model and the benefit of +the online stabilization procedure was demonstrated. +We find the approach to be a promising new way to build hybrid surrogates which therefore merits further research +on a number of fronts. The current architecture is not by construction concerned with enforcing unconditional ther- +modynamic consistency or other physical constraints of interest. Although we do find empirically that well-trained +surrogates with thermodynamically consistent decoders tend to perform well, some constitutive models might not be +suitable for having their properties evolve in time. Fortunately, the framework can cope with extra constraints with- +out necessarily giving up on its flexibility, by enforcing them locally through online retraining. Although training +exclusively on monotonic paths already allows for path dependency to be fairly well captured, some decoders might +perform better in extrapolation if trained with a (small) number of extra non-monotonic and non-proportional strain +paths — for instance when encoder and decoder can each explain the same phenomenon on their own (e.g. pressure +dependency in the model by Melro et al.). We also foresee combining the present approach with the one in [46] into a +unified family of flexible hybrid surrogates with a range of possible combinations of feature extractors for physics-rich +time convolution, fixed-property models with learned strain distributions and evolving material models. +24 + +Acknowledgements +The authors gratefully acknowledge the TU Delft AI Initiative for their support through the SLIMM AI Lab. FM also +acknowledges financial support from the Netherlands Organization for Scientific Research (NWO) under Vidi grant +nr. 16464. +References +[1] Siddhant Kumar, Stephanie Tan, Li Zheng, and Dennis M. Kochmann. Inverse-designed spinodoid metamateri- +als. npj Computational Materials, 6(1):1–10, June 2020. +[2] Miguel A. Bessa, Piotr Glowacki, and Michael Houlder. Bayesian Machine Learning in Metamaterial Design: +Fragile Becomes Supercompressible. Advanced Materials, 31(48):1904845, 2019. +[3] Silvan Gantenbein, Chiara Mascolo, Caroline Houriet, Robert Zboray, Antonia Neels, Kunal Masania, and +Andr´e R. Studart. Spin-Printing of Liquid Crystal Polymer into Recyclable and Strong All-Fiber Materials. +Advanced Functional Materials, 31(52):2104574, 2021. +[4] Wilhelm Woigk, Yannick Nagel, Silvan Gantenbein, Fergal B. Coulter, Kunal Masania, and Andr´e R. Studart. +Flax-based natural composites hierarchically reinforced by cast or printed carbon fibres. 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Computer Methods in Applied Mechanics and Engineering, 198(9):1006–1016, February 2009. +28 + diff --git a/kdFRT4oBgHgl3EQfYDcO/content/tmp_files/load_file.txt b/kdFRT4oBgHgl3EQfYDcO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..48f551f58dd7050a1fa0d8ea6e5bf1dabb3eebd5 --- /dev/null +++ b/kdFRT4oBgHgl3EQfYDcO/content/tmp_files/load_file.txt @@ -0,0 +1,1354 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf,len=1353 +page_content='Machine learning of evolving physics-based material models for multiscale solid mechanics I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Rocha1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Kerfriden2, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' van der Meer1 1Delft University of Technology, Faculty of Civil Engineering and Geosciences, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Box 5048, 2600GA Delft, The Netherlands 2Mines Paris, PSL University, Centre des mat´eriaux, 63-65 Rue Henri-Auguste Desbrueres BP87, F-91003 ´Evry, France Abstract In this work we present a hybrid physics-based and data-driven learning approach to construct surrogate models for concurrent multiscale simulations of complex material behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We start from robust but inflexible physics-based constitutive models and increase their expressivity by allowing a subset of their material parameters to change in time according to an evolution operator learned from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This leads to a flexible hybrid model combining a data-driven encoder and a physics-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Apart from introducing physics-motivated bias to the resulting surrogate, the internal variables of the decoder act as a memory mechanism that allows path dependency to arise naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We demonstrate the capabilities of the approach by combining an FNN encoder with several plasticity decoders and training the model to reproduce the macroscopic behavior of fiber-reinforced composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The hybrid models are able to provide reasonable predictions of unloading/reloading behavior while being trained exclusively on monotonic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Furthermore, in contrast to traditional surrogates mapping strains to stresses, the specific architecture of the hybrid model allows for lossless dimensionality reduction and straightforward enforcement of frame invariance by using strain invariants as the feature space of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Keywords: Concurrent multiscale (FE2) modeling, Surrogate modeling, Hybrid learning 1 Introduction Recent advances in materials science and manufacturing techniques are paving the way for the design of materials with highly-tailored microstructures, including metamaterials [1, 2], novel composite material systems [3, 4], printed cementitious materials [5] and multifunctional living materials [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The common thread in these new developments is a shift from traditional design focused on tailoring structures to material constraints towards tailoring material microstructures to macroscopic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This shift in turn requires the development of highly-detailed models of material behavior across spatial scales and a shift to virtual structural certification, as trial-and-error design becomes infeasible [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Scale bridging has been traditionally performed through a bottom-up approach: physics-based constitutive models at smaller scales are calibrated using experiments and used to perform numerical simulations (using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' the Finite Element (FE) method) on representative lower-scale domains from which higher-scale physics-based models can be calibrated [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' However, physics-based constitutive models come with a priori assumptions that often fail to reproduce complex lower-scale behavior [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The alternative is to opt for an FE2 (or Computational Homogenization) approach: lower-scale FE models are embedded at every Gauss point of a higher-scale model and material behavior is directly upscaled with no constitutive assumptions at the higher scale [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Yet, the computational cost associated with repeatedly solving a large number of micromodels quickly becomes a bottleneck, in particular for many-query procedures such as design exploration and optimization that require several higher-scale simulations to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Since the bottleneck of FE2 lies in computing lower-scale models, a popular approach to reduce computational effort is to substitute the original FE micromodels with either structure-preserving reduced-order models [15–21] or purely data-driven surrogates [22–27] trained offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' More recently, Recurrent Neural Networks (RNN) have become the model of choice especially for strain path-dependent materials, with a large body of literature dedicated to their use and tuning to different applications [28–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' RNNs can reproduce complex long-term time dependencies in material behavior by learning latent representations of the material state, making them fast and flexible surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' However, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='13547v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='NA] 31 Jan 2023 these learned representations are not a priori related to actual thermodynamic internal state variables and the model is therefore poorly interpretable (see [35] for an interesting discussion on the subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Furthermore, training for path dependency requires sampling from a potentially infinite-dimensional space of arbitrarily-long strain paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This means training RNNs to reproduce complex material behavior often requires an inordinate amount of data (curse of dimensionality) and their purely data-driven nature limits their ability to extrapolate away from paths seen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In order to address these drawbacks, a growing number of recent works are shifting focus to models with a fusion of data-driven and physics-based components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Inspired by physics-informed neural networks ([36]), the authors in [37] opt for data-driven models with physics-inspired bias by enforcing thermodynamic principles in a weak sense through an augmented loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In a similar vein, the model in [38] learns hyperelasticity by linking together several carefully crafted neural nets to represent quantities with clear physical meaning, improving the interpretability of the resulting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In [39] the authors extend a similar hyperelastic surrogate with a network that learns plastic flow direction and the evolution of a yield surface parametrized by a level set function, resulting in a hyperelastic-plastic model with superior extrapolation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' A common thread in the aforementioned approaches, however, is that their learning architectures are heavily dependent on the type of model being learned (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' hyperelasticity, plasticity), making extensions to other models a convoluted task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In contrast, the authors in [40, 41] propose a surrogate for heterogeneous micromodels constructed by directly employing unmodified versions of the constitutive models used for the micro constituents and using a customized network architecture to infer a homogenization operator from data that combines their responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, the method employs a highly-specialized iterative online prediction routine requiring extra implementation effort and with increased computational overhead when compared to that of traditional surrogates mapping strains to stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, in [42–44] a dictionary of candidate physics-based models is assumed and the role of machine learning shifts instead to that of performing model selection and/or design of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In this work we explore an alternative approach for constructing hybrid surrogate models for path-dependent multiscale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We start from the premise that existing physics-based models — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' the ones used to describe microscale constituents — are not flexible enough to reproduce macroscale behavior but nonetheless encapsulate crucial physical features such as frame invariance and loading/unloading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' It is our aim to avoid learning these features directly from data, as that would require either an excessively large dataset or a highly-specialized learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We therefore opt for keeping the constitutive model as intact as possible and instead increasing flexibility by allowing some (or all) of its material parameters to evolve in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The resulting model can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1: a data-driven encoder that learns the evolution of a set of material properties is linked to a physics-based material model decoder that maps strains to stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In contrast to other strategies in literature, we keep the architecture as general as possible: a general feature extractor parses macroscopic strains into features for the encoder — which can be as simple as the strains themselves or other derived quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' strain invariants) — and any type of constitutive model can in principle act as decoder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' hyperelasticity, plasticity, damage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' By relegating stress computations to the decoder, we effectively introduce physics-based bias to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 Furthermore, by letting the material model handle the evolution of its own internal variables, the model benefits from a recurrent component with interpretable memory structure that allows path dependency to arise naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The strategy we explore here is related to the one we propose in [46], but in that work we let an encoder learn local strain distributions for several virtual material points with fixed properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We see the two approaches as being complementary, and therefore with potential for being used in combination to form a flexible range of hybrid surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The remainder of the work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Section 2 contains a primer on concurrent multiscale (FE2) modeling and discusses the difficulties of training purely data-driven surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In Section 3, we particularize the model of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 to the case of a feedforward neural network encoder and discuss aspects related to offline training and online numerical stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In Section 4 we assess the performance of the hybrid model in reproducing the behavior of fiber-reinforced composites using different encoder features and decoder models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, some concluding remarks and future research directions are discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1In purely data-driven surrogates, we accept some bias in exchange for reduced variance — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' by employing regularization or adopting prior distributions for model parameters [45] — in order to counter overfitting and improve generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' But in that case the bias is merely a way to reduce complexity, with no physical interpretation and no a priori impact on the extrapolation capabilities of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2 F φt D θt M εΩ t Gauss point σΩ t Gauss point εΩ t Gauss point material model feature extractor data-driven model macroscale FE model Gauss point features evolving material properties encoder decoder αΩ t−1 internal variables αΩ t Figure 1: The hybrid surrogate combining a data-driven encoder for material parameters and a physics-based material model decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2 Concurrent multiscale (FE2) modeling In this section we present a short discussion on FE2 modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The goal is not to be comprehensive — the inter- ested reader is referred to [13, 14] for detailed discussions on the subject — but rather to expose the computational bottleneck associated with the method and pinpoint where surrogate models can be used to alleviate the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then demonstrate how a Recurrent Neural Network (RNN) can be used as surrogate model and showcase some of the difficulties associated with their training and their extrapolation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 Scale separation and coupling In FE2 we assume the problem being solved can be split into a homogeneous macroscopic domain Ω and a heteroge- neous microscopic domain ω ≪ Ω where small-scale geometric features are resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Here we opt for a first-order homogenization approach assuming the displacements on both scales can be related by: uω = εΩxω + �u (1) where microscopic displacements uω are split into a linear contribution proportional to the macroscopic strains εΩ and a fluctuation term �u that accounts for microscopic heterogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Since εΩ varies throughout the macroscopic domain, a micromodel for ω is embedded at each Gauss point in Ω and a microscopic boundary-value equilibrium problem assuming small displacements and strains is solved: ∇ · σω = 0 εω = 1 2 � ∇uω + (∇uω)T� (2) microscopic stress σω is related to microscopic strain εω with traditional physics-based constitutive models for each phase in the heterogeneous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In the general case where the material models feature internal variables α, we can write the constitutive update for the microscale domain as: Mω � αω t = A � εω t , αω t−1, θω� σω t = S (εω t , αω t , θω) (3) where θω are the material parameters of the microscopic constituents, the operators A and S can be split into an arbitrary number of blocks with different models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' elasticity, elastoplasticity, damage) for the different material phases, and αω is a concatenation of the internal variables of every microscopic Gauss point and therefore fully describes the path-dependent state of the microscopic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In order to determine the strains εΩ that serve as boundary conditions for the micromodels, a macroscopic small- strain equilibrium problem is solved: ∇ · σΩ = 0 εΩ = 1 2 � ∇uΩ + � ∇uΩ�T� (4) 3 but this time no constitutive assumptions are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Macroscale stresses are instead directly homogenized from the microscopic response: σΩ = 1 |ω| � ω σωdω (5) which couples the macroscopic strain εΩ with the microscopic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (1) also couples the solutions in the opposite direction, a bidirectional coupling is formed which requires the two-scale equilibrium problem to be solved iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 Data-driven surrogate modeling The coupled problem of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 is extremely computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The lower-scale domain ω usually features complicated geometric features and must therefore be modeled with dense FE meshes in order to ensure accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Worse yet, an independent microscopic problem must be solved at every integration point in Ω for every iteration of every time step of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This nested nature quickly forms a computational bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Since the bulk of the computational effort lies in solving the micromodels, a popular approach to make multiscale analysis viable for practical applications is to substitute the microscopic FE models by data-driven surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The idea is to perform a number of micromodel simulations under representative boundary conditions and use the resulting stress-strain pairs to train a machine learning model to be deployed when performing the actual two-scale simulations of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Naturally, the approach tacitly assumes that the number of offline micromodel computations required to train the model is much smaller than the number of times the microscopic behavior will be computed online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In the following, we use a simple example to demonstrate a number of difficulties associated with training such a model to reproduce path-dependent material behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='3 Example: A one-dimensional RNN surrogate For this demonstration, we train a Long Short-term Memory (LSTM) network [47] to reproduce one-dimensional (single stress/strain component) elastoplasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The architecture of the model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2a and is implemented in PyTorch [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In order to minimize the risk of overfitting, a pragmatic model selection procedure is performed by first training the model with several non-monotonic strain paths and gradually increasing cell size until reasonable accuracy is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This leads to a parsimonious model with a single LSTM cell with 5 latent units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' At this point it is interesting to draw a parallel between the network and the micromodel whose behavior is being re- produced: the concatenation of the hidden state h and cell state c of the LSTM cell can be seen as a lower-dimensional surrogate for the set of microscopic internal variables αω of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' However, in contrast to the variables in α, the latent variables h and c have no physical interpretation and evolve purely according to heuristic memory mechanisms that mimic patterns inferred during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' First, we train the LSTM using only monotonic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Since only one strain component is being modeled, this initial dataset is composed simply of one strain path in tension and one in compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The trained model is then used to predict a tension path with one unloading-reloading cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Having never seen unloading during training, the network reverses course and unloads on top of its loading path (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This result is hardly surprising, but sheds light on the potentially deceiving nature of the training procedure: even though we are only concerned with a single strain component, predictions actually take place in an augmented space that describes strain paths in time which can be arbitrarily high-dimensional (as paths can be arbitrarily long).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We can further demonstrate this manifestation of the curse of dimensionality with the two additional examples of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3a we train the network with two unloading paths and it fails to predict a third one at an intermediate strain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Here it can be deceiving to assume the third path can be interpolated from the other two: in the 48- dimensional space of strain paths (we use paths with 48 time steps each) the network is actually operating far away from training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3b the network tries to reproduce a path seen during training but we first let the material rest at zero strain for five time steps before loading starts and for another five time steps at the end of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' With purely data-driven latent dynamics, the initial rest disturbs the memory structure of the network and causes large deviations for a large portion of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For the rest at the end of the path, we see that the surrogate fails to predict the characteristic that the stress does not change upon constant deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Training data-driven models to accurately reproduce path dependency is therefore not straightforward: their latent representations of material state are not interpretable and even phenomena as trivial as resting at zero strain must be learned from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' At the core of successful applications of RNNs to this task are either extensive datasets obtained 4 εΩ t LSTM σΩ t ht−1 ct−1 ht ct (a) Model architecture 0 10 20 30 40 0 20 40 60 80 100 120 Time step [-] Stress [MPa] RVE RNN (unseen) (b) Failure to predict unseen unloading Figure 2: An LSTM recurrent neural network as surrogate for 1D path-dependent material behavior trained with only monotonic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 0 50 100 150 Strain [-] Stress [MPa] RVE RNN (trained) RNN (unseen) (a) Unloading at new strain level 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 0 50 100 150 200 Strain [-] Stress [MPa] RVE RNN (trained) RNN (unseen) (b) Stopping for 5 steps at ε = 0 and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 Figure 3: 1D LSTM surrogate trained with unloading/reloading and used to predict unseen unloading paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5 εΩ t−1 φt−1 θt−1 αΩ t−1 σΩ t−1 εΩ t−1 αΩ t−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' feature extractor (fixed) encoder (learned) decoder (fixed) εΩ t φt θt αΩ t σΩ t εΩ t εΩ t+1 φt+1 θt+1 αΩ t+1 αΩ t+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' σΩ t+1 εΩ t+1 Figure 4: Graph representation of the hybrid model architecture combining a data-driven encoder and a physics-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Filled circles represent observable variables and hollow circles represent latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' with carefully crafted sampling strategies [33, 49] or highly tailored datasets for specific macroscopic problems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Alternatively, active learning frameworks may be used to skip offline training altogether [50, 51], but at the cost of producing slower surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3 A hybrid surrogate model In this work we attempt to avoid the curse of dimensionality by relegating to a physics-based material model some of the tasks the RNN of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='3 has to explicitly learn from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In this section, we further formalize the hy- brid approach of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 by looking at the roles of each model component and their dependencies in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then particularize the model for the case of a feedforward neural network (FNN) encoder and discuss feature selection and numerical stabilization strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 Evolving material parameters Physics-based material models are traditionally formulated with a fixed set of parameters θ either directly computed from a specific set of (numerical) experiments or indirectly from stress-strain measurements in a Maximum Likelihood Estimation (MLE) approach2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Here we start from the premise that letting (part of) θ evolve in time increases flexibility and allows the model to capture more complex material behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Conversely, keeping the remainder of the model intact improves interpretability and provides physics-based bias to the data-driven model tasked to learn this evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4, the hybrid model of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 is unrolled in time for a number of consecutive time steps and represented as a graph showing the dependencies between variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Filled and hollow nodes represent observed and latent variables, respectively, and are color coded to represent the different model components in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Similar to the microscale models of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (3), we assume the constitutive behavior at the macroscale is given by a physics-based material model: MΩ � � � αΩ t = A � εΩ t , αΩ t−1, θΩ t � σΩ t = S � εΩ t , αΩ t , θΩ t � (6) but now with time-dependent parameters θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Note that the model response at time t depends on the material state at time t−1 through a set of internal variables αΩ t−1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This gives the model a recurrent nature not unlike that of the RNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2a with its state variables c and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The advantage here is that α has clear physical interpretation (plastic strains, damage variables, etc) and its evolution is handled by the fixed operator A composed of clearly interpretable algorithmic steps grounded in physics and/or classical material phenomenology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' a return mapping algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2The parameters θ can also be estimated through Bayesian inference and would therefore be described by a multivariate probability density instead of a fixed set of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Regardless, that density would still be stationary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 6 On the encoder side, we let the material properties θ evolve according to an evolution operator D whose shape is learned from data: θt = D (ϕt) (7) as a function of a set of features ϕ that are themselves obtained from the macroscopic strains through a feature extractor F: ϕt = F � εΩ t � (8) where ϕt could be simply the strains themselves or other quantities derived from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' More importantly, note that θt depends only on the current features ϕt and we therefore assume the encoder is not recurrent (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This choice effectively limits the flexibility of D and makes the hybrid surrogate fully rely on the more robust model MΩ to explain path-dependent phenomena, helping counter the curse of dimensionality associated with sampling strain paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For instance, it opens up the possibility to train the surrogate exclusively with monotonic data, as we will demonstrate in the examples of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In the following sections, we particularize the model for the case of D being a fully-connected neural network and for specific choices of F and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, the general architecture of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 and 4 is meant to be as flexible as possible: The nature and dimensionality of ϕ is not tied to that of εΩ since strains are also given directly to MΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Other machine learning models for regression can also be used as D, and it could in principle be split into different models handling the evolution of different subsets of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Any number of model parameters may also be left out of θ and either fixed as constants or optimized to constant values during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' No assumption is made on the form of MΩ or the nature or dimensionality of αΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Instead of a single model, it could also for instance be a mixture of physics-based models combined with analytical homogenization tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 Feature extractors A pragmatic choice for F is to simply assume ϕ is the macroscopic strain vector εΩ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' It is also a familiar one, as we can then relate the resulting model to conventional surrogates mapping strains to stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' However, since macroscopic strains are also directly passed on to the decoder, the architecture gives us the freedom to experiment with different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5 shows the two model architectures we explore in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For the two variants in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5a we either use εΩ itself or a set of small-strain invariants of the macroscopic strain tensor of increasing dimensionality: IΩ ε = �Iε 1 � or IΩ ε = �Iε 1 Iε 2 � (9) where the variants are given by the well-known expressions: Iε 1 = tr (ε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Iε 2 = 1 2 � tr (ε)2 − tr � ε2�� (10) Additionally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' since the current study focus on elastoplasticity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' it is also interesting to explore feature spaces including invariants from the deviatoric strain tensor: IΩ ε = �Jε 2 � or IΩ ε = �Iε 1 Jε 2 � (11) where: Jε 2 = 1 3 (Iε 1)2 − Iε 2 (12) By using features based on invariants and since the decoder material model is itself already frame invariant for small strains,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' it follows that the resulting surrogate will naturally inherit this beneficial characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This stands in contrast with traditional black-box surrogates mapping strains to stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Furthermore, opting for invariant-based features can be seen as a physics-based dimensionality reduction operation that can potentially reduce the amount of data needed to train the hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 7 · · εΩ or IΩ ε · · · · · · · · · · M σΩ, DΩ α Feedforward NN θ εΩ (a) Macroscopic strains or invariants directly used as fea- tures · · α or I¯σ · · · · · · · · · · M σΩ, DΩ α Feedforward NN θ M α εΩ Precalibrated model εΩ (b) Features extracted from an informative microscopic constitutive model Figure 5: The two types of FNN-based model architectures explored in this work, with different feature extraction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We also investigate the possibility of extracting features from the outputs of a precalibrated physics-based material model M subjected to the same strain path seen at the macroscale (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Note that this specific architecture introduces an additional recurrent component to the model through the set α of internal variables of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' From a machine learning perspective, the role of M would be analogous to that of a temporal convolution operator or an RNN cell appended to the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The key difference, however, is that M is fixed a priori and therefore should not require extra sampling effort with respect to the more straightforward extractor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Naturally, different choices for M yield models with distinct learning capabilities, and we therefore assume M encapsulates relevant information about not only the current values of εΩ but also of its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In the present scenario where the data is coming from micromodel computations, we opt for the intuitive choice of having M be one of the known constitutive models used to describe the microscopic material phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We can therefore conceptually see M as an imaginary representative material point at the microscale that is always subjected to the average micromodel strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then use either a subset of its internal variables α or a set of invariants I¯σ of its stress outputs as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='3 Neural network encoder For simplicity, we opt for modeling the evolution of θ using classical feedforward neural networks with fully- connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' As both architectures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5 ultimately compute macroscopic stresses given macroscopic strains, we can use supervised learning to train the model with a straightforward Maximum Likelihood approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Gather- ing the complete set of network weights in a vector w and seeing the complete surrogate as a monolithic model that computes an approximation �σ for stresses, we adopt the following observation model for the snapshot stresses σ: σ = �σ (ε, w) + ξ, ξ ∼ N � ξ|0, β−1I � (13) where the superscript Ω is dropped for convenience, I is an identity matrix, and ξ is an additive Gaussian noise3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Under the assumption of a squared loss, maximizing the likelihood of a training dataset with N observations amounts to minimizing the loss function [45]: L = 1 2 N � n=1 ∥σn − �σ (εn, w)∥2 (14) with the variance of the noise that explains data misfit being simply β = N/2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The resulting loss function is the same one used for conventional data-driven surrogates and is therefore straightforward to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, it is worth noting that since we cannot directly observe θ, computing the gradients of L with respect to w involves backpropagating derivatives through the decoder M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Furthermore, since w affects the evolution of the internal variables α, backpropagation in time becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (14) and walking back through 3Even though our observations come from a computer model and can be considered noiseless, the surrogate �σ is in general not arbitrarily flexible and the random variable ξ is therefore still necessary to explain why the model does not exactly fit every single observation in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 8 the graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4, the gradient of the loss at time step t of a given strain path is given by: ∂Lt ∂w = ∂L ∂�σt � � � ∂�σt ∂θt ∂θt ∂w + ∂�σt ∂αt ∂αt ∂θt ∂θt ∂w + ∂�σt ∂αt 1 � ¯t=t−1 � � � � ¯t+1 � ˜t=t ∂α˜t ∂α˜t−1 � � ∂α¯t ∂θ¯t ∂θ¯t ∂w � � � � � (15) where the remaining gradient chain ∂θ/∂w is computed with conventional backpropagation through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' If M is implemented in a code base that allows for automatic differentiation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' in PyTorch), these time dependencies are naturally taken into account as long as a persistent gradient tape is used within each strain path4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In this work we instead implement network training directly into an existing FE code, and therefore opt for the pragmatic approach of computing all partial derivatives of quantities derived from M using finite differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, in order to enforce upper and lower bounds for θ and avoid unphysical parameter values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' negative elasticity moduli), we apply sigmoid activation to the final layer of the network and scale the parameters back from a [0, 1] range using predefined bounds: θi = θlow i + θσ i � θupp i − θlow i � (16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='4 Material decoders As previously mentioned, any constitutive model can in principle be used as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For the present study we focus on reproducing elastoplasticity and therefore narrow our choices down to the following set of potential decoders with increasing levels of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The simplest one is a linear-elastic isotropic material with no internal variables: σij = Dijklεkl with Dijkl = G (δijδkl + δilδjk) + � K − 2 3G � δijδkl (17) where index notation is used for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For this model, θ comprises only the bulk and shear moduli K and G, or equivalently the Young’s modulus E the Poisson’s ratio ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The second decoder option is a simple plasticity model with J2 (von Mises) flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The stress update in this case becomes: σij = Dijkl � εij − εp ij � (18) where strain is additively decomposed into elastic and plastic (εp) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The yield criterium and plastic flow rule are given by: φ = � 3Jσ 2 − σy ≤ 0 and ∆εp ij = ∆γ � 3 2 Sij ∥Sij∥F (19) where S is the deviatoric part of the stresses, γ is a plastic multiplier, σy is a yield stress parameter and we write the Frobenius norm as ∥·∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In order to keep the model as simple as possible, we assume σy is a material constant and therefore end up with a perfectly-plastic model with associative flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The internal variables of this model are components of the plastic strain vector εp and the only new material parameter is the yield stress σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, we also consider the more complex pressure-dependent, non-associative plasticity model proposed by Melro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Stress update is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (18), but yield surface and plastic flow are given by: φ = 6Jσ 2 + 2Iσ 1 (σc − σt) − 2σcσt ≤ 0 and ∆εp ij = ∆γ � 3Sij + 1 − 2νp 1 + νp Iσ 1 δij � (20) where δij is the Kronecker delta, σt and σc are yield stresses in tension and compression, respectively, and νp is a new parameter controlling plastic contraction and allowing for compressible plastic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Hardening can be described by making the yield stresses general functions of εp, but when used as a decoder we assume σt and σc do not depend on εp and instead let the decoder D describe their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The model by Melro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' [52] is also the one used to describe the microscopic material phase responsible for the nonlinear behavior observed when homogenizing micromodel response, and can therefore be seen as the natural choice for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, the other two decoders can provide interesting insights on the effect of introducing different levels of bias to the hybrid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4This is already the case for RNNs, so switching from RNNs to the present model should require little to no changes to the way training is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='5 Online predictions and inherited stability The architecture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 is developed to be minimally intrusive and allow for existing material models to be used as decoders with minimum effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We therefore implement the online routine of the model as a wrapper around an existing implementation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The basic structure of the wrapper can be seen in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The hybrid nature of the model allows for a robust approach that ensures the numerical stability of the original model M is inherited by the surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This is achieved by only updating θ at the end of each time step, after the global implicit Newton- Raphson scheme converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Material properties are therefore fixed while the global solver is iterating, and that means the tangent stiffness D comes directly from M and inherits its stability features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Algorithm 1: Material wrapper implementing the online component of the hybrid surrogate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Input: strain εΩ new at macroscopic Gauss point Output: stress σΩ and stiffness DΩ at macroscopic Gauss point 1 use nested model with converged parameters and internal state: � σΩ, DΩ, αnew � ← M � εΩ new, αold, θ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 2 if global solver has converged : 3 store latest converged strain: εold ← εnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4 commit material history: αold ← αnew;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 5 compute new features: ϕnew ← F (εnew);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 6 update model parameters for the upcoming time step: θ ← D (ϕnew);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 7 if first global iteration of time step and Gauss point is unstable : 8 stabilize encoder: D ← stabilizeNetwork � εΩ new � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 9 recompute features: ϕold ← F � εΩ old � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 10 recompute model parameters for the current time step: θ ← D (ϕold);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 11 return σΩ, DΩ As an example, the J2 plasticity model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (19) is unconditionally stable as long as its hardening modulus h ≥ 0 for any � εΩ t , αΩ t � , which is the case for the perfectly-plastic version we consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' It then follows that any hybrid surrogate with J2 decoder is also unconditionally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Note that this is only possible because strains are directly passed on to the decoder and would therefore not be an option for conventional surrogates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' the RNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For those surrogates, the tangent stiffness would come directly from the jacobian of a highly-flexible data-driven model, often at the cost of numerical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='6 Numerical stabilization Nevertheless, the decoder M may be inherently unstable even with fixed material constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This is for instance the case for the model by Melro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' [52]: the non-associative flow rule of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (20) can cause the tangent stiffness DΩ to lose positive definiteness under certain strain conditions and for certain combinations of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To accommodate such a scenario and open up the possibility for online model adaptivity in other contexts, we propose a scheme for updating the encoder D on the fly in order to enforce extra constraints locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Back to Algorithm 1, at the beginning of a new time step we keep θ fixed to the one obtained with converged strains from the previous step and let the solver make a first strain prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' After this first iteration, a stability criterion is checked and used to define a new loss function that can be used to update network weights in case instability is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Here we employ the determinant of the acoustic tensor Q: Q = nT d DΩnd (21) where nd is the vector normal to the strain localization direction creating the instability, which we find through an angle sweep procedure as in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then use det (Q) as a metric of stability and trigger a retraining procedure in case a negative value is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then introduce a new loss function: LQ = −⟨det (Q)⟩− det (Q0) (22) 10 Linear-elastic fibers Elastoplastic matrix Figure 6: The micromodel used in the examples of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' where ⟨·⟩− extracts the negative part of its operand and Q0 is a reference value for the acoustic tensor computed at the start of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We minimize this new loss at every unstable point for a small number of epochs with low learning rate, and to discourage significant drifts from the original model we finish the stabilization procedure by updating the network using the original loss of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (14) for a single minibatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, θ is updated using the retrained model and is kept fixed for the remaining iterations5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Note that the local constraint of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (22) is therefore only enforced in a soft way and remaining instabilities might still cause the global solver to diverge, in which case we cancel the current increment, go back to the beginning of the time step and allow for the procedure to be triggered again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4 Numerical examples The proposed model was implemented in an in-house Finite Element code developed using the open-source C++ numerical analysis library Jem/Jive [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' In order to allow for seamless online retraining, network training was also implemented within the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We start this section by describing the datasets and model selection strategies used to build the surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then investigate the performance of the approach under several choices of encoders and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Finally, we use the model within an FE2 simulation and demonstrate the online stabilization procedure of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' All simulations are performed on cluster nodes equipped with Xeon E5-2630V4 processors and 128 GB RAM running CentOS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 Data sampling and model selection Models are trained to reproduce the behavior of the fiber-reinforced composite micromodel shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Fibers are modeled as linear-elastic and the matrix is described by the pressure-dependent non-associative elastoplastic model by Melro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' [52] (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Microscale material properties are adopted from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The microscopic geometry shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 6 results from an RVE study performed in [10] and is therefore considered representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Following the discussion in Section 3, our aim is to investigate up to which extent it is possible to circumvent the curse of dimen- sionality associated with path dependency by training surrogates exclusively on monotonic strain paths and having time-dependent behavior arise naturally from a physics-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We therefore limit ourselves to monotonic paths for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For consistency, we also employ exclusively motononic data to perform model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For efficiency, we limit the present investigation to 2D simulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' three strain components) in the plane perpendicular to the fibers, but nevertheless expect the discussion and conclusions to generalize to 3D simulations as long as appropriate orthotropic decoders are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Datasets with 2000 monotonic strain paths are generated under both plane strain and plane stress assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 7 shows the complete plane strain dataset, with a similar one also being generated for plane stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Each path is generated with an FE2 simulation of a single macroscopic element under displacement control along a fixed direction in strain space sampled from a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To circumvent convergence issues, we employ an adaptive time stepping technique that progressively reduces time step size when the simulation does not converge and gradually increases it back for subsequent increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The simulations are stopped once a strain norm of 10 % is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' As the adaptive scheme leads to paths with different numbers of 5Changing D and therefore θ after every iteration would not work in favor of improving stability, but rather have the opposite effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 εxx [-] εyy [-] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 εxx [-] γxy [-] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 εyy [-] γxy [-] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −600 −400 −200 0 200 εxx [-] σxx [MPa] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −600 −400 −200 0 200 εyy [-] σyy [MPa] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −100 −50 0 50 100 γxy [-] τxy [MPa] Figure 7: The complete plane strain dataset used to train the surrogates, comprising 2000 monotonic strain-stress paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' A similar dataset is generated under plane stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' time increments, we balance the dataset by ensuring every path is composed of 30 steps with strain norms as equally spaced as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To keep model selection straightforward and avoid the need for cumbersome k-fold cross validation or bootstrap- ping, we train a preliminary model with enough flexibility and an extensive training dataset and gradually increase the size of the validation set until the validation error converges to a good estimate of the expected prediction error [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This results in validation sets with 500 paths selected at random from the original datasets, leaving 1500 paths to be used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We then perform model selection by gradually increasing the complexity of our FNN encoders until the validation error stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' From experimenting with different architectures, we find that encoders with 5 hidden layers of 50 units each with Scaled Exponential Linear Unit (SELU) [56] activation provide enough flexibility for all the examples treated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To ensure enough regularization when computing learning curves with small datasets, we employ Bernoulli dropout layers with a rate of 1 % after every hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Networks are trained for 20 000 epochs and the model with lowest historical validation error is kept after training, further reducing the risk of overfitting on small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To assess the capabilities of the trained surrogates, we compute an additional test dataset comprising 50 monotonic, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0 50 100 150 Strain [-] Stress [MPa] xx yy xy (a) Monotonic −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 −200 −100 0 Strain [-] Stress [MPa] xx yy xy (b) Unloading-reloading −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −200 −100 0 100 Strain [-] Stress [MPa] xx yy xy (c) Slow cycling Figure 8: Examples from a test dataset with 50 paths of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' They are not used to train any of the networks or perform model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='100 120 140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Number of training paths [] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Mean validation error [MPa] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='[εxx εyy γxy] → FNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1/Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='→ FNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='[εxx εyy γxy] → Elastic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1/Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='→ Elastic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='(a) Expectations over 50 datasets of each size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Number of training paths [] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Mean validation error [MPa] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='[εxx εyy γxy] → FNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1/Iε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='→ Elastic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='(b) Detailed comparison including standard devia- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='tions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='Figure 9: Learning curves of models with elastic decoders and conventional FNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Mean error over the 500 validation monotonic paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 50 unloading-reloading and 50 slow cycling paths, examples of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To keep the comparisons fair, none of these paths are used to perform model selection and are therefore only considered after the surrogates are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We will use example curves like those from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 8 for visual inspection of the model performance, but also the complete sets of 50 curves each for more rigorous statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='2 Elastic decoder It is interesting to first consider the simple linear-elastic decoder of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (17), as it has no internal variables and therefore leads to a surrogate model comparable in nature to a conventional FNN trained on stress-strain pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' As we will demonstrate, however, the limited physical bias provided by such simple model already proves advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Here we let both elastic properties be controlled by the learned encoder: θ = �E ν� (23) where the bounds 101 < E < 105 and 0 < ν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='5 are enforced as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We first perform a feature selection study and investigate how efficiently the model learns as the size of the dataset is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' From the original plane strain training dataset of 1500 monotonic strain paths, we draw datasets with sizes ranging between 1 and 150 paths without replacement and use them to train networks with different encoder features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' To get a reliable estimate of the expected prediction error, we repeat this process 50 times for each dataset size and encoder type, and for comparison we also do the same for conventional FNNs trained directly on stress targets (keeping the same architecture but going directly from the final hidden layer to stresses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' This amounts to a total of 3400 trained networks from which we can compute an estimate of the prediction error by averaging ∥σ − �σ∥ over the 500 paths left for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 9a plots averages of the validation error over the 50 training datasets used for each size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Although the hybrid architecture does not show an advantage over the FNN when the encoder is trained on strain features, there is a clear gain in learning speed when using only the two first strain invariants as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Apart from accelerating learning and resulting in lossless dimensionality reduction, using invariants also results in a surrogate which is frame invariant under small strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' For comparison, we also train a conventional FNN on the same set of features, but those are unsurprisingly not enough to describe general strain states and much of the material response is interpreted by the FNN as observation noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' We zoom into the first part of the learning curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 9b, this time also showing single standard deviation uncertainty bands coming from the variance among the 50 training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' The hybrid network outperforms conventional FNNs in the low data regime and tends to be less sensitive to changes in dataset starting from about 20 training paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, the extra flexibility of conventional FNNs allow them to achieve lower validation errors if significantly more training paths are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Training the invariant-based hybrid network with the complete dataset of 1500 curves leads to surrogates with validation errors of about 4 MPa, accurately representing the monotonic behavior of the original micromodel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 10 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 0 50 100 150 Strain [-] Stress [MPa] xx yy xy Micromodel I1/I2 →Elastic (a) Monotonic −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 −300 −200 −100 0 Strain [-] Stress [MPa] xx yy xy Micromodel I1/I2 →Elastic (b) Unloading-reloading −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 −200 −100 0 100 Strain [-] Stress [MPa] xx yy xy Micromodel I1/I2 →Elastic (c) Slow cycling Figure 10: Performance of the elastic decoder model for different test scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='02 0 −400 −200 0 Strain [-] Stress [MPa] xx yy xy Micromodel I1/I2 → Elastic −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content='1 −200 −100 0 100 Strain [-] Stress [MPa] xx yy xy Micromodel I1/I2 → Elastic Figure 11: Predicting unloading with a linear-elastic decoder through history-aware feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' shows representative predictions of this model for paths from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' As expected, this surrogate with no internal variables is not capable of predicting non-monotonic strain paths, and effectively behaves like a hyperelastic material model just as the conventional FNN would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' Nevertheless, the flexible and interpretable encoder-decoder architecture of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 1 allows for new creative ap- proaches in feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' As a demonstration, we keep the trained network of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFRT4oBgHgl3EQfYDcO/content/2301.13547v1.pdf'} +page_content=' 10 intact and only modify its feature extractor to introduce a simple path-dependent mechanism: ϕT ≡ � I ε 1 I ε 2 � T = argmax 0 0.85 when +the initial test environment was used with the ratio method +(AVG). In those cases, the stochasticity is masked by the high +impact of the mutations and all trained agents behave similarly. +Thus, generating multiple test environments allowed us to see +7 + +that those mutations might not be much of interest. Indeed, +they are rather trivial to detect and so can probably be ignored. +Secondly, we see the test environments generated on Cart- +Pole are relatively more likely to catch mutations with most of +the mutations being killed by at least half the test environments +compared to the ones generated on LunarLander. While the +number of test environments generated might play a role, it is +likely because LunarLander is a more complex environment +than CartPole, so mutations might become harder to detect as +the environment complexity increases. For instance, CartPole +seems to be less sensitive to the MTS or PAC mutations, +i.e., removing the terminal state of an episode or changing +the activation function of the policy network does not seem +to affect much the agents trained on CartPole contrary to +LunarLander. +RQ2-1 Contrary to using only the initial test environ- +ment, generating additional test environments allows +us to roughly evaluate which mutations might be trivial +and which are more interesting based on the number +of environments killing them. We also found that it +appears that, the more complex the environments the +more likely the mutation is not to be found. +In a second step, it is possible to go beyond the raw number +of test environments killing a certain mutation and, instead, to +inspect which test environment kills the mutation. Indeed, as +we generated test environments through the modification of +the initial environment parameters, this can allow us to shed +some light on which parameter a mutation might be more +easily sensitive to. By doing so, we can determine which test +environments can help identify certain potential faults, thus +leading to some sort of fault-detection method. Because of +space constraints, we will report one example using R and we +refer the reader to our implementation repository [25] for all +the raw results. In the case of LunarLander/PPO/PAC ReLU, +in Figure 1, test environments with lower (absolute) gravity +compared to the initial environment kill the mutation while the +ones with higher (absolute) gravity do not. Thus, the mutation +seems to be affected somehow by the gravity parameter. When +the gravity is close to the one in the initial environment, higher +engine power will be crucial to kill the mutation. +Test environments are depicted as orange/blue dots, de- +pending on whether or not they kill the mutation, and we +can see they can be easily separated linearly following the +previous description. In particular, we can see that both test +environments P1 and P2 seem to be close to some frontiers +for this mutation since, while being close in parameters space, +they lead to an opposite decision on the mutation. Note that +the red frontier is arbitrary as we do not have access to the +exact frontier, as it would be potentially too computationally +expensive to find it as we explained in Section III-C. Thus, +it could be possible to find environments between P3 and +the initial environment that could still kill the mutation. +We just know P3 is an environment for which the healthy +agents’ reward distribution is at the limit of being different +14 +12 +10 +8 +6 +4 +2 +0 +gravity +0 +1 +2 +3 +4 +5 +6 +side engine power +P1 +P2 +P3 +Fig. 1. Generated test environments for LunarLander/PPO/PAC ReLU and +a potential way to separate them based on whether or not they killed the +mutation. Orange points kill the mutation while the Blue ones don’t. The +origin is centered on the initial environments. +from the distribution observed in the initial environment, but +it gives no information on the distribution of the mutated +agents. While it might not be possible to draw meaningful +information for all mutations, especially on a reduced set of +test environments, it shows nonetheless that generating test +environments in that way also allows us to explore a potential +link between parameters of the environments and their impact +on the mutation. +RQ2-2 By mapping, for a given mutation, which gen- +erated test environments kill it or not, we can analyze +which of the parameters of the test environments affect +the decision to kill the mutation. This outlines some +form of fault-detection method. +3) RQ3. Properties of generated HOM: Following RQ2, we +gathered for each environment/algorithms/killing definition the +non-trivial FOM (i.e., nor killed by all test environments nor +killed by none), see Table IV. To generate HOM, we need +at least two FOM as we stick to HOM of order 2. We then +trained new mutated agents based on the gathered HOM in the +same way as FOM. Finally, mutated agents were evaluated +using the previously generated test environments depending +on the mutation killing definition (R or DtR), and the type of +HOM was analyzed based on the classification we introduced +in Section II-B. Results are presented in Table V. +As we can see, following the procedure mentioned for RQ2, +we do not end up with a high number of non-trivial FOM +and so the pool of generated HOM is relatively small with +even some configurations not yielding any. Nonetheless, we +managed to obtain 12 HOM when the R mutation killing +definition was used and 23 with DtR but only in LunarLander. +Among those, Non-Subsuming (NS) constitutes 50% of HOM +using R method but only 30% using DtR, the difference +between the two methods potentially being explained by the +increased sensitivity of DtR. The remaining Subsuming HOM +are mostly of the Weakly Subsuming Coupled (WSC) types +8 + +TABLE IV +NUMBER OF TEST ENVIRONMENTS KILLING EACH FOM. GREEN CELLS ARE FOM THAT WILL BE USED TO GENERATE HOM. +Environment +DRL +Killing +Mutations +Algorithm +Criteria +ILF +M-1.0 +R-1.0 +Ra-1.0 +RN-1.0 +NDF +NR +MSU +MTS +PAC-ReLU +PAC-Sigmoid +POC-SGD +CartPole +PPO +R +4/4 +4/4 +0/4 +4/4 +3/4 +4/4 +4/4 +4/4 +3/4 +3/4 +4/4 +4/4 +DtR +4/4 +4/4 +3/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +A2C +R +4/4 +4/4 +0/4 +4/4 +4/4 +3/4 +4/4 +4/4 +3/4 +4/4 +2/4 +4/4 +DtR +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +4/4 +DQN +R +4/4 +4/4 +4/4 +4/4 +0/4 +4/4 +- +4/4 +4/4 +- +4/4 +4/4 +DtR +4/4 +4/4 +4/4 +4/4 +3/4 +4/4 +- +4/4 +4/4 +- +4/4 +4/4 +LunarLander +PPO +R +8/8 +8/8 +0/8 +8/8 +1/8 +8/8 +8/8 +8/8 +0/8 +4/8 +8/8 +8/8 +DtR +6/8 +6/8 +2/8 +6/8 +3/8 +6/8 +6/6 +6/6 +3/6 +3/6 +6/6 +6/6 +A2C +R +9/9 +9/9 +0/9 +9/9 +0/9 +6/9 +9/9 +9/9 +2/9 +3/9 +5/9 +9/9 +DtR +9/9 +9/9 +2/9 +9/9 +4/9 +7/9 +9/9 +9/9 +4/9 +7/9 +6/9 +9/9 +DQN +R +9/9 +9/9 +8/9 +9/9 +0/9 +9/9 +- +9/9 +0/9 +- +0/9 +9/9 +DtR +9/9 +9/9 +9/9 +9/9 +4/9 +9/9 +- +9/9 +6/9 +- +4/9 +9/9 +TABLE V +TYPES OF HOM GENERATED USING RQ2 NON-TRIVIAL FOM. IF NO +HOM WAS GENERATED, THEN A “-” IS USED. NS: NON SUBSUMING, +WSC: WEAKLY SUBSUMING COUPLED, WSD: WEAKLY SUBSUMING +DECOUPLED, SSC: STRONGLY SUBSUMING COUPLED. +Environment +DRL +Killing +HOM types +Algorithm +Criteria +HOM +NS +WSC +WSD +SSC +CartPole +PPO +R +3 +1 +0 +0 +2 +DtR +- +- +- +- +- +A2C +R +3 +2 +1 +0 +0 +DtR +- +- +- +- +- +DQN +R +- +- +- +- +- +DtR +- +- +- +- +- +LunarLander +PPO +R +1 +1 +0 +0 +0 +DtR +6 +1 +5 +0 +0 +A2C +R +5 +2 +3 +0 +0 +DtR +14 +6 +8 +0 +0 +DQN +R +- +- +- +- +- +DtR +3 +0 +3 +0 +0 +and generally compose more than half the generated HOM +for each configuration, which is similar to results obtained +by Jia et al [4] with HOM in some traditional software +programs. Interestingly, we found 2 HOM that fit the type +Strongly Subsuming Coupled (SSC), that is, HOM for which +test environments killing said HOM also kill its constituent +FOM. Aside from those two, no other SSC and no WSD were +found. Even in Jia et al. case, SSC represents a rare occurrence +(< 1% of Subsuming HOM) and so the fact we found none is +not too surprising judging by our limited number of HOM/test +environments. Nonetheless, we showed that Subsuming HOM +(even if Weakly ones), which are more complex and subtle +mutants, could be generated and make up for more than half +the HOM generated. +RQ3 HOM generated from non-trivial FOM, while +few, are in the majority Subsuming HOM, which de +facto makes them more interesting cases to use as we +pointed out in Section II-B. If more subtle Subsuming +HOM such as SSC are not as widely represented, +the fact that some can be generated shows that such +property is reachable in RL too. +V. THREATS TO VALIDITY +Construct validity: The design choices of Mutation Testing +could affect our results. Since our main goal was not to +design a new mutation killing definition for Deep Learning +but rather to adapt and assess how existing approaches fared, +all the mutation killing definitions used are based on existing +approaches. Moreover, while the hyper-parameters used could +play a role in declaring a mutation killed (p-value, threshold +θ...), we preferred to stick to the hyper-parameters given in +each original implementation of the killing definitions and +leave to future work the study of the influence of hyper- +parameters. The way we searched the parameters space to +generate test environments could also have impacted our +results. Nonetheless, the goal here was simply to get a simple +and effective approach that could be automated and serve as a +basic heuristic for future work. The rest of our methodology, +such as the definition of the properties of HOM, is grounded +in the scientific literature and is taken as they were originally +defined. +Internal validity: Mutation operators chosen could affect +our results. While we can not be exhaustive, we made sure +to create operators based on existing faults or taxonomy +and to provide sufficient diversity in terms of the effect +on the code (agent-based, environment-based, policy-based). +For the environment-based operators, while the probability of +application of 1.0 might not be real to simulate sensor defaults, +it was necessary to avoid potential bias over which step would +be affected by the mutation when the probability is set to +< 1.0. The environment parameters modified to generate the +test environments could also impact the results. However, the +goal was to show we could generate automatically simple and +effective test environments that could be leveraged to analyze +Mutation Testing in RL. As such, the choice of the parameters +on its own is not as relevant. +External validity: The choice of environment types/algo- +rithms could impact the generalization of our experiments. +We made sure to select two environments as well as three +algorithms that are generally used as a benchmark in RL. +Reliability validity: Implementing the mutations could lead +to some unintended faults. We used the Stable-Baselines +9 + +framework as a basis to implement our mutations to lower the +risk. We also make our implementation and artifacts available +[25]. +VI. RELATED WORK +Gur et al. [32] proposed an agent which produces envi- +ronments depending on the learning agent’s level of skill. +Their approach allows for more methodological training of +agents and increases their robustness and ability to generalize. +While interesting, their approach jointly trains the environment +generator with the agents which is not doable within the +Mutation Testing scope. To combat the problem of overfitting, +Cobbe et al. [33] introduced the Procgen benchmark. This +benchmark comprises 16 procedurally generated environments +that allow practitioners to test the generalization of the agents. +Their idea allows for testing agents yet the main focus of +Mutation Testing is to evaluate test set quality. Nonetheless, it +would be interesting in future work to evaluate RLMutation on +their environments, as they are designed to push the general- +ization property of the algorithm and so might induce different +behavior over the mutations. In another direction, a meta- +heuristic-based algorithm could be another venue to improve +the generation of relevant test environments [34]. Finally, +Biagiola et al. [28] focused on generating test environments +through a combination of binary/exponential search, to map +the adaptation/anti-regression performance of an agent through +the lens of continual learning. In our work, we similarly +generate test environments by acting on their parameters, yet +we do not test for the adaptation nor do we retrain the agent +with continual learning. If anything, we similarly test for anti- +regression of our healthy agents on both initial/generated test +environments. Nonetheless, their approach could be used to +improve our test generation part. +As mentioned in Introduction and Section II, some frame- +works applied Mutation Testing to Supervised Learning [6]– +[8]. While we reuse some of their mutation operators/design +choices, we specifically tailored our approach for RL. Lu et +al. [12] also tackled Mutation Testing in RL and studied how +well-crafted environments could be used to reveal a particular +mutation which is more akin to the fault detection method. +On the contrary, we focus on generating, in an automatic way, +simple yet effective test environments that can be used to +reveal more complex faults, i.e., HOM. We also show their +Mutation Testing design choice could be a limiting factor +because of the stochasticity of RL. Shen et al. [35] used +decision boundaries of Supervised Learning models in order +to find a subset of the test set that is more likely to trigger +the mutation. In a sense, our method is similar to theirs but +from an RL perspective: we aim to find test environments that +are on some boundaries. However, contrary to them, we do +not have an actual test set to work on and have to generate +the test environments. Moreover, we do not assume that the +test environments on this boundary are more likely to reveal +the mutation. Heuristically, we just assume it’s a potential +point of interest, better than any random point if we are +constrained by the number of test environments to be generated +and tested. Finally, Tambon et al [36] study more in-depth the +effect that the choice of Deep Learning model instances has +over the result of Mutation Testing in Supervised Learning +using DeepCrime’s approach. They showed that this choice +can affect the outcome of the Mutation Testing and that using +the Bayesian approach can mitigate this issue. In our approach, +we do not account for this effect and just focused on existing +killing definitions and generation of test environments, since +the main goal was to analyze FOM and subsequent HOM in +RL. Nonetheless, it would be interesting to verify that a similar +behavior also exists in RL. +VII. CONCLUSION +In this paper, we have presented RLMutation, a framework +for Mutation Testing in RL. We have defined three distinct +categories of FOM based on real faults that can happen during +the design and training of deep RL agents. We compared dif- +ferent mutation killing definition choices based on the previous +framework applying Mutation Testing to Deep Learning. We +also evaluated HOM which are a combination of FOM and can +prove to be more complex to kill which makes for interesting +corner faults to investigate. The FOM used in the HOM +generation were selected based on their relevance to generated +parameterized test environments. We have tested our approach +on a set of state-of-the-art RL algorithms (DQN, A2C, and +PPO) over two benchmark environments (LunarLander and +CartPole). The results of our study show that the mutation +killing definition choice is important when it comes to killing +mutations (ratio vs. distribution of rewards). We demonstrate +that by testing the agents in modified environments, we can +detect non-trivial FOM which are not detected by testing the +agents in the environments they were trained in. Finally, we +show that HOM generated from non-trivial FOM possess in the +majority the interesting property of being subsuming, meaning +that they are harder to kill than their constituent FOM and so +more interesting from a testing perspective. +Overall this study showed that, while Mutation Testing +applied to RL raises numerous challenges to consider from +the mutation killing definition design to the generation of +relevant test environments and HOM, it is an interesting venue +to improve testing of RL-based software systems. As such, we +believe that further research on this topic should focus on those +issues to enhance Mutation Testing applied to RL. +ACKNOWLEDGMENT +This work was supported by: Fonds de Recherche du +Qu´ebec (FRQ), the Canadian Institute for Advanced Research +(CIFAR) as well as the DEEL project CRDPJ 537462-18 +funded by the National Science and Engineering Research +Council of Canada (NSERC) and the Consortium for Research +and Innovation in Aerospace in Qu´ebec (CRIAQ), together +with its industrial partners Thales Canada inc, Bell Textron +Canada Limited, CAE inc and Bombardier inc. +10 + +REFERENCES +[1] Y. Jia and M. Harman, “An analysis and survey of the development of +mutation testing,” IEEE Transactions on Software Engineering, vol. 37, +no. 5, pp. 649–678, 2011. +[2] M. Papadakis, M. Kintis, J. Zhang, Y. Jia, Y. Le Traon, and M. Harman, +“Mutation testing advances: an analysis and survey,” in Advances in +Computers, vol. 112, pp. 275–378, Elsevier, 2019. +[3] A. J. 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Antoniol, “A probabilistic framework for +mutation testing in deep neural networks,” Information and Software +Technology, vol. 155, p. 107129, 2023. +11 + diff --git a/mNE5T4oBgHgl3EQfiw-y/content/tmp_files/load_file.txt b/mNE5T4oBgHgl3EQfiw-y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee4e85d4e1178ce99959a3cf3dc5b760377a676d --- /dev/null +++ b/mNE5T4oBgHgl3EQfiw-y/content/tmp_files/load_file.txt @@ -0,0 +1,1040 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf,len=1039 +page_content='Mutation Testing of Deep Reinforcement Learning Based on Real Faults Florian Tambon§,Vahid Majdinasab§, Amin Nikanjam, Foutse Khomh, Giuliano Antoniol Polytechnique Montr´eal, Canada {florian-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='tambon, vahid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='majdinasab, amin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='nikanjam, foutse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='khomh, giuliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='antoniol}@polymtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='ca Abstract—Testing Deep Learning (DL) systems is a complex task as they do not behave like traditional systems would, notably because of their stochastic nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, being able to adapt existing testing techniques such as Mutation Testing (MT) to DL settings would greatly improve their potential verifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While some efforts have been made to extend MT to the Supervised Learning paradigm, little work has gone into extending it to Reinforcement Learning (RL) which is also an important component of the DL ecosystem but behaves very differently from SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This paper builds on the existing approach of MT in order to propose a framework, RLMutation, for MT applied to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Notably, we use existing taxonomies of faults to build a set of mutation operators relevant to RL and use a simple heuristic to generate test cases for RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This allows us to compare different mutation killing definitions based on existing approaches, as well as to analyze the behavior of the obtained mutation operators and their potential combinations called Higher Order Mutation(s) (HOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We show that the design choice of the mutation killing definition can affect whether or not a mutation is killed as well as the generated test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Moreover, we found that even with a relatively small number of test cases and operators we manage to generate HOM with interesting properties which can enhance testing capability in RL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Index Terms—Reinforcement Learning, Deep Learning, Muta- tion Testing, Real Faults I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' INTRODUCTION Mutation Testing is a white box testing method that aims to inject artificial changes based on real faults in order to evaluate a test suite’s capability to reveal faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mutation Testing has been extensively studied and used in traditional software engineering [1], [2] to assess the quality of test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' A fundamental hypothesis of Mutation Testing is the Coupling Effect hypothesis which posits that “complex mutants are coupled to simple mutants in such a way that a set of test cases that detects all simple mutants in a program will detect a large percentage of the complex mutants” [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such uncovering such unkilled complex mutants is of interest, with one way of generating such complex mutants being to combine simple mutants, called First Order Mutation(s) (FOM), together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This is the concept of Higher Order Mutation(s) (HOM) introduced by Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Such established testing techniques and concepts could be useful for Deep Learning systems, in an effort to increase their reliability, since such systems are notoriously hard to test because of their peculiar nature [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Because of the paradigm §Equal contribution differences between Deep Learning-based systems and tradi- tional software systems, it is only recently that researchers have started proposing Mutation Testing frameworks tailored for Deep Learning-based systems, in particular Supervised Learning [6]–[8], to assess the quality of test dataset at revealing faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Yet, Supervised Learning is not the only sub-paradigm in Machine Learning, and (deep) Reinforcement Learning (RL), one of the other main sub-paradigms with a wide range of applications [9], [10] is increasingly being adopted in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RL differs deeply from Supervised Learning: while in Super- vised Learning a model learns from a training dataset in order to generalize to any new data from the input distribution, RL is based on the idea of training an agent using its interaction with an environment through a feedback system [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, a robot (agent) evolving in a room (environment) with a goal to go from A to B with some traps on the way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, previ- ously introduced frameworks might present several limitations if applied to RL, for instance, the mutation operators defined in Supervised Learning to obtain mutant models might not apply to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In parallel, to the best of our knowledge, [12] is the only research work that tackled Mutation Testing in RL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' proposing a fault detection approach that is based on the manual crafting of relevant environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, they introduced the idea that traditional test cases used in Mutation Testing applied to traditional software systems could be translated to the notion of test environments in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, their study is limited to only one type of RL algorithm and does not explore real fault-based operators nor the potential usefulness of combining existing operators to form HOM which could prove useful to assess test environments’ capacity to find subtle faults in RL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In this paper, we propose a framework, RLMutation, for Mu- tation Testing of deep RL programs leveraging HOM adapted to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We defined mutation operators for RL motivated by existing taxonomized faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We then analyzed how they fared on different RL environments and algorithms by using and comparing a number of mutation killing definitions adapted from previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In order to leverage HOM power to highlight more complex faults, we adapt existing work on HOM [4] to the RL task specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Namely, we conceive a simple heuristic tailored to the RL problem to systematically generate some test environments in order to obtain test cases to assess HOM usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, we aim to provide some insights into how Mutation Testing could be applied to RL 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05651v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='LG] 13 Jan 2023 while acknowledging existing differences with Supervised Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Our contribution is the proposed RL framework composed of the following: 11 mutation operators based on real taxono- mized faults, a comparison of the impact of mutation killing definition design over the FOM killed, and a heuristic to generate relevant test environments to study both FOM and HOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The remainder of the paper is structured as follows: Section II gives relevant background knowledge about Muta- tion Testing, HOM, and RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Section III presents the mutation operators introduced, as well as the procedure to determine how to generate relevant HOM when using Mutation Testing for RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Section IV reports about our experiments and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Section V discusses threats to the validity of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Section VI reviews the related literature, while Section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' BACKGROUND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Reinforcement Learning RL is a Machine Learning sub-paradigm in which an agent learns from interacting with an environment [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The main task consists of learning how to map states to actions by max- imizing the long-term reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' So, there are 3 main components in an RL problem: environment, agent, and a learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Formally, at each time step t, the agent perceives the state of the environment it is in (st ∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' After doing so, the agent takes an action (at ∈ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Upon taking the action, the agent receives a reward (rt ∈ R), and transitions to the next state (st+1 ∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The symbols S, A, and R denote the state, action, and reward spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We denote the (st, at, rt, st+1) tuple, which contains a record of the agent’s interaction with the environment as an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The agent interacts with the environment and collects observations until the environment reaches a terminal state where the environment ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' An episode indicates the length of the experiences the agent collects starting from an (initial) state and ending in a terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The agent aims to learn a policy π ∈ Π (the policy space) which maximizes the expected cumulative reward or return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As the agent interacts with the environment and collects rewards, it needs to learn the trade-off between short-term and long-term rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To facilitate this concept, the discount factor (γ), a hyperparameter between 0 and 1, is used during the calculation of the cumulative rewards [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Equation 1 shows how the value of a policy in a state is calculated: V π(s) = E[ Rt|st = s ], with Rt = ∞ � k=0 γkrt+k+1 (1) Recently, researchers have successfully integrated Deep Learning methods in RL to solve some challenging sequential decision-making problems [14] by employing Deep Neural Networks to learn the policy effectively [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mutation Testing and Higher Order Mutation Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [4] studied the different types of HOM, by classifying them based on two main properties: 1) a HOM is Coupled if, for a given set of test cases killing the HOM TH ̸= ∅ and a given union of sets of test cases killing its constituent FOM ∪ i Ti, we have TH ∩ ∪ i Ti ̸= ∅, and 2) HOM can be Subsuming if |TH| < | ∪ i Ti| where | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' | is the cardinal of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The latter can be further refined between Strongly and Weakly Subsuming, with Strongly Subsuming HOM being defined as TH ⊂ ∩ i Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' From the definitions, one can see that Subsuming HOM are of particular interest as they lead to mutations that turn out to be more complex to be detected than their constituents FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, focusing on HOM means fewer and more subtle mutants to use in Mutation Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mutation Testing of Deep Learning Applying Mutation Testing to Deep Learning, similarly to how it is done in traditional software engineering, raises several questions with the most prominent one being: Is a test case killing a mutation, or is it just an artifact of the stochasticity of the model’s training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Indeed, given a test case x, a neural network N and a mutant M, having N(x) ̸= M(x) does not necessarily mean that x killed the mutation M, as the mutation being killed could only be due to the stochasticity inherent to the training of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, training two neural networks, N1 and N2, on the same data and specification does not guarantee they will agree on a test case, as multiple sources of stochasticity might make them diverge from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, previous researchers in Supervised Learning made an argument in favor of considering not just one instance of a model but rather a group of instances when using Mutation Testing [6], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' DeepMutation [6] introduced the idea of averaging the kill ratio of multiple versions of a model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', neural network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Jahangirova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [18] and then Humbatova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [8] proposed to consider Mutation Testing through the lens of statistical testing over the accuracy of a distribution of instances that compares n non-mutant instances against n mutant instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, for a given test set T and the sets of accuracy over T of non-mutated models (AN1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', ANn) and mutated models with a given mutation operator (AM1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', AMn), they defined the following test function: MTT = � 1 if p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05 and effectSize ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='5 0 else (2) where the p-value is obtained by using Generalised Linear Model [19] and the effectSize is calculated using Cohen’s d [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While not directly part of the test function, they consider the power analysis to exclude mutations for which the statistical power of the test is too low (with the threshold β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mutation Testing has also been applied to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Authors in [12] proposed an approach to evaluate a crafted environment’s ability for revealing mutants based on mutation operators 2 designed for RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' They defined a mutant as killed if, for a healthy agent A and a mutated agent AM, the ratio pM/p of their average rewards over n episodes on a given environment E are inferior to a given threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, one can see that just like with Supervised Learning, there can be cases where (A1,AM1) would reveal the mutation, yet (A2,AM2) would not because of RL’s inherent stochasticity, which can deeply change the results among trained agents using the same environment and hyperparameter configurations but with different seeds [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Hence, two users training two agents on the same specification would end up with two different test results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' MUTATION OPERATOR FOR (DEEP) REINFORCEMENT LEARNING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' First Order Mutation Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' introduced several mutation operators applica- ble to RL [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, their operators present several shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' First, some can not be generalized to any RL algorithm which limits their relevance, for instance, mutations based on the epsilon parameter are limited to a subset Off- policy algorithms such as DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Secondly, some operators such as removing/adding a neuron on a particular layer will most likely lead to a crash, as the fault itself leads to cascading changes in the model architecture as pointed out in DeepCrime [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, some of the mutation operators are not justified since they are not based on real faults which undermine the usefulness of the operator, for instance, shuffling the replay priority in the replay buffer is not motivated by any real fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, to obtain relevant FOMs, we started from operators defined in [12] as a basis and further extracted existing taxonomies reporting on bugs affecting RL or Deep Learning models [23] [24] or directly adapting existing mutation op- erators [18] [8] used in Supervised Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We obtained a (non-exhaustive) list of mutation operators that we divided into three categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', environment, agent, and policy) based on what the mutation is affecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Due to space constraints, we only briefly describe the mutation operators in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' A comprehensive description can be found in our replication package [25] with a reference to the specifics that motivated each operator along with a comparison with operators defined in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Environment-level: The environment-level mutations are meant to simulate the faults that can happen when an agent receives observations as it interacts with the environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' receiving an incorrect observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' These faults can, for instance, be the results of faulty sensors, faults in environment design, or even nefarious attacks [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Each of the operators also has the probability of being applied to a given step, with 100% probability meaning that the mutation is applied to every step of the agent’s training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Reward Noise (RN): Adds a (Gaussian) noise to the true reward that the agent was meant to receive and returns it to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mangled (M): Damages the correlation between collected experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This operator returns a random st+1 and rt which are not the state and reward the agent should receive according to its current state and the taken action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Random (Ra): Similar to the mangled mutation operator, the Ra mutation returns a (st, at, r′ t, s′ t+1) tuple to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, unlike the M operator where s′ t+1 and r′ t are selected randomly and are not associated with each other, the random operator returns some s′ t+1 and r′ t to the agent which were sampled from the same experience tuples but have no association with st and at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Repeat (R): Returns the previous observation to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' If the agent has two consecutive experiences in the form of (st, at, rt, st+1) and (st+1, at+1, rt+1, st+2), this operator returns (st+1, at+1, rt, st+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Agent-level: As shown in [23] many of the issues faced by developers in implementing RL algorithms, are the result of incorrect coding of RL concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The agent-level mutations are meant to simulate the faults that can happen when a developer makes mistakes in implementing the concepts of an RL-based agent in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' No Discount Factor (NDF): Removes the discount factor γ from the reward calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Missing Terminal State (MTS): Removes the terminal state of an episode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', reaching the goal, or falling into a trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' No Reverse (NR): Wrongly reverses the order of the received rewards and therefore makes the agent learn an incorrect association between the experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Missing State Update (MSU): Removes the state update after the agent takes an action, meaning the agent will always see the same state in the experience tuple e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', (st, at+1, rt+1, st+2) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Incorrect Loss Function (ILF): Modifies the loss function used in the neural network that learns the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Policy-level: This category contains mutations affecting the policy of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In general, neural networks being used, these mutations are similar to mutations that were previously defined in Supervised Learning [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Policy Activation Change (PAC): Changes the default activation function used in the policy network of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Policy Optimizer Change (POC): Modifies the default optimizer of the algorithm while keeping the original learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' High Order Mutations Based on our previously defined FOMs, we set to evaluate HOM in RL as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To identify interesting HOMs, we follow a similar procedure to what was introduced in Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [4], presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Starting from Line 2, the algorithm describes how we implement in our case the heuristic of [4] to determine which FOM to be considered for the HOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We aim to determine which FOM are not trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', not killed by all test environments or by none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To evaluate if a mutation is killed by an environment, we use a distribution test over 3 Algorithm 1: HOM generation algorithm Input : m FOM F = F OM1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', F OMm, n healthy agents A = A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', An, for each FOM the relevant mutated agents AF OMi = A1,F OMi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', An,F OMi, initial environment E0, parameters to modify params Output: The set of FOMs F∗ to consider for the HOM 1 E = {E0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', Ep} ← GenerateBoundsEnvironments (A, params, E0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 2 F∗ ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 3 foreach f ∈ F do 4 i ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 5 foreach e ∈ E do 6 if IsDifferent (A, e, Af , e) then 7 i++;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 8 end 9 if i ̸= |E| and i ̸= 0 then 10 F∗ ← F∗ ∪ f 11 end the rewards of healthy agents A evaluated on environment e and mutated agents Af evaluated on the same environment (function IsDifferent in Algorithm 1), using, for instance, the distribution test mentioned in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, to apply the heuristic, different test environments are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, we need a way to simply and effectively generate more test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Generating simple test environment for Mutation Testing One way to generate new test environments in a relatively straightforward way that can be automated is to modify certain properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', parameters) of said environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, we define a test environment Ei to be dependent on its physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, the CartPole environment [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='] consists of a cart with a pole connected to it through a pivot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Two parameters of the environment are the cart’s and pole’s masses, which can be varied and then influence the agent’s decision which in turn can reveal faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finding diverse and non-trivial test cases which can have a different impact on the FOM is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Indeed, an exhaustive search is not practical because of the high number of parameter combinations and the computational cost of eval- uating the agent’s behavior, mutated or not, in each candidate test environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' At the same time, a random search might lead to many trivial test environments and similarly can be computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, leveraging some form of heuristic, even simplistic, is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Intuitively, environments close to the initial one have a high chance of yielding the same decision concerning the mutation, which would limit the relevance of such environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' At the same time, going too far away from the initial environment might unexpectedly affect the behavior of any agent and will also lower the relevance of the test environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Consequently, finding the right balance between the two extremes is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This, however, assumes a certain continuity of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, other works using a similar method such as Biagiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [28] suggest such an approach can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Therefore, we propose the following: by using only healthy agents, to decrease the computational cost, we can look for “frontier” environments, that is, environments that are different enough from the initial environment in terms of behavior but not too different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' One way to find such a tipping point is to leverage binary search between healthy agents’ reward on the Algorithm 2: Generate Bounds Environments Input : n healthy agents A = A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', An, initial environment E0, parameters to modify params Output: The set of boundary environments E 1 E ← {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 2 foreach p ∈ params do // Test environments on the upper boundaries 3 if E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p ̸= p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='lupper then 4 Ec ← E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 5 Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p ← p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='lupper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 6 if not IsDifferent (A, Ec, A, E0) then 7 E ← E ∪ Ec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 8 else 9 Eb ← E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 10 while CheckPrecision (Ec, Eb) do 11 pm ← Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p+Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 12 if not IsDifferent (A, Ec, A, E0) then 13 Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p = pm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 14 else 15 Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p = pm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 16 end 17 end 18 E ← E ∪ Eb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 19 end 20 else 21 E ← E ∪ E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 22 end 23 end 24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' /* Similar for lower boundaries / 25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 26 for i ← 1 to depth do 27 Em ← {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 28 for j ← 1 to |E|-1 do 29 if E[j] ̸= E0 AND E[j + 1] ̸= E0 then 30 Ec ← E[j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p+E[j+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 31 if not IsDifferent (A, Ec, A, E0) then 32 E⇕ ← Em ∪ {E[j], Ec, E[j + 1]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 33 else 34 Eb ← E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 35 while CheckPrecision (Ec, Eb) do 36 pm ← Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p+Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 37 if not IsDifferent (A, Ec, A, E0) then 38 Eb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p = pm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 39 else 40 Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='p = pm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 41 end 42 end 43 Em ← Em ∪ {E[j], Eb, E[j + 1]};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 44 end 45 end 46 E ← Em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 47 end 48 if E0 /∈ E then 49 E ← E ∪ E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' initial environment and agents’ reward on the generated can- didate test environment, comparing them with the previously defined mutation killing definition (see Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This is the goal of the function GenerateBoundsEnvironments on Line 1 of Algorithm 1 and presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The method first looks for test environments along the parameter axes by applying a binary search over only one parameter between the initial environment (E0) parameters and the defined search limits of the parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='l (Line 2- 25 Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In each step of the search, for the healthy agent, the distribution of reward obtained on environment Ec is then tested against the distribution obtained on the initial environment E0 (function IsDifferent in Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The search stops when a certain precision is reached between the lower/upper boundaries of the environment’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Then (Line 26-46), the algorithm will loop for a certain number of pre-defined depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' It searches for test environments in- between the environments that were calculated previously by 4 iteratively applying the same method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Note that the algorithm was implemented for a 2-D space, since it is what was used in our experiments, yet it can be easily extended without loss of generality to n-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, for the 2-D case, from 4 points, the algorithm will yield 8 after the first depth, and 16 after the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In the end, the set of test environments is returned in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' EXPERIMENTAL DESIGN AND RESULTS In this section, we introduce our research questions and describe the implementation of studied environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Fur- thermore, we describe our RL training process, experimental design, mutation killing definitions used, and obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Research questions To assess the effectiveness of Mutation Testing for RL, we formulate the following research questions (RQ): RQ1: What are the limitations of existing mutation killing definitions when applied to RL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ2: How are different agents and environments affected by the different mutations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ3: Do the HOM generated from our FOM possess the subsuming property similarly to traditional software en- gineering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 1) RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Limitations of current mutation killing definitions: As we detailed in Section II, Mutation Testing for RL can be applied in, at least, two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The first definition is proposed in [12], where a mutation is considered killed if the ratio of the average rewards over n episodes gained by the healthy agent to the average of the mutated one is lower than a threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The second definition is a distribution- wise statistical test as leveraged for instance in DeepCrime [8], which in RL’s case can be based on the reward over n episodes instead of the accuracy over the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, the first definition is limited because of the stochasticity of RL while the second one is a straightforward adaptation of a Mutation Testing application for Supervised Learning, which might not be completely valid for RL as the reward is not necessarily equal to accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This question aims to evaluate the relevance of both those mutation killing definitions in RL over the mutation operators we defined previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To have a fair comparison between the two definitions, we will need to modify the first approach (ratio of the average rewards over n episodes gained by the healthy agent to the average of the mutated one) as it does not account for N multiple agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We will count the number of times the ratio is lower than a certain threshold among N agents trained independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The choice of the threshold θ can have an impact on the number of mutations found, as the higher the threshold is, the more likely it is that a mutation can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Yet at the same time, a too high θ might reduce the usefulness of the ratio in the case a mutated agent behaves similarly to a healthy agent in the test environment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' an agent recovering from the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [12] chose a threshold of θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We instead chose a more conservative value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='9, which would make the test more likely to find a mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We will consider the mutation killed if at least 80% of the N ratios are lower than the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='9, which is roughly the power of the statistical test used for the distribution-wise test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The second method will use a statistical test over the reward of N agents similar to the original implementation in DeepCrime [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 2) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Behaviors of FOM on generated test environments: The goal of this RQ is to investigate the behaviors of FOM with regard to the type of mutations used and the generated test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Test environments were generated following the procedure detailed in Section III-C with a depth of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Depth was chosen to keep a low number of environments for computation not to be too expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Increasing it would increase the number of environments generated and so, likely, the potential number of relevant FOM at the cost of increased processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The procedure allows us to obtain a finer grain analysis of the FOM operators by getting the number of test environments killing a given FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Moreover, it is then possible to analyze the parameters of the test environments in order to assess what parameters’ set is more likely to trigger a certain mutation (for instance, for CartPole, if some mutations are more likely to be triggered when the mass of the cart is lowered or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, we will leverage FOM as presented in Section III-C, to deduce the interesting FOM to generate potential subsuming HOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The test environments will be generated by altering two parameters of each environment: for CartPole, the mass of the cart and of the pole will be modified, while for LunarLander, the gravity and the side engine power of the spacecraft will be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We refer the readers to our replication package [25] for the initial parameters as well as the search boundaries used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 3) RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Properties of generated HOM: Finally, similar to RQ2, in this RQ we aim to analyze the HOM generated in the previous experiments, by reusing our test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The goal is to understand which property the said generated HOM possesses, following the description briefly presented in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, we aim to see if we can generate Subsuming HOM from our defined FOM and test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Using chosen FOM from RQ2, we generate HOM that will in turn be used to train N agents for each environment/algo- rithm/mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We will then evaluate them on the previously generated test environments and compare obtained results with those obtained from RQ2’s FOM to deduce their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Implementation and models We used Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='8 and Stable Baselines 3 [29] framework version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='2 as a basis to implement our RL models and mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The motivation to use this framework is double: first, it allows us to evaluate some mutations that could directly affect the users of such a framework, for instance, a wrong activation or optimizer for the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Secondly, it serves as a solid basis to implement mutations affecting the potential customized code of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Indeed, mutation can be introduced this way by overriding the base code by modifying 5 TABLE I AVERAGE REWARDS ACROSS THE 20 AGENTS WITH THE STANDARD DEVIATION IN PARENTHESIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' PPO A2C DQN CartPole 500 (0) 500 (0) 414 (143) LunarLander 262 (16) 141 (56) 155 (84) only the relevant part, which ensures our code is less prone to unintended errors and allows better control over the mutation implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To test our mutations, we chose two well-known envi- ronments in the RL community: CartPole and LunarLander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The CartPole environment [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='] consists of a cart with a pole connected to it through a pivot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The goal of the agent is to stabilize the pole and prevent it from falling by applying an appropriate amount of horizontal force by moving the cart left or right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' LunarLander is a more complex environment that represents a spaceship that must land on a surface delimited by two flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The agent can control the spaceship by throttling it in three directions (left, right, and down) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We used three deep RL algorithms: similarly to the previous paper [12], we use Deep Q-Network (DQN) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' On top of that, instead of using plain Q-learning, which is a relatively simplistic algorithm, we will consider two other algorithms namely Advantage Actor-Critics (A2C) and Proximal Policy Optimization (PPO), which are classical algorithms in deep RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This allows us to compare Off-Policy algorithms (DQN) with On-Policy algorithms (PPO, A2C), which are two major approaches in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We trained for each environment-algorithm- mutation, N = 20 agents with different seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We report for each algorithm/environment the average accumulated reward and standard deviation across the agents in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We also remind the readers of the mutations used in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Some mutations are reliant on some parameters and so they will be defined with both the mutation operator identifier and the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, Policy Activation Change requires defining which new activation will be used, such as Stochastic Gradient Descent (SGD) and so will be noted as PAC SGD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Environment-level mutations are based on some probability of being applied for each given step, and so M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 means that the mutation operator Mangled will be used with a 100% probability at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' If no parameters are needed, then just the identifier is used, for instance, No Reverse will simply be noted as NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' All mutations are used for all agents, except NR and PAC-ReLU for DQN, the first one not applying to DQN while the second one being the default activation of DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Results 1) RQ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Existing limitations of current mutation killing definitions: Results of FOM for a given algorithm/environment and killing definition are given in Table III with AVG being the method using the average of the ratio and R being the method using reward-wise statistical comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As one can see,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' the limitations of AVG we briefly introduced in Section II-C are shown: for multiple mutation operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' no matter the environment/algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' the agents evaluated by the ratio might ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='TABLE II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='MUTATION OPERATORS SUMMARY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Category ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Operator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Environment-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Reward noise (RN) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Adding noise to the reward the agent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='receives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Mangled (M) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Returning next state and reward which ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='are not related to each other ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Random (Ra) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Returning next state and reward that are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='related to each other but not related to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='the action taken ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Repeat (R) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Returning next state and reward from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='previous observation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Agent-level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='No discount factor (NDF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='No discount factor during calculating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='cumulative rewards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='No reverse (NR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Not reversing the order of the received ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='rewards during calculating cumulative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='rewards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Missing state update (MSU) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Not updating agent’s observations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Missing terminal state (MTS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Failing to save the terminal state obser- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='vation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Incorrect loss function (ILF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='Defining an incorrect loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='(wrong formula,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=') Policy-level Policy activation change (PAC) Different activation for agent’s neural network Policy optimizer change (POC) Different optimizer for agent’s neural network yield a very different result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' If some mutation operators such as ILF or POC-SGD seem to be killed by the test environment on all healthy/mutated pairs, many mutations show a relatively low number of pairs declaring them killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' It is even possible that, for mutations with relatively small effects, the mutated agent ends up with a higher reward than the healthy agent if the initialization was not favorable for a particular healthy agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, it seems that the stochasticity of the training process greatly influences the ratio obtained, leading to mutation not being killed across multiple test runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' On the other hand, using a reward distribution statistical test (R) leads to more mutations being killed in all cases except for LunarLander/DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, this method allows mutations to be killed while AV G only had a handful of agents pair declaring said mu- tations killed using the same test environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, CartPole/A2C/RN-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 where only 25% of ratios revealed the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ1-1 Previously introduced mutation killing defini- tion in RL based on a ratio between healthy/mutated agents is not sufficient as it might miss a high num- ber of mutations because of the stochasticity of the training process, particularly for mutations with low effect on the training which are harder to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Using distribution-wise statistical tests based on the reward improves this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, for the same test environment, the distribution- wise test R still does not allow the detection of a high number of mutations, even mutations for which there are a sizable amount of ratios declaring the mutation killed with the AVG definition, such as LunarLander/A2C/NDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In those cases, while some mutated agents can exhibit a lower reward than some healthy agents, both healthy and mutated agents’ reward distribution might not exhibit a statistically significant 6 TABLE III FOM MUTATION TEST RESULTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' ✓ MEANS MUTATION IS KILLED WHILE \x17 MEANS MUTATION IS NOT KILLED OR THE STATISTICAL POWER OF THE TEST IS TOO LOW (INCONCLUSIVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' THE KILLING CRITERIA USED ARE AVG: AVERAGE REWARD MUTANT/HEALTHY WITH THE VALUE IN PARENTHESES BEING THE PROPORTION OF RATIOS BELOW THE THRESHOLD θ, R: REWARD-BASED STATISTICAL TEST, AND DTR: DISTANCE TO HEALTHY REWARD STATISTICAL TEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' “-” MEANS MUTATION IS NOT APPLICABLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Environment DRL Killing Mutations Algorithm Criteria ILF M-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 R-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 Ra-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 RN-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 NDF NR MSU MTS PAC-ReLU PAC-Sigmoid POC-SGD CartPole PPO AVG ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='65) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='25) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) R ✓ ✓ \x17 ✓ \x17 ✓ ✓ ✓ \x17 \x17 ✓ ✓ DtR ✓ ✓ \x17 ✓ \x17 ✓ ✓ ✓ ✓ ✓ ✓ ✓ A2C AVG ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='15) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='25) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='35) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='9) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='2) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) R ✓ ✓ \x17 ✓ ✓ ✓ ✓ ✓ \x17 ✓ ✓ ✓ DtR ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ DQN AVG ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='9) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='3) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='8) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) R ✓ ✓ ✓ ✓ \x17 ✓ ✓ ✓ ✓ ✓ DtR ✓ ✓ ✓ ✓ \x17 ✓ ✓ ✓ ✓ ✓ LunarLander PPO AVG ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='2) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='05) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) R ✓ ✓ \x17 ✓ \x17 ✓ ✓ ✓ \x17 \x17 ✓ ✓ DtR ✓ ✓ \x17 ✓ \x17 ✓ ✓ ✓ ✓ \x17 ✓ ✓ A2C AVG ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='3) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='45) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='45) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='85) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='45) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='2) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='6) ✓ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0) R ✓ ✓ \x17 ✓ \x17 \x17 ✓ ✓ \x17 ✓ \x17 ✓ DtR ✓ ✓ \x17 ✓ ✓ ✓ ✓ ✓ \x17 ✓ \x17 ✓ DQN AVG ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='9) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='6) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) \x17 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='7) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='8) ✓ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='95) R ✓ ✓ ✓ ✓ \x17 ✓ ✓ \x17 \x17 ✓ DtR ✓ ✓ ✓ ✓ \x17 ✓ ✓ \x17 ✓ ✓ difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This can particularly be the case when a few agents end up exhibiting a very different reward because of the effect of the mutation not being overcome by the training or being accentuated by a non-favorable initialization as initial seed can have a high impact on the training [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Removing such data points is not ideal as it masks potentially useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' One way to account for those data is to leverage the fact that we know the variation between healthy/mutated agents but also between healthy agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' And so any potential outlier would lead to a large difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Such variation can be estimated by calculating the Hellinger distance [31] of the rewards between samples of the healthy agents’ distribution (intra distance) and the same distance between samples of the healthy/mutated agents distribution (inter dis- tance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The Hellinger distance is a metric bounded between 0 and 1, with 0 meaning both distributions are the same which thus can be interpreted as a measure of similarity between the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For the discrete case, it is defined for two distributions P and Q as: H(P, Q) = 1 √ 2||P − Q||2 (3) where ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='||2 is the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' By repeating the calculation by sampling from both agents’ reward distribution, one can obtain the distributions of the inter/intra distance and use the same statistical test used in the definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In practice, we will use samples of 10 agents out of 20 to calculate the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The results are presented in Table III in the DtR rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Using this approach improves on using simply the rewards of healthy/mutated agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, it manages to lead to some mutations with relatively low AVG scores to be killed by the same test environment, such as LunarLander/PPO/MTS, as DtR allows to account for the previous potential outlier problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, one can see the method is not perfect as mutations such as LunarLander/DQN/RN end up not being killed while a sizable number of agent pairs are declared mutant by AVG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In those cases, the variation among healthy agents might be too pronounced compared to the ones between healthy and mutated agents, which is why DtR can not catch the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This however highlights an important observa- tion: the choice of the mutation killing definition design in RL is a crucial step to determine if a mutation is killed or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ1-2 More than choosing which mutation operators to consider, careful selection of how to use Mutation Testing when applied to RL is another important parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, using Hellinger distance to healthy reward instead of plain reward allowed for more mutations being killed for the same test envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, investigating different mutation killing definition designs in RL is a crucial step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 2) RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Behaviors of FOM on generated test environments: Following the procedure of Section III-C, we generated several test environments for each algorithm/environment and made a number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Results are presented in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Mutations flagged in green for a given environment/algorithm mean that the mutation is not trivial in that case and so will be relevant when we need to generate HOMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Note that we focus only on distribution-wise tests from this point on, as RQ1 illustrated the drawback of the mutation killing definition in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The first observation we make is that five mutations are killed by all generated test environments, namely ILF, MSU, NR, POC-SGD, M-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 and Ra-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Those mutations have too much of an impact not to be detected, which is also highlighted in Table III, as those mutations yielded a score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='85 when the initial test environment was used with the ratio method (AVG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In those cases, the stochasticity is masked by the high impact of the mutations and all trained agents behave similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, generating multiple test environments allowed us to see 7 that those mutations might not be much of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Indeed, they are rather trivial to detect and so can probably be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Secondly, we see the test environments generated on Cart- Pole are relatively more likely to catch mutations with most of the mutations being killed by at least half the test environments compared to the ones generated on LunarLander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While the number of test environments generated might play a role, it is likely because LunarLander is a more complex environment than CartPole, so mutations might become harder to detect as the environment complexity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For instance, CartPole seems to be less sensitive to the MTS or PAC mutations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', removing the terminal state of an episode or changing the activation function of the policy network does not seem to affect much the agents trained on CartPole contrary to LunarLander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ2-1 Contrary to using only the initial test environ- ment, generating additional test environments allows us to roughly evaluate which mutations might be trivial and which are more interesting based on the number of environments killing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We also found that it appears that, the more complex the environments the more likely the mutation is not to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In a second step, it is possible to go beyond the raw number of test environments killing a certain mutation and, instead, to inspect which test environment kills the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Indeed, as we generated test environments through the modification of the initial environment parameters, this can allow us to shed some light on which parameter a mutation might be more easily sensitive to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' By doing so, we can determine which test environments can help identify certain potential faults, thus leading to some sort of fault-detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Because of space constraints, we will report one example using R and we refer the reader to our implementation repository [25] for all the raw results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In the case of LunarLander/PPO/PAC ReLU, in Figure 1, test environments with lower (absolute) gravity compared to the initial environment kill the mutation while the ones with higher (absolute) gravity do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, the mutation seems to be affected somehow by the gravity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' When the gravity is close to the one in the initial environment, higher engine power will be crucial to kill the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Test environments are depicted as orange/blue dots, de- pending on whether or not they kill the mutation, and we can see they can be easily separated linearly following the previous description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In particular, we can see that both test environments P1 and P2 seem to be close to some frontiers for this mutation since, while being close in parameters space, they lead to an opposite decision on the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Note that the red frontier is arbitrary as we do not have access to the exact frontier, as it would be potentially too computationally expensive to find it as we explained in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Thus, it could be possible to find environments between P3 and the initial environment that could still kill the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We just know P3 is an environment for which the healthy agents’ reward distribution is at the limit of being different 14 12 10 8 6 4 2 0 gravity 0 1 2 3 4 5 6 side engine power P1 P2 P3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Generated test environments for LunarLander/PPO/PAC ReLU and a potential way to separate them based on whether or not they killed the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Orange points kill the mutation while the Blue ones don’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The origin is centered on the initial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' from the distribution observed in the initial environment, but it gives no information on the distribution of the mutated agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While it might not be possible to draw meaningful information for all mutations, especially on a reduced set of test environments, it shows nonetheless that generating test environments in that way also allows us to explore a potential link between parameters of the environments and their impact on the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ2-2 By mapping, for a given mutation, which gen- erated test environments kill it or not, we can analyze which of the parameters of the test environments affect the decision to kill the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This outlines some form of fault-detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' 3) RQ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Properties of generated HOM: Following RQ2, we gathered for each environment/algorithms/killing definition the non-trivial FOM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', nor killed by all test environments nor killed by none), see Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To generate HOM, we need at least two FOM as we stick to HOM of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We then trained new mutated agents based on the gathered HOM in the same way as FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, mutated agents were evaluated using the previously generated test environments depending on the mutation killing definition (R or DtR), and the type of HOM was analyzed based on the classification we introduced in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Results are presented in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As we can see, following the procedure mentioned for RQ2, we do not end up with a high number of non-trivial FOM and so the pool of generated HOM is relatively small with even some configurations not yielding any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, we managed to obtain 12 HOM when the R mutation killing definition was used and 23 with DtR but only in LunarLander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Among those, Non-Subsuming (NS) constitutes 50% of HOM using R method but only 30% using DtR, the difference between the two methods potentially being explained by the increased sensitivity of DtR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The remaining Subsuming HOM are mostly of the Weakly Subsuming Coupled (WSC) types 8 TABLE IV NUMBER OF TEST ENVIRONMENTS KILLING EACH FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' GREEN CELLS ARE FOM THAT WILL BE USED TO GENERATE HOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Environment DRL Killing Mutations Algorithm Criteria ILF M-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 R-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 Ra-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 RN-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='NDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='NR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='MSU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='MTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='PAC-ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='PAC-Sigmoid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='POC-SGD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='CartPole ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='PPO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='3/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='4/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='DtR ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='TYPES OF HOM GENERATED USING RQ2 NON-TRIVIAL FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' IF NO HOM WAS GENERATED, THEN A “-” IS USED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' NS: NON SUBSUMING, WSC: WEAKLY SUBSUMING COUPLED, WSD: WEAKLY SUBSUMING DECOUPLED, SSC: STRONGLY SUBSUMING COUPLED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Environment DRL Killing HOM types Algorithm Criteria HOM NS WSC WSD SSC CartPole PPO R 3 1 0 0 2 DtR A2C R 3 2 1 0 0 DtR DQN R DtR LunarLander PPO R 1 1 0 0 0 DtR 6 1 5 0 0 A2C R 5 2 3 0 0 DtR 14 6 8 0 0 DQN R DtR 3 0 3 0 0 and generally compose more than half the generated HOM for each configuration, which is similar to results obtained by Jia et al [4] with HOM in some traditional software programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Interestingly, we found 2 HOM that fit the type Strongly Subsuming Coupled (SSC), that is, HOM for which test environments killing said HOM also kill its constituent FOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Aside from those two, no other SSC and no WSD were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Even in Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' case, SSC represents a rare occurrence (< 1% of Subsuming HOM) and so the fact we found none is not too surprising judging by our limited number of HOM/test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, we showed that Subsuming HOM (even if Weakly ones), which are more complex and subtle mutants, could be generated and make up for more than half the HOM generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RQ3 HOM generated from non-trivial FOM, while few, are in the majority Subsuming HOM, which de facto makes them more interesting cases to use as we pointed out in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' If more subtle Subsuming HOM such as SSC are not as widely represented, the fact that some can be generated shows that such property is reachable in RL too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' THREATS TO VALIDITY Construct validity: The design choices of Mutation Testing could affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Since our main goal was not to design a new mutation killing definition for Deep Learning but rather to adapt and assess how existing approaches fared, all the mutation killing definitions used are based on existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Moreover, while the hyper-parameters used could play a role in declaring a mutation killed (p-value, threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='), we preferred to stick to the hyper-parameters given in each original implementation of the killing definitions and leave to future work the study of the influence of hyper- parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The way we searched the parameters space to generate test environments could also have impacted our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, the goal here was simply to get a simple and effective approach that could be automated and serve as a basic heuristic for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The rest of our methodology, such as the definition of the properties of HOM, is grounded in the scientific literature and is taken as they were originally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Internal validity: Mutation operators chosen could affect our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While we can not be exhaustive, we made sure to create operators based on existing faults or taxonomy and to provide sufficient diversity in terms of the effect on the code (agent-based, environment-based, policy-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' For the environment-based operators, while the probability of application of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0 might not be real to simulate sensor defaults, it was necessary to avoid potential bias over which step would be affected by the mutation when the probability is set to < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The environment parameters modified to generate the test environments could also impact the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, the goal was to show we could generate automatically simple and effective test environments that could be leveraged to analyze Mutation Testing in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, the choice of the parameters on its own is not as relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' External validity: The choice of environment types/algo- rithms could impact the generalization of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We made sure to select two environments as well as three algorithms that are generally used as a benchmark in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Reliability validity: Implementing the mutations could lead to some unintended faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We used the Stable-Baselines 9 framework as a basis to implement our mutations to lower the risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We also make our implementation and artifacts available [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' RELATED WORK Gur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [32] proposed an agent which produces envi- ronments depending on the learning agent’s level of skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Their approach allows for more methodological training of agents and increases their robustness and ability to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While interesting, their approach jointly trains the environment generator with the agents which is not doable within the Mutation Testing scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' To combat the problem of overfitting, Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [33] introduced the Procgen benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' This benchmark comprises 16 procedurally generated environments that allow practitioners to test the generalization of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Their idea allows for testing agents yet the main focus of Mutation Testing is to evaluate test set quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, it would be interesting in future work to evaluate RLMutation on their environments, as they are designed to push the general- ization property of the algorithm and so might induce different behavior over the mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In another direction, a meta- heuristic-based algorithm could be another venue to improve the generation of relevant test environments [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, Biagiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [28] focused on generating test environments through a combination of binary/exponential search, to map the adaptation/anti-regression performance of an agent through the lens of continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In our work, we similarly generate test environments by acting on their parameters, yet we do not test for the adaptation nor do we retrain the agent with continual learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' If anything, we similarly test for anti- regression of our healthy agents on both initial/generated test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, their approach could be used to improve our test generation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As mentioned in Introduction and Section II, some frame- works applied Mutation Testing to Supervised Learning [6]– [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' While we reuse some of their mutation operators/design choices, we specifically tailored our approach for RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [12] also tackled Mutation Testing in RL and studied how well-crafted environments could be used to reveal a particular mutation which is more akin to the fault detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' On the contrary, we focus on generating, in an automatic way, simple yet effective test environments that can be used to reveal more complex faults, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=', HOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We also show their Mutation Testing design choice could be a limiting factor because of the stochasticity of RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' [35] used decision boundaries of Supervised Learning models in order to find a subset of the test set that is more likely to trigger the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In a sense, our method is similar to theirs but from an RL perspective: we aim to find test environments that are on some boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' However, contrary to them, we do not have an actual test set to work on and have to generate the test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Moreover, we do not assume that the test environments on this boundary are more likely to reveal the mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Heuristically, we just assume it’s a potential point of interest, better than any random point if we are constrained by the number of test environments to be generated and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, Tambon et al [36] study more in-depth the effect that the choice of Deep Learning model instances has over the result of Mutation Testing in Supervised Learning using DeepCrime’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' They showed that this choice can affect the outcome of the Mutation Testing and that using the Bayesian approach can mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' In our approach, we do not account for this effect and just focused on existing killing definitions and generation of test environments, since the main goal was to analyze FOM and subsequent HOM in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Nonetheless, it would be interesting to verify that a similar behavior also exists in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' CONCLUSION In this paper, we have presented RLMutation, a framework for Mutation Testing in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We have defined three distinct categories of FOM based on real faults that can happen during the design and training of deep RL agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We compared dif- ferent mutation killing definition choices based on the previous framework applying Mutation Testing to Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We also evaluated HOM which are a combination of FOM and can prove to be more complex to kill which makes for interesting corner faults to investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The FOM used in the HOM generation were selected based on their relevance to generated parameterized test environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We have tested our approach on a set of state-of-the-art RL algorithms (DQN, A2C, and PPO) over two benchmark environments (LunarLander and CartPole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' The results of our study show that the mutation killing definition choice is important when it comes to killing mutations (ratio vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' distribution of rewards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' We demonstrate that by testing the agents in modified environments, we can detect non-trivial FOM which are not detected by testing the agents in the environments they were trained in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Finally, we show that HOM generated from non-trivial FOM possess in the majority the interesting property of being subsuming, meaning that they are harder to kill than their constituent FOM and so more interesting from a testing perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' Overall this study showed that, while Mutation Testing applied to RL raises numerous challenges to consider from the mutation killing definition design to the generation of relevant test environments and HOM, it is an interesting venue to improve testing of RL-based software systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' As such, we believe that further research on this topic should focus on those issues to enhance Mutation Testing applied to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE5T4oBgHgl3EQfiw-y/content/2301.05651v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported by: Fonds de Recherche du Qu´ebec (FRQ), the Canadian Institute for Advanced Research (CIFAR) as well as the DEEL project CRDPJ 537462-18 funded by the National Science and Engineering Research Council of Canada (NSERC) and the Consortium for Research and Innovation in Aerospace in Qu´ebec (CRIAQ), together with its industrial partners Thales Canada inc, Bell Textron Canada Limited, CAE inc and Bombardier inc.' metadata={'source': 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in +semantic-aware communication systems is presented. By introducing an unobservable semantic source, +we extend the classical results by Csiszar to semantic-aware communication systems. Both upper and +lower bounds of the exponent for the discrete memoryless source-channel pair are established. Moreover, +an extended achievable bound of the excess distortion exponent for MIMO systems is derived. Further +analysis explores how the block fading and numbers of antennas influence the exponent of semantic- +aware MIMO systems. Our results offer some theoretical bounds of error decay performance and can +be used to guide future semantic communications with joint source-channel coding scheme. +Yuxuan Shi and Shuo Shao are with the School of Cyber and Engineering, Shanghai Jiao Tong University, Shanghai 200240, +China (e-mail: ge49fuy@sjtu.edu.cn; shuoshao@sjtu.edu.cn). +Yongpeng Wu and Wenjun Zhang are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai +200240, China (e-mail: yongpeng.wu, zhangwenjun@sjtu.edu.cn). +Xiang-Gen Xia is with the Department of Electrical, and Computer Engineering, University of Delaware, Newark, DE 19716, +USA (e-mail: xianggen@udel.edu). +Chengshan Xiao is with the Department of Electrical, and Computer Engineering, Lehigh University, Bethlehem, PA 18015, +USA (e-mail: xiaoc@lehigh.edu). +arXiv:2301.04357v1 [cs.IT] 11 Jan 2023 + +2 +Index Terms +Semantic-aware communication, Excess distortion exponent, Joint source-channel coding, MIMO +block fading channel +I. INTRODUCTION +As a new paradigm in 6G networks, semantic communication gains significant attention in +recent days, and is expected to become a promising technology in future wireless communi- +cations. According to the definition of semantic information from Weaver and Shannon [1], +this new paradigm, considering the meanings behind symbols instead of pursuing the accurate +reconstructions, is able to transmit the desired semantic information to specific receivers. Con- +sequently, compared with the conventional paradigm, semantic-aware communication systems +can thus compress the source information in a larger extent, and reduce the corresponding +communication cost, such as transmitting power and spectrum resources in wireless systems. +Furthermore, for the future potential scenarios (e.g., smart Cities, IoT, virtual reality, etc.) whose +main purposes are to enable the receiver know the intrinsic meanings and complete the specific +tasks, the studies on semantic communication will be inevitably in full flourish. +A. Related Works +The concept of semantic communication was given by the landmark work [1] in 1950s, in +which the author conceived the communication over semantic level. Hereafter the efforts on +how to model the semantic information in practical communication have been made in the +last seven decades [2–4]. Specifically, Carnap [2] proposed the logical probability measure for +contexts instead of the statistical probability measure in Shannon’s classic theory, Bao [3] stressed +that the background information plays a key role in the semantic communication, and Juba +[4] utilized the feedback/sensing as the intermediate to capture the essence of a message in +a goal-oriented communication system. More recently, Liu, Zhang and Poor [5] proposed a +rate-distortion framework to characterize the semantic information, which models the source as +intrinsic and extrinsic states, and solve the optimization problem in some special cases. Liu et. +al. in [6] extended the rate-distortion function according to the information bottleneck theory, +and realize the semantic-aware image compression. The authors in [7] connected a semantic + +3 +communication layer (SC) on top of the technique communication layer (TC), and proposed +different measures on entropy to enhance the knowledge base. +Besides the aforementioned theoretical works, lots of papers focus on the practical realization +of semantic communication with the help of artificial intelligence (AI). Numbers of frameworks +on semantic communication were proposed to improve the compression or transmission perfor- +mances, based on the machine learning techniques in terms of the texts, audios and images (see +e.g., [8–14] for a few representative works). Among these, joint source-channel coding (JSCC) +based on deep learning (DL) networks is widely applied to improve the semantic communication +performance. More specifically, authors in [9] proposed a general DL-based JSCC framework +for semantic communication systems, which is named as DeepSC. Based on the result in [9], +authors in [10] presented a similar JSCC framework for a speech transmission and recognition. +The authors in [13] and [14] extended the DeepSC framework in more practical scenarios, which +combined the DL-based semantic communication with IoT fog networks and hybrid auto repeat +quires (HARQ), respectively. +B. Motivations and Contributions +Undoubtedly, semantic-aware communication provides a new paradigm of intelligent informa- +tion exchanges in nowadays wireless communication networks. Nevertheless, existing theoretical +works pay more attention to the compression but usually involve (or even not) simple channel +models, which cannot offer meaningful guides for the implementations of JSCC-based semantic +communication in practical 6G networks. Sparked by the above issue, it is natural to investigate +the performance of JSCC-based semantic communication under practical wireless channels, +e.g. multiple-input multiple-output (MIMO) channels with fadings, which shows fundamental +limits for practical semantic communications. As a revolutionary technique in nowadays wireless +networks, MIMO techniques benefit from the space multiplexing and obtain higher channel ca- +pacity. Numbers of researches focus on MIMO communication theory, such as capacities analysis +[15, 16], channel diversity analysis [17–19] and block coding regimes [20, 21]. Moreover, to +verify the superiority of a JSCC scheme, error exponent is chosen as the performance measure, +since separated source- channel coding (SSCC) performs the same as JSCC, in error probability +sense with infinite block length, while JSCC is strictly optimal in error exponent sense. Roughly + +4 +speaking, error exponent is the number E with property that the error probability of a suitable +code is e−En with block length n. Therefore, the error exponent can be used to measure the JSCC- +based semantic communication performance. The explorations on error exponents of channel +and source with fidelity criterion were given by Gallager [22] and Marton [23], respectively. +Furthermore, Csiszar [24, 25] derived the error exponent of JSCC scheme, and presented it +in a divergence form. Zhong [26–28] and Chang [29] extended the conclusion to systems with +continuous alphabet and side information, respectively. The analysis on Gallager’s random coding +bound of MIMO channel exponent was stated in [19, 30]. +Inspired by the framework in [5], this paper considers a point-to-point semantic-aware com- +munication system under JSCC framework. Specifically, following the rate-distortion function on +characterizing the semantic information, we first start from a long Markov chain which consists +of a source pair (S, X), a noisy channel W and the reconstructions ( ˆS, ˆX), in which S represents +the semantic source (intrinsic state) and X stands for the observed source (extrinsic state). It is a +generalized semantic communication model, which is named as semantic-aware communications, +owing to the two necessary distortion constraints on semantic and observed reconstructions. This +is the main difference between the remote source coding problem and our source system model. +Next we emphasize that this model is highly consistent with most of the AI-based semantic +communication works. Among these, some works transmit the extracted semantics and hope to +recover the original texts/images/videos at the receiver [9, 11, 31, 32], which means they consider +the observed recovery ˆX in their loss functions. Some other works execute the feature-specified +tasks [8, 10, 12, 14], e.g., the object detection and image recognition, which means the semantic +recovery ˆS is considered. Then in the second part of this paper, we further generalize the model +to a MIMO case, and obtain an achievable JSCC error exponent for a semantic-aware MIMO +system. This extension enables the application of error exponent-optimal JSCC scheme in 6G +wireless networks. Finally we conclude the main technical problems in the theoretical analysis: +it is hard to characterize the joint typical sets of source sequences when we incorporate an extra +semantic source. This obstacle is solved by introducing a channel coding theorem from [24] to +show the joint typicality among the semantic, observed and received sequences. Moreover, to +obtain the optimal exponent in MIMO systems, the random matrices instead of random scalars +are operated, e.g., the integration of random channel state matrix, which is difficult to calculate. + +5 +Hence, the hypergeometry function is utilized in the statement for further computation. +Under this model, we first investigate the exponential rate of the excess distortion probability +that either the recovered semantic or observed sequences exceed their required distortions (thus +we use the notation “excess distortion exponent” instead of “error exponent” in the following). +Upper and lower bounds of the exponent are presented as optimization problems in a discrete and +memoryless case. We verify that our results can be degenerated to the Csiszar’s JSCC exponent +[25] or Weissman and Merhav’s noisy source coding exponent [33, 34], and a direct conclusion is +obtained that semantic-aware communication enlarges the error exponent in comparison with the +conventional paradigm. Further, under a Gaussian source combined with a MIMO block fading +channel, an achievable excess distortion exponent of JSCC schemes is given. In this case, the +influences from coherence time, correlation coefficient and antennas numbers can be explicitly +discussed. From the achievability bound, a list coding scheme can be designed by combining +the list size with the semantic entropy. Moreover, the bound can extend some existing works on +JSCC scheme for wireless communications, e.g., spatial coupled LDPC or D-polar codes [35, 36] +to the semantic-aware scenarios. Besides, solution of the optimization problem of JSCC exponent +for semantic-aware MIMO systems is offered. Finally, numerical results on the exponent are also +presented to show how the environment parameters affect the exponent. +This paper is organized as follows: in Section II, we give the notations on semantic-aware +communication system, joint source-channel coding scheme and the excess distortion exponent. +In Section III, upper and lower bounds on JSCC excess distortion exponent are presented +in the discrete and memoryless case, as well as the degenerated cases to Csiszar, Weissman +and Merhav’s exponents. In Section IV, a theory on achievable parametric form in a MIMO +communication system and its optimization problem is presented. In Section V, we provide +some examples and plots to illustrate the exponential behaviors of JSCC exponent, and discuss +the influences of the key quantities. +II. PROBLEM FORMULATION +In this section, we present the model of the semantic-aware communication system, including +the definitions of semantic-aware JSCC scheme, the excess distortion event and the excess +distortion exponent. + +6 +Throughout the paper, an upper case letter stands for a random variable, whose realization +is represented by a lower case letter, and its alphabet is a calligraphy letter. For example, x +taking values in X is the realization of random variable X. |X| is the cardinality of X, and +(x)+ denotes max(x, 0). The distribution PX is the probability mass function (pmf) of X if +it has a countable alphabet. Besides, sequences are labeled with its length as superscript, such +as Xn = (X1, X2, · · · , Xn) and its realization xn follows similarly. EP(x)[X] represents the +expectation of random variable X according to distribution P(x), and IP(x,y)(X, Y ) denotes the +mutual information between X and Y in terms of joint distribution P(x, y). C(A → B) denotes +the set of all conditional distributions P(b|a) where a ∈ A and b ∈ B. Moreover, vectors +and matrices are represented by bold letter, and Im is the m × m identity matrix. Superscript +H and operator tr(·) denote the transpose conjugate and trace function, respectively. Finally, +X ∈ Cm×n ∼ MN(M, U, V) means that X follows matrix normal distribution with probability +density function +pX(X) = π−mn det(U)−n det(V)−m exp +� +tr +� +−U−1(X − M)V−1(X − M)H�� +, +where M ∈ Cm×n, 0 < U = UH ∈ Cm×m, 0 < V = VH ∈ Cn×n, and A > 0 means that matrix +A is positive definite. +A. Problem Formulation +S +PX|S +ϕ(X) +PZ|Y +Z +ψS(Z) +ψX(Z) +X +Y +ˆS +ˆX +dX(x, ˆx) ≤ Dx +dS(s, ˆs) ≤ Ds +Fig. 1: A semantic-aware communication system +A semantic-aware communication system is depicted in Fig. 1. A discrete memoryless source +(DMS) is described as a pair of random variables (S, X) with joint distribution PS,X in product +alphabet S × X. In this model, S is considered as the invisible intrinsic state with semantic +information while X is the extrinsic state and appears as the observable information. Moreover, + +7 +a memoryless channel W is defined with input Y ∈ Y, output Z ∈ Z and transition probability +PZ|Y (In the following, we denote the channel WZ|Y for simplicity). To introduce the block coding +scheme, the probability mass function of a k-length independent and identically distributed (i.i.d.) +sequence sk = (s1, · · · , sk) ∈ Sk is hence given by PSk(sk) = �k +i=1 PS(si), and PXk|Sk(xk|sk) = +�k +i=1 PX|S(xi|si). For such a communication system, a joint source-channel code with block +length n and transmission rate t = k +n symbol per channel use for the memoryless source (S, X) +and channel WZ|Y is defined as a tuple of mappings: +ϕn(·) : X k → Yn, ψk +S(·) : Zn → ˆSk, ψk +X(·) : Zn → ˆ +X k. +That is, a k-length information block sk extracted from semantic source is observed as a k-length +observed block xk, and then is encoded through JSCC as a codeword yn = (y1, y2, · · · , yn) += ϕn(xk), transmitted, received as zn = (z1, z2, · · · , zn). Two different decoders decode the +same received block as ˆsk = ψk +S(zn) and ˆxk = ψk +X(zn), corresponding to the desired semantic +and observed information sequences, respectively. To measure the source distortion, we denote +dk +S and dk +X the block-wise distortion measure functions of semantic and observable sources, +dS : S × ˆS → R, +dk +S(sk, ˆsk) ≜ 1 +k +k +� +i=1 +dS(si, ˆsi), +(1) +dX : X × ˆ +X → R, +dk +X(xk, ˆxk) ≜ 1 +k +k +� +i=1 +dX(xi, ˆxi), +(2) +where ˆsk = (ˆs1, ˆs2, · · · , ˆsk) ∈ ˆSk and ˆxk = (ˆx1, ˆx2, · · · , ˆxk) ∈ +ˆ +X k represent the recovered +semantic and observed sequences, respectively. Given a source pair (S, X) and a channel WZ|Y , +and two distortions Ds, Dx ≥ 0 on semantic and observed sequences, respectively, we define the +erroneous set of (sk, xk, zn) that violates the distortion constraints as +E = +�� +sk, xk, zn� +∈ Sk × X k × Zn : dk +S +� +sk, ψk +S (zn) +� +> Ds or dk +X +� +xk, ψk +X (zn) +� +> Dx +� +. +Note that in remote source coding, only the indirect source is concerned, while both indirect and +direct sources are recovered in a semantic-aware system. Hence how semantic distortions affect +the coding scheme performance, and the tradeoff between semantic and observed distortions can +be discussed. Therefore, we define a lossy JSCC scheme for semantic-aware communications + +8 +which is able to recover both semantic and observable information as the following. +Definition 1 (Lossy Joint Source-Channel Code for semantic-aware communications). The tuple +(ϕn, ψk +S, ψk +X) is an (n, k, Ds, Dx) lossy joint source-channel code for semantic source S ∈ S, +observable source X ∈ X and memoryless channel WZ|Y with two distortions Ds, Dx ≥ 0 if +P{E} ≤ ϵ, where ϵ is a sufficient small positive number. The code rate R = 1 +n log |Yn|. +The JSCC excess distortion probability can be stated as +P {E} ≜ +� +xk∈X k +PXk +� +xk� � +sk∈Sk +PSk|Xk +� +sk|xk� +� +zn∈E(sk,xk) +PZn|Y n � +zn|ϕn � +xk�� +, +(3) +where E +� +sk, xk� += +� +zn ∈ Zn : +� +sk, xk, zn� +∈ E +� +. +Here we use summation if the alphabets are finite, for continuous source and channel pairs, Eq. +(3) can be rewritten by replacing the summation with the integration. The following definition +introduces the JSCC excess distortion exponent. +Definition 2. The optimal JSCC excess distortion exponent Eopt +J (PX, PS|X, WZ|Y , Ds, Dx, t) for +any Ds, Dx ≥ 0, is defined as the supremum of the set including all numbers E for which there +exists a sequence of (n, k, Ds, Dx) JSCC scheme such that +E ≤ lim inf +n→∞ +� +−1 +n log P {E} +� +. +(4) +In the following, we try establishing upper and lower bounds on this excess distortion exponent +Eopt +J (PX, PS|X, WZ|Y , Ds, Dx, t) in the case of discrete and memoryless source-channel pair. +III. JOINT SOURCE-CHANNEL CODING EXCESS DISTORTION EXPONENT FOR +SEMANTIC-AWARE COMMUNICATIONS +In this section, we first investigate bounds on JSCC excess distortion exponent of a discrete and +memoryless semantic-aware communication system depicted in Fig. 1. The bounds are composed +of the source exponent and the channel exponents. We then verify that the proposed bounds can +be degenerated to the known results if relax one of the distortion constraints. + +9 +A. Statement of the Main Result +Theorem 1. For a given memoryless observable source X with distribution PX, a conditional +distribution PS|X, a memoryless channel with transition probability WZ|Y , and two distortions +Ds, Dx ≥ 0, which satisfies tR(PX, PS|X, Dx, Ds) ≤ C(WZ|Y ), the excess distortion exponent +Eopt +J +� +PX, PS|X, WZ|Y , Ds, Dx, t) for optimal (n, k, Ds, Dx) JSCC with distortions Ds, Dx and +transmission rate t is bounded by +Eopt +J +� +PX, PS|X, WZ|Y , Ds, Dx, t +� +≤ min +R∈R +� +t �E +�R +t , PX, PS|X +� ++ Esp +� +R, WZ|Y +�� +, +(5) +Eopt +J +� +PX, PS|X, WZ|Y , Ds, Dx, t +� +≥ min +R∈R +� +t �E +�R +t , PX, PS|X +� ++ Eex +� +R, WZ|Y +�� +. +(6) +Herein +R ≜ {R : tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y )}, +�E(r, PX, PS|X) ≜ min +QX +min +US|X∈C(X→S): +R(QX,US|X,Ds,Dx)≥r +� +D(QX||PX) + D(US|X||PS|X|QX) +� +, +(7) +Esp(R, WZ|Y ) ≜ max +PY +min +VZ|Y :IPY ×V (Z;Y )≤R D(VZ|Y ||WZ|Y |PY (QX)), +(8) +Eex(R, WZ|Y ) ≜ max +PY +min +PY � +Y :P � +Y =PY +� +EdWZ|Y (Y, �Y ) + IPY � +Y +� +Y ; �Y +� +− R +� +, +(9) +where D(V ||W|P) ≜ � +y∈Y P(y)D (V (·|y)||W(·|y)) denotes the conditional K-L divergence +and dWZ|Y (y, �y) is named as the Bhattacharya distance between two channel inputs [22, Chp 7]. +C(WZ|Y ) is the channel capacity and the rate distortion function characterizing semantic infor- +mation is given by [5, Thm 1] as +R(PX, PS|X, Ds, Dx) ≜ min +P ˆ +S, ˆ +X|X +I(X; ˆS ˆX), +(10) +s.t. +E[ ˆdS(X, ˆS)] ≤ Ds, +(11) +E[dX(X, ˆX)] ≤ Dx, +(12) +where ˆdS(x, ˆs) = E[dS(S, ˆs)|X = x], while dS(·, ·) and dX(·, ·) denote the component-wise +distortion functions given in Eq. (1) and Eq. (2), respectively. +Proof: See Appendix A. + +10 +(a) +(b) +Fig. 2: Characterization of the excess distortion exponent in a toy case (a) Upper and lower +bounds in Eq. (5) and Eq. (6); (b) Source excess distortion exponent in Eq. (7) +In Theorem 1, upper and lower bounds of the JSCC excess distortion exponent are established. +The upper bound consists of the sphere-packing bound on channel error exponent Esp(R, WZ|Y ) +and the source excess distortion exponent �E(r, PX, PS|X), which considers a new fidelity on +semantic source. Meanwhile the lower bound is composed of the expurgated random coding +bound on channel error exponent Eex(R, WZ|Y ) and the same source exponent. Note that our +result is a generalized form of Csiszar’s error exponent [25] in which a lossy JSCC encodes a +single source X and imposes a unique constraint on it. The basic idea to prove the results in +Theorem 1 is as follows. To obtain the source exponent, we characterize the excess distortion +probability in terms of sources, by counting the numbers of typical semantic and observable +sequences. The joint typicality among the semantic, observed and received sequences is necessary +to be discussed. To obtain the channel exponent, we prove the sphere-packing bound and the +expurgated random coding bound still hold for the semantic-aware transmission in Fig. 1 via +Csiszar’s channel coding theorem [24]. Finally, by minimizing the source and channel exponents +jointly over a group of JSCC schemes, we formulate upper and lower bounds on optimal JSCC +excess distortion exponent in Eq. (5) and Eq. (6), respectively. +We present an example of excess distortion exponent under a semantic-aware communication +system in Fig. 2, in which a toy case is considered that semantic source takes value in S = +{1, 2, 3} with equal probability, X = {0, 1} and a binary symmetric channel (BSC) with flip rate + +Upper bound Eq. (5) +Lower bound Eq. (6) +0.8 +Ej(Px,Psix,Wzly,R,t) +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +4.0 +1.0 +1.5 +1.0 +2.0 +Ds +0.5 +2.5 +0.0 +3.00.5 +E(r,Px,Ps|x) +0.4 +0.3 +0.2 +0.1 +0.0 +4.0 +2.5 +1.0 +1.5 +1.5 ++0 +1.0 +2.0 +Ds +0.5 +2.5 +0.0 +3.011 +p = 0.3. The left hand side one plots the upper and lower bounds of JSCC exponent, while the +right hand side one gives the source exponent in Eq. (7). In this case, it shows that the exponent +turns to be a non-decreasing function over semantic and observed distortions. Nevertheless, the +behavior of the exponent for generalized (S, X) and WZ|Y is unpredictable. +B. Two Degenerated Cases on Semantic Source +In this subsection, we study the special case where Ds = ∞ or Dx = ∞, which means the +constraint on semantic or observed information is relaxed. As follows, we verify that the bounds +in Theorem 1 can be reduced to the known results under simpler settings, i.e., Csiszar’s JSCC +error exponent and Weissman’s error exponent for noisy source coding [33]. +Corollary 1 (Csiszar’s JSCC error exponent). Without the semantic constraint, i.e., Ds = ∞, +the communication system focuses on reconstructing the observed information X. Consequently, +the excess distortion exponent of an optimal (n, k, ∞, Dx)-JSCC scheme, with the absence of +achievable semantic constraint Eq. (11), is reduced to the Csiszar’s JSCC error exponent with +a fidelity criterion [25, Thm 2, Thm 4]. +Proof: This corollary can be obtained intuitively due to the neglect of semantic information. +For a rigorous proof, according to the excess distortion exponent of semantic source given in +Eq. (7), the conditional divergence can be rewritten as +D +� +US|X||PS|X|QX +� += D +� +QX × US|X||QX × PS|X +� += +� +� +� +0, +if QX × US|X = QX × PS|X +∞, +otherwise +, +which implies the infimum of this term can be obtained by choosing US|X = PS|X such that +minUS|X D +� +US|X||PS|X|QX +� += 0. +Comparing Theorem 1 and Corollary 1, it can be observed that the JSCC error exponent for +semantic-aware systems contains an extra non-negative term and is larger than the Csiszar’s +error exponent in non-trivial cases, which is translated as a faster error decay speed owing to +the involvement of the semantic information. +Corollary 2 (Weissman and Merhav’s Error Exponent). Without the constraint on observed infor- +mation, i.e., Dx = ∞, the model is degenerated to an indirect source coding and communication + +12 +system. Then Eq. (7) can be reduced to Weissman and Merhav’s error exponent [33, Th 1]. +Proof: Note that in this case, we focus on a noisy source excess distortion exponent, +where the reconstruct semantic sequence ˆsk is deterministic for fixed xk and the compression +scheme, since ˆsk = ψk +S +� +ϕk � +xk�� +. Given the sequences +� +sk, xk� +and the excess distortion event +{dk +S +� +sk, ˆsk� +> Ds}, the rate-distortion function becomes +R(PX, PS|X, Ds, ∞) ≜ min +P ˆ +S|X +I(X; ˆS), +s.t. +E[ ˆdS(X, ˆS)] ≤ Ds. +The excess distortion is established directly between the transmitted xk and the reconstructed +sequence ˆsk. Thus, given xk of type QX, the test channel US|X can be rewritten as: +US|X = W × V , +W ∈ C +� +X → ˆS +� +: r ≥ IQX×W +� +X, ˆS +� +, +V ∈ C +� +ˆS × X → S +� +: EQX×W ×V dS +� +S, ˆS +� +> Ds, +in order to impose the constraints on mutual information and the distortion threshold, which +directly yields the conclusion in [33]. +IV. ACHIEVABLE EXCESS DISTORTION EXPONENT IN SEMANTIC-AWARE MIMO SYSTEMS +It is observed that Theorem 1 is easy to be extended into different scenarios. In wireless +networks, it may be of interest to consider a MIMO block fading channel rather than a simple +DMC. In this section, we present an achievable statement of JSCC excess distortion exponent for +a semantic-aware MIMO system composed of Gaussian distributed semantic source and block +fading MIMO channel. Furthermore, the optimization problem of exponent is solved in a simple +case for explicit analysis. +A proposition is used to claim the extension of Theorem 1 to a more general case. +Proposition 1. For a communication system, which consists of a joint Gaussian vector pair +(S, X) and a Gaussian channel WZ|Y with arbitrary memory, the lower bound in Eq. (6) holds. +Proof: Note that we considered discrete sources and channel with finite input and output +alphabets before. Nevertheless, in [37, Ch 4] Zhong etc. combines Csiszar’s exponents with + +13 +Gallager’s reliability functions in discrete cases via Fenchel duality, in which Gallager’s state- +ments are verified to be powerful tools on error exponent and are easily extended to the case of +continuous-alphabet with arbitrary memory. It is worth mentioning that in Gaussian case, only +random coding bound in Eq. (6) can be extended and the sphere-packing bound remains unclear. +For details, the reader can turn to [26, 27] for a rigorous proof of JSCC error exponent under the +setting of Gaussian distributed system and a one-order Markovian system, respectively. Hence, +by slightly abusing the notation defined in Sec. II, we can easily obtain this conclusion. +Note that the lower bound in Eq. (6) can be extended into a MIMO communication setting. +However, even not consider the semantic source, the converse proof on error exponent is not easy +to obtain in such a MIMO case. Among these proofs, Fano inequality and hypothesis testing show +unavailable bounds on exponent, while sphere-packing bound in Eq. (5) though yields a tight +bound in single antenna case, it is difficult to be extended into the multi-antennas case due to the +following two aspects. First, for the sphere-packing of codeword, the solid angle of the Voronoi +regions for matrices in continuous alphabet is difficult to characterize, hence the overall cone +is not a circular cone. Second, the strong converse for JSCC in Lemma 1 does not necessarily +hold in a MIMO communication system, since we cannot use the weak law of large numbers +(WLLN) for memory case [27, Appendix 1]. In summary, we present an achievability bound +as follows, which reveals an achievable excess distortion exponent of JSCC for semantic-aware +MIMO systems. +Theorem 2. For the above semantic-aware communication system, the observed source X follows +a Gaussian vector distribution N(0q, ΣX), where ΣX is an q × q positive semi-definite matrix. +Meanwhile the ℓ-length semantic source S is given by +S = hX + N, +where h is an ℓ × q matrix, and N is a random vector follows N(0ℓ, ΣN), which is independent +of X. The quadratic distortion measures become dS(s,ˆs) = tr{(s − ˆs)(s − ˆs)H} and dS(x, ˆx) = +tr{(x−ˆx)(x−ˆx)H}, where ˆs and ˆx refer to the recovered source vectors. Furthermore, a MIMO +communication system contains nT transmit and nR receive antennas, where the block fading +channel WZ|Y remains invariant for Nc symbols in each coherence time. In each observation +composed of Nb independent coherence intervals, which amount to NbNc symbols, the received + +14 +matrix Zi ∈ CnR×Nc at the i-th interval can be formulated as +Zi = HiYi + Wi, +i = 1, 2, · · · , Nb, +where Yi ∈ CnT ×Nc is the channel input matrix, Hi is the channel state matrix and Wi is the +additive white Gaussian noise matrix, namely Wi ∼ MN(0nR×Nc, NwInR, INc). Here Nw is the +noise coefficient. The transition probability with perfect CSI at the receiver can be stated as +p(Z|Y, H) = (πNw)−nRNc exp +� +− 1 +Nw +(Z − HY)(Z − HY)H +� +, +(13) +where the subscript i is dropped for simplicity since the channel is memoryless for each coherence +interval. Note that Y denotes the power constrained input as +1 +NcE[tr{YYH}] = tr{Q} ≤ P. Then, +there exists an (n, k, Ds, Dx)-lossy JSCC for this semantic-aware MIMO communication system +with excess distortion exponent +Eopt +J +� +PX, PS|X, WZ|Y, Ds, Dx, t +� += min +R∈R EJ +� +PX, PS|X, WZ|Y, R, t +� += min +R∈R +� +t �EG +�R +t , PX, PS|X +� ++ EMIMO � +R, WZ|Y +�� +, +(14) +where +�EG(r, PX, PS|X) = +min +∆:O≺∆⪯ΣX , +tr{∆}≤Dx +min +A∈Cq×q: +det(A)=det(∆)e2r +min +B∈Cℓ×ℓ: +tr{B}=Ds−tr{hH ∆h} +� +1 +2 log det(ΣX) det(ΣN) +det(∆)e2r det(B) ++ tr +� +Σ−1 +X A + Σ−1 +N B + +� +Σ−1 +N B − Iℓ +� +hHΣXh +� +� +− ℓ − q, +(15) +EMIMO � +R, WZ|Y +� += max +0≤ρ≤1 +� +max +δ≥0 Eex(Q, ρ, δ, Nc) − ρR +� +, +(16) +Eex (Q, ρ, δ, Nc) =2δρP +− 1 +Nc +ln EH +� +det +� +QA +� +InT − Q +�HHHA−1HHH +16N 2 +wρ2 +− HHH +4Nwρ + δ +���−Ncρ� +, +(17) +A =δInT − Q−1 − +1 +4NwρHHH. +(18) + +15 +Proof: See Appendix B. +In this theorem, we present an achievable JSCC excess distortion exponent in a semantic- +aware MIMO communication system. Specifically, we consider a jointly Gaussian distributed +source pair, where an ℓ-length semantic vector S is combined with a q-length observed vector X. +Furthermore, the quadratic distortion measure and a MIMO system with block fading channel +are also considered. The optimal exponent Eopt +J +is composed of two parts, namely the source +exponent and the expurgated random coding exponent for MIMO systems. From the achievable +bound, the ergodic capacity and cut-off rate of the above semantic-aware MIMO communication +system can be obtained, by setting ρ = 0 or 1. We note that the generalized vector nature +complicates the statement of the exponent, which makes the optimization in Eq. (14) difficult. +Hence a simple case is discussed as follows, which enables us to further analyze and reveal some +insights on the JSCC scheme design in such a semantic-aware MIMO communication system, +for optimal excess distortion exponent. +Corollary 3 (Excess distortion exponent for semantic-aware MIMO systems in specific case). +Under the same setups in Theorem 2, we further assume X ∼ N(0q, σ2 +XIq), N ∼ N(0ℓ, σ2 +NIℓ) +and H ∼ MN(0nR×nT , InR, InT ) (for simplicity we set nR ≤ nT), Q = +P +nT InT due to the equal +power assignment on the transmitting antennas, and denote SNR = +P +Nw , then the following +equations hold: +(a) For the expurgated random coding bound for MIMO channel in Eq. (18), the derivatives +are calculated as +∂Eex (Q, ρ, δ, Nc) +∂δ +=2ρP − 2ρnTSNR +1 − δ SNR + SNR +Nc +tr +� +K−1(ρ, δ)∂K(ρ, δ) +∂δ +� +(19) +∂Eex (Q, ρ, δ, Nc) +∂ρ +=2δP + 2nT ln(1 − δ SNR) − 1 +Nc +tr +� +K−1(ρ, δ)∂K(ρ, δ) +∂ρ +� +(20) +where K(ρ, δ) is a Hankel matrix with size nT × nT whose (i, j)-th entry follows hypergeo- +metric function +(nT − nR + i + j − 2)!2F0 +� +nT − nR + i + j − 1, Ncρ; − +SNR +2(1 − δ SNR)ρ +� +(b) Given tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y), the JSCC excess distortion exponent is convex +in terms of code rate R. + +16 +(c) Given the above source-channel pair, and the transmission rate t, the optimal achievable +code rate R⋆ can be formulated by +R⋆ = 2t ln +tρ⋆ + 2 +2 min +� +σ2 +X +σ2 +Xtr{hT h}+σ2 +N (Ds − σ2 +N), +1 +σ2 +X Dx +� +(21) +where ρ⋆ satisfies +ρ⋆ = arg max +0≤ρ≤1 {Eex(Q, ρ, δ, Nc) − ρR} +according to Eq.(19) and Eq.(20). +Proof: Given Q = +P +nT InT and Gaussian distributed random matrix H, (a) is obtained by +Eex +� P +nT +InT , ρ, δ, Nc +� += 2δρP − 2ρnT ln(1 − δ SNR) − 1 +Nc +ln det(K(ρ, δ)) +K +(22) +where K is given above and K = �nT +i=1(nR − i)!(i − 1)! according to [19, Cor 4]. This corollary +enables us to obtain the expectation in Eq. (18) via the computable generalized hypergeometric +function 2F0(·, ·; ·) (given by [38, Eq (3)]). Hence the partial derivatives are stated in Eq. (19) +and Eq. (20). For (b), the conclusion is also direct since the second partial derivative over code +rate R is positive when the code rate lies in the interval tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y). +For (c), we obtain the maximization of the excess distortion exponent EMIMO � +R, WZ|Y +� += +Eex (Q, ρ⋆, δ, Nc) − ρ⋆R over δ and ρ successively, and solve +∂ �EG( R +t , PX, PS|X) + EMIMO � +R, WZ|Y +� +∂R += 0 +Herein, we analyze the properties of the semantic-ware JSCC excess distortion exponent. Due +to the matrix essence, we evaluate the expectations on coefficient matrices H via the generalized +hypergeometry function. Moreover, the partial derivatives are derived in order to solve the +optimization problem on JSCC exponent. Note that the statement of exponent is formulated +in the form of an optimization problem over coding rate R. The solution explicitly presents the +optimal design of JSCC scheme for semantic-aware MIMO systems in error exponent sense. + +17 +Fig. 3: An illustration of the JSCC excess distortion exponent with some key quantities +V. NUMERICAL RESULTS +In this section, theoretical bounds of excess distortion exponent in semantic-aware MIMO +communication systems are presented. From the simulations, we verify the convexity and show +the key quantities like rate-distortion function and ergodic capacity. Then, we explore the in- +fluences of MIMO communication systems on semantic information reconstructions, such as +coherence time, exponential correlation coefficient and the numbers of antennas. Finally, we +also discuss how the optimal code rate R⋆ behaviors in terms of different transmission rate t. +A. Experiments Setups +We consider the above semantic-aware MIMO communication system in Corollary. 3, which +consists of a joint Gaussian source pair (S, X) with ΣS = 3Iℓ and ΣX = 4Iq. Moreover, we +assume the same number of transmit and receive antennas nT = nR, and the channel state matrix +HHH ∼ Q(InT , GT, GR) (given by [15]), in which we adopt exponential correlation matrices +GT = {α|i−j| +T +}, GR = {α|i−j| +R +} (Simply assuming αT = αR = α ∈ [0, 1)) to model the spatial +correlation. Besides, the signal-to-noise ratio can be calculated by SNR = +P +Nw . +B. Convexity of JSCC Exponent over Code Rate R for Semantic-Aware MIMO Systems +In Fig. 3, the JSCC excess distortion exponent for a MIMO system EJ +� +PX, PS|X, WZ|Y, R, t +� +is plotted against the code rate R. The distortions Ds = 2, Dx = 1, coherence time Nc = 1, + +EMIMO(R,WzIy) in [39, Prop. 1] +3.5 +EMIMO(R,WzY) in Eq. (16) +3.0 +Achievable exponent +Ej(Px,Psix,Wziy,R,t) +2.5 +E(r,Px,PsIx) +2.0 +1.5 +1.0 +Fopt +0.5 +R(Px,PsIX,Ds,Dx)1.8 +C(WzIY)=5.1 +0.0 +R +0 +1 +2 +3 +4 +5 +6 +R18 +nT = nR = 3, correlation coefficient α = 0.3, SNR= 15, and the transmission rate t = 2 +symbol/channel use. Note that the rate-distortion function characterizing the semantic information +R(PX, PS|X, Ds, Dx) = 1.8 and the ergodic capacity C(WZ|Y) = 5.1, which is marked in this +figure. The solid line represents JSCC exponent function, which consists of source exponent +in the dashed dotted line and MIMO channel exponent given in the expurgated bound Eq. (17) +marked by the star labels. In comparison, we also present the Gallager’s random coding bound of +MIMO channel [39] in dashed line to verify its suboptimality, since the ’bad’ JSCC codewords +are expurgated for a better error probability. Due to the convexity of the JSCC exponent, we +focus on the minimum of EJ, which is labeled by Eopt +J +and the optimal JSCC code rate R⋆. +C. Optimal JSCC Excess Distortion Exponent for Semantic-Aware MIMO Systems +In this subsection, we provide plots on the exponential behaviors of the optimal JSCC excess +distortion probability for semantic-aware MIMO systems. We investigate how the source and +channel key quantities influence the best exponent performance. +Excess Distortion Exponent against Semantic Distortions: Based on the depicted system, the +excess distortion exponents against semantic distortion are plotted in Fig. 4(a) in terms of different +observed distortions, which intuitively illustrates the tradeoff between Ds and Dx. The star, +triangle and diamond lines represent the optimal performance with Dx = 1, 1.25, 1.5, respectively. +Obviously both the increase of Ds and Dx leads to the increase of the exponents, but the curves +remain invariant when the semantic distortion becomes inactive, since the observed constraint +is more demanding. Under the aforementioned experiment setups, the achievable optimal JSCC +excess distortion exponent attains its limit around 0.24 when Dx = 1.5. +Excess Distortion Exponent against SNR: We first compare the optimal excess distortion expo- +nent in terms of MIMO systems with different numbers of antennas, namely nT = nR = 2, 3, 4. +From Fig. 4(b), the exponent increases with the number of antennas dramatically. Specifically, +under a higher SNR environment, a semantic-aware MIMO system with a 4 × 4 array obtains +Eopt +J +≥ 1, while a 2 × 2 system only has 1/10 performance on the exponential probability. This +trend demonstrates the compatibility of semantic-aware communication systems and the massive +MIMO techniques, with an even larger nT. +In Fig. 4(c), we explore how the coherence time Nc in MIMO system affects the exponent. The +star, triangle and diamond curves stand for the optimal exponent ranges from 1 to 3, respectively. + +19 +(a) +(b) +(c) +(d) +Fig. 4: Excess distortion exponent in terms of: (a) observed distortions, (b) number of +antennas, (c) coherence time, (d) correlation coefficient +It can be observed though the longer coherence time results in longer block length, the optimal +exponent decreases with Nc at arbitrary SNR. In Fig. 4(d), the plot shows the performance +in terms of exponential coefficient with α = 0.3, 0.6, 0.9, respectively. The optimal exponent +increases with α, since the larger α means the better channel transmission. +D. Achievable Optimal Code Rate R⋆ in terms of MIMO Key Quantities +In this subsection, we investigate the optimal JSCC code design for the semantic-aware MIMO +systems in error exponent sense. Fixing Dx = 1.5, Ds = 2 and α = 0.3, the ratio t = k +n is plotted +against the optimal code rate R⋆ in Fig. 5(a). The transmission rate t increases with the optimal +code rate and turns a slightly decrease with coherence time Nc. In Fig. 5(b), the exponential +correlated coefficient α shows a positive effect on transmission rate t. Note that given t and R⋆, +we obtain the optimal JSCC scheme for the semantic-aware MIMO system. + +0.20 +0.15 +0.10 +Dx= 1 +0.05 +E +Dx = 1.25 +=1.5 +0.00 +1.0 +1.5 +2.0 +2.5 +3.0 +Dsnt=2 +1.0 +nT=3 +nt=4 +0.8 +0.6 +0.4 +0.2 +E +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +SNR0.7 +Nc=1 +Nc =2 +0.6 +Nc=3 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +SNRα= 0.3 +0.8 +α= 0.6 +0.9 +α=( +0.6 +0.4 +0.2 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +SNR20 +(a) +(b) +Fig. 5: Different JSCC Schemes against Optimal Code Rate R⋆ in terms of (a) different +coherence time Nc (b) different correlation coefficient +VI. CONCLUSION +In this paper, we obtained upper and lower bounds of JSCC excess distortion exponent in +semantic-aware communication systems. We concluded that the participation of semantic source +enlarges the JSCC excess distortion exponent. In a semantic-aware MIMO system, we presented +an achievable bound for JSCC excess distortion exponent, which extends the conclusion to a +practical communication scenario. As a result, based on the achievability bound, we solved the +optimization problem in a simple case and discussed the design of the optimal JSCC scheme in +terms of the code rate and transmission rate. Finally, from the numerical results, we show the +tradeoff between the two distortions and demonstrated that more antennas and larger correlation +efficient lead to a better JSCC excess distortion exponent, while longer coherence time reduces +its performance. +In future works, a tight converse bound for JSCC exponent in semantic-aware MIMO com- +munication system will be considered. Furthermore, the analysis for finite block length case will +be also studied for designing practical coding schemes with advantages over the conventional +non-semantic-aware communication systems. + +7 +Nc=1 +6 +Nc=2 +Nc=3 +5 +4 +R +3 +2 +1 ++0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Transmission rate t10 +α=0.3 +α=0.6 +8 +α=0.9 +6 +R +4 +2 +0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Transmission rate t21 +APPENDIX A +PROOF OF THEOREM 1 +Lemma 1 (JSCC Theorem [40] in Semantic-aware Scenario). Given a memoryless source pair +(S, X) and a memoryless channel WZ|Y , where tR(PX, PS|X, Ds, Dx) > C(WZ|Y ), then for any +(n, k, Ds, Dx) JSC code we have +lim +n→∞ PJ +� +PS|X, PX, WZ|Y , n, k +� += 1. +(23) +Lemma 2 (c.f.[41, Theorem 2] [24, p. 175]). We assume a channel W of capacity C(W) and +an n-length list code (ϕ, ψ) has a range of decoded sequences of cadinality l, which we call the +list size. An erroneous event of list codes occurs when a true message is not on the decoding +list. Let pe(n, R, L) denote the minimal error probability, and pe(n, R, L) be the maximal error +probability for such an n-length list code with rate R and list size L. Then for positive ˜ϵ1(n) → 0 +and ˜ϵ2(n) → 0 as n → ∞, and any R > C(W) + L where L = 1 +n log l, we have +pe(n, R, L) ≥ exp {−n (Esp(R − L, W) + ˜ϵ1(n))} . +(24) +pe(n, R, L) ≤ exp {−n (Eex(R − L, W) − ˜ϵ2(n))} . +(25) +Lemma 2 is a channel coding theorem based on list codes, which characterizes the error +probability via sphere-packing bound and expurgated bound. Note that expurgated bound is a +refined version of random coding bound, which drops the bad codewords beyond the Bhattacharya +distance [22, Chap 7]. +The outline of the proof is the following. We start by presenting a strong converse for JSCC +coding theorem under semantic-aware communications. To obtain the excess distortion expo- +nent, it is reasonable to model the non-trivial source-channel pairs of tR(PX, PS|X, Ds, Dx) ≤ +C(WZ|Y ), which is given in Lemma 1. Here, we focus on investigating the optimal JSCC scheme +of code rate within this interval. Then, as mentioned above, the excess distortion probability, +which refers to the ratio of over-distorted sequences to overall sequences, can be bounded by +computing the sizes of typical sets of sources and channel. Given the source pair (sk, xk), the +excess distortion probability from the over-distorted codewords is bounded in this appendix +via Lemma 2. Moreover, the average numbers of the corrupted sequences from the semantic + +22 +and observable sources are stated as well, by investigating the typical sequences xk and the +conditional typical sequences sk. Finally, by combining these results from source with channel +parts and minimizing the sum in terms of code rate R, we obtain final results. +According to Lemma 1, we investigate the excess distortion performance of a group of JSCC +schemes with code rate tR(PX, PS|X, Ds, Dx) ≤ R ≤ C(WZ|Y ). Now we recall the overall +excess distortion probability defined in Eq. (3), and rewrite +P {E} = +� +xk∈X k +PXk(xk) +� +sk∈Sk +PSk|Xk(sk|xk)pc(sk, xk), +(26) +where we use pc(sk, xk) ≜ � +zn∈E(sk,xk) PZn|Y n(zn|ϕn(xk)). For the excess distortion probability +pc(sk, xk), let TQ denote the typical set of sequences xk ∈ X k of type QX, TU be the joint typical +set of sequences (sk, xk) ∈ Sk×X k, and TU(xk) = +� +sk : (sk, xk) ∈ TU +� +is the conditional typical +set of sk for a given xk, in which the conditional empirical distribution is US|X. Moreover, the +conditional typical set TV (zn) = {(sk, xk) : (sk, xk, zk) ∈ TV } based on the joint typical set TV +composed of sequence tuple (sk, xk, zk). Note that the size of the conditional typical set is +|TV (zn)| ≤ exp {kH(S⋆, X⋆|Z⋆)} = exp {k (H(S⋆, X⋆) − I(S⋆, X⋆; Z⋆))} +≤ exp +� +k +� +H(QX, US|X) − R(QX, US|X, Ds, Dx) +�� +, +where S⋆, X⋆ and Z⋆ are three arbitrary auxiliary random variables characterizing the joint +distribution PS⋆X⋆Z⋆, which is a possible joint type of sequences xk ∈ TQ, sk ∈ TU(xk) and +zn ∈ Zn within the distortion constraints. Next, the number of all possible joint types is upper +bounded by (k + 1)|S||X||Z| via the type counting lemma. Hence the list size can be bounded by +l ≤ (k + 1)|S||X||Z| exp +� +k +� +H(QX, US|X) −R(QX, US|X, Ds, Dx) +�� += exp +� +k +� +H(QX, US|X +� +−R(QX, US|X, Ds, Dx) + ˆϵ1(k) +�� +, +where k = nt, ˆϵ1(k) = 1 +k log(k + 1)|S||X||Z|. Note that +lim +k→∞ ˆϵ1(k) = lim +k→∞ +1 +k log(k + 1)|S||X||Z| (a) += |S||X||Z| lim +k→∞ +1 +k log(k + 1) = 0, +(27) +where equality (a) is because the product of alphabet cardinalities, |S||X||Z|, is a finite constant. + +23 +Thus, the parameter L is +L = 1 +n log l ≤ t +� +H(QX, US|X) − R(QX, US|X, Ds, Dx) + ˆϵ1(k) +� +. +(28) +Since the size of the message set satisfies +exp{nR} ≤ exp{kH(PS,X)} = exp{ntH(QX, US|X)}, +(29) +by substituting Eq. (28) and Eq. (29) into Eq. (24), we have +pc(sk, xk) = exp +� +−n +� +Esp(tR(QX, US|X, Ds, Dx) − tˆϵ1(k), WZ|Y ) + ˜ϵ1(n) +�� +(b) +≥ exp +� +−n +� +Esp(tR(QX, US|X, Ds, Dx), WZ|Y ) + ϵ1(n) +�� +, +(30) +where the inequality (b) holds since Esp(R, W) is a non-increasing function in R, and ϵ1(n) = +tˆϵ1(n) + ˜ϵ1(n) → 0 as n → ∞ . +Now given an observed sequence xk, we define pa(xk) = � +sk∈Sk PSk|Xk(sk|xk)pc(sk, xk), +and +pa(xk) = +� +US|X∈U +� +sk∈TU(xk) +PSk|Xk(sk|xk)pc(sk, xk) +(c) += +� +US|X∈U +� +sk∈TU(xk) +� +(a,b)∈(S×X) +PS|X(a|b)N((a,b)|(sk,xk))pc(sk, xk) += +� +US|X∈U +��TU +� +xk��� exp +� +−kEQX×US|X +� +− log PS|X(S|X) +�� +pc(sk, xk) +(d) += +� +US|X∈U +��TU +� +xk��� exp +� +−kH(US|X|QX) +� +exp +� +−kD +� +US|X +�� PS|X +�� QX +�� +pc(sk, xk) +(e) +≥ +� +US|X∈U +(k + 1)−|S||X| exp +� +−kD +� +US|X +�� PS|X +�� QX +�� +pc(sk, xk). +(31) +where Q denotes the set of all types QX and U denotes the set of conditional types US|X. In (c) +we apply the empirical count function N((a, b)|(sk, xk)) on the conditional probability, in (d) +we use the definition on the conditional divergence defined in Eq. (1), and in (e) the following + +24 +result is used ([24, Lemma 2.3]): +(k + 1)−|S||X| ≤ +��TU +� +xk��� exp +� +−nH(US|X|QX) +� +≤ 1. +To characterize the possible conditional types US|X, given the observable sequences xk ∈ +TQ, we use the conclusion that the code rate is upper bounded by rate-distortion function +tR(QX, US|X, Ds, Dx) ≥ R. Thus all possible US|X should be restricted in the following set: +U ≜ +� +US|X ∈ C(X → S) : R(QX, US|X, Ds, Dx) ≥ r +� +. +(32) +Following Eq. (26), the overall excess distortion probability can be stated in Eq. (33), +P {E} = +� +QX∈Q +� +xk∈TQ +PXk(xk)pa(xk) +≥ exp +� +−k +� +min +QX∈Q +� +D(QX||PX) + min +US|X∈U D(U||PS|X|QX) + ϵ3(k) +��� +pc(sk, xk) += exp +� +−k +� +�E(r, PX, PS|X) + ϵ3(k) +�� +exp +� +−n +� +Esp(R, WZ|Y ) + ϵ1(n) +�� +, +(33) +where ϵ1(n) is given in Eq. (30) and +ϵ3(k) = −1 +k log +� +|Q||U|(k + 1)−|S||X|−|X|� (f) +≤ 1 +k log +� +(k + 1)|X|� +. +Inequality (f) holds since the number of all possible joint types of xk and sk is upper bounded +by1 |Q||U| ≤ (k + 1)|S||X|, hence ϵ3(k) → 0 as k → ∞. Combining Eq. (33) with the definition +of �E(r, PX, PS|X) in Eq. (7), we state the upper bound on the exact excess distortion exponent +as +�EJ(R) ≜ lim inf +n→∞ +� +−1 +n log P {E} +� +≤ k +n +�E(r, PX, PS|X) + Esp(R, WZ|Y ). +(34) +With the achievability Eq. (25) in Lemma 2, we can get the lower bound similarly on �EJ(R) +�EJ(R) ≥ k +n +�E(r, PX, PS|X) + Eex(R, WZ|Y ). +(35) +1More specifically, U ⊆ U ⋆ = {US|X : US|X ∈ C(X → S)}, and the number of all joint types is bounded by type counting +lemma [24, Lemma 2.2] as |Q||U| ≤ |Q||U⋆| ≤ (k + 1)|S||X|. + +25 +By substituting t = k +n and r = R +t in Eq. (34) and Eq. (35), respectively, we finally obtain upper +and lower bounds of excess distortion exponent in Eq. (5) and Eq. (6), respectively. Note that +the excess distortion probability is stated as a function of n, k but the exponent of the optimal +JSCC only concerns the transmission rate t symbol/channel use. +APPENDIX B +PROOF OF THEOREM 2 +Lemma 3. [42, Appendix B] For every pair of Hermitian positive definite matrices A ∈ +Cm×m, C ∈ Cn×n, and denote etr(·) = exp{tr(·)}, then for any matrices B, D ∈ Cm×n, we +have: +� +Cn×m etr +� +−π +� +AUHCU + BHU + UHD +�� +dU = det +� +A⊤ ⊗ C +�−1 etr +� +πA−1BC−1DH� += det(A)−n det(C)−m etr +� +πA−1BC−1DH� +This section shows how to derive Eq. (14) from Eq. (6) under a semantic-aware MIMO +communication system. Specifically, we derive the explicit forms of the source excess distortion +exponent �E(r, PX, PS|X), and the expurgated random coding bound Eex(R, WZ|Y) on channel +excess distortion exponent under MIMO communication systems. The basic idea of the proof is, +for the excess distortion exponent of source pairs, we rewrite the K-L divergence according to +the generalized rate-distortion function into a computable optimization problem. For the MIMO +channel expurgated random coding exponent, we utilize the equivalence between Csiszar’s form +and Gallager’s form by Fenchel duality. +We start from a jointly Gaussian distributed vector pair (S = hX+N, X), and the reconstructed +source vectors ˆS, ˆX. The minimum of Eq. (7) is presented with two distributions QX and +US|X subject to R(QX, US|X, Ds, Dx) ≥ r. Note that under the Gaussian vector assumption, +the semantic-aware rate-distortion function can be rewritten as +R(QX, US|X, Ds, Dx) = min 1 +2 log +� det(A) +det(∆) +� +(36) +s.t. +O ≺ ∆ ⪯ A, +(37) +tr +� +h∆hH� +≤ Ds − tr{B}, +(38) +tr {∆} ≤ Dx, +(39) + +26 +Moreover, in Eq. (7), the first divergence can be computed as +D(QX||PX) = EQX [log QX − log PX] += EQX +�1 +2 log det(ΣX) +det(A) + 1 +2xH � +Σ−1 +X − A−1� +x +� += 1 +2 log det(ΣX) +det(A) − q + tr +� +Σ−1 +X A +� +(40) +where the auxiliary multivariate Gaussian distribution QX(x) = +1 +(2π) +q +2 det +1 +2 (A) exp{1 +2xHA−1x}. +Similarly, the conditional divergence in Eq. (7) can be stated as +D(US|X||PS|X|QX) = EQX×US|X[log US|X − log PS|X] += 1 +2 log det(ΣN) +det(B) + tr +�� +ΣN − B−1� +hHΣXh + Σ−1 +N B +� +− ℓ +(41) +where the distribution US|X(s|x) = +1 +(2π) +l +2 det +1 +2 (B) exp{1 +2 (s − hx)H B−1 (s − hx)}. Finally, combin- +ing Eq. eqref41 and Eq. (41) with the constraints from Eq. (37)-(39), we obtain the semantic- +aware source excess distortion exponent as Eq. (15). +In Theorem 1, we state the channel exponent in Csiszar’s form (Eq. (9)) in light of the sim- +plicity on statement, but it is hard to be computed. Notably, Zhong [37] combined the Csiszar’s +exponent with Gallager’s reliability function via Fenchel duality and proved the equivalence, in +which the later is easy to be extended and analyzed. The reader can turn to [37, Chapter 4] [22, +Chapter 7] for more details. +Under the MIMO system, Gallager’s random coding bound is given by [39, Prop 1]. For an +expurgated bound, which is derived from a codebook expurgating the bad codewords, and hence +performs better than random coding bound at lower code rate, Alfano [30, Thm 3.2] evaluated it +under simple assumptions. Herein we present the expurgated bound on error exponent in terms +of random matrices. From [22], the expurgated exponent is stated as +Eex = − 1 +Nc +ln +� +H +pH(H) +�� +Y˜Y +pY˜Y(Y˜Y) exp +� +δ +� +tr +� +YYH + ˜Y˜Y +H� +− 2P +�� +w(Y, ˜Y, Z) +1 +ρ dY˜Y +�ρ +dH, +(42) +where w(Y, ˜Y, Z) = +� +Z +� +p(Z|Y, H)p(Z|˜Y, H)dZ and − ln w(Y, ˜Y, Z) is the Bhattacharya +distance between channel input matrices Y and ˜Y while Z is the channel output. Next we + +27 +first process the aforementioned integral with the transition probability (13) of MIMO system as +w(Y, ˜Y, Z) = (πNw)−nRNc exp +� +− 1 +2Nw +tr +� +HYYHHH + H˜Y˜Y +HHH�� +× +� +Z +exp +� +− 1 +2Nw +tr +� +2ZZH − ZYHHH − HYZH − Z˜Y +HHH − H˜YZH�� +dZ += exp +� +− 1 +4Nw +tr +� +H +� +Y − ˜Y +� � +Y − ˜Y +�H HH�� +(43) +where (43) follows from Lemma 3. Moreover, by assuming a capacity achieving input distribution +on matrix ˆY, we obtain +� +˜Y +p˜Y(˜Y) exp +� +δtr +� +˜Y˜Y +H�� +w(Y, ˜Y, Z) +1 +ρd˜Y +=π−nT Nc det(Q)−Nc +� +˜Y +exp +� +tr +�� +δInT − Q−1 − +1 +4NwρHHH +� +˜Y˜Y +H +− +1 +4Nwρ +˜Y +HHHHY − +1 +4NwρYHHHH˜Y +�� +d˜Y += det(Q)−Nc det(A)−Nc exp +� +tr +� +1 +16N 2 +wρ2A−1YHHHHHHHY +�� +. +(44) +By applying Lemma 3 again and A = δInT − Q−1 − +1 +4NwρHHH, we achieve equation (44). The +expectation on input matrix Y can be formulated as +det(QA)−Nc +� +Y +pY(Y) exp +� +δtr +� +YYH − 2P +�� +exp +� +tr +� +1 +16N 2 +wρ2A−1YHHHHHHHY +�� +dY += det(QA)−Nc exp{−2δNCP} +� +Y +pY(Y) exp +� +tr +�HHHA−1HHH +16N 2 +wρ2 +− HHH +4Nwρ + δ +� +YYH +� +dY += exp{−2rNCP} det(QA)−Nc det +� +InT − Q +�HHHA−1HHH +16N 2 +wρ2 +− HHH +4Nwρ + δ +��−Nc +(45) +Finally substituting (45) into (42) yields the expurgated bound on the error exponent in (17). +In conclusion, we state the source exponent as piecewise function Eq. (16). 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Taricco, “Asymptotic mutual information statistics of separately correlated Rician fading +MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3490–3504, Aug. 2008. + diff --git a/ndE3T4oBgHgl3EQfKwmr/content/tmp_files/load_file.txt b/ndE3T4oBgHgl3EQfKwmr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..25528f62ec85b5d2c0b80637457455a0fbda41ef --- /dev/null +++ b/ndE3T4oBgHgl3EQfKwmr/content/tmp_files/load_file.txt @@ -0,0 +1,1031 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf,len=1030 +page_content='1 Excess Distortion Exponent Analysis for Semantic-Aware MIMO Communication Systems Yuxuan Shi, Shuo Shao, Member, IEEE, Yongpeng Wu, Senior Member, IEEE, Wenjun Zhang, Fellow, IEEE, Xiang-Gen Xia, Fellow, IEEE, Chengshan Xiao, Fellow, IEEE Abstract In this paper, the analysis of excess distortion exponent for joint source-channel coding (JSCC) in semantic-aware communication systems is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' By introducing an unobservable semantic source, we extend the classical results by Csiszar to semantic-aware communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Both upper and lower bounds of the exponent for the discrete memoryless source-channel pair are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, an extended achievable bound of the excess distortion exponent for MIMO systems is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Further analysis explores how the block fading and numbers of antennas influence the exponent of semantic- aware MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Our results offer some theoretical bounds of error decay performance and can be used to guide future semantic communications with joint source-channel coding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Yuxuan Shi and Shuo Shao are with the School of Cyber and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: ge49fuy@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' shuoshao@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Yongpeng Wu and Wenjun Zhang are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: yongpeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='wu, zhangwenjun@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Xiang-Gen Xia is with the Department of Electrical, and Computer Engineering, University of Delaware, Newark, DE 19716, USA (e-mail: xianggen@udel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Chengshan Xiao is with the Department of Electrical, and Computer Engineering, Lehigh University, Bethlehem, PA 18015, USA (e-mail: xiaoc@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='04357v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='IT] 11 Jan 2023 2 Index Terms Semantic-aware communication, Excess distortion exponent, Joint source-channel coding, MIMO block fading channel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' INTRODUCTION As a new paradigm in 6G networks, semantic communication gains significant attention in recent days, and is expected to become a promising technology in future wireless communi- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' According to the definition of semantic information from Weaver and Shannon [1], this new paradigm, considering the meanings behind symbols instead of pursuing the accurate reconstructions, is able to transmit the desired semantic information to specific receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Con- sequently, compared with the conventional paradigm, semantic-aware communication systems can thus compress the source information in a larger extent, and reduce the corresponding communication cost, such as transmitting power and spectrum resources in wireless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, for the future potential scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', smart Cities, IoT, virtual reality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=') whose main purposes are to enable the receiver know the intrinsic meanings and complete the specific tasks, the studies on semantic communication will be inevitably in full flourish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Related Works The concept of semantic communication was given by the landmark work [1] in 1950s, in which the author conceived the communication over semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hereafter the efforts on how to model the semantic information in practical communication have been made in the last seven decades [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Specifically, Carnap [2] proposed the logical probability measure for contexts instead of the statistical probability measure in Shannon’s classic theory, Bao [3] stressed that the background information plays a key role in the semantic communication, and Juba [4] utilized the feedback/sensing as the intermediate to capture the essence of a message in a goal-oriented communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' More recently, Liu, Zhang and Poor [5] proposed a rate-distortion framework to characterize the semantic information, which models the source as intrinsic and extrinsic states, and solve the optimization problem in some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Liu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' in [6] extended the rate-distortion function according to the information bottleneck theory, and realize the semantic-aware image compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The authors in [7] connected a semantic 3 communication layer (SC) on top of the technique communication layer (TC), and proposed different measures on entropy to enhance the knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Besides the aforementioned theoretical works, lots of papers focus on the practical realization of semantic communication with the help of artificial intelligence (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Numbers of frameworks on semantic communication were proposed to improve the compression or transmission perfor- mances, based on the machine learning techniques in terms of the texts, audios and images (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', [8–14] for a few representative works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Among these, joint source-channel coding (JSCC) based on deep learning (DL) networks is widely applied to improve the semantic communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' More specifically, authors in [9] proposed a general DL-based JSCC framework for semantic communication systems, which is named as DeepSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Based on the result in [9], authors in [10] presented a similar JSCC framework for a speech transmission and recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The authors in [13] and [14] extended the DeepSC framework in more practical scenarios, which combined the DL-based semantic communication with IoT fog networks and hybrid auto repeat quires (HARQ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Motivations and Contributions Undoubtedly, semantic-aware communication provides a new paradigm of intelligent informa- tion exchanges in nowadays wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Nevertheless, existing theoretical works pay more attention to the compression but usually involve (or even not) simple channel models, which cannot offer meaningful guides for the implementations of JSCC-based semantic communication in practical 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Sparked by the above issue, it is natural to investigate the performance of JSCC-based semantic communication under practical wireless channels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' multiple-input multiple-output (MIMO) channels with fadings, which shows fundamental limits for practical semantic communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' As a revolutionary technique in nowadays wireless networks, MIMO techniques benefit from the space multiplexing and obtain higher channel ca- pacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Numbers of researches focus on MIMO communication theory, such as capacities analysis [15, 16], channel diversity analysis [17–19] and block coding regimes [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, to verify the superiority of a JSCC scheme, error exponent is chosen as the performance measure, since separated source- channel coding (SSCC) performs the same as JSCC, in error probability sense with infinite block length, while JSCC is strictly optimal in error exponent sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Roughly 4 speaking, error exponent is the number E with property that the error probability of a suitable code is e−En with block length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Therefore, the error exponent can be used to measure the JSCC- based semantic communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The explorations on error exponents of channel and source with fidelity criterion were given by Gallager [22] and Marton [23], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, Csiszar [24, 25] derived the error exponent of JSCC scheme, and presented it in a divergence form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Zhong [26–28] and Chang [29] extended the conclusion to systems with continuous alphabet and side information, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The analysis on Gallager’s random coding bound of MIMO channel exponent was stated in [19, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Inspired by the framework in [5], this paper considers a point-to-point semantic-aware com- munication system under JSCC framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Specifically, following the rate-distortion function on characterizing the semantic information, we first start from a long Markov chain which consists of a source pair (S, X), a noisy channel W and the reconstructions ( ˆS, ˆX), in which S represents the semantic source (intrinsic state) and X stands for the observed source (extrinsic state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' It is a generalized semantic communication model, which is named as semantic-aware communications, owing to the two necessary distortion constraints on semantic and observed reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This is the main difference between the remote source coding problem and our source system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Next we emphasize that this model is highly consistent with most of the AI-based semantic communication works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Among these, some works transmit the extracted semantics and hope to recover the original texts/images/videos at the receiver [9, 11, 31, 32], which means they consider the observed recovery ˆX in their loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Some other works execute the feature-specified tasks [8, 10, 12, 14], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', the object detection and image recognition, which means the semantic recovery ˆS is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then in the second part of this paper, we further generalize the model to a MIMO case, and obtain an achievable JSCC error exponent for a semantic-aware MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This extension enables the application of error exponent-optimal JSCC scheme in 6G wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally we conclude the main technical problems in the theoretical analysis: it is hard to characterize the joint typical sets of source sequences when we incorporate an extra semantic source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This obstacle is solved by introducing a channel coding theorem from [24] to show the joint typicality among the semantic, observed and received sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, to obtain the optimal exponent in MIMO systems, the random matrices instead of random scalars are operated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', the integration of random channel state matrix, which is difficult to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 5 Hence, the hypergeometry function is utilized in the statement for further computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Under this model, we first investigate the exponential rate of the excess distortion probability that either the recovered semantic or observed sequences exceed their required distortions (thus we use the notation “excess distortion exponent” instead of “error exponent” in the following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Upper and lower bounds of the exponent are presented as optimization problems in a discrete and memoryless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We verify that our results can be degenerated to the Csiszar’s JSCC exponent [25] or Weissman and Merhav’s noisy source coding exponent [33, 34], and a direct conclusion is obtained that semantic-aware communication enlarges the error exponent in comparison with the conventional paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Further, under a Gaussian source combined with a MIMO block fading channel, an achievable excess distortion exponent of JSCC schemes is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In this case, the influences from coherence time, correlation coefficient and antennas numbers can be explicitly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' From the achievability bound, a list coding scheme can be designed by combining the list size with the semantic entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, the bound can extend some existing works on JSCC scheme for wireless communications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', spatial coupled LDPC or D-polar codes [35, 36] to the semantic-aware scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Besides, solution of the optimization problem of JSCC exponent for semantic-aware MIMO systems is offered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, numerical results on the exponent are also presented to show how the environment parameters affect the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This paper is organized as follows: in Section II, we give the notations on semantic-aware communication system, joint source-channel coding scheme and the excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Section III, upper and lower bounds on JSCC excess distortion exponent are presented in the discrete and memoryless case, as well as the degenerated cases to Csiszar, Weissman and Merhav’s exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Section IV, a theory on achievable parametric form in a MIMO communication system and its optimization problem is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Section V, we provide some examples and plots to illustrate the exponential behaviors of JSCC exponent, and discuss the influences of the key quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we present the model of the semantic-aware communication system, including the definitions of semantic-aware JSCC scheme, the excess distortion event and the excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 6 Throughout the paper, an upper case letter stands for a random variable, whose realization is represented by a lower case letter, and its alphabet is a calligraphy letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For example, x taking values in X is the realization of random variable X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' |X| is the cardinality of X, and (x)+ denotes max(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The distribution PX is the probability mass function (pmf) of X if it has a countable alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Besides, sequences are labeled with its length as superscript, such as Xn = (X1, X2, · · · , Xn) and its realization xn follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' EP(x)[X] represents the expectation of random variable X according to distribution P(x), and IP(x,y)(X, Y ) denotes the mutual information between X and Y in terms of joint distribution P(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' C(A → B) denotes the set of all conditional distributions P(b|a) where a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, vectors and matrices are represented by bold letter, and Im is the m × m identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Superscript H and operator tr(·) denote the transpose conjugate and trace function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, X ∈ Cm×n ∼ MN(M, U, V) means that X follows matrix normal distribution with probability density function pX(X) = π−mn det(U)−n det(V)−m exp � tr � −U−1(X − M)V−1(X − M)H�� , where M ∈ Cm×n, 0 < U = UH ∈ Cm×m, 0 < V = VH ∈ Cn×n, and A > 0 means that matrix A is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Problem Formulation S PX|S ϕ(X) PZ|Y Z ψS(Z) ψX(Z) X Y ˆS ˆX dX(x, ˆx) ≤ Dx dS(s, ˆs) ≤ Ds Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 1: A semantic-aware communication system A semantic-aware communication system is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' A discrete memoryless source (DMS) is described as a pair of random variables (S, X) with joint distribution PS,X in product alphabet S × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In this model, S is considered as the invisible intrinsic state with semantic information while X is the extrinsic state and appears as the observable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, 7 a memoryless channel W is defined with input Y ∈ Y, output Z ∈ Z and transition probability PZ|Y (In the following, we denote the channel WZ|Y for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To introduce the block coding scheme, the probability mass function of a k-length independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=') sequence sk = (s1, · · · , sk) ∈ Sk is hence given by PSk(sk) = �k i=1 PS(si), and PXk|Sk(xk|sk) = �k i=1 PX|S(xi|si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For such a communication system, a joint source-channel code with block length n and transmission rate t = k n symbol per channel use for the memoryless source (S, X) and channel WZ|Y is defined as a tuple of mappings: ϕn(·) : X k → Yn, ψk S(·) : Zn → ˆSk, ψk X(·) : Zn → ˆ X k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' That is, a k-length information block sk extracted from semantic source is observed as a k-length observed block xk, and then is encoded through JSCC as a codeword yn = (y1, y2, · · · , yn) = ϕn(xk), transmitted, received as zn = (z1, z2, · · · , zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Two different decoders decode the same received block as ˆsk = ψk S(zn) and ˆxk = ψk X(zn), corresponding to the desired semantic and observed information sequences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To measure the source distortion, we denote dk S and dk X the block-wise distortion measure functions of semantic and observable sources, dS : S × ˆS → R, dk S(sk, ˆsk) ≜ 1 k k � i=1 dS(si, ˆsi), (1) dX : X × ˆ X → R, dk X(xk, ˆxk) ≜ 1 k k � i=1 dX(xi, ˆxi), (2) where ˆsk = (ˆs1, ˆs2, · · · , ˆsk) ∈ ˆSk and ˆxk = (ˆx1, ˆx2, · · · , ˆxk) ∈ ˆ X k represent the recovered semantic and observed sequences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Given a source pair (S, X) and a channel WZ|Y , and two distortions Ds, Dx ≥ 0 on semantic and observed sequences, respectively, we define the erroneous set of (sk, xk, zn) that violates the distortion constraints as E = �� sk, xk, zn� ∈ Sk × X k × Zn : dk S � sk, ψk S (zn) � > Ds or dk X � xk, ψk X (zn) � > Dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that in remote source coding, only the indirect source is concerned, while both indirect and direct sources are recovered in a semantic-aware system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hence how semantic distortions affect the coding scheme performance, and the tradeoff between semantic and observed distortions can be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Therefore, we define a lossy JSCC scheme for semantic-aware communications 8 which is able to recover both semantic and observable information as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Definition 1 (Lossy Joint Source-Channel Code for semantic-aware communications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The tuple (ϕn, ψk S, ψk X) is an (n, k, Ds, Dx) lossy joint source-channel code for semantic source S ∈ S, observable source X ∈ X and memoryless channel WZ|Y with two distortions Ds, Dx ≥ 0 if P{E} ≤ ϵ, where ϵ is a sufficient small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The code rate R = 1 n log |Yn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The JSCC excess distortion probability can be stated as P {E} ≜ � xk∈X k PXk � xk� � sk∈Sk PSk|Xk � sk|xk� � zn∈E(sk,xk) PZn|Y n � zn|ϕn � xk�� , (3) where E � sk, xk� = � zn ∈ Zn : � sk, xk, zn� ∈ E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Here we use summation if the alphabets are finite, for continuous source and channel pairs, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (3) can be rewritten by replacing the summation with the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The following definition introduces the JSCC excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The optimal JSCC excess distortion exponent Eopt J (PX, PS|X, WZ|Y , Ds, Dx, t) for any Ds, Dx ≥ 0, is defined as the supremum of the set including all numbers E for which there exists a sequence of (n, k, Ds, Dx) JSCC scheme such that E ≤ lim inf n→∞ � −1 n log P {E} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (4) In the following, we try establishing upper and lower bounds on this excess distortion exponent Eopt J (PX, PS|X, WZ|Y , Ds, Dx, t) in the case of discrete and memoryless source-channel pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' JOINT SOURCE-CHANNEL CODING EXCESS DISTORTION EXPONENT FOR SEMANTIC-AWARE COMMUNICATIONS In this section, we first investigate bounds on JSCC excess distortion exponent of a discrete and memoryless semantic-aware communication system depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The bounds are composed of the source exponent and the channel exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We then verify that the proposed bounds can be degenerated to the known results if relax one of the distortion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Statement of the Main Result Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For a given memoryless observable source X with distribution PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' a conditional distribution PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' a memoryless channel with transition probability WZ|Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' and two distortions Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' which satisfies tR(PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds) ≤ C(WZ|Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' the excess distortion exponent Eopt J � PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' t) for optimal (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx) JSCC with distortions Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx and transmission rate t is bounded by Eopt J � PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' t � ≤ min R∈R � t �E �R t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X � + Esp � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) Eopt J � PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' t � ≥ min R∈R � t �E �R t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X � + Eex � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) Herein R ≜ {R : tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y )}, �E(r, PX, PS|X) ≜ min QX min US|X∈C(X→S): R(QX,US|X,Ds,Dx)≥r � D(QX||PX) + D(US|X||PS|X|QX) � , (7) Esp(R, WZ|Y ) ≜ max PY min VZ|Y :IPY ×V (Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='Y )≤R D(VZ|Y ||WZ|Y |PY (QX)), (8) Eex(R, WZ|Y ) ≜ max PY min PY � Y :P � Y =PY � EdWZ|Y (Y, �Y ) + IPY � Y � Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' �Y � − R � , (9) where D(V ||W|P) ≜ � y∈Y P(y)D (V (·|y)||W(·|y)) denotes the conditional K-L divergence and dWZ|Y (y, �y) is named as the Bhattacharya distance between two channel inputs [22, Chp 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' C(WZ|Y ) is the channel capacity and the rate distortion function characterizing semantic infor- mation is given by [5, Thm 1] as R(PX, PS|X, Ds, Dx) ≜ min P ˆ S, ˆ X|X I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ˆS ˆX), (10) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' E[ ˆdS(X, ˆS)] ≤ Ds, (11) E[dX(X, ˆX)] ≤ Dx, (12) where ˆdS(x, ˆs) = E[dS(S, ˆs)|X = x], while dS(·, ·) and dX(·, ·) denote the component-wise distortion functions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 10 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 2: Characterization of the excess distortion exponent in a toy case (a) Upper and lower bounds in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (b) Source excess distortion exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7) In Theorem 1, upper and lower bounds of the JSCC excess distortion exponent are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The upper bound consists of the sphere-packing bound on channel error exponent Esp(R, WZ|Y ) and the source excess distortion exponent �E(r, PX, PS|X), which considers a new fidelity on semantic source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Meanwhile the lower bound is composed of the expurgated random coding bound on channel error exponent Eex(R, WZ|Y ) and the same source exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that our result is a generalized form of Csiszar’s error exponent [25] in which a lossy JSCC encodes a single source X and imposes a unique constraint on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The basic idea to prove the results in Theorem 1 is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To obtain the source exponent, we characterize the excess distortion probability in terms of sources, by counting the numbers of typical semantic and observable sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The joint typicality among the semantic, observed and received sequences is necessary to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To obtain the channel exponent, we prove the sphere-packing bound and the expurgated random coding bound still hold for the semantic-aware transmission in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 1 via Csiszar’s channel coding theorem [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, by minimizing the source and channel exponents jointly over a group of JSCC schemes, we formulate upper and lower bounds on optimal JSCC excess distortion exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We present an example of excess distortion exponent under a semantic-aware communication system in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 2, in which a toy case is considered that semantic source takes value in S = {1, 2, 3} with equal probability, X = {0, 1} and a binary symmetric channel (BSC) with flip rate Upper bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) Lower bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='8 Ej(Px,Psix,Wzly,R,t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Ds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 E(r,Px,Ps|x) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 +0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Ds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='011 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The left hand side one plots the upper and lower bounds of JSCC exponent, while the right hand side one gives the source exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In this case, it shows that the exponent turns to be a non-decreasing function over semantic and observed distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Nevertheless, the behavior of the exponent for generalized (S, X) and WZ|Y is unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Two Degenerated Cases on Semantic Source In this subsection, we study the special case where Ds = ∞ or Dx = ∞, which means the constraint on semantic or observed information is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' As follows, we verify that the bounds in Theorem 1 can be reduced to the known results under simpler settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', Csiszar’s JSCC error exponent and Weissman’s error exponent for noisy source coding [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Corollary 1 (Csiszar’s JSCC error exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Without the semantic constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', Ds = ∞, the communication system focuses on reconstructing the observed information X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Consequently, the excess distortion exponent of an optimal (n, k, ∞, Dx)-JSCC scheme, with the absence of achievable semantic constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (11), is reduced to the Csiszar’s JSCC error exponent with a fidelity criterion [25, Thm 2, Thm 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proof: This corollary can be obtained intuitively due to the neglect of semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For a rigorous proof, according to the excess distortion exponent of semantic source given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7), the conditional divergence can be rewritten as D � US|X||PS|X|QX � = D � QX × US|X||QX × PS|X � = � � � 0, if QX × US|X = QX × PS|X ∞, otherwise , which implies the infimum of this term can be obtained by choosing US|X = PS|X such that minUS|X D � US|X||PS|X|QX � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Comparing Theorem 1 and Corollary 1, it can be observed that the JSCC error exponent for semantic-aware systems contains an extra non-negative term and is larger than the Csiszar’s error exponent in non-trivial cases, which is translated as a faster error decay speed owing to the involvement of the semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Corollary 2 (Weissman and Merhav’s Error Exponent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Without the constraint on observed infor- mation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=', Dx = ∞, the model is degenerated to an indirect source coding and communication 12 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7) can be reduced to Weissman and Merhav’s error exponent [33, Th 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proof: Note that in this case, we focus on a noisy source excess distortion exponent, where the reconstruct semantic sequence ˆsk is deterministic for fixed xk and the compression scheme, since ˆsk = ψk S � ϕk � xk�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Given the sequences � sk, xk� and the excess distortion event {dk S � sk, ˆsk� > Ds}, the rate-distortion function becomes R(PX, PS|X, Ds, ∞) ≜ min P ˆ S|X I(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ˆS), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' E[ ˆdS(X, ˆS)] ≤ Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The excess distortion is established directly between the transmitted xk and the reconstructed sequence ˆsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Thus, given xk of type QX, the test channel US|X can be rewritten as: US|X = W × V , W ∈ C � X → ˆS � : r ≥ IQX×W � X, ˆS � , V ∈ C � ˆS × X → S � : EQX×W ×V dS � S, ˆS � > Ds, in order to impose the constraints on mutual information and the distortion threshold, which directly yields the conclusion in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ACHIEVABLE EXCESS DISTORTION EXPONENT IN SEMANTIC-AWARE MIMO SYSTEMS It is observed that Theorem 1 is easy to be extended into different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In wireless networks, it may be of interest to consider a MIMO block fading channel rather than a simple DMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In this section, we present an achievable statement of JSCC excess distortion exponent for a semantic-aware MIMO system composed of Gaussian distributed semantic source and block fading MIMO channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, the optimization problem of exponent is solved in a simple case for explicit analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' A proposition is used to claim the extension of Theorem 1 to a more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For a communication system, which consists of a joint Gaussian vector pair (S, X) and a Gaussian channel WZ|Y with arbitrary memory, the lower bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proof: Note that we considered discrete sources and channel with finite input and output alphabets before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Nevertheless, in [37, Ch 4] Zhong etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' combines Csiszar’s exponents with 13 Gallager’s reliability functions in discrete cases via Fenchel duality, in which Gallager’s state- ments are verified to be powerful tools on error exponent and are easily extended to the case of continuous-alphabet with arbitrary memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' It is worth mentioning that in Gaussian case, only random coding bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) can be extended and the sphere-packing bound remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For details, the reader can turn to [26, 27] for a rigorous proof of JSCC error exponent under the setting of Gaussian distributed system and a one-order Markovian system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hence, by slightly abusing the notation defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' II, we can easily obtain this conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that the lower bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) can be extended into a MIMO communication setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' However, even not consider the semantic source, the converse proof on error exponent is not easy to obtain in such a MIMO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Among these proofs, Fano inequality and hypothesis testing show unavailable bounds on exponent, while sphere-packing bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) though yields a tight bound in single antenna case, it is difficult to be extended into the multi-antennas case due to the following two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' First, for the sphere-packing of codeword, the solid angle of the Voronoi regions for matrices in continuous alphabet is difficult to characterize, hence the overall cone is not a circular cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Second, the strong converse for JSCC in Lemma 1 does not necessarily hold in a MIMO communication system, since we cannot use the weak law of large numbers (WLLN) for memory case [27, Appendix 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In summary, we present an achievability bound as follows, which reveals an achievable excess distortion exponent of JSCC for semantic-aware MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For the above semantic-aware communication system, the observed source X follows a Gaussian vector distribution N(0q, ΣX), where ΣX is an q × q positive semi-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Meanwhile the ℓ-length semantic source S is given by S = hX + N, where h is an ℓ × q matrix, and N is a random vector follows N(0ℓ, ΣN), which is independent of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The quadratic distortion measures become dS(s,ˆs) = tr{(s − ˆs)(s − ˆs)H} and dS(x, ˆx) = tr{(x−ˆx)(x−ˆx)H}, where ˆs and ˆx refer to the recovered source vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, a MIMO communication system contains nT transmit and nR receive antennas, where the block fading channel WZ|Y remains invariant for Nc symbols in each coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In each observation composed of Nb independent coherence intervals, which amount to NbNc symbols, the received 14 matrix Zi ∈ CnR×Nc at the i-th interval can be formulated as Zi = HiYi + Wi, i = 1, 2, · · · , Nb, where Yi ∈ CnT ×Nc is the channel input matrix, Hi is the channel state matrix and Wi is the additive white Gaussian noise matrix, namely Wi ∼ MN(0nR×Nc, NwInR, INc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Here Nw is the noise coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The transition probability with perfect CSI at the receiver can be stated as p(Z|Y, H) = (πNw)−nRNc exp � − 1 Nw (Z − HY)(Z − HY)H � , (13) where the subscript i is dropped for simplicity since the channel is memoryless for each coherence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that Y denotes the power constrained input as 1 NcE[tr{YYH}] = tr{Q} ≤ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' there exists an (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx)-lossy JSCC for this semantic-aware MIMO communication system with excess distortion exponent Eopt J � PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' t � = min R∈R EJ � PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' t � = min R∈R � t �EG �R t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X � + EMIMO � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (14) where �EG(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' PS|X) = min ∆:O≺∆⪯ΣX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' tr{∆}≤Dx min A∈Cq×q: det(A)=det(∆)e2r min B∈Cℓ×ℓ: tr{B}=Ds−tr{hH ∆h} � 1 2 log det(ΣX) det(ΣN) det(∆)e2r det(B) + tr � Σ−1 X A + Σ−1 N B + � Σ−1 N B − Iℓ � hHΣXh � � − ℓ − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (15) EMIMO � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' WZ|Y � = max 0≤ρ≤1 � max δ≥0 Eex(Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Nc) − ρR � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (16) Eex (Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Nc) =2δρP − 1 Nc ln EH � det � QA � InT − Q �HHHA−1HHH 16N 2 wρ2 − HHH 4Nwρ + δ ���−Ncρ� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (17) A =δInT − Q−1 − 1 4NwρHHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (18) 15 Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In this theorem, we present an achievable JSCC excess distortion exponent in a semantic- aware MIMO communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Specifically, we consider a jointly Gaussian distributed source pair, where an ℓ-length semantic vector S is combined with a q-length observed vector X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, the quadratic distortion measure and a MIMO system with block fading channel are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The optimal exponent Eopt J is composed of two parts, namely the source exponent and the expurgated random coding exponent for MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' From the achievable bound, the ergodic capacity and cut-off rate of the above semantic-aware MIMO communication system can be obtained, by setting ρ = 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We note that the generalized vector nature complicates the statement of the exponent, which makes the optimization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (14) difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hence a simple case is discussed as follows, which enables us to further analyze and reveal some insights on the JSCC scheme design in such a semantic-aware MIMO communication system, for optimal excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Corollary 3 (Excess distortion exponent for semantic-aware MIMO systems in specific case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Under the same setups in Theorem 2, we further assume X ∼ N(0q, σ2 XIq), N ∼ N(0ℓ, σ2 NIℓ) and H ∼ MN(0nR×nT , InR, InT ) (for simplicity we set nR ≤ nT), Q = P nT InT due to the equal power assignment on the transmitting antennas, and denote SNR = P Nw , then the following equations hold: (a) For the expurgated random coding bound for MIMO channel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (18), the derivatives are calculated as ∂Eex (Q, ρ, δ, Nc) ∂δ =2ρP − 2ρnTSNR 1 − δ SNR + SNR Nc tr � K−1(ρ, δ)∂K(ρ, δ) ∂δ � (19) ∂Eex (Q, ρ, δ, Nc) ∂ρ =2δP + 2nT ln(1 − δ SNR) − 1 Nc tr � K−1(ρ, δ)∂K(ρ, δ) ∂ρ � (20) where K(ρ, δ) is a Hankel matrix with size nT × nT whose (i, j)-th entry follows hypergeo- metric function (nT − nR + i + j − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2F0 � nT − nR + i + j − 1, Ncρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' − SNR 2(1 − δ SNR)ρ � (b) Given tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y), the JSCC excess distortion exponent is convex in terms of code rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 16 (c) Given the above source-channel pair, and the transmission rate t, the optimal achievable code rate R⋆ can be formulated by R⋆ = 2t ln tρ⋆ + 2 2 min � σ2 X σ2 Xtr{hT h}+σ2 N (Ds − σ2 N), 1 σ2 X Dx � (21) where ρ⋆ satisfies ρ⋆ = arg max 0≤ρ≤1 {Eex(Q, ρ, δ, Nc) − ρR} according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Proof: Given Q = P nT InT and Gaussian distributed random matrix H, (a) is obtained by Eex � P nT InT , ρ, δ, Nc � = 2δρP − 2ρnT ln(1 − δ SNR) − 1 Nc ln det(K(ρ, δ)) K (22) where K is given above and K = �nT i=1(nR − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (i − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' according to [19, Cor 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This corollary enables us to obtain the expectation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (18) via the computable generalized hypergeometric function 2F0(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' ·) (given by [38, Eq (3)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hence the partial derivatives are stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (19) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For (b), the conclusion is also direct since the second partial derivative over code rate R is positive when the code rate lies in the interval tR(PX, PS|X, Dx, Ds) ≤ R ≤ C(WZ|Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For (c), we obtain the maximization of the excess distortion exponent EMIMO � R, WZ|Y � = Eex (Q, ρ⋆, δ, Nc) − ρ⋆R over δ and ρ successively, and solve ∂ �EG( R t , PX, PS|X) + EMIMO � R, WZ|Y � ∂R = 0 Herein, we analyze the properties of the semantic-ware JSCC excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Due to the matrix essence, we evaluate the expectations on coefficient matrices H via the generalized hypergeometry function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, the partial derivatives are derived in order to solve the optimization problem on JSCC exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that the statement of exponent is formulated in the form of an optimization problem over coding rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The solution explicitly presents the optimal design of JSCC scheme for semantic-aware MIMO systems in error exponent sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 3: An illustration of the JSCC excess distortion exponent with some key quantities V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, theoretical bounds of excess distortion exponent in semantic-aware MIMO communication systems are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' From the simulations, we verify the convexity and show the key quantities like rate-distortion function and ergodic capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then, we explore the in- fluences of MIMO communication systems on semantic information reconstructions, such as coherence time, exponential correlation coefficient and the numbers of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, we also discuss how the optimal code rate R⋆ behaviors in terms of different transmission rate t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Experiments Setups We consider the above semantic-aware MIMO communication system in Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 3, which consists of a joint Gaussian source pair (S, X) with ΣS = 3Iℓ and ΣX = 4Iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, we assume the same number of transmit and receive antennas nT = nR, and the channel state matrix HHH ∼ Q(InT , GT, GR) (given by [15]), in which we adopt exponential correlation matrices GT = {α|i−j| T }, GR = {α|i−j| R } (Simply assuming αT = αR = α ∈ [0, 1)) to model the spatial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Besides, the signal-to-noise ratio can be calculated by SNR = P Nw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Convexity of JSCC Exponent over Code Rate R for Semantic-Aware MIMO Systems In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 3, the JSCC excess distortion exponent for a MIMO system EJ � PX, PS|X, WZ|Y, R, t � is plotted against the code rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The distortions Ds = 2, Dx = 1, coherence time Nc = 1, EMIMO(R,WzIy) in [39, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 1] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 EMIMO(R,WzY) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Achievable exponent Ej(Px,Psix,Wziy,R,t) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 E(r,Px,PsIx) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Fopt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 R(Px,PsIX,Ds,Dx)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='8 C(WzIY)=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 R 0 1 2 3 4 5 6 R18 nT = nR = 3, correlation coefficient α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3, SNR= 15, and the transmission rate t = 2 symbol/channel use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that the rate-distortion function characterizing the semantic information R(PX, PS|X, Ds, Dx) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='8 and the ergodic capacity C(WZ|Y) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='1, which is marked in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The solid line represents JSCC exponent function, which consists of source exponent in the dashed dotted line and MIMO channel exponent given in the expurgated bound Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (17) marked by the star labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In comparison, we also present the Gallager’s random coding bound of MIMO channel [39] in dashed line to verify its suboptimality, since the ’bad’ JSCC codewords are expurgated for a better error probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Due to the convexity of the JSCC exponent, we focus on the minimum of EJ, which is labeled by Eopt J and the optimal JSCC code rate R⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Optimal JSCC Excess Distortion Exponent for Semantic-Aware MIMO Systems In this subsection, we provide plots on the exponential behaviors of the optimal JSCC excess distortion probability for semantic-aware MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We investigate how the source and channel key quantities influence the best exponent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Excess Distortion Exponent against Semantic Distortions: Based on the depicted system, the excess distortion exponents against semantic distortion are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 4(a) in terms of different observed distortions, which intuitively illustrates the tradeoff between Ds and Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The star, triangle and diamond lines represent the optimal performance with Dx = 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Obviously both the increase of Ds and Dx leads to the increase of the exponents, but the curves remain invariant when the semantic distortion becomes inactive, since the observed constraint is more demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Under the aforementioned experiment setups, the achievable optimal JSCC excess distortion exponent attains its limit around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='24 when Dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Excess Distortion Exponent against SNR: We first compare the optimal excess distortion expo- nent in terms of MIMO systems with different numbers of antennas, namely nT = nR = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 4(b), the exponent increases with the number of antennas dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Specifically, under a higher SNR environment, a semantic-aware MIMO system with a 4 × 4 array obtains Eopt J ≥ 1, while a 2 × 2 system only has 1/10 performance on the exponential probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' This trend demonstrates the compatibility of semantic-aware communication systems and the massive MIMO techniques, with an even larger nT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 4(c), we explore how the coherence time Nc in MIMO system affects the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The star, triangle and diamond curves stand for the optimal exponent ranges from 1 to 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 19 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 4: Excess distortion exponent in terms of: (a) observed distortions, (b) number of antennas, (c) coherence time, (d) correlation coefficient It can be observed though the longer coherence time results in longer block length, the optimal exponent decreases with Nc at arbitrary SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 4(d), the plot shows the performance in terms of exponential coefficient with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The optimal exponent increases with α, since the larger α means the better channel transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Achievable Optimal Code Rate R⋆ in terms of MIMO Key Quantities In this subsection, we investigate the optimal JSCC code design for the semantic-aware MIMO systems in error exponent sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Fixing Dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5, Ds = 2 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3, the ratio t = k n is plotted against the optimal code rate R⋆ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The transmission rate t increases with the optimal code rate and turns a slightly decrease with coherence time Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 5(b), the exponential correlated coefficient α shows a positive effect on transmission rate t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that given t and R⋆, we obtain the optimal JSCC scheme for the semantic-aware MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='10 Dx= 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='05 E Dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='25 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Dsnt=2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 nT=3 nt=4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 SNR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='7 Nc=1 Nc =2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 Nc=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 SNRα= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='8 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='9 α=( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 SNR20 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 5: Different JSCC Schemes against Optimal Code Rate R⋆ in terms of (a) different coherence time Nc (b) different correlation coefficient VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' CONCLUSION In this paper, we obtained upper and lower bounds of JSCC excess distortion exponent in semantic-aware communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We concluded that the participation of semantic source enlarges the JSCC excess distortion exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In a semantic-aware MIMO system, we presented an achievable bound for JSCC excess distortion exponent, which extends the conclusion to a practical communication scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' As a result, based on the achievability bound, we solved the optimization problem in a simple case and discussed the design of the optimal JSCC scheme in terms of the code rate and transmission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, from the numerical results, we show the tradeoff between the two distortions and demonstrated that more antennas and larger correlation efficient lead to a better JSCC excess distortion exponent, while longer coherence time reduces its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In future works, a tight converse bound for JSCC exponent in semantic-aware MIMO com- munication system will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Furthermore, the analysis for finite block length case will be also studied for designing practical coding schemes with advantages over the conventional non-semantic-aware communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 7 Nc=1 6 Nc=2 Nc=3 5 4 R 3 2 1 +0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Transmission rate t10 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='6 8 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='9 6 R 4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='0 Transmission rate t21 APPENDIX A PROOF OF THEOREM 1 Lemma 1 (JSCC Theorem [40] in Semantic-aware Scenario).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Given a memoryless source pair (S, X) and a memoryless channel WZ|Y , where tR(PX, PS|X, Ds, Dx) > C(WZ|Y ), then for any (n, k, Ds, Dx) JSC code we have lim n→∞ PJ � PS|X, PX, WZ|Y , n, k � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (23) Lemma 2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' [41, Theorem 2] [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 175]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We assume a channel W of capacity C(W) and an n-length list code (ϕ, ψ) has a range of decoded sequences of cadinality l, which we call the list size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' An erroneous event of list codes occurs when a true message is not on the decoding list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Let pe(n, R, L) denote the minimal error probability, and pe(n, R, L) be the maximal error probability for such an n-length list code with rate R and list size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then for positive ˜ϵ1(n) → 0 and ˜ϵ2(n) → 0 as n → ∞, and any R > C(W) + L where L = 1 n log l, we have pe(n, R, L) ≥ exp {−n (Esp(R − L, W) + ˜ϵ1(n))} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (24) pe(n, R, L) ≤ exp {−n (Eex(R − L, W) − ˜ϵ2(n))} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (25) Lemma 2 is a channel coding theorem based on list codes, which characterizes the error probability via sphere-packing bound and expurgated bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that expurgated bound is a refined version of random coding bound, which drops the bad codewords beyond the Bhattacharya distance [22, Chap 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The outline of the proof is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We start by presenting a strong converse for JSCC coding theorem under semantic-aware communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To obtain the excess distortion expo- nent, it is reasonable to model the non-trivial source-channel pairs of tR(PX, PS|X, Ds, Dx) ≤ C(WZ|Y ), which is given in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Here, we focus on investigating the optimal JSCC scheme of code rate within this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Then, as mentioned above, the excess distortion probability, which refers to the ratio of over-distorted sequences to overall sequences, can be bounded by computing the sizes of typical sets of sources and channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Given the source pair (sk, xk), the excess distortion probability from the over-distorted codewords is bounded in this appendix via Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, the average numbers of the corrupted sequences from the semantic 22 and observable sources are stated as well, by investigating the typical sequences xk and the conditional typical sequences sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, by combining these results from source with channel parts and minimizing the sum in terms of code rate R, we obtain final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' According to Lemma 1, we investigate the excess distortion performance of a group of JSCC schemes with code rate tR(PX, PS|X, Ds, Dx) ≤ R ≤ C(WZ|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Now we recall the overall excess distortion probability defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (3), and rewrite P {E} = � xk∈X k PXk(xk) � sk∈Sk PSk|Xk(sk|xk)pc(sk, xk), (26) where we use pc(sk, xk) ≜ � zn∈E(sk,xk) PZn|Y n(zn|ϕn(xk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For the excess distortion probability pc(sk, xk), let TQ denote the typical set of sequences xk ∈ X k of type QX, TU be the joint typical set of sequences (sk, xk) ∈ Sk×X k, and TU(xk) = � sk : (sk, xk) ∈ TU � is the conditional typical set of sk for a given xk, in which the conditional empirical distribution is US|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, the conditional typical set TV (zn) = {(sk, xk) : (sk, xk, zk) ∈ TV } based on the joint typical set TV composed of sequence tuple (sk, xk, zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that the size of the conditional typical set is |TV (zn)| ≤ exp {kH(S⋆, X⋆|Z⋆)} = exp {k (H(S⋆, X⋆) − I(S⋆, X⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Z⋆))} ≤ exp � k � H(QX, US|X) − R(QX, US|X, Ds, Dx) �� , where S⋆, X⋆ and Z⋆ are three arbitrary auxiliary random variables characterizing the joint distribution PS⋆X⋆Z⋆, which is a possible joint type of sequences xk ∈ TQ, sk ∈ TU(xk) and zn ∈ Zn within the distortion constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Next, the number of all possible joint types is upper bounded by (k + 1)|S||X||Z| via the type counting lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Hence the list size can be bounded by l ≤ (k + 1)|S||X||Z| exp � k � H(QX, US|X) −R(QX, US|X, Ds, Dx) �� = exp � k � H(QX, US|X � −R(QX, US|X, Ds, Dx) + ˆϵ1(k) �� , where k = nt, ˆϵ1(k) = 1 k log(k + 1)|S||X||Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that lim k→∞ ˆϵ1(k) = lim k→∞ 1 k log(k + 1)|S||X||Z| (a) = |S||X||Z| lim k→∞ 1 k log(k + 1) = 0, (27) where equality (a) is because the product of alphabet cardinalities, |S||X||Z|, is a finite constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 23 Thus, the parameter L is L = 1 n log l ≤ t � H(QX, US|X) − R(QX, US|X, Ds, Dx) + ˆϵ1(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (28) Since the size of the message set satisfies exp{nR} ≤ exp{kH(PS,X)} = exp{ntH(QX, US|X)}, (29) by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (28) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (29) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (24), we have pc(sk, xk) = exp � −n � Esp(tR(QX, US|X, Ds, Dx) − tˆϵ1(k), WZ|Y ) + ˜ϵ1(n) �� (b) ≥ exp � −n � Esp(tR(QX, US|X, Ds, Dx), WZ|Y ) + ϵ1(n) �� , (30) where the inequality (b) holds since Esp(R, W) is a non-increasing function in R, and ϵ1(n) = tˆϵ1(n) + ˜ϵ1(n) → 0 as n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Now given an observed sequence xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' we define pa(xk) = � sk∈Sk PSk|Xk(sk|xk)pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' and pa(xk) = � US|X∈U � sk∈TU(xk) PSk|Xk(sk|xk)pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk) (c) = � US|X∈U � sk∈TU(xk) � (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='b)∈(S×X) PS|X(a|b)N((a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='b)|(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='xk))pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk) = � US|X∈U ��TU � xk��� exp � −kEQX×US|X � − log PS|X(S|X) �� pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk) (d) = � US|X∈U ��TU � xk��� exp � −kH(US|X|QX) � exp � −kD � US|X �� PS|X �� QX �� pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk) (e) ≥ � US|X∈U (k + 1)−|S||X| exp � −kD � US|X �� PS|X �� QX �� pc(sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (31) where Q denotes the set of all types QX and U denotes the set of conditional types US|X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In (c) we apply the empirical count function N((a, b)|(sk, xk)) on the conditional probability, in (d) we use the definition on the conditional divergence defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (1), and in (e) the following 24 result is used ([24, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='3]): (k + 1)−|S||X| ≤ ��TU � xk��� exp � −nH(US|X|QX) � ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' To characterize the possible conditional types US|X, given the observable sequences xk ∈ TQ, we use the conclusion that the code rate is upper bounded by rate-distortion function tR(QX, US|X, Ds, Dx) ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Thus all possible US|X should be restricted in the following set: U ≜ � US|X ∈ C(X → S) : R(QX, US|X, Ds, Dx) ≥ r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (32) Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (26), the overall excess distortion probability can be stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (33), P {E} = � QX∈Q � xk∈TQ PXk(xk)pa(xk) ≥ exp � −k � min QX∈Q � D(QX||PX) + min US|X∈U D(U||PS|X|QX) + ϵ3(k) ��� pc(sk, xk) = exp � −k � �E(r, PX, PS|X) + ϵ3(k) �� exp � −n � Esp(R, WZ|Y ) + ϵ1(n) �� , (33) where ϵ1(n) is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (30) and ϵ3(k) = −1 k log � |Q||U|(k + 1)−|S||X|−|X|� (f) ≤ 1 k log � (k + 1)|X|� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Inequality (f) holds since the number of all possible joint types of xk and sk is upper bounded by1 |Q||U| ≤ (k + 1)|S||X|, hence ϵ3(k) → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (33) with the definition of �E(r, PX, PS|X) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7), we state the upper bound on the exact excess distortion exponent as �EJ(R) ≜ lim inf n→∞ � −1 n log P {E} � ≤ k n �E(r, PX, PS|X) + Esp(R, WZ|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (34) With the achievability Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (25) in Lemma 2, we can get the lower bound similarly on �EJ(R) �EJ(R) ≥ k n �E(r, PX, PS|X) + Eex(R, WZ|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (35) 1More specifically, U ⊆ U ⋆ = {US|X : US|X ∈ C(X → S)}, and the number of all joint types is bounded by type counting lemma [24, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2] as |Q||U| ≤ |Q||U⋆| ≤ (k + 1)|S||X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 25 By substituting t = k n and r = R t in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (34) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (35), respectively, we finally obtain upper and lower bounds of excess distortion exponent in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that the excess distortion probability is stated as a function of n, k but the exponent of the optimal JSCC only concerns the transmission rate t symbol/channel use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 2 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' [42, Appendix B] For every pair of Hermitian positive definite matrices A ∈ Cm×m, C ∈ Cn×n, and denote etr(·) = exp{tr(·)}, then for any matrices B, D ∈ Cm×n, we have: � Cn×m etr � −π � AUHCU + BHU + UHD �� dU = det � A⊤ ⊗ C �−1 etr � πA−1BC−1DH� = det(A)−n det(C)−m etr � πA−1BC−1DH� This section shows how to derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (14) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (6) under a semantic-aware MIMO communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Specifically, we derive the explicit forms of the source excess distortion exponent �E(r, PX, PS|X), and the expurgated random coding bound Eex(R, WZ|Y) on channel excess distortion exponent under MIMO communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The basic idea of the proof is, for the excess distortion exponent of source pairs, we rewrite the K-L divergence according to the generalized rate-distortion function into a computable optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For the MIMO channel expurgated random coding exponent, we utilize the equivalence between Csiszar’s form and Gallager’s form by Fenchel duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' We start from a jointly Gaussian distributed vector pair (S = hX+N, X), and the reconstructed source vectors ˆS, ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The minimum of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7) is presented with two distributions QX and US|X subject to R(QX, US|X, Ds, Dx) ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Note that under the Gaussian vector assumption, the semantic-aware rate-distortion function can be rewritten as R(QX, US|X, Ds, Dx) = min 1 2 log � det(A) det(∆) � (36) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' O ≺ ∆ ⪯ A, (37) tr � h∆hH� ≤ Ds − tr{B}, (38) tr {∆} ≤ Dx, (39) 26 Moreover, in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7), the first divergence can be computed as D(QX||PX) = EQX [log QX − log PX] = EQX �1 2 log det(ΣX) det(A) + 1 2xH � Σ−1 X − A−1� x � = 1 2 log det(ΣX) det(A) − q + tr � Σ−1 X A � (40) where the auxiliary multivariate Gaussian distribution QX(x) = 1 (2π) q 2 det 1 2 (A) exp{1 2xHA−1x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Similarly, the conditional divergence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (7) can be stated as D(US|X||PS|X|QX) = EQX×US|X[log US|X − log PS|X] = 1 2 log det(ΣN) det(B) + tr �� ΣN − B−1� hHΣXh + Σ−1 N B � − ℓ (41) where the distribution US|X(s|x) = 1 (2π) l 2 det 1 2 (B) exp{1 2 (s − hx)H B−1 (s − hx)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, combin- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' eqref41 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (41) with the constraints from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (37)-(39), we obtain the semantic- aware source excess distortion exponent as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In Theorem 1, we state the channel exponent in Csiszar’s form (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (9)) in light of the sim- plicity on statement, but it is hard to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Notably, Zhong [37] combined the Csiszar’s exponent with Gallager’s reliability function via Fenchel duality and proved the equivalence, in which the later is easy to be extended and analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The reader can turn to [37, Chapter 4] [22, Chapter 7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Under the MIMO system, Gallager’s random coding bound is given by [39, Prop 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' For an expurgated bound, which is derived from a codebook expurgating the bad codewords, and hence performs better than random coding bound at lower code rate, Alfano [30, Thm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content='2] evaluated it under simple assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Herein we present the expurgated bound on error exponent in terms of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' From [22], the expurgated exponent is stated as Eex = − 1 Nc ln � H pH(H) �� Y˜Y pY˜Y(Y˜Y) exp � δ � tr � YYH + ˜Y˜Y H� − 2P �� w(Y, ˜Y, Z) 1 ρ dY˜Y �ρ dH, (42) where w(Y, ˜Y, Z) = � Z � p(Z|Y, H)p(Z|˜Y, H)dZ and − ln w(Y, ˜Y, Z) is the Bhattacharya distance between channel input matrices Y and ˜Y while Z is the channel output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Next we 27 first process the aforementioned integral with the transition probability (13) of MIMO system as w(Y, ˜Y, Z) = (πNw)−nRNc exp � − 1 2Nw tr � HYYHHH + H˜Y˜Y HHH�� × � Z exp � − 1 2Nw tr � 2ZZH − ZYHHH − HYZH − Z˜Y HHH − H˜YZH�� dZ = exp � − 1 4Nw tr � H � Y − ˜Y � � Y − ˜Y �H HH�� (43) where (43) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Moreover, by assuming a capacity achieving input distribution on matrix ˆY, we obtain � ˜Y p˜Y(˜Y) exp � δtr � ˜Y˜Y H�� w(Y, ˜Y, Z) 1 ρd˜Y =π−nT Nc det(Q)−Nc � ˜Y exp � tr �� δInT − Q−1 − 1 4NwρHHH � ˜Y˜Y H − 1 4Nwρ ˜Y HHHHY − 1 4NwρYHHHH˜Y �� d˜Y = det(Q)−Nc det(A)−Nc exp � tr � 1 16N 2 wρ2A−1YHHHHHHHY �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (44) By applying Lemma 3 again and A = δInT − Q−1 − 1 4NwρHHH, we achieve equation (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' The expectation on input matrix Y can be formulated as det(QA)−Nc � Y pY(Y) exp � δtr � YYH − 2P �� exp � tr � 1 16N 2 wρ2A−1YHHHHHHHY �� dY = det(QA)−Nc exp{−2δNCP} � Y pY(Y) exp � tr �HHHA−1HHH 16N 2 wρ2 − HHH 4Nwρ + δ � YYH � dY = exp{−2rNCP} det(QA)−Nc det � InT − Q �HHHA−1HHH 16N 2 wρ2 − HHH 4Nwρ + δ ��−Nc (45) Finally substituting (45) into (42) yields the expurgated bound on the error exponent in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' In conclusion, we state the source exponent as piecewise function Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Finally, combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' (16) with the statement of expurgated random coding bound of channel exponent, we complete the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' 28 REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} +page_content=' Shannon, “A mathematical theory of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfKwmr/content/2301.04357v1.pdf'} diff --git a/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/2301.13016v1.pdf.txt b/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/2301.13016v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..52cffdcafe1f37a209899fe9ede528a22caec43f --- /dev/null +++ b/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/2301.13016v1.pdf.txt @@ -0,0 +1,723 @@ +Exploring the Versal AI engines for accelerating stencil-based +atmospheric advection simulation +Nick Brown +n.brown@epcc.ed.ac.uk +EPCC at the University of Edinburgh +Edinburgh, UK +ABSTRACT +AMD Xilinx’s new Versal Adaptive Compute Acceleration Platform +(ACAP) is an FPGA architecture combining reconfigurable fabric +with other on-chip hardened compute resources. AI engines are +one of these and, by operating in a highly vectorized manner, they +provide significant raw compute that is potentially beneficial for +a range of workloads including HPC simulation. However, this +technology is still early-on, and as yet unproven for accelerating +HPC codes, with a lack of benchmarking and best practice. +This paper presents an experience report, exploring porting of +the Piacsek and Williams (PW) advection scheme onto the Versal +ACAP, using the chip’s AI engines to accelerate the compute. A +stencil-based algorithm, advection is commonplace in atmospheric +modelling, including several Met Office codes who initially devel- +oped this scheme. Using this algorithm as a vehicle, we explore +optimal approaches for structuring AI engine compute kernels and +how best to interface the AI engines with programmable logic. +Evaluating performance using a VCK5000 against non-AI engine +FPGA configurations on the VCK5000 and Alveo U280, as well as a +24-core Xeon Platinum Cascade Lake CPU and Nvidia V100 GPU, +we found that whilst the number of channels between the fabric +and AI engines are a limitation, by leveraging the ACAP we can +double performance compared to an Alveo U280. +KEYWORDS +Versal ACAP, AI engines, FPGAs, stencil based algorithms, VCK5000, +atmospheric advection, HPC +1 +INTRODUCTION +The Versal Adaptive Compute Acceleration Platform (ACAP) is a +new type of FPGA which combines Programmable Logic (PL) with +other facets including CPU-based Programmable Subsystem (PS) +and AI engines [4]. These AI Engines, or AIEs and we use these +two terms interchangeably throughout this paper, are of specific +interest here as they are designed to accelerate highly-parallel vec- +tor operations. The Versal AI-series contains up to 400 engines +running between 1 and 1.2 GHz, and each engine follows a Very +Long Instruction Word (VLIW) design, capable of issuing seven +instructions per cycle. AI engines are capable of undertaking 8-way +vectorized single-precision floating point operations and up to 128 +8 bit fixed point arithmetic operations per cycle. +The large amount of raw compute provided by the AIEs is inter- +esting for High Performance Computing (HPC) workloads, where +the ability to use the Versal’s PL to tailor memory accesses bespoke +to an application and the AI engines to accelerate the compute has +potential. To date there have been a very limited number of prelimi- +nary AIE studies [5] [10], and-so an important outstanding question +is whether these engines can be effectively leveraged for real world +HPC kernels. In this work we use the atmospheric advection kernel +of the Met Office NERC Cloud model (MONC) [3], which is an open +source high resolution atmospheric modelling framework, as a ve- +hicle to explore the AI engines. Following a stencil-based compute +pattern, which is very common in HPC codes, in this short paper +we explore how to best map this compute pattern onto the AIEs and +how performance compares against other approaches. This paper +is structured as follows, in Section 2 we explore the background to +this work before summarising the experimental setup in Section +3. Section 4 explores structuring our AIE kernel(s) and interfacing +these with the PL, before undertaking a performance comparison +against other hardware in Section 5. We then conclude and discuss +recommendations in Section 6. +The novel contributions of this paper are 1) An exploration of +techniques to most effectively structure AIE kernels 2) An initial +performance comparison between the AIEs and other hardware 3) +Highlighting some of the limitations of the current AIE technology +that one must consider when working with the hardware. +2 +BACKGROUND +2.1 +The Versal AI engines +The VLIW design of Xilinx’s new AI engines is such that, per cycle, +each engine is capable of issuing a maximum of two loads, one +store, one scalar operation, one fixed point or floating point vector +operation, and two move instructions. The vector unit is of size +256 bits, and focusing on single precision floating point arithmetic +in this paper, each engine is capable of undertaking up to eight +single precision floating point calculations per cycle. Consequently +it is important to ensure code is correctly vectorized to obtain +best performance on the AIEs. Based on 400 AI engines running +at 1.2GHz on the VCK5000, there is a theoretical single precision +floating point performance of 3.6 TFLOPS. +AI engines are arranged in a 2D array, with engines connected to +their neighbours in both dimensions. Each engine contains 16KB of +program memory and 32KB of local data memory and, for the later, +is able to directly access the memories of three of its neighbours +providing a total of 128KB contiguous addressable data memory +[8]. Furthermore, each engine has two 32 bit input streams and two +32 bit output streams which are combined with a FIFO to provide +128 bit access every four clock cycles. Lastly, AI engines connect to +one of their neighbours via a cascade stream which is 384 bits wide +and designed to allow arithmetic operations to be chained. +AIE code comprises two parts, kernels which will be mapped to +AI engines and a graph description which connects kernels together +via their streams and memories, as well as to the PL. Programmati- +cally there are two ways in which data can enter or leave a kernel, +arXiv:2301.13016v1 [cs.DC] 28 Dec 2022 + +Nick Brown +windows and streams [9]. Windows provide a buffer, where the +current data position in the window is tracked. For input windows +data is consumed from this buffer by the kernel, for output windows +data is written. The other approach, a stream, provides an infinite +number of scalars and vectors that can be read and written by the +kernel. There underlies an important difference between these two +approaches, where a window of data will only progress to the next +window between outer iterations of the kernel, as driven by the AIE +graph, whereas streams can continually be read from and written to +inside the kernel. Consequently, with windows one must frequently +start and stop their kernels to refresh the window data, which is +not required with streams. +Whilst the AI engines are the major focus in this paper, it is +also important to highlight the general architectural improvements +that Xilinx have made to the PL in their Versal series. Built on a +7nm process technology, numerous components including DDR +controllers and PCIe interface have been hardened compared to +previous generations [1]. Furthermore, a dedicated Network on +Chip (NoC) is provided which not only connects the PL with the +AIEs, but can also be used between IP blocks on the PL. +2.2 +Piacsek and Williams advection kernel +Advection is the movement of values through the atmosphere due +to wind and, at around 40% of the runtime, is the single longest +running piece of functionality in the MONC model [3]. The code +loops over three fields; U, V and W, representing wind velocity in +the x, y and z dimensions respectively. This Piacsek and Williams +(PW) [6] advection scheme is called each timestep of the model +and calculates advection results, otherwise known as source terms, +for each field. This advection scheme is a stencil based algorithm, +of depth one, where calculating the value of a grid cell requires +contributions from neighbouring values across all three dimensions. +In previous work [2] this kernel was ported to an Alveo U280 +using High Level Synthesis (HLS) and leveraging the dataflow +HLS pragma to run multiple components concurrently. The struc- +ture of this HLS kernel is illustrated in Figure 1, where the boxes +are dataflow regions and arrows between these are internal HLS +streams. 3D shift buffers provide a bespoke memory solution which +is capable of delivering all 27-stencil values per cycle to the advec- +tion compute stages, which was found to be the optimal approach +even though not all 27 neighbouring stencil values are required by +the advection calculations. Given this existing structure it was our +hypothesis that we could replace the advection calculation stages +with streams to and from the AI engines, still leveraging the exist- +ing tailoring of memory accesses on the PL that worked well in [2], +with the raw compute power of the AI engines. +3 +EXPERIMENTAL SETUP +In this work we are using a Xilinx VCK5000 containing a Versal +VC1902 ACAP and 16GB of DDR4-DRAM. All VCK5000 runs are +built using Vitis 2022.1, the PL is running at 300MHz, and the +VC1902 contains 400 AI engines running at 1.2GHz. We compare +against an Alveo U280 which contains 8GB of HBM2, is also running +at 300MHz, and Alveo kernels are built using Vitis 2021.1. Both the +VCK5000 and Alveo U280 are PCIe based cards hosted by a machine +containing a 32-core AMD EPYC 7502 processor and 256GB DRAM. +Figure 1: Dataflow design of HLS advection kernel from [2] +All reported results are averaged over five runs and performance +results are reported as useful FLOPS, which is the number of floating +point operations undertaken that contribute to the calculation’s +result. Our performance numbers measure device-side execution +time only and do not include the time taken to copy input data +to, or result data from, the host and device. This is because we +are most interested in the performance of the AIEs in this work, +and device-side performance therefore provides a clearer picture +when comparing against other technologies that exhibit different +host-device data transfer overheads. +4 +AIE PORTING AND OPTIMISATION +We started by decomposing the advection stencil-based calculation +into constituent operations, resulting in, for each grid cell, the +code undertaking six additions, followed by six multiplications, +then four subtractions and finally an addition reduction to sum +these subtractions together. A floating point vector of size six is +not supported by the tooling and-so we pad with an additional +two empty values to make a vector of size eight. This is why we +report useful, rather than total, FLOPS, as useful FLOPS ignores the +processing of these empty values by only considering those floating +point operations that actually contribute to the advection result. +The structure of this kernel is illustrated in Figure 2, with the +first 8-way vector addition requiring sixteen floating point numbers +comprising the operands. The multiplication requires an additional + +HBM2 or +DDR-DRAM +Read data +U +V +W +3D shift +3D shift +3D shift +buffer +buffer +buffer +Replicate +Replicate +Replicate +Advection +Advection +Advection +calculation U +calculation V +calculation W +su +SV +sW +Write data +HBM2 or +DDR-DRAMExploring the Versal AI engines for accelerating stencil-based atmospheric advection simulation +Figure 2: Illustration of AIE calculations per grid cell, with +the numbers representing the number of single precision +floating point numbers provided. +eight input numbers which are multiplied by the result of the pre- +ceding addition. We packaged this as a single AIE kernel and Listing +1 provides a partial sketch of the code. In order to prepare for the +vector addition, streams of four numbers are read and loaded into +the appropriate locations of the lhs and rhs vectors in lines 11 to 14. +These vectors are then provided as arguments to the aie::add method +at line 16, which undertakes the vectorized addition. Multiplication, +subtraction, and reductions operations are handled similarly and +omitted for brevity. It can be seen at line 6 that we are looping over +grid cells, and the directives at lines 7 and 8 instruct the AIE com- +piler to undertake software pipelining where possible, attempting +to keep the VLIW slots filled as per Xilinx’s best practice [8]. +1 +void cell_advection(input_stream ∗ __restrict in_A, +input_stream ∗ __restrict in_B, output_stream< +float> ∗ __restrict out) { +2 +aie::vector in_data; +3 +in_data=readincr_v<4>(in_A); +4 +5 +int32 cells=(int32) in_data.get(0); +6 +for (int i=0;i lhs_nums, rhs_nums; +10 +11 +lhs_nums.insert(0,readincr_v<4>(in_A)); +12 +lhs_nums.insert(1,readincr_v<4>(in_A)); +13 +rhs_nums.insert(0,readincr_v<4>(in_B)); +14 +rhs_nums.insert(1,readincr_v<4>(in_B)); +15 +16 +aie::vector vadd=aie::add(lhs_nums,rhs_nums); +17 +.... +18 +} +Listing 1: Sketch of AIE advection kernel code +The AIE API provides adaptive dataflow graphs which enables +parameters to be dynamically set at runtime. However this is not +supported by the VCK5000 shell and as such an alternative was +required for setting the number of loop iterations at line 6. This +is the reason that, for lines 2 to 5 in Listing 1, four floating point +numbers are read from the in_A stream and the first of these is +extracted, cast to an integer, and used as the number of grid cells to +loop over (this corresponding value has been streamed from the PL +on start up). We must read the number of cells as a float because +there are a maximum of two inputs and two outputs per kernel, +and both inputs are required for the loading of operands. +1 +class simpleGraph : public graph { +2 +private: +3 +kernel cell_advection_kernel[3]; +4 +public: +5 +input_plio in_A[3], in_B[3]; +6 +output_plio out[3]; +7 +8 +simpleGraph(){ +9 +cell_advection_kernel[0]=kernel::create(cell_advection); +10 +... +11 +in_A[0] = input_plio::create("krnl_0_in0", plio_128_bits, +"data/input_A.txt"); +12 +in_B[0] = input_plio::create("krnl_0_in1", plio_128_bits, +"data/input_B.txt"); +13 +out[0] = output_plio::create("krnl_0_out1", plio_32_bits, +"data/output_0.txt"); +14 +... +15 +for (int i=0;i<3;i++) { +16 +connect(in_A[i].out[0], +cell_advection_kernel[i].in[0]); +17 +connect(in_B[i].out[0], +cell_advection_kernel[i].in[1]); +18 +connect(cell_advection_kernel[i].out[0], out +[i].in[0]); +19 +}}}; +Listing 2: Sketch of AIE graph building code +This code of Listing 2 builds the high-level AIE graph, mapping +kernels to individual AI engines. Three advection kernels are cre- +ated, one for each field, and lines 11-13 defines the input and output +ports between the PL and AIE for the first kernel (kernels two and +three are omitted for brevity). These AIE ports are connected to the +kernel ports in the loop at lines 15 to 19. As described in Section +2.1, physical connections between AI engines are 32 bits wide, but +it can be seen for input ports we specify plio_128_bits at lines 11 +and 12. This directs the AIE compiler that streams on the PL are +128 bits wide (of type qdma_axis<128,0,0,0>) and therefore data will +arrive in packets of 128 bits and be unpackaged into four 32 bit +stream values. The reason for this is performance, where the PL is +running much slower (in our case 300MHz) compared to the AIEs +(1.2 GHz) and consequently in one clock cycle the PL is providing +four 32 bit numbers which the AIE will then unpack per cycle. 128 +bits is the maximum width supported, and this is why in Listing +1 the eight numbers comprising either side of the calculation are +read via two readincr_v calls of size four at lines 11-12 and 13-14. +It can be seen in Listing 2 that there is a separate kernel instance +created for each of the three fields, with each of these running +on a separate AI engine. Whilst the calculations for each field are +different, this difference lies in the specific stencil locations that +are used, and the underlying arithmetic operations are the same. +Consequently we are able to reuse the same kernel code, but provide +different values to these from the PL side per field. The performance +of this version is reported in Table 1 by the initial row, and it can +be seen that this was significantly slower compared to instead +undertaking all arithmetic operations on the PL (PL-only (no AIEs)). + +16 +8 +8 +4 +Reduction +1 +Input +Addition +Multiply +Subtract +Result +(add) +8 +InputNick Brown +Version +Performance +(GFLOPS) +Compared +to PL-only +PL-only (no AIEs) +14.32 +- +Initial +1.99 +14% +Multi-kernel +4.06 +28% +Cascade stream +2.78 +19% +Cascade multiplex +3.87 +27% +Multi-kernel windows +0.91 +6% +Chunking windows +10.32 +72% +Reduction on host +16.13 +113% +Double vectorization +18.48 +129% +Table 1: Compute performance of different versions of AIE +design compared against PL-only non-AIE implementation. +All runs undertaken in single precision floating point on Xil- +inx VCK5000 using a problem size of 67 million grid points. +4.1 +Optimising the data transfer +The maximum 128 bit width of data between the AIEs and PL was +a major reason for the poor performance of our initial version +reported in Table 1. This was because, per cycle, the PL was only +able to stream four single precision floating point numbers per +stream to the AIE, whereas 24 were required (16 for the addition +and 8 for the multiplication). The number of inputs to an AIE kernel +is limited to two, therefore meaning that the PL could provide a +maximum of eight values per cycle. Consequently three writes on +each stream were required per grid cell and this conflict resulted in +an initiation interval of three in our HLS code on the PL. +To address this we experimented with alternative kernel struc- +tures and, as illustrated by Figure 3, split the code into multiple +kernels each corresponding to a specific operation. By splitting +apart the addition and multiplication, so each handles four of the +eight calculations, we were able to increase the overall number of +streams to six (two per kernel). There is a downside, as each individ- +ual kernel is now under utilised because it is now only undertaking +four vectorized operations per cycle rather than eight, but this split- +ting results in six, rather than two, 128 bit streams connecting the +PL to AIE kernel inputs. Consequently the HLS kernel running on +the PL is able to stream the entirety of a grid cell’s required data +each cycle, reducing the initiation interval to one. +The performance of this approach is reported by the multi-kernel +row of Table 1, and whilst this doubled performance compared +to the initial version, it was still slower than the PL-only imple- +mentation. When undertaking profiling of our multi-kernel code +using Vitis analyzer, we discovered that kernels were stalling on +stream reads for over 60% of the time. This is because, as described +in Section 2.1, the physical streams between AI engines are 32 bits +wide whereas per vectorized operation the kernel is generating 128 +bits. Consequently the kernels were stalling waiting for the arrival +of this data before operating upon it. +Connecting AIE kernels via cascade streams is an alternative +approach and these, unlike the normal 32 bit streams, are 384 bits +wide. We packed the 128 bit results into the cascade stream’s accfloat +type, and streamed the entirety of the required data in one cycle. +However, the limitation with cascade streams is that they physically +Figure 3: Multi-kernel design, with constituent operations +running across AIEs and connected by streams. Blue arrows +are internal streams, green arrows are external streams be- +tween the AIEs and PL. +connect between AIE cores by travelling in a horizontal manner, +and when reaching the edge of a row connecting to the core above. +Consequently their connection is inflexible, with each AIE core +capable of only consuming cascade stream input from a single +predefined neighbour and providing cascade stream output to its +other neighbour. This is a problem for our multi-kernel design +illustrated in Figure 3 as the subtraction-reduction kernel requires +inputs from two kernels, effectively requiring two cascade streams +to feed into an AIE which is not possible on this architecture. +Therefore, to experiment whether cascade streams would im- +prove performance, we adopted the design illustrated in Figure +4, where one addition kernel undertakes all eight addition oper- +ations, and a separate kernel then undertakes the multiplication, +subtraction, and reduction. The performance of this configuration is +reported by cascade stream in Table 1, and the major reason for the +poor performance is that the initiation interval on the PL increased +to two as streams to the addition kernel require two writes per PL +cycle as all eight pairs of operands are required by the single kernel. +To address this we multiplexed the cascade streaming approach, +with two separate copies on the AI engines such that, on average, +over two clock cycles each AIE configuration receives its data. Per- +formance of this approach is reported by the cascade multiplex row +in Table 1, which improved performance but was still slower than +that obtained by the non-AIE PL-only approach. Incidentally we +also experimented with 4-way and 8-way multiplexing but this had +no measurable improvement on performance. +Figure 4: Cascade streaming approach, Red arrow is cascade +stream, green arrows are external streams to/from the PL. +To this point we have explored connecting kernels and the PL +via streams, however it is also possible to use windows which +provide buffers. Importantly, an AIE can read up to 256 bits per +cycle from memory compared to 32 bits from streams. Therefore we + +Addition +Multiplicatior +Subtraction +Reduction +Addition +MultiplicatiorMultiplicatior +Addition +Subtraction +ReductionExploring the Versal AI engines for accelerating stencil-based atmospheric advection simulation +reverted to our multi-kernel design of Figure 3 and used windows +instead of streams between the kernels as well as to drive input +and output data between the PL. This is illustrated in Figure 5 and +the performance is reported as multi-kernel windows in Table 1. It +can be seen that the performance was extremely poor and this is +because we were operating the windows on a grid cell by grid cell +basis. This meant that there was no longer a pipelined loop within +each kernel because between each grid cell the kernel was stopped +and restarted by the AIE graph to fill and empty the windows as +required by the AIE tooling. +Figure 5: Multi-kernel windowing approach, coloured +squares are the windows, blue connects kernels internally, +and green arrows are external streams to/from the PL. +We modified our windowing approach to work in chunks, where +data for a number of grid cells is buffered into the windows and +these operate ping-pong fashion where one copy is filled with data +from the producer (either the PL or another AIE kernel) whilst +the other window copy is being consumed, with these switched +between outer iterations of the AIE graph. Consequently our ad- +dition, multiplication, and subtraction-reduction AIE kernels are +concurrently processing different chunks of grid cells based upon +the data available, effectively operating as a pipeline. An added com- +plication was that because AIE kernels are operating out of sync, +for example the multiplication kernels are one chunk behind their +corresponding addition kernels, this stalled the PL. This was be- +cause when streaming data to the AIEs, writes to the multiplication +streams are blocked waiting for the window to become free, but this +waiting on the PL also blocks writes to the addition streams which +are required to progress the addition AIE kernel which will unlock +its multiplication kernel. The solution was to implement explicit +ping-pong buffering on the PL for the multiplication streams, with +a dataflow region working in sizes of chunk which is concurrently +filling a buffer with the current chunk’s data and streaming out the +previous chunks data to the AIE kernel. +Performance is reported by chunking windows in Table 1, where +it can be seen that this approach has significantly increased per- +formance on the AIEs, however it is still slightly slower than the +PL-only. Based on profiling via Vitis analyzer we found that the +subtraction and reduction kernel was taking around double the +execution time of the other kernels and this imbalance of work was +causing additional stalling. Consequently we modified the kernel +to perform subtraction only and streamed back to the PL 4 floating +point numbers which the PL then adds together. This is reported by +the row reduction on host in Table 1 which outperforms the PL-only. +As described previously, in this multi-kernel approach each ker- +nel is only working with vector sizes of four whereas the hardware +is capable of undertaking eight single precision floating point oper- +ations per cycle. Working with windows, it was trivial to read two +grid cells concurrently, placing the first in the lower portion of the +vector and the second grid cell in the higher portion. Consequently +this meant that vector operations were now running over eight +operands, effectively processing two grid cells per AIE vectorized +operation. This is reported by double vectorization in Table 1 and +resulted in a performance improvement, albeit modest as we are +still limited to streaming data for only one grid cell between the PL +and AIEs per cycle due to the maximum of port width of 128 bits. +5 +MULTIPLE HLS COMPUTE UNITS +In Section 4 we focused on a single PL HLS Compute Unit (CU). +By decomposing across the advection problem’s grid space, we can +scale to multiple HLS CUs, all with a separate 3D part of the grid +and working independently, connected to their own AIEs. Using +our optimised AIE approach, which requires fifteen AIEs per HLS +CU, we compared performance against other hardware options and +Table 2 reports these results. +The advection kernel running on the AI engines of the VCK5000 +is reported by the row VCK5000 AIEs in Table 2. Whilst not doc- +umented directly, there are a maximum of 78 128-bit PLIO input +streaming interfaces that only become apparent during compilation +as we scaled. This is because AIE tile contains eight 32-bit AI Engine +to AXI4-Stream channels [7] and there are 39 tiles. Consequently, +there are a total of 312 32-bit channels connecting the AIEs to the +PL, or a maximum of 78 128-bit channels as each of these is built +using four 32-bit links. Incidentally AIEs accessing DRAM directly, +without the PL, would also encounter this limitation as the data +still needs to traverse these same physical links. +With six input streams per field, and three fields per CU, this +results in a maximum of four HLS CUs. We are therefore using 60 +AIEs in total, and up to four CUs the performance scales well. Con- +sequently this hardware restriction is a major limitation because, if +we were able to scale to a greater number of CUs, then performance +would likely increase significantly. The importance of streaming an +entire grid cell per cycle between the PL and AIEs was highlighted +in Section 4, and out of the two AIE kernel designs which enable +this, multi-kernel and multiplexed cascade stream, the multi-kernel +is preferable in this regard as it requires six input streams per field +compared to eight for the multiplexed cascade stream. +The Alveo U280 was configured with six HLS CUs, which is +the maximum number that can fit due to limits on the number of +ports in the Alveo shell. It can be seen that performance on the +Alveo U280 is similar to that obtained on the VCK5000 using AI +engines, even though there are only four CUs on the VCK5000. This +is especially impressive considering that the U280 contains external +HBM2 memory whereas the VCK5000 only has DDR4. We are able +to fit eight CUs onto the PL-only VCK5000 configuration, which +does not suffer from limitations on the number of ports due to the +Versal containing a NoC which HLS kernels are connected to. The +VCK5000 combined result reports performance for a combination + +Addition +Multiplicatiol +Subtraction +Reduction +Addition +MultiplicatiolNick Brown +of the four AIE CUs with six PL-only CUs on the VCK5000, and +this combined approach which leverages both the AIEs and PL for +calculations delivers double the performance of the U280. +By comparison, the scheme running over the 24-core Cascade +Lake Xeon Platinum CPU, which was threaded via OpenMP and +compiled using GCC version 10.2 performs poorly compared with +every other hardware technology. The V100 GPU version is imple- +mented using OpenACC and version 20.9 of the Nvidia compiler, +and this out-performs all other CPU and FPGA configurations, +which is largely in agreement with [2]. Whilst the Versal has closed +the gap with the GPU, it is unfortunate that AIE hardware restric- +tions ultimately limit the number of AIE CUs to four. +Description +Performance (GFLOPS) +VCK5000 AIEs (4 CUs) +68.73 +VCK5000 PL-only (8 CUs) +101.78 +VCK5000 combined (4 and 6 CUs) +145.11 +Alveo U280 (6 CUs) +72.32 +24-core Xeon Platinum CPU +23.52 +V100 GPU +227.89 +Table 2: Compute performance of FPGA AIE and PL-only ap- +proaches compared to 24-core Cascade Lake CPU and Nvidia +V100 GPU. All runs undertaken in single precision floating +point using a problem size of 67 million grid points +6 +CONCLUSIONS AND RECOMMENDATIONS +In this paper we have explored porting of the PW atmospheric +advection scheme to the Versal, utilising the PL for tailoring mem- +ory accesses via a 3D shift-buffer and the AIEs for undertaking +computation. Representative of a much wider class of stencil-based +algorithms, which are popular in HPC workloads, we found that +the major challenge was being able to most effectively interface +the PL and AIEs to ensure data continually flows between the +two. There are several possible approaches, and we have explored +how hardware and tooling limitations drive specific choices and +the performance impact of these. Ultimately, we found that the +most effective approach was to use windows in a ping-pong fash- +ion, working on chunks of data within the AIE kernels which rely +on software pipelined loops and fully filled 8-way vectorization. +Comparing against other hardware options, we found that a major +limitation in obtaining performance was in the total number of +streams between the PL and AIEs, which meant we were unable to +scale beyond four HLS CUs. However four CUs using AIEs on the +VCK5000 performed comparatively to six CUs on the Alveo U280 +with the later benefiting from HBM2. The PL-only approach on the +VCK5000 delivered impressive performance against the other FP- +GAs and CPU, which was largely due to being able to fit eight HLS +CUs onto the PL, and when combining the AIEs and PL for compute +we were able to deliver a significant improvement in performance +compared to other FPGA approaches and the CPU. Therefore, AIEs +aside, our PL-only experiments demonstrate that the Versal is a +powerful architecture and improves on the Alveo. +From a development perspective there are many advantages in +using the AIEs, and this will likely make the ACAP more accessi- +ble to software developers compared to traditional FPGAs. These +include the overall compilation being much quicker, the ability to +undertake much of the development exploration using simulation +which itself is fast, no need to rebuild the PL if the interfaces be- +tween the PL and AIEs have not changed (which means bitstream +regeneration takes around a minute), and the rich profiling tooling +to provide insights where bottlenecks lie in the code. However it +is crucial to match the workload to the architecture, and given the +bandwidth between the PL and AIEs those kernels which have a +higher FLOP to byte ratio than the stencil computation described +in this paper will likely suit the AIEs much better. Therefore, an +important lesson from this work is to focus primarily on those +kernels that will not be limited by the current generation’s PL to +AIE interface, and algorithms with a high FLOP to byte ratio are +likely where we will see the greatest benefit from this architecture. +Considering future enhancements to the Versal, in future AIE +versions it would be beneficial if Xilinx were to make the physical +streams between AIEs wider than 32 bits and increase the PL to AIE +memory size from 128 to 256 bits, as well as supporting a larger num- +ber of PLIO streams. Increased flexibility around vector sizes would +also be useful, for instance it is not possible to have a single preci- +sion floating point vector of numerous sizes including six, which +required us to pad with empty values to eight, and this increased +the amount of data transferred between PL and AIE. Considering +the wider Vitis technology, within HLS it is not possible to create +arrays of external AXI streams (e.g. of type qdma_axis<128,0,0,0>) +and this resulted in messy code when experimenting with multi- +plexing. This is important because interfacing with AIEs will likely +require a greater number of AXI streams compared to what is cur- +rently most common in HLS, and-so improved flexibility would be +advantageous. +7 +ACKNOWLEDGEMENTS +The authors would like to thank the ExCALIBUR H&ES CGRA +project who funded this work. We also acknowledge the ExCAL- +IBUR H&ES FPGA testbed and AMD Xilinx HACC program for +access to compute resource used in this work, the later who also +kindly provided comments and technical advice. For the purpose of +open access, the author has applied a Creative Commons Attribu- +tion (CC BY) licence to any Author Accepted Manuscript version +arising from this submission. +REFERENCES +[1] Sagheer Ahmad, Sridhar Subramanian, Vamsi Boppana, Shankar Lakka, Fu-Hing +Ho, Tomai Knopp, Juanjo Noguera, Gaurav Singh, and Ralph Wittig. 2019. Xilinx +first 7nm device: Versal AI core (VC1902). In 2019 IEEE Hot Chips 31 Symposium +(HCS). IEEE Computer Society, 1–28. +[2] Nick Brown. 2021. Accelerating advection for atmospheric modelling on Xilinx +and Intel FPGAs. In 2021 IEEE International Conference on Cluster Computing +(CLUSTER). IEEE, 767–774. +[3] Nick Brown et al. 2015. A highly scalable Met Office NERC Cloud model. In Pro- +ceedings of the 3rd International Conference on Exascale Applications and Software. +University of Edinburgh, 132–137. +[4] Brian Gaide, Dinesh Gaitonde, Chirag Ravishankar, and Trevor Bauer. 2019. Xilinx +adaptive compute acceleration platform: VersalTM architecture. In Proceedings +of the 2019 ACM/SIGDA International Symposium on Field-Programmable Gate +Arrays. 84–93. + +Exploring the Versal AI engines for accelerating stencil-based atmospheric advection simulation +[5] David Lee, Gregory Allen, Matthew Cannon, Hunter Earnest, Paul Thelen, +Nathaniel Dodds, Jeffrey McCasland, and Carol Chen. 2021. Preliminary Re- +sults from Heavy-Ion Irradiation of the Xilinx Versal ACAP. Technical Report. +Sandia National Lab.(SNL-NM), Albuquerque, NM (United States). +[6] Steve A Piacsek and Gareth P Williams. 1970. Conservation properties of con- +vection difference schemes. J. Comput. Phys. 6, 3 (1970), 392–405. +[7] Xilinx. 2021. Versal ACAP AI Engine Architecture Manual (AM009). +https: +//docs.xilinx.com/r/en-US/am009-versal-ai-engine +[8] Xilinx. 2022. AI Engine Kernel Coding Best Practices Guide (UG1079)). https: +//docs.xilinx.com/r/en-US/ug1079-ai-engine-kernel-coding +[9] Xilinx. 2022. Versal ACAP AI Engine Programming Environment User Guide +(UG1076). https://docs.xilinx.com/r/en-US/ug1076-ai-engine-environment +[10] Chengming Zhang, Tong Geng, Anqi Guo, Jiannan Tian, Martin Herbordt, Ang +Li, and Dingwen Tao. 2022. H-GCN: A Graph Convolutional Network Accelerator +on Versal ACAP Architecture. arXiv preprint arXiv:2206.13734 (2022). + diff --git a/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/load_file.txt b/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b26cfce29b32e73b4f4773a9c72bd9775d7cb644 --- /dev/null +++ b/ndFPT4oBgHgl3EQfJzQg/content/tmp_files/load_file.txt @@ -0,0 +1,302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf,len=301 +page_content='Exploring the Versal AI engines for accelerating stencil-based atmospheric advection simulation Nick Brown n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='brown@epcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='uk EPCC at the University of Edinburgh Edinburgh, UK ABSTRACT AMD Xilinx’s new Versal Adaptive Compute Acceleration Platform (ACAP) is an FPGA architecture combining reconfigurable fabric with other on-chip hardened compute resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' AI engines are one of these and, by operating in a highly vectorized manner, they provide significant raw compute that is potentially beneficial for a range of workloads including HPC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' However, this technology is still early-on, and as yet unproven for accelerating HPC codes, with a lack of benchmarking and best practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This paper presents an experience report, exploring porting of the Piacsek and Williams (PW) advection scheme onto the Versal ACAP, using the chip’s AI engines to accelerate the compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' A stencil-based algorithm, advection is commonplace in atmospheric modelling, including several Met Office codes who initially devel- oped this scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Using this algorithm as a vehicle, we explore optimal approaches for structuring AI engine compute kernels and how best to interface the AI engines with programmable logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Evaluating performance using a VCK5000 against non-AI engine FPGA configurations on the VCK5000 and Alveo U280, as well as a 24-core Xeon Platinum Cascade Lake CPU and Nvidia V100 GPU, we found that whilst the number of channels between the fabric and AI engines are a limitation, by leveraging the ACAP we can double performance compared to an Alveo U280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' KEYWORDS Versal ACAP, AI engines, FPGAs, stencil based algorithms, VCK5000, atmospheric advection, HPC 1 INTRODUCTION The Versal Adaptive Compute Acceleration Platform (ACAP) is a new type of FPGA which combines Programmable Logic (PL) with other facets including CPU-based Programmable Subsystem (PS) and AI engines [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' These AI Engines, or AIEs and we use these two terms interchangeably throughout this paper, are of specific interest here as they are designed to accelerate highly-parallel vec- tor operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The Versal AI-series contains up to 400 engines running between 1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='2 GHz, and each engine follows a Very Long Instruction Word (VLIW) design, capable of issuing seven instructions per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' AI engines are capable of undertaking 8-way vectorized single-precision floating point operations and up to 128 8 bit fixed point arithmetic operations per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The large amount of raw compute provided by the AIEs is inter- esting for High Performance Computing (HPC) workloads, where the ability to use the Versal’s PL to tailor memory accesses bespoke to an application and the AI engines to accelerate the compute has potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' To date there have been a very limited number of prelimi- nary AIE studies [5] [10], and-so an important outstanding question is whether these engines can be effectively leveraged for real world HPC kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' In this work we use the atmospheric advection kernel of the Met Office NERC Cloud model (MONC) [3], which is an open source high resolution atmospheric modelling framework, as a ve- hicle to explore the AI engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Following a stencil-based compute pattern, which is very common in HPC codes, in this short paper we explore how to best map this compute pattern onto the AIEs and how performance compares against other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This paper is structured as follows, in Section 2 we explore the background to this work before summarising the experimental setup in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Section 4 explores structuring our AIE kernel(s) and interfacing these with the PL, before undertaking a performance comparison against other hardware in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' We then conclude and discuss recommendations in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The novel contributions of this paper are 1) An exploration of techniques to most effectively structure AIE kernels 2) An initial performance comparison between the AIEs and other hardware 3) Highlighting some of the limitations of the current AIE technology that one must consider when working with the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 2 BACKGROUND 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='1 The Versal AI engines The VLIW design of Xilinx’s new AI engines is such that, per cycle, each engine is capable of issuing a maximum of two loads, one store, one scalar operation, one fixed point or floating point vector operation, and two move instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The vector unit is of size 256 bits, and focusing on single precision floating point arithmetic in this paper, each engine is capable of undertaking up to eight single precision floating point calculations per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Consequently it is important to ensure code is correctly vectorized to obtain best performance on the AIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Based on 400 AI engines running at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='2GHz on the VCK5000, there is a theoretical single precision floating point performance of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='6 TFLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' AI engines are arranged in a 2D array, with engines connected to their neighbours in both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Each engine contains 16KB of program memory and 32KB of local data memory and, for the later, is able to directly access the memories of three of its neighbours providing a total of 128KB contiguous addressable data memory [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Furthermore, each engine has two 32 bit input streams and two 32 bit output streams which are combined with a FIFO to provide 128 bit access every four clock cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Lastly, AI engines connect to one of their neighbours via a cascade stream which is 384 bits wide and designed to allow arithmetic operations to be chained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' AIE code comprises two parts, kernels which will be mapped to AI engines and a graph description which connects kernels together via their streams and memories, as well as to the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Programmati- cally there are two ways in which data can enter or leave a kernel, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='13016v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='DC] 28 Dec 2022 Nick Brown windows and streams [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Windows provide a buffer, where the current data position in the window is tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' For input windows data is consumed from this buffer by the kernel, for output windows data is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The other approach, a stream, provides an infinite number of scalars and vectors that can be read and written by the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' There underlies an important difference between these two approaches, where a window of data will only progress to the next window between outer iterations of the kernel, as driven by the AIE graph, whereas streams can continually be read from and written to inside the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Consequently, with windows one must frequently start and stop their kernels to refresh the window data, which is not required with streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Whilst the AI engines are the major focus in this paper, it is also important to highlight the general architectural improvements that Xilinx have made to the PL in their Versal series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Built on a 7nm process technology, numerous components including DDR controllers and PCIe interface have been hardened compared to previous generations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Furthermore, a dedicated Network on Chip (NoC) is provided which not only connects the PL with the AIEs, but can also be used between IP blocks on the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='2 Piacsek and Williams advection kernel Advection is the movement of values through the atmosphere due to wind and, at around 40% of the runtime, is the single longest running piece of functionality in the MONC model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The code loops over three fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' U, V and W, representing wind velocity in the x, y and z dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This Piacsek and Williams (PW) [6] advection scheme is called each timestep of the model and calculates advection results, otherwise known as source terms, for each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This advection scheme is a stencil based algorithm, of depth one, where calculating the value of a grid cell requires contributions from neighbouring values across all three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' In previous work [2] this kernel was ported to an Alveo U280 using High Level Synthesis (HLS) and leveraging the dataflow HLS pragma to run multiple components concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The struc- ture of this HLS kernel is illustrated in Figure 1, where the boxes are dataflow regions and arrows between these are internal HLS streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 3D shift buffers provide a bespoke memory solution which is capable of delivering all 27-stencil values per cycle to the advec- tion compute stages, which was found to be the optimal approach even though not all 27 neighbouring stencil values are required by the advection calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Given this existing structure it was our hypothesis that we could replace the advection calculation stages with streams to and from the AI engines, still leveraging the exist- ing tailoring of memory accesses on the PL that worked well in [2], with the raw compute power of the AI engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 3 EXPERIMENTAL SETUP In this work we are using a Xilinx VCK5000 containing a Versal VC1902 ACAP and 16GB of DDR4-DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' All VCK5000 runs are built using Vitis 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='1, the PL is running at 300MHz, and the VC1902 contains 400 AI engines running at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='2GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' We compare against an Alveo U280 which contains 8GB of HBM2, is also running at 300MHz, and Alveo kernels are built using Vitis 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Both the VCK5000 and Alveo U280 are PCIe based cards hosted by a machine containing a 32-core AMD EPYC 7502 processor and 256GB DRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Figure 1: Dataflow design of HLS advection kernel from [2] All reported results are averaged over five runs and performance results are reported as useful FLOPS, which is the number of floating point operations undertaken that contribute to the calculation’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Our performance numbers measure device-side execution time only and do not include the time taken to copy input data to, or result data from, the host and device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This is because we are most interested in the performance of the AIEs in this work, and device-side performance therefore provides a clearer picture when comparing against other technologies that exhibit different host-device data transfer overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 4 AIE PORTING AND OPTIMISATION We started by decomposing the advection stencil-based calculation into constituent operations, resulting in, for each grid cell, the code undertaking six additions, followed by six multiplications, then four subtractions and finally an addition reduction to sum these subtractions together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' A floating point vector of size six is not supported by the tooling and-so we pad with an additional two empty values to make a vector of size eight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' This is why we report useful, rather than total, FLOPS, as useful FLOPS ignores the processing of these empty values by only considering those floating point operations that actually contribute to the advection result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The structure of this kernel is illustrated in Figure 2, with the first 8-way vector addition requiring sixteen floating point numbers comprising the operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' The multiplication requires an additional HBM2 or DDR-DRAM Read data U V W 3D shift 3D shift 3D shift buffer buffer buffer Replicate Replicate Replicate Advection Advection Advection calculation U calculation V calculation W su SV sW Write data HBM2 or DDR-DRAMExploring the Versal AI engines for accelerating stencil-based atmospheric advection simulation Figure 2: Illustration of AIE calculations per grid cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' with the numbers representing the number of single precision floating point numbers provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' eight input numbers which are multiplied by the result of the pre- ceding addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' We packaged this as a single AIE kernel and Listing 1 provides a partial sketch of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' In order to prepare for the vector addition, streams of four numbers are read and loaded into the appropriate locations of the lhs and rhs vectors in lines 11 to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' These vectors are then provided as arguments to the aie::add method at line 16, which undertakes the vectorized addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' Multiplication, subtraction, and reductions operations are handled similarly and omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' It can be seen at line 6 that we are looping over grid cells, and the directives at lines 7 and 8 instruct the AIE com- piler to undertake software pipelining where possible, attempting to keep the VLIW slots filled as per Xilinx’s best practice [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 1 void cell_advection(input_stream ∗ __restrict in_A, input_stream ∗ __restrict in_B, output_stream< float> ∗ __restrict out) { 2 aie::vector in_data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 3 in_data=readincr_v<4>(in_A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 4 5 int32 cells=(int32) in_data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='get(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content=' 6 for (int i=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndFPT4oBgHgl3EQfJzQg/content/2301.13016v1.pdf'} +page_content='i 5 Å. The electron localization results in incomplete +breakup of the P3̅m matrix, but without the formation of the full SoD motif. The light red profile in Fig. 2d and +the intensity band in Fig. 1e at ~3.3 Å remain broad, consistent with incoherent distortions from weakly +developed fluctuating charge correlations. In this diluted limit local distortions were approximated as P3̅ +nanoscale inclusions, Fig. 2g, within the undistorted P3̅m matrix. This model (cyan traces in Fig. 2c (top), Rw=8%) +accounts for the misfit, resulting in ~20 wt % of P3̅ distortions involving heavily puckered hexagonal Ta discs +surrounding central tantalum. (This is consistent with ~10 wt% of locally distorted environments in the 1T-layers +in the polymorphic regime above 630 K discussed in the previous section). +ICCDW regime – The growth of charge correlations and structural coherence of associated distortions results in +their signature extending over a longer length-scale (Fig. 1c). Stacks of experimental PDFs for 15-500 K (Fig. 2d,e) +show that short length-scale peaks (r < 5 Å) sharpen gradually on cooling. However, the data still does not show +a clear SoD motif (Fig. 2d). The data at ~3.5 Å are broad and unresolved, and at ~3.75 Å are rather featureless. +Conversely, peaks at higher distances show the opposite temperature trend. In the light red stack, Fig. 2e, the +signal at ~23.5 Å is the sharpest just below the M-IC transition, portraying high symmetry, and systematically +broadens with intensity reduction as temperature decreases, evidencing growth of broken-symmetry +correlations. The change in slope of Te U11(T) at 550 K (Fig. 1b) indicates an increased localized charge density. +Model-derived Ta-Ta distances for T > 350 K shown in Fig. 3a portray substantially distorted hexagonal SoD +cluster cores comprised of short Ta-Ta contacts (Fig. 3d,f). The distorted fraction increases to ~30 wt% compared +to the M state (Fig. 3c). The PDF peaks which are sensitive to interlayer stacking are very broad, as shown in Fig. +4 a,c, implying inhomogeneous stacking. The distribution centroids of the closest interlayer separation, Fig. 4a, +are shifted to the left, well below 5.9 Å, indicating local compression of layers which are, on average, closer +together than they would be in the undistorted 1T structure. The observation of negative thermal expansion +jumps along the stacking axis is relevant in this context55 (see Supplement for discussion). +NCCDW regime – Significant local structure changes occur at the IC-NC transition. The distortion fingerprint, Fig. +1c, grows dramatically and PDF intensities vividly redistribute below 5 Å (Figs. 1f and 2d). The Ta-Ta peak at 3.25 +Å, which is broad in the IC phase, splits into an incompletely resolved triplet of unequal intensity peaks at ~3.15 +Å, ~3.3 Å and ~3.8 Å, marking the formation of well-defined SoD clusters. The feature at ~3.15 Å corresponds to +the Ta13 SoD intra-cluster configuration, whereas the other two features depict the inter-cluster correlations +(Fig. 3). Residual intensity at ~3.4 Å is associated with discommensuration regions of approximately P3̅ m +character, separating hexagonal domains of well-ordered SoD clusters within the Ta layers56,57, well observable +by STM. Multiphase modeling with coexisting P3̅ and P3̅m phases confirms this picture (Fig. 3c inset, see +Methods). As SoD clusters emerge, the distortion magnitude increases (Fig. 3a,b), consistent with increasingly +resolved signal seen for 3 Å < r < 4 Å shown in Fig 2d. Importantly, the star vertices exhibit sudden asymmetric +contraction, accompanied by substantial puckering (Fig. 3d,f) and SoD twisting (Fig. 3e). The distorted P3̅ phase +becomes the dominant fraction below 350 K, at ~60 wt% (Fig. 3c). This is consistent with the patchy nature of +the discommensurate NCCDW phase. Narrow thermal hysteresis, shown in Figs. 3 b-e, reflects the 1st order +character of the IC-NC transition. As SoD local order is established in the NC state, they simultaneously form well +defined bilayer correlations, Figs. 4 a,c: The SoDs in adjacent layers couple and align along the c-axis. In contrast, +inter-bilayer correlations are still fuzzy: different bilayers exhibit appreciable stacking disorder, evident from +inter-bilayer sensitive peak, Figs. 4b,d, which is still considerably broad in the NC state. +CCDW regime – At the NC-C transition, the PDF intensity at ~3.4 Å diminishes, implying the gradual +disappearance of the ordered domain walls, while the intra-SoD-cluster peak sharpens (Fig. 2d), consistent with +increased SoD density and additional charge localization. Similar changes are observed in the far-r limit (Fig. 2e), + +as a homogeneous crystal structure forms in the commensurate state. Closer inspection of the data unveils an +additional restructuring below ~50 K, embodied in subtle peak shifts and sharpening observed in the dark blue +traces in Figs. 2d and 4a. Modeling shows that the distribution of Ta-Ta distances depicting intra- and inter-SoD +Ta-Ta contacts changes character in the C phase, with tightly bound SoDs structurally further regularizing below +50 K (Fig. 3a,b). The intra-SoD angle  reduces, approaching 120o at base temperature, Fig 3e. The intra-star +twist angle  increases, approaching 90o below 50 K (Fig. 3d), while the local inter-star angle  reaches 150o as +SoDs align and form fully commensurate order (Fig. 3f) and the undistorted phase diminishes, consistent with +vanishing domain walls (Fig 3c). Simultaneously, well defined inter-bilayer correlations develop (Fig 4 b,d). The +most dramatic changes are observed at ~13.1 Å where PDF intensity redistributes and sharpens, indicating an +offset of pairs of SoDs, going from one bilayer to another, by one P3̅m lattice spacing along both planar lattice +vectors, Fig 4e. Notably, there are 6 choices of such offsets, consistent with 13x stacking period and helical +stacking arrangement suggested in some works36. Further qualitative changes are seen below 50 K in the PDF +peak which includes the nearest neighbor intra-bilayer stacking peak just below 6 Å, whose intensity distinctly +and abruptly shifts to lower distance (Figs. 4a,c). In contrast, the inter-bilayer peak at ~13.1 Å sharpens, but does +not shift (Figs. 4b,d). This could imply enhanced intra-bilayer coupling, with neighboring bilayers presumably +adjusting to preserve the inter-bilayer spacings.36 +Order parameter. On the basis of a symmetry analysis of the PDF displacements we can identify the irreducible +representations 𝐴2𝑢 and 𝐸𝑢 that correspond to acoustic mode displacements in the parent P-3m structure +space group, which gives the symmetry of the order parameter(s) in the condensed polaron states. The +displacements are described by two modes with wavevectors of the form 𝒒1 = (𝑎, 𝑏, 0) and 𝒒2 = (−𝑏, 𝑎 + +𝑏, 0) at 60° to 𝒒1. In the C phase, 𝒒1 = +3 +13 𝒂∗ + +1 +13 𝒃∗, and 𝒒2 = − +1 +13 𝒂∗ + +4 +13 𝒃∗, where 𝒂∗ and 𝒃∗ are the +reciprocal lattice vectors of the undistorted lattice. It is possible to decompose the atomic displacements of the +Ta atoms at any temperature in terms of the amplitudes of the 𝐴2𝑢 and 𝐸𝑢 modes described by 𝒒1 and 𝒒𝟏 + 𝒒2. +The displacements at 15 K are shown in the insert to Figure 3f. In the NC phase at room temperature36, aNC = +0.2448(2) and bNC = 0.0681(2), which is close to the C values (𝑎𝐶 = 3/13 and 𝑏𝐶 = 1/13), while in the IC phase, +aIC = 0.283(2), and b = 0, where aIC is close to 2/7=0.2857. Note that our PDF analysis does not reveal any +information about long range order, and 𝒒1 and 𝒒2 should not be interpreted to imply its existence. Rather, they +describe the symmetry of the lattice distortions associated with the twists and displacements of the Ta atoms in +the distorted SoD polaron structures. A detailed description of the frozen phonon mode analysis at different +temperatures and the atomic displacements is given in the supplement. +DISCUSSION +The local structure data in 1T-TaS2 reveals symmetry-specific local structural fingerprints of polaron formation +whose ordering evolves with temperature as shown schematically in Fig. 4f. Surprisingly, the polaron fingerprints +are already visible in the polymorphic regime (associated with individual 1T-monolayers), in the metallic phase +(> 600 K), then successively through the IC, NC and C ordering transitions, and eventually in the non-QSL-like +regime below 50 K, where – remarkably – SoD symmetry is restored in a layer-dimerized undistorted in-plane +SoD structure. An important fundamental question arises in the context of conventional CDW viewpoint +regarding the lattice distortions in the high-temperature M phase: Can the deformations be understood in terms +of a uniform reduced modulation amplitude in the IC-like phase, or in terms of dynamical polaron fluctuations? +Figure 1d indicates the range of the structural correlations. Two differential PDF traces (data at T minus data) +are shown: the black trace shows a difference across the IC-M transition, while the red trace is a difference +corresponding to two PDFs in the same (M) phase separated by the same Δ𝑇. This comparison shows that +dominant changes across IC-M occur at 𝑟 > 5 Å, with smaller changes for 𝑟 < 5 Å. This implies that on going +from M to IC phase the number of distorted sites grows and the extent of IC spatial correlations grows. The M +phase thus looks like a sparse polaron gas, whereas the IC phase is a polaron crystal (incommensurate with the +lattice). We can now consider the three possible scenarios for the IC melting transition based on the PDF data: + +In the first one, in the M phase, the long range IC-CDW becomes dynamic, with amplitude and phase fluctuations +on a timescale faster than the lattice can respond. In the second scenario, all relevant charges become fully +itinerant on all length scales in the M phase without lattice distortions. In the third, polaronic, scenario the IC- +CDW breaks up in the M regime but the lattice follows the localized charge fluctuations, which is equivalent to +thermally activated polaron hopping. In all three scenarios, we would see a disappearance of the reflections at +the IC modulation wave vector on going from IC to M in Bragg reflections, with the third scenario leaving some +diffuse scattering in the M phase. In PDF data, the first and second scenarios would result in no local distortion +in the M phase. The fact that we do see local distortions implies that the polaron fluctuation scenario is relevant +in 1T-TaS2. The persistence of polarons in the polymorphic regime, Fig. 2b,c, and the change from 20 wt% to 10 +wt% on the transition from uniform 1T to 6R structure suggests that polarons exist only in individual 1T +monolayers. These may be expected to appear also in free-standing monolayers or monolayers on substrates, +provided that strain and interaction with the substrate do not interfere. So far, they have been observed in free- +standing bilayers by TEM. +In the CCDW range (<200K), the local structural data are consistent with un-dimerized SoD distortions in adjacent +layers. It is interesting to consider the symmetry of the magnetic structure of the SoDs, compatible with the two +possible in-plane magnetic space groups P3̅ (#147.13) and P3̅’ (147.15). Two (ferro)magnetic P3̅m subgroups +can be derived from the analysis of the symmetric irreducible representation at the Γ point of the trigonal +Brillouin zone of the 1T stack. The choice of the directions of the in-plane components of the magnetic moment +on different atoms is arbitrary, so it is possible to choose the moment along a direction pointing towards the +central Ta1 atom. For P3̅𝑚, inversion does not flip the magnetic moment, allowing a magnetic moment +component along 𝑐 for Ta1 and no constraints for the magnetic moments of the Ta2 and Ta3 atoms. In contrast, +for the P3̅’m magnetic space group, inversion reverses time and thus flips the magnetic moment, the magnetic +moments of the Ta atoms of the same kind are not flipped by the symmetry operators. Choosing different +components for Ta2 and Ta3 magnetic moments it is possible to create different motifs maintaining the same +symmetry. Figs. 4g,h show the motifs compatible with the symmetry operations of the two magnetic groups for +in-plane chiral and non-chiral arrangements of magnetic moments respectively. An in-plane QSL-like state is +consistent with fluctuating in-plane moments, and c-axis antiparallel moments in the P3̅’ magnetic group, +consistent with dimerized stacking of CDW orders in the C state. A remarkable feature of the data is that local +in-plane mirror symmetries appear to be restored below 50 K. More specifically, 𝛼 = 120° and 𝛽 ≃ 150° imply +that a higher-symmetry state is restored. The data below 50 K are strongly suggestive of structural dimerization +along the 𝑐 axis, consistent with inter-plane spin singlet formation and spin gap formation that could explain the +apparent cross-over from a ~𝑇2 QSL-like spin relaxation above 50 K to a T dependence more characteristic of a +gapped spin system below this temperature2. Regarding the origin of the apparent symmetry restoration at low +temperature, the transition from the NC to C state is associated with the disappearance of discommensurations +at TNC-C. However, their existence in the metastable ‘hidden’ phase down to the lowest temperatures39 suggests +that remnant domain walls may also persist deep into the thermodynamic C phase. Even if they are very sparse, +they are topologically protected, preventing interlayer dimerization due to the lateral shift in the charge ordering +that they introduce in individual layers. The restoration of SoD symmetry at lowest temperatures may thus be +associated with the ultimate disappearance of DWs. +The polaron crystallization paradigm58 has the benefit of revealing detailed physics behind the unusual behavior +of this material, while providing a common framework for the understanding the common features not only in +the equilibrium phases, but also the metastable phases of the CDW states in TMDs18. The local structure +symmetry analysis is particularly relevant for revealing the origin of the still poorly understood spin polaron +structure and reconciles the observation of QSL-like behavior with theoretical band structure modelling, which +commonly suggests a spin-paired dimerized ground state. + + +METHODS +Crystal growth and analysis. The 1T-TaS2 samples were grown using iodine vapor transport reaction, grown at +850 °𝐶, and quenched to room temperature from the growth temperature. Single crystal XRD shows a pure +single 1T phase is retained after the quench, with Ta:S composition determined by EDS to be 33: 66 ± 1 𝑎𝑡 %. +Synchrotron X-ray atomic pair distribution function (PDF) measurements: Temperature dependent X-ray total +scattering data were collected at 28-ID-1 beamline of the National Synchrotron Light Source II (NSLS II) at +Brookhaven National Laboratory. Finely ground powders of 1T-TaS2 were sealed in 1mm (outer diameter) +polyimide capillary (referred to as experiment 1 or exp 1) and 1.5 mm (outer diameter) quartz capillary +(experiment 2 or exp 2) within a glovebox under light vacuum. Measurements were carried out in capillary +transmission geometry using a 2D PerkinElmer amorphous silicon area detector (2048 × 2048 pixels with 200 +μm2 pixel size) placed ~204 mm downstream of the sample. The setup utilized a monochromatic X-ray beam +with 74.5 keV energy (λ = 0.1665 Å). Sample temperature control was achieved using a Cryo Industries of America +cryostat (exp 1, 15 K  T  500 K) and a FMB Oxford Hot Air Blower model GSB1300 (exp 2, 300 K  T  915 K). +Data for each experiment were collected in 5 K steps using 5 K/minute temperature ramp and 2 minutes +thermalization at each temperature. In exp1 data in temperature range 15 K  T  300 K were collected on +cooling, while these in temperature range 300 K  T  500 K were collected on warming and cooling. In exp 2 +data were collected on warming. + PDF data processing and analysis: Calibrations of the experimental geometry, momentum transfer range, +and detector orientation were carried out by utilizing nickel standard measurements performed under the same +conditions. Appropriate masking of the beam-stop shadow, inactive and outlier pixels, and subsequent +azimuthal integration of the 2D images to obtain 1D diffraction patterns of intensity versus Q data were done +using pyFAI software package59,60 Standardized corrections to the data for experimental effects to obtain the +reduced total scattering structure function, F(Q), and the subsequent sine Fourier transforms to obtain +experimental PDFs, G(r), with Qmax=25 Å-1 (exp 1) and Qmax=20 Å-1 (exp 2) were carried out using the PDFgetX3 +program within the xPDFsuite software package61 The PDF analysis was carried out using the PDFgui62 modeling +platform. The PDF, derived from powder diffraction-based Bragg and diffuse scattering data collected over a +broad range of momentum transfer, describes direction-averaged distribution of atom pairs in a material as a +function of interatomic distance, r, and provides structural information across different length scales. Due to +large X-ray scattering contrast (Z(Ta)=73, Z(S)=16) the dominant contribution to total PDF (Fig. 1e) originates +from Ta-Ta pairs (red trace), followed by the Ta-S pairs’ contribution (blue trace). The S-S (green trace) +contribution is about 20 times weaker than the Ta-Ta contribution and 4 times weaker than that of Ta-S pairs. + Estimate of distorted fraction and structure modeling in polymorphic regime: By using a calculated PDF based +on undistorted P3̅m model as a reference, we form difference PDFs for all T>600 K data. Such differences are +then integrated over a 2.6-4 Å range (highlighted region on abscissa in Fig. 2b), and normalized to the 600 K +value to quantify the relative change with temperature of the fraction of distorted sites across the polymorphic +transformations, as shown in inset to Fig. 2b (for illustration see Supplement). Above 730 K the short-range PDF +data can be explained by a 6R-TaS2-like model (R3m symmetry) featuring alternating 1T and 1H layers in equal +abundance, Fig. 2c (bottom). Although the exact stacking and long-range character are likely more complex, as +the model does not fully explain the data over longer length-scales, local analysis is rather robust. Importantly, +the 6R-TaS2 model features undistorted 1T & 1H layers, hence not accounting for local distortions responsible +for nontrivial nearest neighbor Ta-Ta distance distribution. +Multiphase treatment: The fits using only the P3̅ phase in the NC regime were noticeably worse compared to +these in the C state, with observable increase of fit residual, as shown in inset to Fig. 3c. This is consistent with +the presence of discommensuration domain walls and lower SoD density in the NC state whose atomic structure +is less distorted and not accounted for in our simple distorted P3̅ model. We approximated domain wall +contribution by adding an undistorted P3̅m structure as a secondary (minority) phase. We utilized this approach + +to fit the PDF data in the IC and M regimes as well, where the distorted and undistorted phases swap roles and +the distorted phase becomes a minority phase. The local models were fit to the data over 1 nm range. + +ACKNOWLEDGMENTS +We wish to acknowledge Igor Vaskivskyi, Jaka Vodeb and Viktor Kabanov for valuable discussions. Work at +Brookhaven National Laboratory is supported by the Office of Basic Energy Sciences, Materials Sciences, and +Engineering Division, U.S. Department of Energy (DOE) under Contract No. DE-SC0012704. This research used +beamline 28-ID-1 of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of +Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract +No. DE-SC0012704. DM and PS wish to acknowledge funding from ARRS. The work at the Jozef Stefan Institute +was supported by the Slovenian Research Agency (P1-0040, ). +AUTHOR CONTRIBUTIONS +EB and MA designed and conducted the X-ray experiments, EB performed PDF analysis, GB and SC performed +group theoretical analysis. EB and DM wrote the paper. PS synthesized the crystals. +REFERENCES + +1. Law, K. T. & Lee, P. A. 1T-TaS2 as a quantum spin liquid. Proc National Acad Sci 114, 6996–7000 +(2017). +2. Klanjsek, M. et al. A high-temperature quantum spin liquid with polaron spins. Nat Phys 13, 1130– +1134 (2017). +3. Darancet, P., Millis, A. J. & Marianetti, C. A. Three-dimensional metallic and two-dimensional +insulating behavior in octahedral tantalum dichalcogenides. Phys Rev B 90, 045134 (2014). +4. Wilson, J. A., Salvo, F. J. D. & Mahajan, S. Charge-density waves and superlattices in the metallic +layered transition metal dichalcogenides. Adv Phys 24, 117–201 (1975). +5. Kogar, A. et al. 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Nanoscale manipulation of the Mott insulating state coupled to charge order in 1T- +TaS2. Nat Commun 7, 10453 (2016). +58. Fratini, S. & Quémerais, P. Polaron Crystallization and Melting: Effects of the Long-Range +Coulomb Forces. Mod Phys Lett B 12, 1003–1012 (1998). +59. Ashiotis, G. et al. The fast azimuthal integration Python library: pyFAI. J Appl Crystallogr 48, 510– +519 (2015). +60. Kieffer, J., Valls, V., Blanc, N. & Hennig, C. New tools for calibrating diffraction setups. J +Synchrotron Radiat 27, 558–566 (2020). + +61. Juhás, P., Davis, T., Farrow, C. L. & Billinge, S. J. L. PDFgetX3: a rapid and highly automatable +program for processing powder diffraction data into total scattering pair distribution functions. J +Appl Crystallogr 46, 560–566 (2013). +62. Farrow, C. L. et al. PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals. +J Phys Condens Matter 19, 335219 (2007). + + + + + +Fig. 1 Electronic transitions & lattice symmetry breaking in 1T-TaS2. (a) Temperature dependent resistivity, +adopted from Sipos et al.8, highlighting electronic phases in 1T-TaS2. The insert shows the undistorted P3̅m +lattice structure. (b) Temperature evolution of in-plane ADP of Ta from P3̅m model fit to PDF data over 20-60 Å +range. Red color indicates heating, blue color indicates cooling. Sloping black dotted line is a reference. Inset: Fit +residual, Rw, implicates IC-M and IC-NC transitions. (c) P3̅m model fit to 600 K data. Green difference trace (offset +for clarity) reveals short range deviations. Stack of difference curves for the same model against data at different +temperatures (as indicated) is shown underneath. Red arrow indicates a length-scale coarsely corresponding to +the SoD polaron and the in-plane nearest neighbor inter-star center-to-center distance (inset). (d) Comparison +of difference PDF between experimental data that are 10 K apart within metallic phase (T > 550 K) and across +the IC-M transition, implicating existence of local distortions in the M phase over at least a radius r. (e) Calculated +total PDF and partial pair-contributions in the P3̅ m structure. (f) False color plot of PDF intensity versus +temperature over a short r-range. + +a +c +DISTORTED +UNDISTORTED +e +CCDW +CCDW +ICCDW +M +5 +0 +10-1 +a +NC +a +G (A-2) +total +10-2 +0 +Atomic PDF, +(A-2) +G +AG +M +600K +Ta-Ta +ICCDW +500K +-10 +10-3 +Ta-S +ICCDW +400K +S-S +-10 +NCCDW +300K +NCCDW +225K +0 +5 +10 +15 +b +CCDW +100K +Interatomic distance,r (A) +15 +009 +CCDW +15K +2.7 +500 +exp 2 +3 +(A2) +exp 1 +(A-2) +Temperature (K) +400 +2 +×100 +6 +10 +20 +30 +40 +50 +60 +2 +Interatomic distance, r (A) +G +(%) +d +200300 +PDF, +U11 +Rw +5 +acrossTiccDw-M +2 +Atomic +400.500 +dxe +100 +L- +2.4 +T (K) +△G +-0.1 +△T=10K +inMphase +2 +100200300400500 +0 +5 +10 +15 +20 +25 +30 +2 +3 +4 +5 +Temperature(K) +Interatomic distance,r (A) +Interatomic distance, r (A) +Fig. 2 Temperature progression of atomic structure in 1T-TaS2. (a) Stack of diffraction patterns revealing a +polymporphic transformation on warming at ~630 K. Inset: intensity collapse of the (012) P3̅m Bragg reflection +during the restructuring from pure 1T to a hybrid 1T & 1H coordination. Further restructuring above 880 K results +in additional Bragg reflections, marked by asterisk. (b) Corresponding PDFs over a narrow r-range. Data feature +isosbestic points (circled) - nodes of temperature-invariant PDF intensity. Label 1T* indicates presence of local +distortions in the 1T-type layers. Vertical blue arrow (1T*) marks polaronic distortion intensity. Inset: Change of +fraction of distorted 1T* environments, relative to 600 K, estimated over the shaded region on the abscissa. (c) +Polaron signature, seen in the M phase (top), persists in the restacked polymorph structure (bottom), and is +associated with the 1T-type layers. Adding distortions, described in (g), as nanoscale inclusions results in +improved fit (cyan trace). In (d) and (e) a stack of PDF data on different length scales is shown over the 15 K – +500 K range, color-coded according to the electronic phase they belong to. At high temperature longer length- +scale peaks sharpen (indicated by vertical arrows) due to apparent average symmetry increase, despite expected +broadening due to ADP increase at elevated temperature and in contrast to the shorter length-scale behavior. +Features exhibiting negative thermal expansion (NTE) are indicated by block arrows. (f) At 15 K clear distortions +associated with the star of David (SoD) are seen in the local structure, incompatible with the P3̅m symmetry +(top) but explained well (bottom) by a simple P3̅ broken symmetry model, depicted in (g). Colored arrows show +the Ta displacements. Out-of-plane degrees of freedom provide for puckering distortions. + +a +b +d +T<630K +4 +e +1T*+1H +1500 +transient +Cn +1T +T<50K +undistorted +Intensity (arb. units) +T>730K +(arb. +17 +1T + (A2) +CCDW +(A-2) +5 +NCCDW +IMAX +,G +600800 +ICCDW +G +012 +Atomic PDF, +PDF, +T (K) +Atomic PDF, +012 +400 +600 +800 +ITE +Atomic +T (K) +900K +900K +T<50K +- +CCDW +NCCDW +2 +ICCDW +1 +2 +3 +4 +5 +6 +2 +3 +4 +5 +6 +2 +3 +4 +5 +22 +23 +24 +Q (A-1) +Interatomic distance, r (A) +Interatomic distance, r (A) +Interatomic distance, r (A) +c +polaron +g +disc +vertices +P3m (1T) +P3m +5 +M +5 +(1T) +C +2 +0 + (A2) +G +5 ++P3distortion o0 +600K +PDF, +Atomic PDF, +Atomic +R3m（1T+1H） +intra +5 +P3 +inter +SoD +C +5 +750K +2 +4 +6 +8 +10 +2 +4 +6 +8 +10 +Interatomicdistance.r(A) +Interatomicdistance.r(A) +Fig. 3 Local structure quantification from distorted local SoD model. (a) Local Ta-Ta distances from a P3̅ +model fits to PDF data, obtained on cooling, over 1 nm length-scale in 15-500 K range. For reference, dashed +horizontal gray line marks the undistorted P3̅m value at 300 K. Intra- and inter-star distances are color coded as +indicated in the legend. (b) Closeup of C-phase behavior (left) and hysteretic response across the IC-NC transition +(right). (c) In NC and IC phases secondary undistorted P3̅m phase was necessary to reduce fit residual, Rw (inset). +Distorted phase fraction, displaying thermal hysteresis consistent with one seen in resistivity, is calculated from +atomic weight-based phase content obtained from the fits. (d)-(f) show selected intra-SoD and inter-SoD bond +angles, as indicated in the sketch. In the IC-phase the SoD motif is heavily distorted, with local distortions +resembling discs. Upon entering the NC phase, the SoD distortion becomes better differentiated, with stars +exhibit twists, and with interatomic angles beginning to evolve towards ideal values. In the C-phase twists start +to diminish. Below ~50 K a sharp regularization of stars is observed, with intra-star Ta-Ta distances becoming +equal, intra-star angles approaching ideal values, and no twisting. Colored arrows in (b)-(f) indicate data +collection on warming (red) and on cooling (blue) cycles. Inset to (f): displacements of the Ta atoms for the q1 +(red) and q1+q2 (blue) modes from symmetry analysis at 15 K, compared to the parent P3̅m phase (gray). + +a +d +e +Ta2-Ta3 +120 +Ta3-Ta3 +Inter +3.6 +Ta3-Ta3 +06 +Ta2-Ta3 +(degrees) +119 +e (degrees) +Ta2-Ta3 +** +TWIST +3.4 +Intra +Ta2-Ta2 +Interatomic +Ta1-Ta2 +118 +α +8 +2 +3 +117 +Ta1=center +100 +200 +300 +400 +500 +Ta2 = disc +100 +200 +300 +400 +500 +100 +200 +300 +400 +500 +Temperature(K) +Ta3=vertex +Temperature (K) +Temperature(K) +b +c +Ta2-Ta3 +lal +3 +Ta2 +distance,r( +3 +0.8 +Distortedfraction +Ta3 + (degrees) +Ta3-Ta3 +0.6 +CtoNC +T<50K +3.4 +Interatomic +0.4 +R +B +1-phasefit +5 +T>50K +2-phasefit +P3m +0.2 +150 +2 +100 +500 +91 +3 +300 +Ta2-Ta3 +T (K) +b+'b +50 +100 +300350400 +100 +200 +300 +400 +500 +100 +200 +300 +400 +500 +Temperature (K) +Temperature (K) +T>350K +Temperature(K) +Fig. 4 Model independent considerations of stacking correlations. (a) Temperature stack of experimental +PDFs around ~5.9 Å peak containing intra-bilayer correlations of SoDs, indicated in (e) by blue and red double +arrows. (b) Temperature stack of PDFs around ~13.1 Å peak depicting inter-bilayer correlations of SoDs, +indicated in (e) by purple double arrows. In (c) and (d) temperature stacks of differential PDFs are shown for +data in (a) and (b), respectively, with differentials calculated using 500 K reference. Vertical dashed lines mark +positions of apparent maxima in the C-phase. (e) Sketch of two bilayers highlighting correlations discussed in +text. Inter-bilayer SoDs are separated by ~13.1 Å at low temperature, a peak forming on cooling from a broad +distribution at high temperature. This separation results from a SoD-shift by approximately one P3̅m lattice +spacing in each P3̅m lattice direction. Note 6 possible choices of relative positioning of nearest bilayers, indicated +by the purple shaded hexagon. (f) Schematics of distortions across different electronic phases in 1T-TaS2. (g, h) +magnetic moments consistent with the distortions of the SoDs as determined by the PDF analysis for two +different settings. The arrows at the center Ta1 atom pointing along the z axis are shown in green (only in P3̅m). + + +a +C +T<50 K +) - G(500 K) (A-2) +Atomic PDF, G (A-2) +G(T) - G(500 K) (A-2) +2 +CCDW +NCCDW +G +ICCDW +PDF, +0 +15 K +Atomic +NTE +50K +200K +G(T) +2 +2 +350K +500K +4 +5.6 +5.8 +6 +6.2 +13 +13.5 +14 +disordered +high T +5.65.8 +6 +6.2 +12.6 +13 +13.4 +r (A) +r (A) +r (A) +r (A) +e +9 +P3(147.13) +M-IC += +550K +2 +bilayer +TIC-NC=350K +P3'(147.15) +bilayer +NC-C +180h \ No newline at end of file diff --git a/q9E5T4oBgHgl3EQflQ94/content/tmp_files/load_file.txt b/q9E5T4oBgHgl3EQflQ94/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1ecbe6d6104593b8bae95ac27f276985ab512b1 --- /dev/null +++ b/q9E5T4oBgHgl3EQflQ94/content/tmp_files/load_file.txt @@ -0,0 +1,801 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf,len=800 +page_content='HIGH-TEMPERATURE POLARONIC LATTICE DISTORTIONS AND CHARGE ORDERING THROUGH THE CHARGE-DENSITY WAVE AND QUANTUM SPIN LIQUID PHASE TRANSITIONS IN 1T-TAS2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Bozin1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Abeykoon2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Conradson3, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Baldinozzi4, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Sutar3 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Mihailovic3 1 Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, NY 11973, USA 2 Photon Sciences Division, Brookhaven National Laboratory, Upton, NY 11973, USA 3 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' of Complex Matter, Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 4 Centralesupélec, CNRS, SPMS, Université Paris-Saclay, bât Eiffel, Gif-sur-Yvette, Île-de-France, 91190, FRANCE ABSTRACT Interesting emergent behavior in quantum materials arises when the interaction of electrons with the lattice leads to partial localization and ordering of charge at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The triangular lattice of some transition metal dichalcogenides additionally presents an interesting case, where spin order is frustrated, leading to an additional complex interplay of interactions involving spin, charge and lattice degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Here we present a study of local symmetry breaking of the lattice structure in the layered dichalcogenide material 1T- TaS2 using x-ray pair-distribution function measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Remarkably, we observe symmetry-breaking polaronic distortions of the lattice structure around individual localized electrons at temperatures well above any of the known long-range ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' These characteristic polaronic signatures remain on cooling through the spin and charge ordered states, eventually revealing a new transition near 50 K to a state displaying partially restored symmetry and significant inter-layer dimerization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The order parameter associated with the observed local atomic displacements and symmetry-allowed polaron spin structure in the ground state suggests that charge ordering is driven by the crystallization of polarons, rather than conventional Fermi surface nesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Symmetry analysis shows that the distorted structure is consistent with a breakup of the QSL phase at low temperature, concurrent with the disappearance of domains in the charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Quasi-two dimensional quantum-ordered materials display very diverse and often exotic spin1–3 and charge ordering behaviour4, including exciton condensation5 and superconductivity6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' They can also display new phases which can be manipulated by pressure 8,9, light 10–14 or doping15–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' A common feature of these systems is that the strength of the electron phonon interaction (EPI) and electron density control the electron kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Increasing the EPI leads to bandwidth narrowing, enhanced Coulomb-interaction induced correlations, and incipient electron localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This results in polaronic effects, charge-density wave (CDW) formation18, and in some systems high superconducting critical temperatures in the fluctuation-dominated intermediate coupling region19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Recent model calculations 18 spanning a wide range of transition metal dichalcogenide (TMD) materials and comparisons with scanning tunneling microsopy experiments highlighted the importance the Coulomb interaction that lead to some generic features of CDW materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In addition, specific features of the band structure may lead to further interesting effects, such as exciton condensation5 and CDW ordering aided by Fermi surface nesting20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Most recent interest has arisen from metastability and control of quantum orders achieved by light or carrier injection, which is very important both from a fundamental point of view11,13 as well as applications21–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' For a detailed understanding of these phenomena, structural data on local lattice deformations are crucial in elucidating symmetry breaking and structural changes associated with electron ordering in different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' A prototypical example of such systems that has been of interest lately is the layered dichalcogenide 1T-TaS24, a kinetically trapped TaS2 polymorph, with stacked TaS6 octahedra (Fig 1a) 24–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In this material a single Ta metal band crossing the Fermi level is responsible for the formation of a CDW, a low-temperature correlated Mott state, quantum spin ordering and two very different metastable quantum electronic phases11,27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' It shows an incommensurate (IC) CDW along the principal crystal axes already at high temperatures (between 550 and 350 K), with an associated Kohn anomaly at a wavevector 𝑞𝐼𝐶 ≃ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='283, 0,0) in units of the reciprocal lattice vector 𝑎0 ∗ 28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Below 350 K, the IC state breaks up into a patchy periodic network of commensurate islands separated by discommensurations30–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Upon further cooling, this nearly commensurate (NC) state becomes fully commensurate (C) after a first order transition at ~ 180 K, with the appearance of a Mott gap in the charge excitation spectrum observed by single-particle tunneling33 and angle-resolved photoemission20,34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In contrast, gapless spin relaxation in the range 50-200 K is attributed to a possible quantum spin liquid (QSL) phase, arising from the spin frustration on the triangular superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' While band structure calculations3 suggest that the QSL phase may be suppressed by inter-layer coupling35, this cannot explain the marked difference of spin and charge excitation spectra2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' A further intriguing feature is a discontinuity in the magnetic susceptibility and an anomalous peak in the spin-spin relaxation time concurrent with a crossover around ~ 50 K of the spin-lattice relaxation time from a QSL-like, to a gapped T-dependence36,37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Near 50 K, a resistivity upturn from nearly T-independent resistivity to strongly T-dependent variable-range hopping behavior has been commonly reported, suggesting an onset of low-temperature charge localization4,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Finally, significant recent interest in this compound comes from emergent non-equilibrium metastable quantum states11 that are created through topological39 and jamming phase transitions27 that do not have long-range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The common picture of the 1T-TaS2 lattice structure in the C state assumes simple distortions from the average lattice structure (trigonal P3̅m, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1a), whereby 12 Ta atoms are attracted symmetrically towards the central Ta atom that carries an extra electronic charge, forming a star of David (SoD) pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The SoDs are arranged in a √13 × √13 unit cell superlattice structure (P3̅ symmetry40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' At such an electron density of 1/13, the dimensionless ratio 𝑟𝑠 of the Coulomb energy 𝑉 to the kinetic energy T 𝑟𝑠 = 𝑉 𝑇 = 𝑒2𝑚 ℏ2√𝑛 ≃ 70~100, where 𝑒 is the elementary charge, 𝑛 is the (2D) density, and 𝑚~3 is the effective electron mass20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Considering that the Wigner crystal stability limit is 𝑟𝑠 = 31~38 18,41–43, 1T-TaS2 is comfortably in the Wigner crystal regime, which means that electronic superlattice ordering on the basis of dominant Coulomb interactions and tell-tale lattice deformations around localized carriers may be anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Room temperature extended X-ray absorption fine structure (EXAFS)44, high- resolution transmission electron microscopy (HRTEM)45–47 and X-ray structural measurements36 indeed suggest the existence of symmetry-breaking atomic displacements, but studies over a wide range of ordering temperatures have not yet been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Here we present the first systematic measurements of the local lattice structure that covers temperatures from 15 to 915 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The data reveal local symmetry breaking from the established P3̅m symmetry at all measured temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1a, inset), revising our current notions about the origin of charge and spin ordering in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' RESULTS The local structure of 1T-TaS2 is investigated using X-ray PDF analysis48,49 for 15 K \uf0a3 T \uf0a3 915 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This encompasses the known electronically ordered states below 600 K portrayed by the resistivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' temperature plot shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1a8, as well as the sequence of irreversible polytype transformations above 600 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The anisotropic atomic displacement parameters, Uij, of Ta atoms were retrieved from a fit to the established P3̅m structure over a range of 20-60 Å (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' For T < 600 K, the Uij, reveal an anomalously large in-plane component, U11, approximately a factor of 2 larger than the component along the stacking axis, U33 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' at 500 K U11 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='025 Å2 whereas U33 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='01 Å2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This is unusual for a layered system in which a larger out-of-plane component is expected to arise from interlayer disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' At high T, in the metallic (M) phase, the large U11 signifies an apparent intralayer ‘disorder’, unaccounted for by the P3̅m structural model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' At lower temperature, the P3̅m model is inadequate at any temperature, revealed by the difference PDFs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The spatial extent and character of this symmetry breaking changes across the cascade of CDW transitions, tracking complex correlations in different electronic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Intriguingly, as we describe below, nanoscale symmetry breaking is also evident in the polymorphs above 630 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' As an aid to understanding, pair-specific contributions to the total PDF within the P3̅m model are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1e, where Ta-Ta pair contributions are dominant (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the P3̅m structure, the undistorted Ta sublattice features a single-valued nearest-neighbor Ta-Ta distance, seen as a sharp peak at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 Å, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' However, the data reveal a distinct additional shoulder, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1e and 2b for 600 K), which is unexplained by the P3̅m model (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c and 2c (top)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' At high temperature this distortion is most likely dynamic, as expected for symmetry-specific lattice fluctuations associated with carrier localization and polaron formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Very specific distributions of Ta-Ta and Ta-S pairs in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='9 Å range are expected for SoD polaronic distortions40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the C phase with a single layer √13 × √13 supercell P3̅ model the SoD polaron structure is adequately described (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2f) 40,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' However, for all other electronic regimes it was necessary to add P3̅m as a secondary phase to achieve good fits (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3c, inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The temperature dependence of the interatomic distances and bond angles, and deductions resulting from fits to the data are addressed sequentially below, starting from high temperatures, well above any known charge or spin ordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Polymorphic regime – Heating 1T-TaS2 above 600 K results in irreversible polymorphic transformations altering average symmetry and stacking, 26,51where nominally 50% of octahedrally coordinated 1T layers convert into trigonal prismatically coordinated 1H layers 26,51 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2a inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This is accompanied by polymorph-specific stacking of the two layer-types 4,37 and changes of the local Ta environment in the prismatic layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Two such transformations are seen in the PDF data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b), at 630 and 880 K (see Supplement for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Remarkably, a shoulder-signal (~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 Å, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b) associated with local lattice distortions (polarons) is observable for 730 K < T < 915 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distorted fraction does not vanish in the polymorphs but instead drops to half its value observed in the 1T regime (T < 630 K), with no significant reduction through the second transformation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Notably, PDF data above 730 K conform to the 6R-TaS2-like model (R3m symmetry) featuring equally abundant alternating 1T and 1H layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2c, bottom) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The model features undistorted 1T and 1H layers, but a clear distortion signal, albeit weaker, such as is seen at 600 K, persists in the fit differential, implying the existence of polaron-specific deformations in the 1T layers well above 600K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2c, top), with ~10 wt% distorted fraction (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' High-temperature metallic regime – the data at 600 K show multiple components at the position where the P3̅m model predicts a single Ta-Ta peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c and red trace fit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2c, Rw=15%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distortion spans ~5 Å with weaker signal extending up to ~12 Å, depicted by the difference PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Such signatures, also seen in the IC phase, are weaker than in the NC and C regimes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' However, their similarity and continuity implies a common origin and unambiguously reveals the presence of high temperature electron localization, detectable by charge-lattice coupling53,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Despite similar P3̅m model misfits shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c, explicit data differences, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1d, reveal subtle changes across the M-IC transition for r > 5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The electron localization results in incomplete breakup of the P3̅m matrix, but without the formation of the full SoD motif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The light red profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2d and the intensity band in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1e at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='3 Å remain broad, consistent with incoherent distortions from weakly developed fluctuating charge correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In this diluted limit local distortions were approximated as P3̅ nanoscale inclusions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2g, within the undistorted P3̅m matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This model (cyan traces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2c (top), Rw=8%) accounts for the misfit, resulting in ~20 wt % of P3̅ distortions involving heavily puckered hexagonal Ta discs surrounding central tantalum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (This is consistent with ~10 wt% of locally distorted environments in the 1T-layers in the polymorphic regime above 630 K discussed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' ICCDW regime – The growth of charge correlations and structural coherence of associated distortions results in their signature extending over a longer length-scale (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Stacks of experimental PDFs for 15-500 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2d,e) show that short length-scale peaks (r < 5 Å) sharpen gradually on cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' However, the data still does not show a clear SoD motif (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The data at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 Å are broad and unresolved, and at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='75 Å are rather featureless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Conversely, peaks at higher distances show the opposite temperature trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the light red stack, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2e, the signal at ~23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 Å is the sharpest just below the M-IC transition, portraying high symmetry, and systematically broadens with intensity reduction as temperature decreases, evidencing growth of broken-symmetry correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The change in slope of Te U11(T) at 550 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1b) indicates an increased localized charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Model-derived Ta-Ta distances for T > 350 K shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3a portray substantially distorted hexagonal SoD cluster cores comprised of short Ta-Ta contacts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3d,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distorted fraction increases to ~30 wt% compared to the M state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The PDF peaks which are sensitive to interlayer stacking are very broad, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4 a,c, implying inhomogeneous stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distribution centroids of the closest interlayer separation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4a, are shifted to the left, well below 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='9 Å, indicating local compression of layers which are, on average, closer together than they would be in the undistorted 1T structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The observation of negative thermal expansion jumps along the stacking axis is relevant in this context55 (see Supplement for discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' NCCDW regime – Significant local structure changes occur at the IC-NC transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distortion fingerprint, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1c, grows dramatically and PDF intensities vividly redistribute below 5 Å (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1f and 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The Ta-Ta peak at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='25 Å, which is broad in the IC phase, splits into an incompletely resolved triplet of unequal intensity peaks at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='15 Å, ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='3 Å and ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='8 Å, marking the formation of well-defined SoD clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The feature at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='15 Å corresponds to the Ta13 SoD intra-cluster configuration, whereas the other two features depict the inter-cluster correlations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Residual intensity at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 Å is associated with discommensuration regions of approximately P3̅ m character, separating hexagonal domains of well-ordered SoD clusters within the Ta layers56,57, well observable by STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Multiphase modeling with coexisting P3̅ and P3̅m phases confirms this picture (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3c inset, see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' As SoD clusters emerge, the distortion magnitude increases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3a,b), consistent with increasingly resolved signal seen for 3 Å < r < 4 Å shown in Fig 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Importantly, the star vertices exhibit sudden asymmetric contraction, accompanied by substantial puckering (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3d,f) and SoD twisting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The distorted P3̅ phase becomes the dominant fraction below 350 K, at ~60 wt% (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This is consistent with the patchy nature of the discommensurate NCCDW phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Narrow thermal hysteresis, shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3 b-e, reflects the 1st order character of the IC-NC transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' As SoD local order is established in the NC state, they simultaneously form well defined bilayer correlations, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4 a,c: The SoDs in adjacent layers couple and align along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In contrast, inter-bilayer correlations are still fuzzy: different bilayers exhibit appreciable stacking disorder, evident from inter-bilayer sensitive peak, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4b,d, which is still considerably broad in the NC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' CCDW regime – At the NC-C transition, the PDF intensity at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 Å diminishes, implying the gradual disappearance of the ordered domain walls, while the intra-SoD-cluster peak sharpens (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2d), consistent with increased SoD density and additional charge localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Similar changes are observed in the far-r limit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2e), as a homogeneous crystal structure forms in the commensurate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Closer inspection of the data unveils an additional restructuring below ~50 K, embodied in subtle peak shifts and sharpening observed in the dark blue traces in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2d and 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Modeling shows that the distribution of Ta-Ta distances depicting intra- and inter-SoD Ta-Ta contacts changes character in the C phase, with tightly bound SoDs structurally further regularizing below 50 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The intra-SoD angle \uf071 reduces, approaching 120o at base temperature, Fig 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The intra-star twist angle \uf061 increases, approaching 90o below 50 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3d), while the local inter-star angle \uf062 reaches 150o as SoDs align and form fully commensurate order (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3f) and the undistorted phase diminishes, consistent with vanishing domain walls (Fig 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Simultaneously, well defined inter-bilayer correlations develop (Fig 4 b,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The most dramatic changes are observed at ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1 Å where PDF intensity redistributes and sharpens, indicating an offset of pairs of SoDs, going from one bilayer to another, by one P3̅m lattice spacing along both planar lattice vectors, Fig 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Notably, there are 6 choices of such offsets, consistent with 13x stacking period and helical stacking arrangement suggested in some works36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Further qualitative changes are seen below 50 K in the PDF peak which includes the nearest neighbor intra-bilayer stacking peak just below 6 Å, whose intensity distinctly and abruptly shifts to lower distance (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In contrast, the inter-bilayer peak at ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1 Å sharpens, but does not shift (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4b,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This could imply enhanced intra-bilayer coupling, with neighboring bilayers presumably adjusting to preserve the inter-bilayer spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='36 Order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' On the basis of a symmetry analysis of the PDF displacements we can identify the irreducible representations 𝐴2𝑢 and 𝐸𝑢 that correspond to acoustic mode displacements in the parent P-3m structure space group, which gives the symmetry of the order parameter(s) in the condensed polaron states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The displacements are described by two modes with wavevectors of the form 𝒒1 = (𝑎, 𝑏, 0) and 𝒒2 = (−𝑏, 𝑎 + 𝑏, 0) at 60° to 𝒒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the C phase, 𝒒1 = 3 13 𝒂∗ + 1 13 𝒃∗, and 𝒒2 = − 1 13 𝒂∗ + 4 13 𝒃∗, where 𝒂∗ and 𝒃∗ are the reciprocal lattice vectors of the undistorted lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' It is possible to decompose the atomic displacements of the Ta atoms at any temperature in terms of the amplitudes of the 𝐴2𝑢 and 𝐸𝑢 modes described by 𝒒1 and 𝒒𝟏 + 𝒒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The displacements at 15 K are shown in the insert to Figure 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the NC phase at room temperature36, aNC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='2448(2) and bNC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='0681(2), which is close to the C values (𝑎𝐶 = 3/13 and 𝑏𝐶 = 1/13), while in the IC phase, aIC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='283(2), and b = 0, where aIC is close to 2/7=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='2857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Note that our PDF analysis does not reveal any information about long range order, and 𝒒1 and 𝒒2 should not be interpreted to imply its existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Rather, they describe the symmetry of the lattice distortions associated with the twists and displacements of the Ta atoms in the distorted SoD polaron structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' A detailed description of the frozen phonon mode analysis at different temperatures and the atomic displacements is given in the supplement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' DISCUSSION The local structure data in 1T-TaS2 reveals symmetry-specific local structural fingerprints of polaron formation whose ordering evolves with temperature as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Surprisingly, the polaron fingerprints are already visible in the polymorphic regime (associated with individual 1T-monolayers), in the metallic phase (> 600 K), then successively through the IC, NC and C ordering transitions, and eventually in the non-QSL-like regime below 50 K, where – remarkably – SoD symmetry is restored in a layer-dimerized undistorted in-plane SoD structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' An important fundamental question arises in the context of conventional CDW viewpoint regarding the lattice distortions in the high-temperature M phase: Can the deformations be understood in terms of a uniform reduced modulation amplitude in the IC-like phase, or in terms of dynamical polaron fluctuations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Figure 1d indicates the range of the structural correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Two differential PDF traces (data at T minus data) are shown: the black trace shows a difference across the IC-M transition, while the red trace is a difference corresponding to two PDFs in the same (M) phase separated by the same Δ𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This comparison shows that dominant changes across IC-M occur at 𝑟 > 5 Å, with smaller changes for 𝑟 < 5 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This implies that on going from M to IC phase the number of distorted sites grows and the extent of IC spatial correlations grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The M phase thus looks like a sparse polaron gas, whereas the IC phase is a polaron crystal (incommensurate with the lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' We can now consider the three possible scenarios for the IC melting transition based on the PDF data: In the first one, in the M phase, the long range IC-CDW becomes dynamic, with amplitude and phase fluctuations on a timescale faster than the lattice can respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the second scenario, all relevant charges become fully itinerant on all length scales in the M phase without lattice distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the third, polaronic, scenario the IC- CDW breaks up in the M regime but the lattice follows the localized charge fluctuations, which is equivalent to thermally activated polaron hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In all three scenarios, we would see a disappearance of the reflections at the IC modulation wave vector on going from IC to M in Bragg reflections, with the third scenario leaving some diffuse scattering in the M phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In PDF data, the first and second scenarios would result in no local distortion in the M phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The fact that we do see local distortions implies that the polaron fluctuation scenario is relevant in 1T-TaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The persistence of polarons in the polymorphic regime, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b,c, and the change from 20 wt% to 10 wt% on the transition from uniform 1T to 6R structure suggests that polarons exist only in individual 1T monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' These may be expected to appear also in free-standing monolayers or monolayers on substrates, provided that strain and interaction with the substrate do not interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' So far, they have been observed in free- standing bilayers by TEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the CCDW range (<200K), the local structural data are consistent with un-dimerized SoD distortions in adjacent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' It is interesting to consider the symmetry of the magnetic structure of the SoDs, compatible with the two possible in-plane magnetic space groups P3̅ (#147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='13) and P3̅’ (147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Two (ferro)magnetic P3̅m subgroups can be derived from the analysis of the symmetric irreducible representation at the Γ point of the trigonal Brillouin zone of the 1T stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The choice of the directions of the in-plane components of the magnetic moment on different atoms is arbitrary, so it is possible to choose the moment along a direction pointing towards the central Ta1 atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' For P3̅𝑚, inversion does not flip the magnetic moment, allowing a magnetic moment component along 𝑐 for Ta1 and no constraints for the magnetic moments of the Ta2 and Ta3 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In contrast, for the P3̅’m magnetic space group, inversion reverses time and thus flips the magnetic moment, the magnetic moments of the Ta atoms of the same kind are not flipped by the symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Choosing different components for Ta2 and Ta3 magnetic moments it is possible to create different motifs maintaining the same symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4g,h show the motifs compatible with the symmetry operations of the two magnetic groups for in-plane chiral and non-chiral arrangements of magnetic moments respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' An in-plane QSL-like state is consistent with fluctuating in-plane moments, and c-axis antiparallel moments in the P3̅’ magnetic group, consistent with dimerized stacking of CDW orders in the C state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' A remarkable feature of the data is that local in-plane mirror symmetries appear to be restored below 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' More specifically, 𝛼 = 120° and 𝛽 ≃ 150° imply that a higher-symmetry state is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The data below 50 K are strongly suggestive of structural dimerization along the 𝑐 axis, consistent with inter-plane spin singlet formation and spin gap formation that could explain the apparent cross-over from a ~𝑇2 QSL-like spin relaxation above 50 K to a T dependence more characteristic of a gapped spin system below this temperature2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Regarding the origin of the apparent symmetry restoration at low temperature, the transition from the NC to C state is associated with the disappearance of discommensurations at TNC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' However, their existence in the metastable ‘hidden’ phase down to the lowest temperatures39 suggests that remnant domain walls may also persist deep into the thermodynamic C phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Even if they are very sparse, they are topologically protected, preventing interlayer dimerization due to the lateral shift in the charge ordering that they introduce in individual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The restoration of SoD symmetry at lowest temperatures may thus be associated with the ultimate disappearance of DWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The polaron crystallization paradigm58 has the benefit of revealing detailed physics behind the unusual behavior of this material, while providing a common framework for the understanding the common features not only in the equilibrium phases, but also the metastable phases of the CDW states in TMDs18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The local structure symmetry analysis is particularly relevant for revealing the origin of the still poorly understood spin polaron structure and reconciles the observation of QSL-like behavior with theoretical band structure modelling, which commonly suggests a spin-paired dimerized ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' METHODS Crystal growth and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The 1T-TaS2 samples were grown using iodine vapor transport reaction, grown at 850 °𝐶, and quenched to room temperature from the growth temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Single crystal XRD shows a pure single 1T phase is retained after the quench, with Ta:S composition determined by EDS to be 33: 66 ± 1 𝑎𝑡 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Synchrotron X-ray atomic pair distribution function (PDF) measurements: Temperature dependent X-ray total scattering data were collected at 28-ID-1 beamline of the National Synchrotron Light Source II (NSLS II) at Brookhaven National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Finely ground powders of 1T-TaS2 were sealed in 1mm (outer diameter) polyimide capillary (referred to as experiment 1 or exp 1) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 mm (outer diameter) quartz capillary (experiment 2 or exp 2) within a glovebox under light vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Measurements were carried out in capillary transmission geometry using a 2D PerkinElmer amorphous silicon area detector (2048 × 2048 pixels with 200 μm2 pixel size) placed ~204 mm downstream of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The setup utilized a monochromatic X-ray beam with 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 keV energy (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1665 Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Sample temperature control was achieved using a Cryo Industries of America cryostat (exp 1, 15 K \uf0a3 T \uf0a3 500 K) and a FMB Oxford Hot Air Blower model GSB1300 (exp 2, 300 K \uf0a3 T \uf0a3 915 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Data for each experiment were collected in 5 K steps using 5 K/minute temperature ramp and 2 minutes thermalization at each temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In exp1 data in temperature range 15 K \uf0a3 T \uf0a3 300 K were collected on cooling, while these in temperature range 300 K \uf0a3 T \uf0a3 500 K were collected on warming and cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In exp 2 data were collected on warming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' PDF data processing and analysis: Calibrations of the experimental geometry, momentum transfer range, and detector orientation were carried out by utilizing nickel standard measurements performed under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Appropriate masking of the beam-stop shadow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' inactive and outlier pixels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' and subsequent azimuthal integration of the 2D images to obtain 1D diffraction patterns of intensity versus Q data were done using pyFAI software package59,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='60 Standardized corrections to the data for experimental effects to obtain the reduced total scattering structure function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' F(Q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' and the subsequent sine Fourier transforms to obtain experimental PDFs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' G(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' with Qmax=25 Å-1 (exp 1) and Qmax=20 Å-1 (exp 2) were carried out using the PDFgetX3 program within the xPDFsuite software package61 The PDF analysis was carried out using the PDFgui62 modeling platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The PDF, derived from powder diffraction-based Bragg and diffuse scattering data collected over a broad range of momentum transfer, describes direction-averaged distribution of atom pairs in a material as a function of interatomic distance, r, and provides structural information across different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Due to large X-ray scattering contrast (Z(Ta)=73, Z(S)=16) the dominant contribution to total PDF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1e) originates from Ta-Ta pairs (red trace), followed by the Ta-S pairs’ contribution (blue trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The S-S (green trace) contribution is about 20 times weaker than the Ta-Ta contribution and 4 times weaker than that of Ta-S pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Estimate of distorted fraction and structure modeling in polymorphic regime: By using a calculated PDF based on undistorted P3̅m model as a reference, we form difference PDFs for all T>600 K data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Such differences are then integrated over a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='6-4 Å range (highlighted region on abscissa in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b), and normalized to the 600 K value to quantify the relative change with temperature of the fraction of distorted sites across the polymorphic transformations, as shown in inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2b (for illustration see Supplement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Above 730 K the short-range PDF data can be explained by a 6R-TaS2-like model (R3m symmetry) featuring alternating 1T and 1H layers in equal abundance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2c (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Although the exact stacking and long-range character are likely more complex, as the model does not fully explain the data over longer length-scales, local analysis is rather robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Importantly, the 6R-TaS2 model features undistorted 1T & 1H layers, hence not accounting for local distortions responsible for nontrivial nearest neighbor Ta-Ta distance distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Multiphase treatment: The fits using only the P3̅ phase in the NC regime were noticeably worse compared to these in the C state, with observable increase of fit residual, as shown in inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This is consistent with the presence of discommensuration domain walls and lower SoD density in the NC state whose atomic structure is less distorted and not accounted for in our simple distorted P3̅ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' We approximated domain wall contribution by adding an undistorted P3̅m structure as a secondary (minority) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' We utilized this approach to fit the PDF data in the IC and M regimes as well, where the distorted and undistorted phases swap roles and the distorted phase becomes a minority phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The local models were fit to the data over 1 nm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' ACKNOWLEDGMENTS We wish to acknowledge Igor Vaskivskyi, Jaka Vodeb and Viktor Kabanov for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Work at Brookhaven National Laboratory is supported by the Office of Basic Energy Sciences, Materials Sciences, and Engineering Division, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Department of Energy (DOE) under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This research used beamline 28-ID-1 of the National Synchrotron Light Source II, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' DM and PS wish to acknowledge funding from ARRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The work at the Jozef Stefan Institute was supported by the Slovenian Research Agency (P1-0040, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS EB and MA 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' J Phys Condens Matter 19, 335219 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 1 Electronic transitions & lattice symmetry breaking in 1T-TaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (a) Temperature dependent resistivity, adopted from Sipos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='8, highlighting electronic phases in 1T-TaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The insert shows the undistorted P3̅m lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (b) Temperature evolution of in-plane ADP of Ta from P3̅m model fit to PDF data over 20-60 Å range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Red color indicates heating, blue color indicates cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Sloping black dotted line is a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Inset: Fit residual, Rw, implicates IC-M and IC-NC transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (c) P3̅m model fit to 600 K data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Green difference trace (offset for clarity) reveals short range deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Stack of difference curves for the same model against data at different temperatures (as indicated) is shown underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Red arrow indicates a length-scale coarsely corresponding to the SoD polaron and the in-plane nearest neighbor inter-star center-to-center distance (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (d) Comparison of difference PDF between experimental data that are 10 K apart within metallic phase (T > 550 K) and across the IC-M transition, implicating existence of local distortions in the M phase over at least a radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (e) Calculated total PDF and partial pair-contributions in the P3̅ m structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (f) False color plot of PDF intensity versus temperature over a short r-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' a c DISTORTED UNDISTORTED e CCDW CCDW ICCDW M 5 0 10-1 a NC a G (A-2) total 10-2 0 Atomic PDF, (A-2) G AG M 600K Ta-Ta ICCDW 500K 10 10-3 Ta-S ICCDW 400K S-S 10 NCCDW 300K NCCDW 225K 0 5 10 15 b CCDW 100K Interatomic distance,r (A) 15 009 CCDW 15K 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='7 500 exp 2 3 (A2) exp 1 (A-2) Temperature (K) 400 2 ×100 6 10 20 30 40 50 60 2 Interatomic distance, r (A) G (%) d 200300 PDF, U11 Rw 5 acrossTiccDw-M 2 Atomic 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='500 dxe 100 L- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 T (K) △G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1 △T=10K inMphase 2 100200300400500 0 5 10 15 20 25 30 2 3 4 5 Temperature(K) Interatomic distance,r (A) Interatomic distance, r (A) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 2 Temperature progression of atomic structure in 1T-TaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (a) Stack of diffraction patterns revealing a polymporphic transformation on warming at ~630 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Inset: intensity collapse of the (012) P3̅m Bragg reflection during the restructuring from pure 1T to a hybrid 1T & 1H coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Further restructuring above 880 K results in additional Bragg reflections, marked by asterisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (b) Corresponding PDFs over a narrow r-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Data feature isosbestic points (circled) - nodes of temperature-invariant PDF intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Label 1T* indicates presence of local distortions in the 1T-type layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Vertical blue arrow (1T*) marks polaronic distortion intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Inset: Change of fraction of distorted 1T* environments, relative to 600 K, estimated over the shaded region on the abscissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (c) Polaron signature, seen in the M phase (top), persists in the restacked polymorph structure (bottom), and is associated with the 1T-type layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Adding distortions, described in (g), as nanoscale inclusions results in improved fit (cyan trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In (d) and (e) a stack of PDF data on different length scales is shown over the 15 K – 500 K range, color-coded according to the electronic phase they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' At high temperature longer length- scale peaks sharpen (indicated by vertical arrows) due to apparent average symmetry increase, despite expected broadening due to ADP increase at elevated temperature and in contrast to the shorter length-scale behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Features exhibiting negative thermal expansion (NTE) are indicated by block arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (f) At 15 K clear distortions associated with the star of David (SoD) are seen in the local structure, incompatible with the P3̅m symmetry (top) but explained well (bottom) by a simple P3̅ broken symmetry model, depicted in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Colored arrows show the Ta displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Out-of-plane degrees of freedom provide for puckering distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' a b d T<630K 4 e 1T*+1H 1500 transient Cn 1T T<50K undistorted Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' units) T>730K (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 17 1T (A2) CCDW (A-2) 5 NCCDW IMAX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='G 600800 ICCDW G 012 Atomic PDF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' PDF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' T (K) Atomic PDF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 012 400 600 800 ITE Atomic T (K) 900K 900K T<50K CCDW NCCDW 2 ICCDW 1 2 3 4 5 6 2 3 4 5 6 2 3 4 5 22 23 24 Q (A-1) Interatomic distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' r (A) Interatomic distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' r (A) Interatomic distance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' r (A) c polaron g disc vertices P3m (1T) P3m 5 M 5 (1T) C 2 0 (A2) G 5 +P3distortion o0 600K PDF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Atomic PDF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Atomic R3m（1T+1H） intra 5 P3 inter SoD C 5 750K 2 4 6 8 10 2 4 6 8 10 Interatomicdistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='r(A) Interatomicdistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='r(A) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 3 Local structure quantification from distorted local SoD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (a) Local Ta-Ta distances from a P3̅ model fits to PDF data, obtained on cooling, over 1 nm length-scale in 15-500 K range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' For reference, dashed horizontal gray line marks the undistorted P3̅m value at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Intra- and inter-star distances are color coded as indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (b) Closeup of C-phase behavior (left) and hysteretic response across the IC-NC transition (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (c) In NC and IC phases secondary undistorted P3̅m phase was necessary to reduce fit residual, Rw (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Distorted phase fraction, displaying thermal hysteresis consistent with one seen in resistivity, is calculated from atomic weight-based phase content obtained from the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (d)-(f) show selected intra-SoD and inter-SoD bond angles, as indicated in the sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the IC-phase the SoD motif is heavily distorted, with local distortions resembling discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Upon entering the NC phase, the SoD distortion becomes better differentiated, with stars exhibit twists, and with interatomic angles beginning to evolve towards ideal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In the C-phase twists start to diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Below ~50 K a sharp regularization of stars is observed, with intra-star Ta-Ta distances becoming equal, intra-star angles approaching ideal values, and no twisting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Colored arrows in (b)-(f) indicate data collection on warming (red) and on cooling (blue) cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Inset to (f): displacements of the Ta atoms for the q1 (red) and q1+q2 (blue) modes from symmetry analysis at 15 K, compared to the parent P3̅m phase (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' a d e Ta2-Ta3 120 Ta3-Ta3 Inter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='6 Ta3-Ta3 06 Ta2-Ta3 (degrees) 119 e (degrees) Ta2-Ta3 ** TWIST 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 Intra Ta2-Ta2 Interatomic Ta1-Ta2 118 α 8 2 3 117 Ta1=center 100 200 300 400 500 Ta2 = disc 100 200 300 400 500 100 200 300 400 500 Temperature(K) Ta3=vertex Temperature (K) Temperature(K) b c Ta2-Ta3 lal 3 Ta2 distance,r( 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='8 Distortedfraction Ta3 (degrees) Ta3-Ta3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='6 CtoNC T<50K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 Interatomic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 R B 1-phasefit 5 T>50K 2-phasefit P3m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content="2 150 2 100 500 91 3 300 Ta2-Ta3 T (K) b+'b 50 100 300350400 100 200 300 400 500 100 200 300 400 500 Temperature (K) Temperature (K) T>350K Temperature(K) Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' 4 Model independent considerations of stacking correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (a) Temperature stack of experimental PDFs around ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='9 Å peak containing intra-bilayer correlations of SoDs, indicated in (e) by blue and red double arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (b) Temperature stack of PDFs around ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1 Å peak depicting inter-bilayer correlations of SoDs, indicated in (e) by purple double arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' In (c) and (d) temperature stacks of differential PDFs are shown for data in (a) and (b), respectively, with differentials calculated using 500 K reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Vertical dashed lines mark positions of apparent maxima in the C-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (e) Sketch of two bilayers highlighting correlations discussed in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Inter-bilayer SoDs are separated by ~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='1 Å at low temperature, a peak forming on cooling from a broad distribution at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' This separation results from a SoD-shift by approximately one P3̅m lattice spacing in each P3̅m lattice direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' Note 6 possible choices of relative positioning of nearest bilayers, indicated by the purple shaded hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (f) Schematics of distortions across different electronic phases in 1T-TaS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' (g, h) magnetic moments consistent with the distortions of the SoDs as determined by the PDF analysis for two different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' The arrows at the center Ta1 atom pointing along the z axis are shown in green (only in P3̅m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content=' a C T<50 K ) - G(500 K) (A-2) Atomic PDF, G (A-2) G(T) - G(500 K) (A-2) 2 CCDW NCCDW G ICCDW PDF, 0 15 K Atomic NTE 50K 200K G(T) 2 2 350K 500K 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='2 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='5 14 disordered high T 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='6 13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='4 r (A) r (A) r (A) r (A) e 9 P3(147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content="13) M-IC = 550K 2 bilayer TIC-NC=350K P3'(147." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} +page_content='15) bilayer NC-C 180h' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9E5T4oBgHgl3EQflQ94/content/2301.05670v1.pdf'} diff --git a/rdAzT4oBgHgl3EQfO_vG/vector_store/index.faiss b/rdAzT4oBgHgl3EQfO_vG/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..915597e3a9723c2b2be83b308b460e5944fbc678 --- /dev/null +++ b/rdAzT4oBgHgl3EQfO_vG/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d37510ce0c579f32fbb7d7a2c15813db350bbce748f8d02aa53a6e386b3ed1ae +size 1638445 diff --git a/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/2301.00471v1.pdf.txt b/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/2301.00471v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f4a94acb12ceef0168798b8dc4a67726fcb0647 --- /dev/null +++ b/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/2301.00471v1.pdf.txt @@ -0,0 +1,2174 @@ +arXiv:2301.00471v1 [math.OC] 1 Jan 2023 +Null-controllability of underactuated linear +parabolic-transport systems with constant coefficients +Armand Koenig, Pierre Lissy† +January 3, 2023 +Abstract +The goal of the present article is to study controllability properties of mixed systems of lin- +ear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one- +dimensional torus. The equations are coupled by zero or first order coupling terms, with con- +stant coupling matrices, without any structure assumptions on them. The distributed control acts +through a constant matrix operator on the system, so that there might be notably less controls than +equations, encompassing the case of indirect and simultaneous controllability. More precisely, we +prove that in small time, such kind of systems are never controllable in appropriate Sobolev spaces, +whereas in large time, null-controllability holds, for sufficiently regular initial data, if and and only +if a spectral Kalman rank condition is verified. We also prove that initial data that are not regular +enough are not controllable. Positive results are obtained by using the so-called fictitious control +method together with an algebraic solvability argument, whereas the negative results are obtained +by using an appropriate WKB construction of approximate solutions for the adjoint system asso- +ciated to the control problem. As an application to our general results, we also investigate into +details the case of 2 × 2 systems (i.e., one pure transport equation and one parabolic equation). +MSC Classification 93B05, 93B07, 93C20, 35M30. +Keywords Parabolic-transport systems, null-controllability, observability. +1 +Introduction +1.1 +Context and state of the art +Controllability properties of coupled systems of PDEs has attracted a lot of attention this last two +decades, due to their link with real-life models and also the specific mathematical difficulties arising +in this context. An important part of the literature is devoted to systems where all components of the +equations have the same qualitative behaviour (meaning that they are for instance all parabolic, or +all hyperbolic, etc.). However, the case where different dynamics are mixed has been less studied, de- +spite its mathematical interest. Indeed, in this context, the controllability propertiesof each equation +taken separately might be totally different (for instance, the heat equation with distributed control is +controllable in arbitrary small time from any open subset [29, 22], whereas the wave equation with +IMT, +Université +de +Toulouse, +CNRS, +Université +Toulouse +III-Paul +Sabatier +(UPS), +Toulouse, +France +(armand.koenig@math.univ-toulouse.fr) +†CEREMADE, +Université +Paris-Dauphine +& +CNRS +UMR +7534, +Université +PSL, +75016 +Paris, +France +(lissy@ceremade.dauphine.fr). +1 + +distributed control is controllable in large time and under some geometric conditions [6]), so that the +controllability properties of the final coupled system might be difficult to guess. Moreover, when we +are considering underactuated systems (in the sense that there are less controls than equations) as in +the present article, additional mathematical difficulties are appearing, due notably to the algebraic +and analytic effects of the coupling terms, that become predominant in the understanding of the con- +trollability or observability properties of the system under study. Here, in the present article, we aim +to study the indirect controllability properties of a model of coupled parabolic-transport equations as +introduced in [7]. +Let us mention that many realistic models already studied in the literature can be reformulated in +terms of coupled parabolic-transport equations, notably the wave equation with structural damping +[37, 34, 10, 24], the heat equation with memory [26, 23], the 1D-Linearized compressible Navier- +Stokes equations [20, 13, 12, 8], or the Benjamin-Bona-Mahony equation [38]. For more details, we +also refer to [7, §1.4]. This justifies the interest of studying a general version of coupled parabolic- +transport systems as in the present article, that can be seen as an attempt to find a unified framework +in order to encompass many existing results of the literature and to generalize them. Other results +of interest, related to the present work, are [2], where the authors study a one-dimensional system of +one transport equation and one parabolic equation, for which they prove a non-controllability result +in small time by a WKB approach, and [11], where the authors prove a controllability result in large +time for a one-dimensional system of one transport equation and one elliptic equation. +1.2 +Presentation of the parabolic-transport system under study +Let 푇 > 0 some final time , 핋 = ℝ∕(2휋ℤ) the one-dimensional torus,휔 an nonempty open subset of 핋, +푑 ∈ ℕ∗ (which represents the number of equations in our system) , 푚 ∈ {1, … , 푑} (which represents +the number of controls in our system), 퐴, 퐵, 퐾 ∈ ℳ푑(ℝ) (that are some constant coupling matrices), +and 푀 ∈ ℳ푑,푚(ℝ) (that is a constant control operator). Our goal is to study the controllability +properties of the following coupled system of parabolic-transport equations: +{ 휕푡푓 − 퐵휕2 +푥푓 + 퐴휕푥푓 + 퐾푓 = 푀푢1휔 +in (0, 푇) × 핋, +푓(0, ⋅) = 푓0 +in 핋. +(Sys) +Here, the state is 푓∶ [0, 푇] × 핋 → ℝ푑, and the control is 푢∶ [0, 푇] × 핋 → ℝ푚. The exact regularity +chosen for 푓 and 푢 will be made more precise later on. +We assume that +푑 = 푑h + 푑p with 1 ≤ 푑h < 푑, 1 ≤ 푑p < 푑, +(H.1) +퐵 = (0 +0 +0 +퐷) , with 퐷 ∈ ℳ푑p(ℝ), +(H.2) +ℜ(Sp(퐷)) ⊂ (0, +∞). +(H.3) +푑h represents the number of purely hyperbolic equations, whereas 푑p represents the number of +parabolic equations. +Notice that (H.3) is necessary to ensure that the matrix operator 휕푡−퐷∆ is parabolic is the sense of +Petrovskii ([28, Chapter 7, Definition 2]). Introducing the similar block decomposition for the 푑 × 푑 +matrix 퐴 = ( 퐴′ 퐴12 +퐴21 퐴22 +), we make the following hypothesis on the matrix 퐴′ ∈ ℳ푑ℎ(ℝ) +퐴′ is diagonalizable with Sp(퐴′) ⊂ ℝ. +(H.4) +Notice that it is well-known that (H.4) is necessary (and sufficient, see [7, §2.2]) to ensure the well- +posedness of (Sys). +2 + +1.3 +Main results +To state our results, we need to introduce the following notations: +퓁(휔) ∶= sup{|퐼|; 퐼 connected component of 핋 ⧵ 휔}, +(1) +휇∗ ∶= min{|휇|; 휇 ∈ Sp(퐴′)}, +and +푇∗ = 푇∗(휔) ∶= { +퓁(휔) +휇∗ +if 휇∗ > 0, ++∞ +if 휇∗ = 0. +(2) +For 푛 ∈ ℤ, we also set +퐵푛 ∶= −푛2퐵 − i푛퐴 − 퐾 +(3) +and +[퐵푛|푀] ∶= ( +푀 +퐵푛푀 +… +퐵푑−1 +푛 +푀 +) . +(4) +Our main result is the following one. +Theorem 1. Assume that the hypotheses (H.1)–(H.4) hold, that 푇 > 푇∗. +Then, the spectral Kalman rank condition rank([퐵푛|푀]) = 푑 holds for all 푛 ∈ ℤ if and only if +for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔)푚 such that the solution 푓 of the +parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0. +Remark 2. +• Recall that the Kalman rank condition is necessary for the control of ODE sys- +tems [14, Theorem 1.16]. Therefore, writing the parabolic-transport system in Fourier, we im- +mediately find that for every 푇 > 0, the spectral Kalman-rank condition ∀푛 ∈ ℤ, rank([퐵푛|푀]) = +푑 is necessary for the null-controllability of every 퐻푘 initial conditions in time 푇. +• Actually, we prove two slightly stronger versions of this theorem, namely theorems 9 and 12, +that are useful in order to obtain some controllability results under some constraints on Fourier +coefficients of the hyperbolic part of the initial condition (see proposition 20, proposition 21, +proposition 22). +• One can refine a little bit the regularity stated in theorem 1, as follows. Assume that 푇 > 푇∗ +and that for all 푛 ∈ ℤ, the spectral Kalman rank condition rank([퐵푛|푀]) = 푑 holds. Then: +1. for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑ℎ × 퐻4푑(푑−1)−1(핋)푑푝, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔)푚 +such that the solution 푓 of the parabolic-transport system (Sys) with initial condition 푓0 +satisfies 푓(푇, ⋅) = 0. +2. if 퐴12 = 0, for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑ℎ × 퐻4푑(푑−1)−2(핋)푑푝, there exists a control 푢 ∈ +퐿2([0, 푇]×휔)푚 such that the solution 푓 of the parabolic-transportsystem (Sys) with initial +condition 푓0 satisfies 푓(푇, ⋅) = 0. +Indeed, by letting evolve the system freely on a short interval of time, we can show using the +method of lemma 23 that the parabolic component becomes 퐻4푑(푑−1)(핋)푑푝, so that theorem 1 +can be applied, taking into account that the condition 푇 > 푇∗ is open and that the system is +time-invariant. +• The spectral Kalman rank condition rank([퐵푛|푀]) = 푑 was first introduced in [5] for coupled +systems of heat equations with diagonalizable diffusions (see also [33] for non-diagonalizable +diffusions). +3 + +Theorem 3. Let 휇 ∈ Sp(퐴′), 푁 ∈ ℕ and 푇 > 0. Assume that every initial condition 푓0 ∈ 퐿2(핋)푑 ∩ +{∑ +|푛|>푁 푋푛ei푛푥}issteerableto0intime푇 withcontrolin퐿2((0, 푇)×휔). Then, thereexists푉0 ∈ ker(퐴′∗+ +휇) such that 푀∗( 푉0 +0 +) ≠ 0. +Remark 4. Theorems 1, 9 and 12 only ensures null-controllability of smooth enough initial conditions. +Theorem 3 proves that such a regularity condition is needed in general: even if the time is large +enough and if the Kalman rank condition is satisfied for every 푛, it might happen that some 퐿2 initial +condition cannot be steered to 0 with a 퐿2 control. +1.4 +Precise scope and organization of the article +This article can be seen as a continuation of [7], insofar as we generalize the results of the above- +mentioned article, since we are able to treat any matrices 퐴, 퐵, 퐾, 푀 without any restrictions on their +structure. Indeed, in [7], the authors treated the case where 푀 = 퐼푑 (where no Kalman rank condi- +tion is needed), or particular cases where only the parabolic or the hyperbolic parts are controlled, +under strong restrictions on the structure of the coupling matrices 퐴, 퐵 and 퐾 and also on the diffu- +sion matrix 퐵. +Let us mention that our results are sharp in terms of the controllability conditions we obtain. +However, it is very likely that the initial state space (whose choice is determined by technical reasons +coming from the specific strategy we use, that is consuming in terms of regularity, see Section 3.2) is +almost never sharp and depends strongly on the structure of the coupling terms. Finding the exact +“good” state space remains an open problem that seems to be difficult to solve in all generality. +The article is organized as follows. In section 2, we give some notations and we gather some +existing results that will be used in our proof. Section 3 is devoted to proving that the condition +rank([퐵푛|푀]) = 푑 is sufficient in order to obtain our desired controllability result in large time The +argument is based on a fictitious control argument detailed in section 3.1, where we first prove an +auxiliary controllability result, in the case 푀 = 퐼푑, with regular enough controls for regular enough +initial data. Then, in section 3.2, we explain how to obtain a control in the range on 푀 by performing +algebraic manipulations. Notice that the method of fictitious control plus algebraic solvability, that +has been introduced in [16] in the context of the controllability of PDEs, has been successfully used +for various problems [4, 17, 18, 32, 15, 39, 40, 19]. One of the main novelties here is that the algebraic +solvability is not directly performed on the system (or its adjoint as in [19]) but on a projected version +of the system on its Fourier components. Section 4 is devoted to proving some necessary conditions +of controllability. Section 4.1 is devoted to constructing WKB solutions. These solutions are used to +disprove controllability in small time in section 4.2 and to prove theorem 3 in section 4.3. Section 5 +aims to give an application of our results to the particular case of 2 × 2 systems together with some +considerationsabout the sharpness of our regularity assumptions in this precise setting. To conclude, +appendix A proves a general result about a “control up to a finite-dimensional space plus unique +continuation” strategy that is used in section 3.1, in the spirit of [30, 7]. +Acknowledgement Armand Koenig is supported by the ANR LabEx CIMI (under grant ANR- +11-LABX-0040) within the French State Programme “Investissements d’Avenir”. +Pierre Lissy is supported by the Agence Nationale de la Recherche, Project TRECOS, under grant +ANR-20-CE40-0009. +2 +Some notations and preliminary results +We will rely on some basic results on the parabolic-transport system (Sys) that are already known, +see [7]. For the reader convenience, we collect here the notations and results we will use most often, +and we will recall some others along the way as they are used. +4 + +Let ℒ be the unbounded operator on 퐿2(핋)푑 with domain 퐻1(핋)푑h × 퐻2(핋)푑p defined by +ℒ푓 = −퐵휕2 +푥푓 + 퐴휕푥푓 + 퐾푓. +The operator −ℒ generates a strongly continuous semigroup of bounded operators of 퐿2(핋)푑 [7, +Proposition 11]. Every 퐻푘(핋)푑 is stable by e−푡ℒ, and the restriction of e−푡ℒ on 퐻푘(핋)푑 is a strongly +continuous semigroup of bounded operators [7, Remark 13]. We denote by 푆(푇, 푓0, 푢) the solution +at time 푇 of the parabolic-transport system (Sys) with control matrix 푀 = 퐼푑 (the identity matrix of +size 푑, i.e., we control every component with a different control), initial condition 푓0 and control 푢. +Let 푛0 ∈ ℕ to be chosen large enough later on. We denote by 푒푛 ∶ 푥 ∈ 핋 ↦ ei푛푥. We also denote +by 퐸 ∶ ℂ → ℳ푑(ℂ) the following function: +퐸(푧) = 퐵 + 푧퐴 − 푧2퐾. +Let 푟 > 0 small enough. For |푧| < 푟, let 푃h(푧) be the eigenprojection on the sum of eigenspaces +of 퐸(푧) associated to the set of eigenvalues 휆(푧) ∈ Sp(퐸(푧)) such that |휆(푧)| < 푟. According to [7, +Proposition 5], 푃h(푧) satisfies: +• 푃h(0) = ( 퐼 0 +0 0 +); +• 푧 ↦ 푃h(푧) is holomorphic; +• 푃h(푧) is a projection that commutes with 퐸(푧); +• 푃h(푧)퐸(푧) = 푂(푧) as 푧 → 0. +We also set 푃p(푧) = 퐼 − 푃h(푧). This projection 푃p(푧) satisfies similar properties as 푃h(푧) ([7, Proposi- +tions 6]). +Following [7, Proposition 18], we denote by 퐹0 the space of frequencies less than 푛0 and by 퐹h (re- +spectively 퐹p) the space of hyperbolic frequencies greater than 푛0 (respectively the space of parabolic +frequencies greater than 푛0), i.e. +퐹0 = ⨁ +|푛|≤푛0 Span(푒푛); +퐹p = ⨁ +|푛|>푛0 Range(푃p(i∕푛))푒푛; +퐹h = ⨁ +|푛|>푛0 Range(푃h(i∕푛))푒푛. +By [7, Proposition 18], we notably have +퐿2(핋)푑 = 퐹0 ⊕ 퐹p ⊕ 퐹h. +The space 퐹p is stable by the semigroup e−푡ℒ (see the definition of 푃p [7, Proposition 5] and the +definition of 퐹p [7, Proposition 18]). We denote by ℒp the restriction of ℒ to 퐹p. +Similarly, the space 퐹h is stable by the semigroup e−푡ℒ. We denote by ℒh the restriction of ℒ to +퐹h, and −ℒh generates a strongly continuous group of bounded operators on 퐹h [7, Proposition 19]. +Let Π0, Πp, Πh and Π be the projections defined by +퐿2(핋)푑 = 퐹0 ⊕ 퐹p ⊕ 퐹h; +Π0 = 퐼퐹0 + 0 + 0; +Πp = 0 + 퐼퐹p + 0; +Πh = 0 + 0 + 퐼퐹h; +Π = 0 + 퐼퐹p + 퐼퐹h = Πp + Πh. +These projections are bounded operators on 퐿2(핋)푑 [7, Proposition 18] (and also on every 퐻푘(핋)푑, as +one can readily convince by following the proof of [7, Proposition 18]). +5 + +3 +Null controllability of regular initial conditions +3.1 +Regular controls for regular initial conditions +As a technical preparation for the proof of theorem 1, we need some results regarding the regularity +of controls, when the control matrix is 푀 = 퐼푑. +Proposition 5. Assume that 푇 > 푇∗ (as defined in eq. (2)) and that 푀 = 퐼푑. Let 푘, 퓁 ∈ ℕ. For +every 푓0 ∈ 퐻푘(핋)푑, there exists 푢 ∈ 퐻푘 +0 ((0, 푇) × 휔)푑h × 퐻퓁 +0 ((0, 푇) × 휔)푑p such that the solution of the +parabolic-transport system (Sys) with initial condition 푓0 and control 푢 satisfies 푓(푇, ⋅) = 0. +We adapt the proof of the corresponding result when 푘 = 0 [7, Theorem 2]. First, we prove the +following adaptation of [7, Proposition 21]. +Proposition 6. Let 푇′ ∈ (푇∗, 푇) and 푘 ∈ ℕ. If 푛0 (in the definition of 퐹0, see [7, Eq. (40–42)]) is large +enough, there exists a continuous operator +풰h ∶ 퐻푘(핋)푑 × 퐻푘 +0 ((푇′, 푇) × 휔)푑p→ 퐻푘 +0((0, 푇′) × 휔)푑h +(푓0, 푢p) +↦ 푢h, +such that for every (푓0, 푢p) ∈ 퐻푘(핋)푑 × 퐻푘 +0((푇′, 푇) × 휔)푑p (where 푢p is extended by 0 on (0, 푇′) and 푢h +is extended by 0 on (푇′, 푇)), +Πh푆(푇; 푓0, (풰h(푓0, 푢p), 푢p)) = 0. +Proof. As in [7, §4.3.1], the conclusion of proposition 6 is equivalent to the exact controllability of +the system 휕푡푓 + ℒh푓 = Πh(푢, 0) at time 푇′. Since −ℒh generates a strongly continuous group, the +exact controllability at time 푇′ is equivalent to the null-controllability at time 푇′, which is what we +are going to prove. +When 푘 = 0, [7, Proposition 23] is the claimed result. To extend this result to 푘 > 0, we use a +general result of Ervedoza and Zuazua concerning the regularity of controls for regular initial data +in the context of groups of operators [21, Theorem 1.4]. Let ˜휔 an open subset of 핋 such that ˜휔 ⊂ 휔 +and 푇∗(˜휔) < 푇′. Let 휒 ∈ 퐶∞ +푐 (휔) such that 휒 = 1 on ˜휔. Let 휂 ∈ 퐶∞ +0 (0, 푇′). Let 푧0 ∈ 퐻푘(핋)푑 be an +initial condition. Let 푌푇′ as defined by [21, Proposition 1.3] and define the control as +푉(푡) = 휂(푡)휒(푥)푀∗푌(푡), +where 푌 is the solution to +휕푡푌 − 퐵∗휕2 +푥푌 − 퐴∗휕푥푌 + 퐾∗푌 = 0 +associated to the initial condition 푌(푇′) = 푌푇′. According to [21, Proposition 1.3], 푉(푡) is a con- +trol that steers 푧0 to 0 at time 푇′. According to [21, Theorem 1.4], 푌푇′ ∈ 퐻푘(핋)푑 (hence 푉 ∈ +퐿2(0, 푇′; 퐻푘(휔)푑)) and 푉 ∈ 퐻푘(0, 푇′; 퐿2(휔)푑), with estimates of the form +‖푉‖2 +퐿2(0,푇′;퐻푘(휔)푑) + ‖푉‖2 +퐻푘(0,푇′;퐿2(휔)푑) ≤ 퐶푘‖푧0‖2 +퐻푘(핋)푑. +We claim that 퐿2(0, 푇′; 퐻푘 +0(휔))∩퐻푘 +0(0, 푇′; 퐿2(휔)) ⊂ 퐻푘((0, 푇′)×휔). Indeed, for every 휏 ∈ ℝ and +휉 ∈ ℝ, +(1 + 휏2 + 휉2)푘 ≤ 퐶푘 +((1 + 휏2)푘 + (1 + 휉2)푘). +Hence, integrating in Fourier space, +‖푓‖2 +퐻푘(ℝ2) ≤ 퐶푘 +(‖푓‖2 +퐿2(ℝ;퐻푘(ℝ)) + ‖푓‖2 +퐻푘(ℝ;퐿2(ℝ)) +). +6 + +Recall that for Ω ⊂ ℝ푛 convex1, 퐻푘 +0(Ω) is the set of functions whose extension by zero outside Ω +are 퐻푘(ℝ푛). Hence, 퐿2(0, 푇′; 퐻푘 +0 (휔)) ∩ 퐻푘 +0(0, 푇′; 퐿2(휔)) ⊂ 퐻푘((0, 푇′) × 휔) as claimed, so that 푉 ∈ +퐻푘((0, 푇′) × 휔)푑. +Since 휂 ∈ 퐶∞(0, 푇′) and 휒 ∈ 퐶∞ +0 (휔), we conclude that 푉 ∈ 퐻푘 +0((0, 푇′) × 휔)푑. +For the proof of proposition 5, we will also use: +Proposition 7 ([7], proposition 22). Let 푇′ ∈ (푇∗, 푇) and 푘 ∈ ℕ. If 푛0 is large enough, there exists a +continuous operator +풰p ∶ 퐿2(핋)푑 × 퐿2((0, 푇′) × 휔)푑h→ 퐶∞ +푐 ((푇′, 푇) × 휔)푑p +(푓0, 푢h) +↦ 푢p, +(in the sense that for any 푠 ∈ ℕ, 풰p ∶ 퐿2(핋)푑×퐿2((0, 푇′)×휔)푑h → 퐻푠 +0(푇′, 푇)×휔)푑p is continuous for the +natural topologies associated to these spaces) such that for every (푓0, 푢h) ∈ 퐿2(핋)푑 × 퐿2((0, 푇′) × 휔)푑h, +Πp푆(푇; 푓0, (푢h, 풰p(푓0, 푢h)) = 0. +We can now prove proposition 5 by mimicking the proof of the case 푘 = 0 [7, Proposition 20 & +§4.5]. +Proof of proposition 5. Step 1: Control up to final dimensional space. — We claim that there exists +a closed finite codimensional space 풢 of 퐻푘(핋)푑 and a continuous operator 풰 ∶ 풢 → 퐻푘 +0((0, 푇′) × +휔)푑h × 퐶∞ +푐 ((푇′, 푇) × 휔)푑p (in the sense that for any 푠 ∈ ℕ, 풰 ∶ 풢 → 퐻푘 +0((0, 푇′) × 휔)푑h × 퐻푠 +0(푇′, 푇) × +휔)푑p is continuous for the natural topologies associated to these spaces) such that for every 푓0 ∈ 풢, +Π푆(푇, 푓0, 풰푓0) = 0. +The property Π푆(푇, 푓0, (푢h, 푢p)) = 0 holds if +{ +푢h = 풰h(푓0, 푢p) = 풰h +1 (푓0) + 풰h +2 (푢p), +푢p = 풰p(푓0, 푢h) = 풰p +1(푓0) + 풰p +2 (푢h). +(5) +Set 풞 = 풰p +1 + 풰2 +p풰h +1 . Then, the previous relations hold if +풞푓0 = (퐼 − 풰p +2 풰h +2 )푢p. +(6) +Since 풰p +2 is continuous from 퐻푘 +0((푇′, 푇) × 휔)푑p into 퐶푐((푇′, 푇) × 휔)푑p, we deduce that the operator +풞∶ 퐻푘 +0((푇′, 푇) × 휔) +푑p → 퐻푘 +0((푇′, 푇) × 휔) +푑p is compact. Thus, according to Fredholm’s alternative, +the relation (6) holds on a closed finite codimensional space 풢. +Step 2: Conclusion. — Dealing with the finite (co)dimensional spaces 퐹0 and 풢 is a straightforward +adaptation of [7, §4.5]; more specifically, we use proposition 25 proved in Appendix A with 퐻 = 푉 = +퐻푘(핋)푑, 푈푇 = 퐻푘 +0 ((0, 푇) × 휔)푑h × 퐻퓁 +0 ((0, 푇) × 휔), 퐴 = −ℒ, 퐵 = 1휔, 풢 = 풢 and ℱ = 퐹0. The control +up to a finite dimensional space hypothesis is satisfied according to the previous step. The unique +continuation hypothesis is satisfied because every generalised eigenvector is a finite sum of elements +of the form 푋푛ei푛푥 (푋푛 ∈ ℂ푑), and finite linear combinations of 푋푛ei푛푥 have the unique continuation +property thanks to, e.g., Jerison-Lebeau’s spectral inequality (see [30, Theorem 3], or [7, Eq. (90)] for +our specific case). +For technical reasons, we will need the control to be in the form 푃(휕푥)푢, where 푃(휕푥) is a constant +coefficients differential operator to be chosen later on. +1More generally, satisfiying the segment condition, see[1, Definition 3.21 & Theorem 5.29]. +7 + +Proposition 8. Assume that 푇 > 푇∗ (as defined in (2)) and that 푀 = 퐼푑. Let 푘, 퓁 ∈ ℕ. Let 푃 be +a nonzero polynomial with complex coefficients. Assume that 퓁 ⩾ deg(푃). Let 푓0 ∈ 퐻푘(핋)푑 be such +that for every 푛 ∈ ℤ, 푃(i푛) = 0 ⟹ 푐푛(푓0) = 0. Then, there exists 푢 ∈ 퐻푘+deg(푃) +0 +((0, 푇) × 휔)푑h × +퐻퓁 +0 ((0, 푇) × 휔)푑p such that the solution of the parabolic-transport system (Sys) with initial condition 푓0 +and control 푃(휕푥)푢 satisfies 푓(푇, ⋅) = 0. +Proof. 푘, 퓁 ∈ ℕ with 퓁 ⩾ deg(푃).Let 푓0 ∈ 퐻푘(핋)푑 be such that for every 푛 ∈ ℤ, 푃(i푛) = 0 +⟹ +푐푛(푓0) = 0. We define ˜푓0 ∶= 푃(휕푥)−1푓0 by 푐푛( ˜푓0) ∶= 푃(i푛)−1푐푛(푓0) if 푃(i푛) ≠ 0 and 푐푛( ˜푓0) ∶= 0 +if 푃(i푛) = 0. Note that 푃(휕푥) ˜푓0 = 푓0 and that ˜푓0 ∈ 퐻푘+deg(푃) +0 +(휔)푑. Then, applying proposition 5 +to ˜푓0 leads to the fact that there exists ˜푢 ∈ 퐻푘+deg(푃) +0 +((0, 푇) × 휔)푑h × 퐻퓁 +0 ((0, 푇) × 휔)푑p such that the +solution ˜푓 of the parabolic-transport system (Sys) with initial condition ˜푓0 and control ˜푢 satisfies +˜푓(푇, ⋅) = 0. Moreover, since ˜푓0 ∈ 퐻푘+deg(푃) +0 +(휔)푑 and ˜푢 ∈ 퐻푘+deg(푃) +0 +((0, 푇) × 휔)푑h × 퐻퓁 +0 ((0, 푇) × 휔)푑p +with 퓁 ⩾ deg(푃), we notably have ˜푓 ∈ 퐿2((0, 푇); 퐻푘+deg(푃)(핋)). Hence, setting 푓 = 푃(휕푥) ˜푓 and +푢 = 푃(휕푥) ˜푓, and using that 푃(휕푥) has constant coefficients (so that it commutes with the operator +휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾퐼푑)) ensures that 푓 verifies (Sys) with initial condition 푓0 and control 푃(휕푥)푢. +Moreover, since ˜푓(푇, ⋅) = 0, we also have푓(푇, ⋅) = 푃(휕푥) ˜푓(푇, ⋅) = 0, which leads to the desired +result. +3.2 +Algebraic solvability +For 푘 ∈ ℕ, we define +[퐵푛|푀]푘 ∶= ( +푀 +퐵푛푀 +… +퐵푘−1 +푛 +푀 +) . +(7) +We prove the following variant of theorem 1. +Theorem 9. Assume that the hypotheses (H.1)–(H.4) hold, and that 푇 > 푇∗. Let 푘 ∈ ℕ. Assume that +for all |푛| ∈ ℕ large enough, the Kalman rank condition rank([퐵푛|푀]푘) = 푑 holds. Define the following +space of functions +퐸 ∶= {푓 ∈ 퐿2(핋)푑 ∶ ∀푛 ∈ ℤ, 푐푛(푓) ∈ Range([퐵푛|푀])}. +Set, when it is defined, +[퐵푛|푀]+ +푘 ∶= [퐵푛|푀]∗ +푘 +( +[퐵푛|푀]푘[퐵푛|푀]∗ +푘 +)−1 +. +Write [퐵푛|푀]+ +푘 by blocks as +[퐵푛|푀]+ +푘 = +⎛ +⎜ +⎜ +⎝ +퐿h +푛,1 +퐿p +푛,1 +⋮ +⋮ +퐿h +푛,푘 +퐿p +푛,푘 +⎞ +⎟ +⎟ +⎠ +, +where the 퐿h +푛,푗 are of size 푚×푑h and the 퐿p +푛,푗 are of size 푚×푑p. Considering the 퐿h +푛,푗 as rational functions +of 푛, and denoting their degree by deg(퐿h +푛,푗), set +푝 ∶= max +1≤푗≤푘 deg(푛푗−1퐿h +푛,푗) = max +1≤푗≤푘 +(푗 − 1 + deg(퐿h +푛,푗)). +Then, for every 푓0 ∈ 퐻푝(핋)푑 ∩ 퐸, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔) such that the solution 푓 of +the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0. +The idea of the proof is to first choose a “fictitious” control that acts on every components. Then, +we look at the Fourier coefficients of 푓. This transforms the control system (Sys) into a family of +8 + +finite-dimensional control systems. On each of these finite-dimensional system, we perform some +algebraic manipulations, called algebraic solvability, that transform the fictitious control (that acted +on every component) into an “actual” control (that acts only on Range(푀)). +We begin with the algebraic solvability result we will use, which is essentially taken from [32, +§2.1]. +Lemma 10. Let 푘 ∈ ℕ∗. Let ˜퐵 ∈ ℳ푑(ℂ) and ˜ +푀 ∈ ℳ푚,푑(ℝ). Let 푋0 ∈ ℂ푑 and 푤 ∈ 퐻푘−1 +0 +(0, 푇; ℂ푚푘). +Write 푤 by blocks as +푤 = +⎛ +⎜ +⎝ +푤1 +⋮ +푤푘 +⎞ +⎟ +⎠ +, +where 푤푗 ∈ 퐻푘−1 +0 +(핋; ℂ푚), and set 푢 = 푤1 + 푤′ +2 + ⋯ + 푤(푘−1) +푘 +. Let 푋, ˜푋 ∈ 퐶0(0, 푇; ℂ푑) be the solutions +of +푋′ = ˜퐵푋 + [˜퐵| ˜ +푀]푘푤, +˜푋′ = ˜퐵푋 + ˜ +푀푢, +푋(0) = ˜푋(0) = 푋0, +where +[˜퐵| ˜ +푀]푘 ∶= ( ˜ +푀 +˜퐵푀 +… +˜퐵푘−1 ˜ +푀) . +Then 푋(푇) = ˜푋(푇). +Proof. Consider ˜ +ℳ푘 the operator matrix with 푑 + 푚 rows and 푘푚 columns defined by blocks as +˜ +ℳ푘 ∶= ( 0 +− ˜ +푀 +⋯ +− ∑푘−2 +푗=0 휕푗 +푡 ˜퐵푘−2−푗 ˜ +푀 +−퐼 +−휕푡 +⋯ +−휕푘−1 +푡 +) = ( +˜ +ℳ푘,1 +˜ +ℳ푘,2 +) . +Set also +풫∶ (푋, 푊) ∈ 퐻1 +0(0, 푇; ℂ푑) × 퐿2(0, 푇; ℂ푚) → 휕푡푋 − ˜퐵푋 − ˜ +푀푊 ∈ 퐿2(0, 푇; ℂ푑). +We claim that +풫◦ ˜ +ℳ푘 = [˜퐵| ˜ +푀]푘. +(8) +Indeed, we have by blocks 풫◦ ˜ +ℳ푘 = (퐶0 +⋯ +퐶푘−1 +) with +퐶퓁 = −(휕푡 − ˜퐵) +퓁−1 +∑ +푗=0 +휕푗 +푡 ˜퐵퓁−1−푗 ˜ +푀 + ˜ +푀휕퓁 +푡 . +Then, remarking that this is a telescoping sum, +퐶퓁 = − +퓁∑ +푗=1 +휕푗 +푡 ˜퐵퓁−푗 ˜ +푀 + +퓁−1 +∑ +푗=0 +휕푗 +푡 ˜퐵퓁−푗 ˜ +푀 + ˜ +푀휕퓁 +푡 += −휕퓁 +푡 ˜ +푀 + ˜퐵퓁 ˜ +푀 − ˜ +푀휕퓁 +푡 , +which proves the claimed formula (8). +Now, plug eq. (8) into the differential equation 푋′ = ˜퐵푋 + [˜퐵| ˜ +푀]푘푤, which gives +푋′ = ˜퐵푋 + (휕푡 − ˜퐵) ˜ +ℳ푘,1푤 − ˜ +푀 ˜ +ℳ푘,2푤. +With 푌 ∶= 푋 − ˜ +ℳ푘,1푤, and remarking that ˜ +ℳ푘,2푤 = −푢, this can be written as 푌′ = ˜퐵푌 + ˜ +푀푢. Since +푤 ∈ 퐻푘−1 +0 +(0, 푇; ℂ푚푘), ˜ +ℳ푘,1푤(0) = ˜ +ℳ푘,1푤(푇) = 0. Hence 푌(0) = 푋(0) = ˜푋(0) and 푌(푇) = 푋(푇). +Thus 푌 solves the same Cauchy problem as ˜푋. This proves that 푌 = ˜푋, hence ˜푋(푇) = 푌(푇) = +푋(푇). +9 + +We can now prove theorem 9. +Proof of theorem 9. Let 푓0 ∈ 퐻푝(핋)푑. Set 푋푛(푡) = 푐푛(푓(푡, ⋅)) and 푢푛(푡) = 푐푛(푢(푡, ⋅)). The desired +conclusion 푓(푇, ⋅) = 0 reads in Fourier as: ∀푛 ∈ ℤ, 푋푛(푇) = 0. Moreover, 푋푛 satisfies +{ 푋′ +푛(푡) = 퐵푛푋푛(푡) + 푀푢푛(푡), +푡 ∈ (0, 푇), +푋푛(0) = 푐푛(푓0). +(9) +First, let us give the idea of the proof: if 푣 steers 푓0 to 0 when 푀 = 퐼, we want to define 푤푛 by +푐푛(푣(푡, ⋅)) = [퐵푛|푀]푘푤푛 (this is possible for 푛 large enough) and choose 푢푛 ∶= 푤푛1+푤′ +푛2+⋯+푤(푘−1) +푛푘 +. +Then, according to lemma 10, the function 푢푛 steers 푋푛 from 푐푛(푓0) to 0. There are two problems +with this crude choice of 푢푛: this construction only works for 푛 large enough, and more importantly, +we have no guarantee that the support of ∑ 푢푛ei푛푥 is included in [0, 푇] × 휔. +The control strategy is to first bring frequencies less than 푛0 to 0 in time 휀 for some 푛0 > 0 large +enough to be chosen later and 휀 > 0 small enough so that 푇 > 푇∗ + 2휀, and second use a refined +version of the construction outlined above. +Step 1: Control of a finite number of frequencies. — Recall that Π is the projection on frequencies +larger than 푛0 and that 퐸 was defined in the statement of theorem 9. We claim that for any 푛0 ∈ ℕ∗, +휀 > 0 and 푓0 ∈ 퐸 there exists 푢 ∈ 퐿2(0, 휀; 퐻푝 +0 (휔))푚 such that (1 − Π)푆(휀, 푓0, 푀푢) = 0. +This property is equivalent to the null-controllability of the system (Sys) projected on frequencies +less or equal than 푛0. The observability inequality associated with this problem [14, Theorem 2.44] +is: +∃퐶 > 0, ∀푔0 ∈ (1 − Π)퐸, ‖e−휀ℒ∗푔0‖2 +퐻−푝(핋)푑 ≤ 퐶 ∫ +휀 +0 +‖푀∗e−푡ℒ∗푔0‖2 +퐿2(휔)푚 d푡. +Since (1 − Π)퐸 is finite dimensional, this is equivalent to the unique continuation property +∀푔0 ∈ (1 − Π)퐸, +( +푀∗e−푡ℒ∗푔0(푥) = 0 for (푡, 푥) ∈ (0, 휀) × 휔 +) +⟹ 푔0 = 0. +Let us prove this property. Let 푔0 ∈ (1 − Π)퐸 such that 푀∗e−푡ℒ∗푔0(푥) = 0 for (푡, 푥) ∈ (0, 휀) × 휔. +Since finite sums of ei푛푥 have the unique continuation property, we have for every 0 < 푡 < 휀 and +|푛| ≤ 푛0, +푐푛(푀∗e−푡ℒ∗푔0) = 0. +We can rewrite this as +푀∗e−푡퐵∗ +푛푐푛(푔0) = 0. +Differentiating 퓁 times in 푡 and evaluating at 푡 = 0, we get that for all 퓁 ∈ ℕ and |푛| ≤ 푛0, +푀∗(퐵∗ +푛)퓁푐푛(푔0) = 0. +Since we assumed that for |푛| > 푛0, 푐푛(푔0) = 0, this means that 푐푛(푔0) ∈ ker([퐵푛|푀]∗). But, by +definition of 퐸, 푐푛(푔0) ∈ Range([퐵푛|푀]) = ker([퐵푛|푀]∗)⟂. Thus, 푐푛(푔0) = 0 and 푔0 = 0. This proves +the unique continuation property, and the claim. +Step 2: Construction of 푢푛. — We set 푇′ = 푇∗ + 휀 = 푇 − 휀. +Let us write [퐵푛|푀]+ +푘 = 푄(i푛)∕푃(i푛) where 푄 is a polynomial with matrix coefficients, 푃 is a +polynomial (with scalar coefficients). If we denote the adjugate matrix of a matrix 퐶 by Adj(퐶), note +that we may take +푄(i푛) = [퐵푛|푀]∗ +푘 Adj([퐵푛|푀]푘[퐵푛|푀]∗ +푘); +푃(i푛) = det([퐵푛|푀]푘[퐵푛|푀]∗ +푘). +10 + +Increasing 푛0 if necessary, we may assume that for every |푛| > 푛0, 푃(i푛) ≠ 0. We first apply a +control as in step 1: for any 푓0 ∈ 퐸, there exists 푢 ∈ 퐿2(0, 휀; 퐻푝 +0 (휔))푚 such that (1−Π)푆(휀, 푓0, 푀푢) = +0. Then, the resulting solution 푓(휀, ⋅) is such that 푃(i푛) = 0 ⟹ 푐푛(푓(휀, ⋅)) = 0, since 푃(i푛) ≠ 0 for +|푛| > 푛0 and 푐푛(푓(휀, ⋅)) = 0 for |푛| ⩽ 푛0. +We consider this 푓(휀, ⋅) as our new initial condition, that we denote by 푓휀, and we have to steer +it to 0 in time 푇′. Note that since 푓0 ∈ 퐻푝(핋) and 푢 ∈ 퐿2(0, 휀; 퐻푝 +0 (휔))푑, according to Duahmel’s +formula and the fact that the semigroup e−푡ℒ is strongly continuous on 퐻푝(핋)푑, the state 푓휀 also +belongs to 퐻푝(핋)푑. +Let 퓁 ∈ ℕ large enough. According to proposition 8, there exists +푣 ∈ 퐻푝+deg 푃 +0 +((0, 푇′) × 휔)푑h × 퐻퓁 +0 ((0, 푇′) × 휔)푑p +such that 푆(푇′, 푓휀, 푃(휕푥)푣) = 0. Write 푄(i푛) by blocks as: +푄(i푛) = +⎛ +⎜ +⎝ +푄1(i푛) +⋮ +푄푘(i푛) +⎞ +⎟ +⎠ += +⎛ +⎜ +⎝ +푄h +1(i푛) +푄p +1(i푛) +⋮ +⋮ +푄h +푘(i푛) +푄p +푘(i푛) +⎞ +⎟ +⎠ +. +where the 푄푗(i푛) are of size 푚 × 푑, the 푄h +푗 (i푛) are of size 푚 × 푑h and 푄p +푗(i푛) are of size 푚 × 푑p. Notice +that the 퐿h +푛,푗 defined in the statement of theorem 9 are 퐿h +푛,푗 = 푄h +푗 (i푛)∕푃(i푛). Set also +푤푛(푡) ∶= 푄(i푛)푐푛(푣(푡, ⋅)). +Write it by blocks as +푤푛(푡) = +⎛ +⎜ +⎝ +푤푛,1(푡) +⋮ +푤푛,푘(푡) +⎞ +⎟ +⎠ += +⎛ +⎜ +⎝ +푄1(i푛)푐푛(푣(푡, ⋅)) +⋮ +푄푘(i푛)푐푛(푣(푡, ⋅)) +⎞ +⎟ +⎠ +. +Finally, set +푢푛(푡) ∶= 푤푛,1(푡) + 푤′ +푛,2(푡) + ⋯ + 푤(푘−1) +푛,푘 +(푡). +Step 3: Conclusion. — Remark that for every 푛 ∈ ℤ, +[퐵푛|푀]푘푤푛(푡) = [퐵푛|푀]푘푄(i푛)푐푛(푣(푡, ⋅)) += [퐵푛|푀]푘[퐵푛|푀]∗ +푘 Adj([퐵푛|푀]푘[퐵푛|푀]∗ +푘)푐푛(푣(푡, ⋅)) += det([퐵푛|푀]푘[퐵푛|푀]∗ +푘)푐푛(푣(푡, ⋅)) += 푃(i푛)푐푛(푣(푡, ⋅)). +Moreover, since 푆(푇′, 푓휀, 푃(휕푥)푣) = 0, the control ˜푣푛(푡) ∶= 푃(i푛)푐푛(푣(푡, ⋅)) steers 푐푛(푓휀) to 0 for +the system 푋′ +푛 = 퐵푛푋푛 + ˜푣푛 in time 푇′. That is to say, 푤푛 steers 푐푛(푓휀) to 0 for the system 푋′ +푛 = +퐵푛푋푛 + [퐵푛|푀]푘푤푛 in time 푇′. Thus, according to lemma 10, 푢푛 steers 푐푛(푓휀) to 0 for the system (9) +in time 푇′. +Thus, the control 푢 formally defined by 푢 ∶= ∑ +푛∈ℤ 푢푛푒푛 is such that 푆(푓휀, 푇′, 푀푢) = 0. Notice +that the previous sum is well-defined in 퐿2(0, 푇′; 퐿2(핋)). Remark that, if we define 푢 in the sense of +distributions, +푢 = (푄1(휕푥) + 휕푡푄2(휕푥) + ⋯ + 휕푘−1 +푡 +푄푘(휕푥))푣. +Since 푣 is supportedon [0, 푇′]×휔, so is 푢. Considerthe differentialoperator풬 ∶= 푄1(휕푥)+휕푡푄2(휕푥)+ +⋯ + 휕푘−1 +푡 +푄푘(휕푥). We have 푢 = 풬푤. Write this operator by blocks as 풬 = (풬h +풬p). In other words, +풬h ∶= 푄h +1(휕푥) + 휕푡푄h +2(휕푥) + ⋯ + 휕푘−1 +푡 +푄h +푘(휕푥). +11 + +The order of the differential operator 풬h is at most +Order(풬h) ≤ max +1≤푗≤푘(푗 − 1 + deg(푄h +푗 )). +Since 퐿h +푛,푗 = 푄h +푗 (i푛)∕푃(i푛), according to the definition of 푝 (see theorem 9), Order(풬h) ≤ 푝 + deg(푃). +Moreover, recall that 푣 ∈ 퐻푝+deg(푃) +0 +((0, 푇′) × 휔)푑h × 퐻퓁 +0 ((0, 푇′) × 휔)푑p. Thus, if we choose 퓁 ≥ +Order(풬p), 푢 ∈ 퐿2((0, 푇′) × 휔). +3.3 +Upper bound on the loss of regularity +Theorem 9 requires initial condition to be 퐻푝 for some 푝. In this section, we provide an elementary +upper bound on 푝. +Proposition 11. Assume that for |푛| large enough, the Kalman rank condition rank([퐵푛|푀]) = 푑 +holds. Let +푘(푛) ∶= inf{푘∶ rank([퐵푛|푀]푘) = 푑} ∈ {−∞} ∩ ℕ. +Then, thesequence(푘(푛))푛∈ℤ iseventuallyconstantwhen|푛| → +∞. Wewilldenote푘0 ∶= lim|푛|→+∞ 푘(푛). +Proof. The rank condition rank([퐵푛|푀]푘) = 푑 is equivalent to det([퐵푛|푀]푘[퐵푛|푀]∗ +푘) ≠ 0. Let +푃푘(푛) = det([퐵푛|푀]푘[퐵푛|푀]∗ +푘). 푃푘 is a polynomial in 푛, hence if 푃푘(푛0) ≠ 0 for some 푛0, then +푃푘(푛) ≠ 0 for every large enough |푛|. Thus, for every 푛0, there exists 푛1 such that 푘(푛) ≤ 푘(푛0) +whenever |푛| ≥ 푛1. Since 푘(푛) is integer valued, it is eventually constant. +Then, we have the following version of theorem 9. +Theorem 12. Assume that the hypotheses (H.1)–(H.4) hold, that 푇 > 푇∗ and that for all |푛| ∈ ℕ large +enough, the Kalman rank condition rank([퐵푛|푀]) = 푑 holds. Let 푘0 as in proposition 11. Let 퐸 as in +theorem 9. +Then, for every 푓0 ∈ 퐻4푑(푘0−1)(핋)푑∩퐸, there exists a control푢 ∈ 퐿2([0, 푇]×휔) such that the solution +푓 of the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0. +The sufficient part of theorem 1, as stated in the introduction is a special case of this theorem, +since we always have 푘0 ⩽ 푑. Here is the main lemma that allows us to bound the 푝 of theorem 9 +(see also [5, Theorem 2.1] for similar considerations). +Lemma 13. Let 퐴 ∈ ℳ푑(ℂ)푝[푋] a polynomial of degree at most 푝 with 푑 × 푑 matrices coefficients. +Assume that for some 푧0 ∈ ℂ, 퐴(푧0) is invertible. Then, 퐴−1 ∈ ℂ푑×푑 +푝(푑−1)(푋), i.e., the coefficients of +(퐴(푧))−1 are rational functions of 푧 of degree at most 푝(푑 − 1). +Proof. Write +퐴(푧)−1 = +1 +det(퐴(푧)) Adj(퐴(푧)), +where Adj(퐴(푧)) is the adjugate matrix of 퐴(푧). det(퐴(푧)) and Adj(퐴(푧)) are nonzero polynomials in +푧. Moreover, the coefficients of Adj(퐴(푧)) are sums of products on 푑 − 1 coefficients of 퐴(푧). Hence, +they are polynomials of degree at most (푑 − 1)푝. +The case we are interested in is: +12 + +Corollary 14. With 푘0 as in proposition 11, set, when it is defined +[퐵푛|푀]+ +푘0 ∶= [퐵푛|푀]∗ +푘0 +([퐵푛|푀]푘0[퐵푛|푀]∗ +푘0 +)−1. +Then, as a function of 푛, [퐵푛|푀]+ +푘0 ∈ ℂ푑×푑 +2(푘0−1)(2푑−1)(푋). +Proof. We have [퐵푛|푀]푘0 ∈ ℂ푑×푚푘0 +2(푘0−1)[푋], hence +[퐵푛|푀]푘0[퐵푛|푀]∗ +푘0 ∈ ℂ푑×푑 +4(푘0−1). +According to the previous lemma, +([퐵푛|푀]푘0[퐵푛|푀]∗ +푘0 +)−1 ∈ ℂ푑×푑 +4(푘0−1)(푑−1)(푋). +Hence [퐵푛|푀]+ +푘0 ∈ ℂ푑×푑 +푘 +(푋) with 푘 = 4(푘0 − 1)(푑 − 1) + 2(푘0 − 1) = 2(푘0 − 1)(2푑 − 1). +Proof of theorem 12. According to theorem 9, every initial condition in 퐸∩퐻푝(핋)푑 can be steered to 0, +where 푝 = deg([퐵푛|푀]+ +푘0) + 푘0 − 1 (degree as a rational function of 푛). But according to corollary 14, +deg([퐵푛|푀]+ +푘0) ≤ 2(푘0 − 1)(2푑 − 1). Thus 푝 ≤ 4푑(푘0 − 1). Hence, every initial condition in 퐸 ∩ +퐻4푑(푘0−1)(핋)푑 can be steered to 0. +4 +Necessary conditions for null-controllability +4.1 +Construction of WKB solutions +We will give other necessary conditions of null-controllability using so called WKB solutions, that we +construct here. Using these kind of approximate solutions is standard for wave equation (see,e.g., [25, +pp. 426–428] or [31, Appendix B] for a more elementary presentation) or Schrödinger equation (see, +e.g., [35, pp. 16–17]). WKB solutions were also used to disprove observability of some 2×2 parabolic- +transport system with variable coefficients [2, §3] (see also [3, §3] for a Navier-Stokes system with +Maxwell’s law). Our construction is a generalization of their construction for system of arbitrary +size, which brings a few difficulties. For the sake of clarity, we construct WKB solutions only for +systems with constant coefficients, which is enough for our purposes. But it is likely that such a +construction could be adapted to a large class of variable-coefficients parabolic-transport systems of +arbitrary sizes. +To disprove the observability inequality, these WKB solutions ought to be constructed for the +adjoint system. But the parabolic-transport system (Sys) and its adjoint have the same structure, so, +in order to lighten the notations, we construct the WKB solutions for the system (Sys). +Let 휙 ∈ 퐶∞([0, 푇] × 핋; ℂ) such that ℑ(휙) ≥ 0 and 휕푥휙 never vanishes. We search approximate +solutions 푔WKB +ℎ +(푡, 푥) of the parabolic-transport system (Sys) with the following ansatz, where ℎ > 0 +is assumed to be small: +⎧ +⎨ +⎩ +푔WKB +ℎ +(푡, 푥) = 푋ℎ(푡, 푥)ei휙(푡,푥)∕ℎ, +푋ℎ(푡, 푥) ∼ +∑ +푗≥0 +ℎ푗푌푗(푡, 푥). +(10) +13 + +We have +휕푥푔WKB +ℎ += (휕푥푋ℎ + i +ℎ휕푥휙푋ℎ) ei휙∕ℎ, +휕푡푔WKB +ℎ += (휕푡푋ℎ + i +ℎ휕푡휙푋ℎ) ei휙∕ℎ, +휕2 +푥푔WKB +ℎ += (휕2 +푥푋ℎ + 2i +ℎ 휕푥휙휕푥푋ℎ − 1 +ℎ2 (휕푥휙)2푋ℎ + i +ℎ휕2 +푥휙푋ℎ) ei휙∕ℎ. +Assuming that this 푔WKB +ℎ +is solution of the parabolic-transport system (Sys), we get +0 = (휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾) (푋ℎei휙∕ℎ) +=[ (휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾) 푋ℎ + 1 +ℎ +(i휕푡휙 + i퐴휕푥휙 − i퐵휕2 +푥휙 − 2i퐵휕푥휙휕푥 +) 푋ℎ + 1 +ℎ2 퐵(휕푥휙)2푋ℎ]ei휙∕ℎ. +Plugging in the asymptotic expansion of 푋ℎ, we get +0 ∼ +∑ +푗≥−2 +[(휕푥휙)2퐵푌푗+2 + (i휕푡휙 + i퐴휕푥휙 − i퐵휕2 +푥휙 − 2i퐵휕푥휙휕푥 +) 푌푗+1 + (휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾) 푌푗]ℎ푗, +where, by convention, 푌푗 = 0 for 푗 < 0. We want to cancel each of the terms in this sum. Thus, we +are looking for (푌푗)푗≥0 such that for all 푗 ≥ −2, +(휕푥휙)2퐵푌푗+2 + (i휕푡휙 + i퐴휕푥휙 − i퐵휕2 +푥휙 − 2i퐵휕푥휙휕푥 +) 푌푗+1 + (휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾) 푌푗 = 0. +(11) +Solving this induction relation requires us to look at different projections of this equation. From +now on, we will denote +푌푗 = ( +푌h +푗 +푌p +푗 +) +with 푌h +푗 ∈ ℂ푑h and 푌p +푗 ∈ ℂ푑p. +Then, recalling that 퐵 = ( 0 0 +0 퐷 +) and taking the parabolic components of eq. (11) (i.e., the 푑p last +components), we get +(휕푥휙)2퐷푌p +푗 = − (0 +퐼) [(i휕푡휙 + i퐴휕푥휙 − i퐵휕2 +푥휙 − 2i퐵휕푥휙휕푥 +) 푌푗−1 + (휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾) 푌푗−2 +] . +(12) +Since 퐷 is invertible, this formula determines 푌p +푗 as a function of 푌푗−1 and 푌푗−2. +Before looking at the other projections of eq. (11), let us recall that 퐴 = ( 퐴′ 퐴12 +퐴21 퐴22 +). We similarly +write 퐾 in blocks as ( 퐾′ 퐾12 +퐾21 퐾22 +). Then, taking the transport (i.e., the first 푑h) components of eq. (11), +we get +0 = (i휕푡휙 + i휕푥휙퐴′)푌h +푗 + i휕푥휙퐴12푌p +푗 + (퐼 +0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1. +(13) +From now on, we choose 휙 of the form2 +휙(푡, 푥) = 휓(푥 − 휇푡), +(14) +2Equations (12) and (13) with 푗 = 0 implies (휕푡휙 + 휕푥휙퐴′)푌h +0 = 0. If we want a non-trivial 푌h +0 , this imposes 휙 to depend +only on 푥 − 휇푡 for some 휇 ∈ Sp(퐴′). +14 + +where 휇 is an eigenvalue of 퐴′ an 휓′ never vanishes. With this 휙, eq. (13) reads +0 = i휓′(푥 − 휇푡)(퐴′ − 휇)푌h +푗 + i휓′(푥 − 휇푡)퐴12푌p +푗 + (퐼 +0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1 += i휓′(푥 − 휇푡)(퐴′ − 휇)푌h +푗 + i휓′(푥 − 휇푡)퐴12푌p +푗 + (휕푡 + 퐴′휕푥 + 퐾′)푌h +푗−1 + (퐴12휕푥 + 퐾12)푌p +푗−1. +(15) +Denote by 푃′ +휇 the projectionon the eigenspace of 퐴′ associatedwith 휇 along the other eigenspaces. +We consider 푌h +푗,휇 ∈ Range(푃′ +휇) defined by 푌h +푗,휇 = 푃′ +휇푌h +푗 . Similarly, we set 푌h +푗,≠휇 ∈ ker(푃′ +휇) as 푌h +푗,≠휇 = +(퐼 − 푃′ +휇)푌h +푗 . Finally, we write in blocks 퐴′ and 퐾′ along the sum ℝ푑 = Range(푃′ +휇) ⊕ ker(푃′ +휇) as +퐴′ = (휇 +0 +0 +퐴′ +22 +) , +퐾′ = (퐾′ +11 +퐾′ +12 +퐾′ +21 +퐾′ +22 +) , +where 퐴′ +22 ∈ ℒ(ker(푃′ +휇)), 퐾′ +11 = 푃′ +휇퐾′푃′ +휇 ∈ ℒ(Range(푃′ +휇)), 퐾′ +12 ∈ ℒ(ker(푃′ +휇), Range(푃′ +휇)), etc. Then, +projecting eq. (15) on ker(푃′ +휇) along Range(푃′ +휇) (i.e., multiplying by (퐼 − 푃′ +휇)), we get +i휓′(푥 − 휇푡)(퐴′ +22 − 휇)푌h +푗,≠휇 = −(퐼 − 푃′ +휇) +[ +i휓′(푥 − 휇푡)퐴12푌p +푗 + (퐼 +0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1 +] +. +(16) +Since 푃′ +휇 is the projection on the eigenspace of 퐴′ associated with the eigenvalue 휇, 퐴′−휇 is invertible +on ker(푃′ +휇), i.e., 퐴′ +22 − 휇 is invertible. Hence, eq. (16) determines 푌h +푗,≠휇 as a function of 푌p +푗 and 푌푗−1. +Finally, we project eq. (15) on Range(푃′ +휇), we get +0 = (휕푡 + 휇휕푥 + 퐾′ +11)푌h +푗,휇 + 퐾′ +12푌h +푗,≠휇 + 푃′ +휇(퐴12휕푥 + 퐾12)푌p +푗 + i휓′(푥 − 휇푡)푃′ +휇퐴12푌p +푗+1. +(17) +We then use eq. (12) to express 푌p +푗+1 as +푌p +푗+1 = 퐷1푌h +푗 + 퐷2푌p +푗 + 퐷3푌푗−1, with 퐷1 = − +i +휓′(푥 − 휇푡)퐷−1퐴21, +and where 퐷2 and 퐷3 are matrix first or second-order differential operators. Their specific expressions +do not matter for our purpose. Plugging this in eq. (17), we get +(휕푡 + 휇휕푥 + 퐾′ +11 + 푃′ +휇퐴12퐷−1퐴21푃′ +휇)푌h +푗,휇 += −퐾′ +12푌h +푗,≠휇 − 푃′ +휇(퐴12휕푥 + 퐾12)푌p +푗 − i휓′(푥 − 휇푡)푃′ +휇퐴12(퐷1(퐼 − 푃′ +휇)푌h +푗,≠휇 + 퐷2푌p +푗 + 퐷3푌푗−1). +(18) +If we chose an initial condition 푌h +푗,휇,0 for 푌h +푗,휇, eq. (18) determines 푌h +푗,휇 as a function of 푌h +푗,휇,0, 푌h +푗,≠휇, +푌p +푗 and 푌푗−1. +We have seenthat if 휙 is given by eq. (14), the (푌푗)푗∈ℕ that solve the WKB recurrence equation (11) +are given by eqs. (12), (16) and (18). +To be rigorous, we have only proved that if (푌푗)푗푛ℕ solves eq. (11), then 푌p +푗 , 푌h +푗,≠휇 and 푌h +푗,휇 solves +eqs. (12), (16) and (18) respectively, but not the reciprocal (which is what we are actually interested +in). However, we easily rephrase the computations of this section as a sequence of equivalences: +• ∀푗 ≥ −2, 푌푗 solves eq. (11) if and only if; +• ∀푗 ≥ 0, 푌p +푗 solves eq. (12), 푌h +푗,≠휇 solves eq. (16) and 푌h +푗,휇 solves eq. (17) if and only if; +• ∀푗 ≥ 0, 푌p +푗 solves eq. (12), 푌h +푗,≠휇 solves eq. (16) and 푌h +푗,휇 solves eq. (18). +15 + +We summarize the computations of this section in the following proposition: +Proposition 15. Let 휓 ∈ 퐶∞(핋) such that 휓′ never vanishes and ℑ(휓) ≥ 0. Let 휇 ∈ Sp(퐴′) and set 휙 +as in eq. (14). +For every 푗 ≥ 0, let 푌h +푗,휇,0 ∈ 퐶∞(핋; ker(퐴′ − 휇)). Define (푌p +푗 )푗≥−2, (푌h +푗,≠휇)푗≥−2 and (푌h +푗,휇)푗≥−2 with +the following recursive procedure: +• set 푌p +−2 = 푌p +−1 = 0, 푌h +−2,≠휇 = 푌h +−1,≠휇 = 0, 푌h +−2,휇 = 푌h +−1,휇 = 0; +• if 푌p +푘, 푌h +푘,≠휇, 푌h +푘,휇 are defined for −2 ≤ 푘 ≤ 푗 − 1, define 푌p +푗 with eq. (12), 푌h +푗,≠휇 with eq. (16) and +푌h +푗,휇 with eq. (18) with initial condition 푌h +푗,휇,0. +For 푗 ≥ 0, set 푌푗(푡, 푥) = ( +푌h +푗,휇+푌h +푗,≠휇 +푌p +푗 +). Let 푞 ∈ ℕ. Let the function 푔WKB +ℎ +be defined by +푔WKB +ℎ +(푡, 푥) = +푞∑ +푗=0 +ℎ푗푌푗ei휙(푡,푥)∕ℎ. +(19) +Then, defining 푟ℎ by +(휕푡 − 퐵휕2 +푥 + 퐴휕푥 + 퐾)푔WKB +ℎ +(푡, 푥) = 푟ℎ(푡, 푥)ei휙(푡,푥)∕ℎ, +for every 푘 ∈ ℕ, 퓁 ∈ ℕ, 푡 ∈ [0, 푇] and 푥 ∈ 핋, +|휕푘 +푡 휕퓁 +푥푟ℎ(푡, 푥)| ≤ 퐶푘,퓁ℎ푞−1. +Remark 16. Assume that ℎ−1 ∈ ℕ. Then, replacing 휙 by 휙 + 2푘휋 in eq. (10) does not change the +WKB solution 푔WKB +ℎ +. Hence, 휙 can be defined up to a factor 2푘휋. That way, 휙 can be non-periodic, +as long as 휙 mod 2휋 is. Thus, we can choose +휙(푡, 푥) = i휑(푥 − 휇푡) + 푛0(푥 − 휇푡) with 휇 ∈ Sp(퐴′), 휑 ≥ 0, and 푛0 ∈ ℕ ⧵ {0}. +These WKB solutions will be used to disproveobservability inequalities that often feature a projec- +tion on high frequencies. To deal with these projection on high frequencies, we will use the following +lemma. +Lemma 17. Let 푛 ∈ ℤ. Under the assumptions of proposition 15, for every 퓁 ∈ ℕ, we have uniformly +in 0 ≤ 푡 ≤ 푇, in the limit ℎ → 0+, +(푔WKB +ℎ +(푡, ⋅), 푒푛)퐿2 = 푂(ℎ퓁). +Proof. The scalar product (푔WKB +ℎ +(푡, ⋅), ei푛푥)퐿2 can be written as +(푔WKB +ℎ +(푡, ⋅), 푒푛)퐿2 = ∫ +핋 +푤푡,ℎ,푛(푥)ei휓(푥−휇푡)∕ℎ d푥, +where +푤푡,ℎ,푛(푥) ∶= +푞∑ +푗=0 +ℎ푗푌푗(푡, 푥)e−i푛푥. +16 + +Note that 푤푡,ℎ,푛 and its derivative are uniformly bounded for 0 ≤ 푡 ≤ 푇 and ℎ ≤ 1. Consider the +differential operator 퐿 ∶= (i휓′(푥 − 휇푡))−1휕푥. Here, we use the fact that 휓′ never vanishes. This +operator is such that +ℎ퐿ei휓(푥−휇푡)∕ℎ = ei휓(푥−휇푡)∕ℎ. +Thus, denoting 퐿∗ the adjoint of 퐿, by integration by parts, +(푔WKB +ℎ +(푡, ⋅), 푒푛)퐿2 = ℎ푙 ∫ +핋 +(퐿∗)퓁(푤푡,ℎ,푛)(푥)ei휓(푥−휇푡)∕ℎ d푥. +The operator 퐿∗ is a differential operator independent of ℎ. Hence, by definition of 푤푡,ℎ,푛 +(푔WKB +ℎ +(푡, ⋅), 푒푛)퐿2 = 푂(ℎ퓁). +4.2 +The parabolic-transport system is not null controllable in small time +We now prove that the time condition 푇 ⩾ 푇∗ is necessary (remark that the equality case 푇 = 푇∗ +remains an open question). It was already proved to be necessary for the null-controllability of every +퐿2 initial conditions [7]. But this proof did not exclude the null-controllability of every 퐻푘 initial +condition when 푇 < 푇∗. +Proposition 18. Let 푇 > 0 and assume that there exists 푁 ∈ ℕ∗ and 푘 ∈ ℕ such that every initial +conditions in 퐻푘(핋)푑 ∩ {∑ +|푛|>푁 푋푛ei푛푥} for the parabolic-transport system (Sys) can be steered to 0 in +time 푇. Then 푇 ≥ 푇∗. +Proof. Let 휇 ∈ Sp(퐴′) with maximum modulus. By definition, 푇∗ = 퓁(휔)∕|휇|. Let 푇 < 푇∗. +We aim to disprove that the observability inequality associated to the control problem of propo- +sition 18 using the WKB solution constructed above. We claim that this observability inequality is: +there exists 퐶 > 0 such that for every 푔0 ∈ 퐿2(핋)푑, the solution 푔 of +(휕푡 − 퐵∗휕푥 − 퐴∗휕푥 + 퐾∗휕푥)푔(푡, 푥) = 0, +푔(0, 푥) = 푔0(푥) +(20) +satisfies +‖휋푁푔(푇, ⋅)‖퐻−푘(핋) ≤ 퐶‖푀∗푔‖퐿2((0,푇)×휔), +(21) +where 휋푁 ∶ ∑ +푛∈ℤ 푋푛ei푛푥 ∈ 퐿2(핋) ↦ ∑ +|푛|>푁 푋푛ei푛푥. +This is proved using a standard duality lemma, see e.g. [14, Lemma 2.48] with 퐶2 = e−푡ℒ◦휋∗ +푁◦휄푘 +and 퐶1 ∶ 푢 ∈ 퐿2((0, 푇) × 휔) ↦ ∫푇 +0 e−(푇−푡)ℒ푀푢(푡) d푡, where 휄푘 is the injection 퐻푘(핋) → 퐿2(핋). Note +that 휋∗ +푁 is the injection {∑ +|푛|>푁 푋푛ei푛푥} → 퐿2(핋), and that 휄∗ +푘 is a bijective isometry 퐻−푘(핋) → 퐻푘(핋) +([7, Lemma 33]). +Testing this observability inequality on initial conditions of the form 휕푘 +푥푔0 instead of 푔0, we get +‖휋푁푔(푇, ⋅)‖퐿2(핋) ≤ 퐶‖휕푘 +푥푀∗푔‖퐿2((0,푇)×휔), +(22) +Step 1: Construction of the counterexample. — Let 푇 < 푇∗. There exists 푥0 ∉ 휔 such that 푥0 −휇푡 ∉ 휔 +for every 0 ≤ 푡 ≤ 푇. Choose 휑 ∈ 퐶∞(핋) real-valued such that 휑(푥0) = 0, 휑′′(푥0) = 1 and 휑(푥) > 0 +for every 푥 ≠ 푥0. Then, choose 휙(푡, 푥) = i휑(푥 + 휇푡) + (푥 + 휇푡)푛0, as we did in remark 16 (the change +from 휇 to −휇 is because we are considering −퐴∗ instead of 퐴). +This choice of 휙 ensures that whatever the choices of the 푌푗, the WKB solution 푔WKB +ℎ +defined +by eq. (10) stays concentrated around 푥0 + 휇푡. +17 + +Let 푌h +0,휇,0 ∈ 퐶∞(핋; ker(퐴′∗ + 휇)) with 푌h +0,휇,0(푥0) ≠ 0. For 푗 ≥ 1, set 푌h +푗,휇,0 = 0. Let 푞 > 푘 + 1. +Consider the function 푔WKB +ℎ +defined by proposition 15 (where 퐵, and 퐾 are replaced respectively by +퐵∗ and 퐾∗, and where 퐴 is replaced by −퐴∗). +Set also 푔ℎ(푡, 푥) the solution of the adjoint system (20) with initial condition 푔WKB +ℎ +(푡 = 0, ⋅). +Step 2: Estimation of the difference between 푔WKB +ℎ +and 푔ℎ. — According to proposition 15, +(휕푡 − 퐵∗휕2 +푥 − 퐴∗휕푥 + 퐾∗)푔WKB +ℎ += 푂(ℎ푘+1)ei휙(푡,푥)∕ℎ, +Hence, with 푟ℎ ∶= 푔WKB +ℎ +− 푔ℎ, we have 푟ℎ(0, 푥) = 0 and +(휕푡 − 퐵∗휕2 +푥 − 퐴∗휕푥 + 퐾∗)푟ℎ = 푂(ℎ푘+1)ei휙(푡,푥)∕ℎ. +where the 푂 has to be understood in the 퐶∞-topology. Since the parabolic-transport system is well- +posed in 퐻푘(핋)푑, we get that for every 푗 ∈ ℕ, uniformly in 0 < 푡 < 푇, +‖휕푗 +푥(푔WKB +ℎ +(푡, ⋅) − 푔ℎ(푡, ⋅))‖퐿2 ≤ 퐶푗ℎ푘−푗+1. +(23) +Step 3: Upper bound on the right-hand side of the observability inequality. — According to the triangle +inequality, +‖휕푘 +푥푀∗푔ℎ‖퐿2((0,푇)×휔) ≤ ‖휕푘 +푥푀∗푔WKB +ℎ +‖퐿2((0,푇)×휔) + ‖휕푘 +푥푀∗푟ℎ‖퐿2((0,푇)×휔). +According to step 2, the second term of the right-hand side is 푂(ℎ). For the first term of the right- +hand side, we recall that 푔WKB +ℎ += ∑푞 +푗=0 ℎ푗푌푗ei휓(푥−휇푡), and that, thanks to our choice of 휓, ei휓(푥+휇푡) is +exponentially small when 푥 + 휇푡 ≠ 푥0. Therefore, since 푥0 − 휇푡 ∉ 휔 for every 0 ≤ 푡 ≤ 푇, for some +푐 > 0, +‖휕푘 +푥푀∗푔WKB +ℎ +‖퐿2((0,푇)×휔) = 푂(e−푐∕ℎ). +This proves that +‖휕푘 +푥푀∗푔ℎ‖퐿2((0,푇)×휔) = 푂(ℎ). +(24) +Step 4: Lower bound on the left-hand side of the observability inequality. — According to lemma 17, +for any 퓁 ≥ 0, +‖휋푁푔WKB +ℎ +(푇, ⋅)‖퐿2(핋) = ‖푔WKB +ℎ +(푇, ⋅)‖퐿2(핋) + 푂(ℎ퓁). +(25) +Thus, using the inverse triangle inequality, +‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ ‖휋푁푔WKB +ℎ +(푇, ⋅)‖퐿2(핋) − ‖휋푁푟ℎ(푇, ⋅)‖퐿2(핋). +Using the error estimates of step 2, and eq. (25), we get +‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ ‖푔WKB +ℎ +(푇, ⋅)‖퐿2(핋) − 퐶ℎ. +(26) +Thus, we only need to find a lower-bound for ‖푔WKB +ℎ +(푇, ⋅)‖퐿2(핋). We have +‖푔WKB +ℎ +(푇, ⋅)‖2 +퐿2(핋) = ∫ +핋 +|||||||||| +푞∑ +푗=0 +ℎ푗푌푗(푡, 푥) +|||||||||| +2 +e−2휑(푥+휇푇)∕ℎ d푥 = ∫ +핋 +|푌0(푡, 푥)|2e−2휑(푥+휇푇)∕ℎ d푥 + 푂(ℎ). +Recall that 휑(푥0) = 0, that for 푥 ≠ 푥0, 휑(푥) is strictly positive and that 휑′′(푥0) ≠ 0. Then, using +Laplace’s method (see e.g. [36, §2.2] and in particular [36, eq. (2.34)]), we get +‖푔WKB +ℎ +(푇, ⋅)‖2 +퐿2(핋) = 푐 +√ +ℎ + 푂(ℎ3∕2) +18 + +for some 푐 > 0. Plugging this into eq. (26), we get that for ℎ small enough, +‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ 푐 +√ +ℎ. +(27) +Step 5: Conclusion. — Comparing the lower bound (27) and the upper bound (24) and taking ℎ small +enough, we see that the observability inequality (21) cannot hold if 푇 < 푇∗, hence the parabolic- +transport system (Sys) with initial conditions in 퐻푘 ∩ 휋푁(퐿2(핋)) is not null-controllable in time 푇 < +푇∗. +4.3 +Rough initial conditions are not null-controllable +We now give necessary conditions for every 퐿2 initial condition to be steerable to 0. To do this, we +only need the first term of the WKB expansion of proposition 15. By analyzing higher-order terms +of the WKB expansion, it is likely that we could get necessary conditions for the null-controllability +of every 퐻푘 initial conditions. But doing this analysis in general seems hard, and we leave this for +future work, or on a case-by-case basis. In fact, we will prove the following statement, which is a +refined version of theorem 3. +Proposition 19. Let 휇 ∈ Sp(퐴′), 푁 ∈ ℕ and 푇 > 0. Let 푃′ +휇 be the projection on the eigenspace of 퐴′ +associated to 휇. Write 퐾 in blocks as ( 퐾′ 퐾12 +퐾21 퐾22 +), with 퐾′ ∈ ℳ푑ℎ(ℝ). Set +퐾∗ +휇 ∶= (푃′ +휇)∗ ((퐾′)∗ + 퐴∗ +21(퐷∗)−1퐴∗ +12 +) (푃′ +휇)∗ +Assume that every initial condition 푓0 ∈ 퐿2(핋)푑∩{∑ +|푛|>푁 푋푛ei푛푥} is steerable to 0 in time 푇 with control +in 퐿2((0, 푇) × 휔). Then, for every 휇 ∈ Sp(퐴′) and for every non-zero subspace 푆 ⊂ Range((푃′ +휇)∗) that is +stable by 퐾∗ +휇, there exists 푉0 ∈ 푆 such that 푀∗( 푉0 +0 +) ≠ 0. +Proof. Step 1: Observability inequality. — Using a standard duality lemma [14, Lemma 2.48], and +as in the proof of proposition 18, we get an observability inequality that is equivalent to the null- +controllability of the system (Sys) with initial conditions in 퐿2(핋)푑 ∩ {∑ +|푛|>푁 푋푛ei푛푥}. This observ- +ability inequality is: there exists 퐶 > 0 such that for every 푔0 ∈ 퐿2(핋)푑, the solution 푔 of +(휕푡 − 퐵∗휕2 +푥 − 퐴∗휕푥 + 퐾∗)푔(푡, 푥) = 0, +푔(0, 푥) = 푔0(푥) +(28) +satisfies +‖휋푁푔(푇, ⋅)‖퐿2(핋) ≤ 퐶‖푀∗푔‖퐿2((0,푇)×휔), +(29) +where, as in the proof of proposition 18, 휋푁 ∶ ∑ +푛∈ℤ 푋푛ei푛푥 ∈ 퐿2(핋) ↦ ∑ +|푛|>푁 푋푛ei푛푥. +Step 2: Construction of the counterexample. — Let 푉0 ∈ 푆 ⧵ {0}. Set 휑 ∶== 0 and let 휙(푡, 푥) = +푛0(푥 − 휇푡) as in remark 16. Set 푌h +0,휇,0(푥) ∶= 푉0. For 푗 > 0, set 푌h +푗,휇,0 ∶= 0. Let 푔WKB +ℎ +be defined by +proposition 15 with 퐵 and 퐾 replaced respectively by 퐵∗ and 퐾∗ and 퐴 by −퐴∗, and with 푞 ≥ 2. Let +푔ℎ be the solution of the parabolic-transport system (Sys) with initial condition 푔WKB +ℎ +(0, ⋅). +Remark that according to proposition 15, and in particular eq. (18), +(휕푡 − 휇휕푥 + 퐾∗ +휇)푌h +0,휇 = 0. +Thus, 푌h +0,휇(푡, 푥) = e−푡퐾∗ +휇푉0. In particular, since 푆 is stable by 퐾∗ +휇, 푌h +0,휇(푡, 푥) ∈ 푆 for all 푡, 푥. +Step 3: Error estimate between 푔WKB +ℎ +and 푔ℎ. — Set 푟ℎ ∶= 푔ℎ−푔WKB +ℎ +. Then 푟ℎ(0, 푥) = 0, and according +to proposition 15, +(휕푡 − 퐵∗휕2 +푥 − 퐴∗휕푥 + 퐾∗)푟ℎ = 푂(ℎ)e푖휙(푡,푥)∕ℎ. +19 + +Since the parabolic-transport system is well-posed in 퐿2(핋)푑, uniformly in 0 ≤ 푡 ≤ 푇, +‖푟ℎ(푡, ⋅)‖퐿2(핋) ≤ 퐶ℎ. +Step 4: Upper bound of the right-hand side of the observability inequality. — Using the error estimate +of the previous step, the right-hand side of the observability inequality (29) satisfies +‖푀∗푔ℎ‖2 +퐿2((0,푇)×휔) ≤ ‖푀∗푔WKB +ℎ +‖2 +퐿2((0,푇)×휔) + 퐶ℎ +≤ ‖푀∗푌h +0ei휙∕ℎ‖2 +퐿2((0,푇)×휔) + 퐶ℎ += +‖‖‖‖‖‖‖‖‖ +푀∗ (푌h +0,휇 +0 ) +‖‖‖‖‖‖‖‖‖ +2 +퐿2((0,푇)×휔) ++ 퐶ℎ += 2휋 ∫ +푇 +0 +||||||||| +푀∗ (e−푡퐾∗ +휇푉0 +0 +) +||||||||| +2 +d푡 + 퐶ℎ, +(30) +where we used the definition of 푔WKB +ℎ +for the last three inequalities. +Step 5: Lower-bound of the left-hand side of the observability inequality. — Using again the error +estimate of step 3, the left-hand side of the observability inequality (29) satisfies +‖휋푁푔ℎ(푇, ⋅)‖2 +퐿2 ≥ ‖휋푁푔WKB +ℎ +(푇, ⋅)‖2 +퐿2 − 퐶ℎ. +Then, using the estimate on low frequencies of 푔WKB +ℎ +(lemma 17) +‖휋푁푔ℎ(푇, ⋅)‖2 +퐿2 ≥ ‖푔WKB +ℎ +(푇, ⋅)‖2 +퐿2 − 퐶ℎ. +Now, using the definition of 푔WKB +ℎ +, and the fact that |ei휙| = 1, +‖휋푁푔ℎ(푇, ⋅)‖2 +퐿2 ≥ ‖푌h +0,휇(푇, ⋅)‖2 +퐿2 − 퐶ℎ. += 2휋|e−푇퐾∗ +휇푉0|2 − 퐶ℎ. +(31) +Step 6: Conclusion. — Comparing the upper bound on the right-hand side of the observability in- +equality (eq. (30)) and the lower bound on the left-hand side (eq. (31)), we see that 푀∗e−푡퐾∗ +휇푉0 cannot +vanish for every 0 ≤ 푡 ≤ 푇. Since e−푡퐾∗ +휇푉0 ∈ 푆 for every 푡, this proves the proposition. +5 +Systems of two equations +We apply the general theorems of the previous sections on 2 × 2 systems. Some of these results are +not new (see, e.g., [13]). Our goal here is only to check whether our results are optimal, at least in +this setting. +5.1 +Control properties of 2 × 2 systems: statements +Here, we consider the parabolic transport-system (Sys) with +퐵 = (0 +0 +0 +푑) , +퐴 = ( 푎′ +푎12 +푎21 +푎22) , +퐾 = (푘11 +푘12 +푘21 +푘22) , +푀 = (푚1 +푚2) . +(32) +20 + +where all lower-case letters are real numbers, with 푑 > 0 and 푎′ ≠ 0. Here, we assume that 푀 has +rank one. We do not need to treat the case where rank(푀) = 2, because it is already covered with +the general theorem where there is a control on every component (see [7, Theorem 2] or theorem 12 +with 푘 = 1): every initial condition in 퐿2(핋)푑 is null-controllable in time 푇 > 푇∗. In the following +three propositions, we detail the applications of our general theorem to eleven cases, showcasing the +variety of phenomena that can appear depending on the values of every coefficients. The proofs are +given in the next subsections. +Proposition 20. Assume that 퐵, 퐴, 퐾, 푀 are given by eq. (32). Assume that (푚1, 푚2) = (1, 0). If +(푎21, 푘21) = (0, 0), the parabolic-transport system (Sys) is not null-controllable, whatever the time 푇 is. +Let 푇 > 퓁(휔)∕|푎′| (where 퓁(휔) is defined in eq. (1)). +• If 푘21 ≠ 0, every initial condition in 퐿2(핋)2 for the system (Sys) can be steered to 0 in time 푇 with +퐿2 controls. +• If 푎21 ≠ 0 and 푘21 = 0, every initial condition 푓0 = (푓h +0, 푓p +0) in 퐿2(핋)2 such that ∫핋 푓p +0 = 0 for the +system (Sys) can be steered to 0 in time 푇 with 퐿2 controls. +Proposition 21. Assume that 퐵, 퐴, 퐾, 푀 are given by eq. (32). Assume that (푚1, 푚2) = (0, 1). If +(푎12, 푘12) = (0, 0), the parabolic-transport system (Sys) is not null-controllable, whatever the time 푇 is. +Let 푇 > 퓁(휔)∕|푎′|. +• If 푎12 ≠ 0 and 푘12 ≠ 0, every initial condition in 퐻1(핋)×퐿2(핋) for the system (Sys) can be steered +to 0 in time 푇 with 퐿2 controls. +• If 푎12 ≠ 0 and 푘12 = 0, every initial condition 푓0 = (푓h +0, 푓p +0) in 퐻1(핋)×퐿2(핋) such that ∫핋 푓h +0 = 0 +for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls. +• If 푎12 = 0 and 푘12 ≠ 0, every initial condition in 퐻2(핋)×퐿2(핋) for the system (Sys) can be steered +to 0 in time 푇 with 퐿2 controls. +In every cases, there exists an initial condition 푓0 in 퐿2(핋) such that ∫핋 푓0 = 0 that cannot be steered to +0 in time 푇 with 퐿2 controls. +In the case where 푎21 = 0 and 푘21 ≠ 0, there is a gap in the regularity condition that is sufficient +for the null controllability (i.e., 퐻2 × 퐿2), and the lack of null-controllability of 퐿2 × 퐿2 initial condi- +tions. Are every 퐻1 × 퐿2 initial conditions steerable to 0? We conjecture that this is not the case, but +theorem 3 is not enough to prove so. We would need to look at the second term in the WKB expan- +sion to find out, or use another method; maybe using a refined version of regularization properties +of lemma 23. +We do not detail in general the case where 푚1 ≠ 0 and 푚2 ≠ 0. Let us just mention that there +is no regularity condition for null-controllability to hold. But depending on whether the solution of +det([퐵푛, 푀]) = 0 (which is a quadratic equation in 푛) are integer, there might be a condition on at +most two fourier components for an initial condition to be steerable to 0. We only detail the following +case that is about the simultaneous control of a transport and a parabolic equation. +Proposition 22. Assume that 퐵, 푀 are given by eq. (32). Assume that 퐴 = ( 푎′ +0 +0 푎22 +) and 퐾 = ( 푘11 +0 +0 +푘22 +). +Assume that (푚1, 푚2) = (1, 1). Let 푇 > 퓁(휔)∕|푎′|. +• If 푎′ ≠ 푎22 and 푘11 = 푘22, every initial condition 푓0 = (푓h +0, 푓p +0) ∈ 퐿2(핋)2 such that ∫푇 푓h +0 = ∫핋 푓p +0 +can be steered to zero with controls in 퐿2. +• If 푎′ ≠ 푎22 and 푘11 ≠ 푘22, every initial condition in 퐿2(핋)2 can be steered to zero with controls in +퐿2. +21 + +• If 푎′ = 푎22 and +√ +(푘22 − 푘11)∕푑 ∉ ℕ, every initial condition in 퐿2(핋)2 can be steered to zero with +controls in 퐿2. +• If 푎′ = 푎22 and 푛0 ∶= +√ +(푘22 − 푘11)∕푑 ∈ ℕ, every initial condition 푓0 = (푓h +0, 푓p +0) ∈ 퐿2(핋)2 such +that 푐±푛0(푓h +0) = 푐±푛0(푓p +0) can be steered to zero with controls in 퐿2. +The case 푎′ ≠ 푎22 and 푘11 = 푘22 is not new, at least in spirit: the simultaneous controllability +(equivalently, additive observability) of a heat equation and a wave equation has been studied by +Zuazua [41, §2.1–2.2]. +5.2 +Regularity of the free equation +We will use some basic regularity results. +Lemma 23. Let 푓0 ∈ 퐻1(핋)푑h × 퐿2(핋)푑p. For every 푡 > 0, e−푡ℒ푓0 ∈ 퐻1(핋)푑. +Assume in addition that 퐴12 = 0, and that 푓0 ∈ 퐻2(핋)푑h × 퐿2(핋)푑p. For every 푡 > 0, e−푡ℒ푓0 ∈ +퐻2(핋)푑. +To prove it, we will use the following (sub)lemma: +Lemma 24. Consider ℒp and 퐹p as defined in section 2 (or [7, §4.1]). For every 푡 > 0, 푘 ∈ ℕ and +푓0 ∈ 퐹p, e−푡ℒp푓0 ∈ 퐻푘(핋)푑. +Proof. Set 푓(푡) = e−푡ℒp푓0. Denote the first 푑h components of 푓(푡) by 푓h(푡) and the last 푑p compo- +nents of 푓(푡) by 푓p(푡) (and similarly for 푓0). +We will use some simple tools from [7, §4.4.1]. For the sake of readability, we redo the proof in +full here. +Step 1: Computing 푓h(푡) as a function of 푓p(푡). — Since 푓(푡) ∈ 퐹p, by definition of 퐹p (section 2), for +every |푛| > 푛0, +푃p(i∕푛)푐푛(푓(푡)) = 푐푛(푓(푡)). +Writing 푃p(푧) by blocks as ( 푝11(푧) 푝12(푧) +푝21(푧) 푝22(푧) +), and taking the first 푑h components, +푝11(i∕푛)푐푛(푓h(푡)) + 푝12(i∕푛)푐푛(푓p(푡)) = 푐푛(푓h(푡)). +Since 푃p(0) = ( 0 0 +0 퐼 +), 푝11(0) = 0 and for 푧 small enough, |푝11(푧)| < 1. Then, increasing푛0 if necessary, +for |푛| > 푛0, +푐푛(푓h(푡)) = (퐼 − 푝11(i∕푛))−1푝12(i∕푛)푐푛(푓p(푡)). +For 푧 ∈ ℂ small enough, let 퐺(푧) = (퐼 − 푝11(푧))−1푝12(푧). Then, 퐺 depends holomorphically in 푧 +small enough, and for |푛| > 푛0 푐푛(푓h(푡)) = 퐺(i∕푛)푐푛(푓p(푡)). +Step 2: Conclusion. — Define 풟 the unbounded operator on 퐿2(핋)푑p with domain 퐻2(핋)푑p by +풟( +∑ +푛 +푋푛ei푛푥) ∶= +∑ +푛 +(푛2퐷 − i푛퐴22 − 퐾22 − 퐺(i∕푛)(i푛퐴21 + 퐾21))푋푛ei푛푥. +Recall that +(휕푡 − 퐷휕2 +푥 + 퐴22휕푥 + 퐾22)푓p(푡) + (퐴21휕푥 + 퐾21)푓h(푡) = 0. +Since 푐푛(푓h(푡)) = 퐺(i∕푛)푐푛(푓p(푡)), this can be written as (휕푡 + 풟)푓p(푡) = 0. Hence, +푓p(푡) = e−푡풟푓p +0 = +∑ +|푛|>푛0 +e−푡( +푛2퐷+i푛퐴22+퐾22+퐺(i∕푛)(i푛퐴21+퐾21)) +푐푛(푓p +0). +22 + +Since ℜ(Sp(퐷)) ⊂ (0, +∞), 푓p(푡) is in every 퐻푘(핋)푑p. Since the first 푑h components of 푓(푡) are +푓h(푡) = +∑ +|푛|>푛0 +퐺(i∕푛)푐푛(푓p(푡))푒푛, +and since 퐺(i∕푛) is bounded as |푛| → +∞, 푓h(푡) also belongs in every 퐻푘(핋)푑h. +Proof of lemma 23. The proof consists in looking at the projection on hyperbolic (respectively parabolic) +components of e−푡ℒ푓0, using the asymptotics for the hyperbolic projection. As in the previous proof, +we denote the first 푑h components of 푓0 by 푓h +0 and the last 푑p components by 푓p +0. +Let us also recall that according to [7, §4.1], +e−푡ℒ푓0 = e−푡ℒ0Π0푓0 + e−푡ℒhΠh푓0 + e−푡ℒpΠ0푓p. +(33) +Step 1: Asymptotics for the hyperbolic projection. — We use the notations 푃p(푧), 푃h(푧) defined in [7, +Proposition 5–6]. Using the series for the perturbation of the total eigenprojections [27, Ch. II, eq. (2.14)], +we get +푃h(푧) = (퐼 +0 +0 +0) − 푧 ((퐼 +0 +0 +0) 퐴 (0 +0 +0 +퐷−1) + (0 +0 +0 +퐷−1) 퐴 (퐼 +0 +0 +0)) + 푂(푧2) += (퐼 +0 +0 +0) − 푧 ( +0 +퐴12퐷−1 +퐷−1퐴21 +0 +) + 푂(푧2). +Thus, +Πh푓0 = +∑ +|푛|>푛0 +[(푐푛(푓h +0) +0 +) − i +푛 (퐴12퐷−1푐푛(푓p +0) +퐷−1퐴21푐푛(푓h +0)) + 푂(푛−2푐푛(푓0))] ei푛푥. +(34) +Step 2: Case where 푓0 ∈ 퐻1 × 퐿2. — Since Π0푓0 is a finite sum of ei푛푥, it is in every 퐻푘, and so +is e−푡ℒ0Π0푓0. According to the regularity of the parabolic frequencies (lemma 24), e−푡ℒpΠp푓0 is in +every 퐻푘. +Since 푓h +0 ∈ 퐻1(핋)푑h, (푐푛(푓h +0))푛 ∈ 퓁2(ℤ; 1 + 푛2) (the 퓁2 space with weight 1 + 푛2). Since 푓p +0 ∈ +퐿2(핋)푑p, (푐푛(푓p +0))푛 ∈ 퓁2(ℤ). Hence, +(푐푛(푓h +0) − i +푛퐴12퐷−1푐푛(푓p +0)) +|푛|>푛0 +∈ 퓁2(|푛| > 푛0; 1 + 푛2), +and +(퐷−1퐴21푐푛(푓h +0)) +|푛|>푛0 ∈ 퓁2(|푛| > 푛0; 1 + 푛2). +Hence, according to the asymptotics for Πh of eq. (34), Πh푓0 ∈ 퐻1(핋)푑. Since e−푡ℒh is continuous +on every 퐻푘, e−푡ℒhΠh푓0 ∈ 퐻1. +Step 3: Case where 푓0 ∈ 퐻2 × 퐿2 and 퐴12 = 0. — The asymptotics (34) reads +Πh푓0 = +∑ +|푛|>푛0 +[(푐푛(푓h +0) +0 +) − i +푛 ( +0 +퐷−1퐴21푐푛(푓h +0)) + 푂(푛−2푐푛(푓0))] ei푛푥. +(35) +The rest of the proof is very similar to the previous case: e−푡ℒ0Π0푓0 and e−푡ℒpΠp푓0 are in every 퐻푘, +while the asymptotics (35) proves that Πℎ푓0 “gains” two derivatives compared to 푓p +0. +23 + +5.3 +Control properties of 2 × 2 systems: proofs +Proof of proposition 20. In this case, +[퐵푛|푀] = (1 +i푛푎′ + 푘22 +0 +i푛푎21 + 푘21) . +In particular, det([퐵푛|푀]) = i푛푎21 +푘21. We see that if (푎21, 푘21) = (0, 0), the Kalman rank condition +never holds, whatever 푛 is. Hence, according to remark 2, item 1, null-controllability does not hold, +whatever 푇 is. +Note that in our case, [퐵푛|푀]+ = [퐵푛|푀]−1 (when the right-hand side exists). Hence, +[퐵푛|푀]−1 = +1 +i푛푎21 + 푘21 +(i푛푎21 + 푘21 +−i푛푎′ − 푘22 +0 +1 +) . +In particular, with the notations of theorem 9 with 푘 = 2, 퐿h +푛,1 = 1 and 퐿h +푛,2 = 0. Thus, 푝 = 0. +If 푘21 ≠ 0, det([퐵푛|푀]) = i푛푎21 + 푘21 never vanishes. In this case, 퐸 (as defined in theorem 9) is +퐸 = 퐿2(핋)2. Hence, according to theorem 9, every 퐿2(핋)2 can be steered to 0 with 퐿2 controls in time +푇 > 퓁(휔)∕|푎′| +If 푎21 ≠ 0 and 푘21 = 0, the Kalman rank condition holds for every 푛 ≠ 0. For 푛 = 0, according +to the formula for [퐵푛|푀], rank([퐵0|푀]) = ℂ × {0}. Thus, 퐸 = {(푓h +0, 푓p +0) ∈ 퐿2(핋)2, ∫핋 푓p +0 = 0}. +Therefore, according to theorem 9, every initial condition (푓h +0, 푓p +0) ∈ 퐿2(핋)2 such that ∫핋 푓p +0 = 0 can +be steered to 0 with controls in 퐿2 in time 푇 > 퓁(휔)∕|푎′|. +Proof of proposition 21. In this case, +[퐵푛|푀] = (0 +i푛푎12 + 푘12 +1 +−푛2푑 + i푛푎22 + 푘22) . +In particular, det([퐵푛|푀]) = −i푛푎12 − 푘12. We see that if (푎12, 푘12) = (0, 0), the Kalman rank condi- +tion never holds, whatever 푛 is. Hence, according to remark 2 item 1, null-controllability does not +hold, whatever 푇 is. +As in the previous proof, [퐵푛|푀]+ = [퐵푛|푀]−1. Hence, +[퐵푛|푀]−1 = +1 +−i푛푎12 − 푘12 +(−푛2푑 + i푛푎22 + 푘22 +−i푛푎12 − 푘12 +−1 +0 +) . +In particular, with the notations of theorem 9 with 푘 = 2, 퐿h +푛,1 = −(−푛2푑 +i푛푎22 +푘22)∕(i푛푎12 +푘12) +and 퐿h +푛,2 = 1∕(i푛푎12 + 푘12). In particular, if 푎12 ≠ 0, 푝 = max(1, 1 − 1) = 1. And if 푎12 = 0 and +푘12 ≠ 0, 푝 = max(2, 1 + 0) = 2. +Step 1: Case 푎12 ≠ 0 and 푘12 ≠ 0. — The Kalman rank condition holds for every 푛. Hence, with the +notations of theorem 9, 푝 = 1 and 퐸 = 퐿2(핋)2, and every initial condition in 퐻1(핋)2 can be steered +to 0 with controls in 퐿2 in time 푇 > 퓁(휔)∕|푎′|. +The strategy to control initial conditions in 퐻1 × 퐿2 is first to let the solution evolve freely during +an arbitrarily small time, which gives a 퐻1(핋)2 state (lemma 23), that we can steer to 0 according to +the previous discussion. +Step 2: Case 푎12 ≠ 0 and 푘12 = 0. — The case is almost the same as the previous one, except that +the Kalman rank condition is not satisfied for 푛 = 0 (and only for 푛 = 0). We have rank([퐵0|푀]) = +24 + +{0} × ℂ and 퐸 = {(푓h +0, 푓p +0) ∈ 퐿2(핋)2, ∫핋 푓h +0 = 0}. We still have 푝 = 1. Hence, we can steer every +initial condition (푓h +0, 푓p +0) ∈ 퐻1(핋)2 such that ∫핋 푓h +0 = 0 an be steered to 0 with controls in time +푇 > 퓁(휔)∕|푎′|. +As in the previous case, to control initial conditions in 퐻1 × 퐿2, we let the solution evolve freely, +which gives a 퐻1(핋)2 state, and preserves the property ∫핋 푓h +0 = 0. Then, we can steer this state in +time 푇 > 퓁(휔)∕|푎′|. +Step 3: Case 푎12 = 0 and 푘12 ≠ 0. — In this case, the Kalman rank condition is satisfied for every 푛, +and 푝 = 2. Hence, according to theorem 9, we can steer every 퐻2(핋)2 initial condition to 0 in time +푇 > 퓁(휔)∕|푎′| with controls in 퐿2. +Again, to control an initial condition in 퐻2 × 퐿2, we let the solution evolve freely for a small time, +which gives a 퐻2(핋)2 state (lemma 23), that we can steer to 0 in time 푇 > 퓁(휔)∕|푎′|. +Step 4: Lack of null-controllability of 퐿2 initial conditions. — We have 푀∗( 1 +0 +) = 0. Hence, according +to theorem 3, (recall that 퐴′ has size 1 × 1), there exists a 퐿2(핋)2 initial condition with zero average +that cannot be steered to 0. +Proof of proposition 22. We have +[퐵푛|푀] = (1 +i푛푎′ + 푘11 +1 +−푑푛2 + i푛푎22 + 푘22) . +In particular, det([퐵푛|푀]) = −푑푛2 + i푛(푎22 − 푎′) + 푘22 − 푘11. We see that for 푛 large enough, this +determinant is non zero. In fact, taking the real and imaginary parts, +det([퐵푛|푀]) = 0 ⇔ { −푑푛2 + 푘22 − 푘11 = 0 +푛(푎22 − 푎′) = 0 +(36) +Moreover, +[퐵푛|푀]+ = [퐵푛|푀]−1 = +1 +det([퐵푛|푀]) (−푑푛2 + i푛푎22 + 푘22 +−i푛푎′ − 푘11 +−1 +1 +) . +Thus, +퐿h +푛,1 = −푑푛2 + 푂(푛) +−푑푛2 + 푂(푛), +and +퐿h +푛,2 = +−1 +−푑푛2 + 푂(푛). +Thus, 푝 = max(0, 1 − 2) = 0. +Step 1: Case 푎′ ≠ 푎22 and 푘11 = 푘22. — According to eq. (36), the Kalman condition is satisfied for +푛 ≠ 0. Moreover, for 푛 = 0, Range([퐵0|푀]) = ℂ푀, thus 퐸 = {(푓h +0, 푓p +0) ∈ 퐿2(핋)2, ∫핋 푓h +0 = ∫핋 푓p +0}. +The theorem 9 gives the claimed controllability result. +Step 2: Case 푎′ ≠ 푎22 and 푘11 ≠ 푘22. — According to eq. (36), the Kalman condition is satisfied for +every 푛 ∈ ℤ. The theorem 9 gives the claimed controllability result. +Step 3: Case 푎′ = 푎22 and +√ +(푘22 − 푘11)∕푑 ∉ ℕ. — As in the previous case, according to eq. (36), the +Kalman condition is satisfied for every 푛 ∈ ℤ. The theorem 9 gives the claimed controllability result. +Step 4: Case 푎′ = 푎22 and 푛0 ∶= +√ +(푘22 − 푘11)∕푑 ∈ ℕ. — According to eq. (36), the Kalman condition +is satisfied for 푛 ≠ ±푛0. For 푛 = ±푛0, Range([퐵±푛0|푀]) = ℂ푀. The theorem 9 gives the claimed +controllability result. +25 + +A +A finite dimension-uniqueness principle for the null-controllability +In the null controllability of parabolic-transport systems, we sometimes prove null-controllability +“up to a finite dimensional space”, and then use functional analysis arguments to deal with the finite- +dimensional spaces that are left [30, 7]. In the previous articles, this was not stated as a general result. +This is the purpose of this appendix. +Proposition 25. Let 푇0 > 0. Let 퐻 be a complex Hilbert space. Let 퐴 be an unbounded operator on +퐻 that generates a strongly continuous semigroup of bounded operator on 퐻. Let 푈 be another Hilbert +space and let 퐵∶ 푈 → 퐻 a bounded control operator. For every 푇 > 0, let 푈푇 be a Hilbert space that is +a subspace of 퐿2(0, 푇; 푈) with continuous and dense injection that satisfies the following “extension by +0 property”:3 if 푢 ∈ 푈푇, 푎, 푏 > 0, then the function ˜푢 defined by ˜푢(푡) = 0 for 0 < 푡 < 푎, ˜푢(푡) = 푢(푡 − 푎) +for 푎 < 푡 < 푇 + 푎, and ˜푢(푡) = 0 for 푇 + 푎 < 푡 < 푇 + 푎 + 푏 is in 푈푇+푎+푏. +Assume that there exists a finite dimensional space ℱ of 퐻 that is stable by the semigroup e푡퐴 and a +closed finite codimensional space4 풢 of 퐻 such that: +• (control up to finite dimension) for every 푓0 ∈ 풢, there exists 푢 ∈ 푈푇0 such that the solution 푓 of +푓′ = 퐴푓 + 퐵푢 satisfies 푓(푇0) ∈ ℱ, +• (unique continuation) for every 휀 > 0 and for every finite linear combination of generalized eigen- +functions 푔0 ∈ 퐻 of 퐴∗, we have 퐵∗(e푡퐴∗푔0) = 0 on 푡 ∈ (0, 휀) ⟹ 푔 = 0. +Then, for every 푇 > 푇0 and every 푓0 ∈ 퐻, there exists 푢 ∈ 푈푇 such that the solution 푓 of 푓′ = +퐴푓 + 퐵푢, 푓(0) = 푓0 satisfies 푓(푇) = 0. +Remark 26. +• In this proposition, we can weaken the hypothesis “퐵 bounded” into “퐵 admissi- +ble” (see [14, §2.3]), but in this article, 퐵 is always bounded. +• If the assertion “(푔0 ∈ 퐻 is a finite linear combination of generalized eigenfunctions of 퐴∗ +and 퐵∗푔0 = 0) ⟹ +푔0 = 0” holds, the unique continuation hypothesis is satisfied by well- +posedness. +Proof. Step 1: We may assume that ℱ ⊂ 풢. — We prove that if we replace 풢 by ℱ +풢, the hypotheses +are still satisfied. Let 푓0 ∈ ℱ + 풢. We write 푓0 = 푓ℱ + 푓풢. According to the hypotheses, there exists +푢 ∈ 푈푇0 such that the solution 푓 of 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓풢 is such that 푓(푇0) ∈ ℱ. Then, the +solution ˜푓 of ˜푓′ = 퐴 ˜푓 + 퐵푢, ˜푓(0) = 푓0 is such that +˜푓(푇0) = e푇0퐴푓ℱ +⏟ ⏟ ⏟ +∈ℱ ++ 푓(푇0) +⏟⏟⏟ +∈ℱ +. +Note that if we replace 푇0 by any 푇1 > 푇0, the hypotheses are still satisfied. +Step 2: For 푇 > 푇0, the control 푢 ∈ 푈푇 such that 푓(푇) ∈ ℱ may be chosen linearly and continuously in +푓0 ∈ 풢. — This is a standard proof of control theory. For 푓0 ∈ 풢, set +푉(푓0) ∶= {푢 ∈ 푈푇 ∶ 푓(푇) ∈ ℱ, 푓 solves 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0}. +Since 퐴 generates a strongly continuous semigroup, 푉(푓0) is a closed affine subspace of 푈푇. Then, +we can define 풰(푓0) as the orthogonal projection of 0 onto 푉(푓0) for the 푈푇-norm. Using the charac- +terization of orthogonal projection on closed convex set, we see that 풰 is linear. Using the fact that 퐴 +3In the application we use here, 푈 = 퐿2(휔) and 푈푇 = 퐻푘 +0 ((0, 푇) × 휔). The hypotheses of proposition 25 are tailored to +allow this situation. +4We do not require 풢 to be stable by e푡퐴. +26 + +generates a strongly continuous semigroup, the characterization of the projection on closed convex +subsets and the closed graph theorem, we see that 풰 is bounded. +For the rest of the proof we set 풰푇 ∶ 풢 → 푈푇 such a map. We also set +풩푇 ∶= {푓0 ∈ 퐻 ∶ ∃푢 ∈ 푈푇, 푓(푇) = 0, 푓 solves 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0}. +(37) +Step 3: For 푇 ≥ 푇0, 풩푇 is a closed finite codimensional subspace of 퐻. — Set 푆0(푡) the semigroup e푡퐴 +restricted to ℱ. Since ℱ is finite dimensional, 푆0(푡) can be written as e푡퐴0, where 퐴0 is a bounded +operator of ℱ. Moreover, 퐴0 = 퐴|ℱ. In particular, 푆0 is actually a group of bounded operators. +For 푓0 ∈ 풢, and 푓′ = 퐴푓 + 퐵풰푇푓0, 푓(0) = 푓0, we have 푓(푇) ∈ ℱ, which allows us to define +풦 ∶ 푓0 ∈ 풢 ↦ −푆0(−푇)푓(푇) ∈ ℱ +The range of this operator 풦 satisfies Range(풦) ⊂ ℱ. Hence, 풦 has finite rank and is compact. +Thus, according to Fredholm’s alternative, (퐼 + 풦)풢 is a closed subspace of 풢 of finite codimension. +Moreover, for every 푓0 ∈ 풢, the solution ˜푓 of ˜푓′ = 퐴 ˜푓 + 퐵풰푇푓0, ˜푓(0) = 푓0 + 풦푓0 satisfies +˜푓(푇) = 푓(푇) + e푇퐴풦푓0 = 푓(푇) − 푆0(푇)푆0(−푇)푓(푇) = 0. +Thus, (퐼 + 풦)풢 ⊂ ℱ푇. According to [9, Proposition 11.5], this proves that 풩푇 is closed and has finite +codimension in 퐻. +Step 4: There exists 훿 > 0 such that for every 푇, 푇′ ∈ (푇0, 푇0 + 훿), 풩푇 = 풩푇′. — Assume 푇0 < 푇 < 푇′. +If 푢 ∈ 풩푇, and if we extend푢 by 0 on (푇, 푇′), we have have 푢 ∈ 풩푇′. Thus codim(풩푇′) ≤ codim(풩푇). +Since codim(풩푇) is an integer, the discontinuities of 푇 ↦ codim(풩푇) are isolated, which proves the +claim. +From now on, we choose 휀 ∈ (0, 훿∕2) arbitrarily small and we set 푇1 = 푇0 + 휀. +Step 5: For 푡 ∈ (0, 휀), (e푡퐴∗풩⊥ +푇1)⊥ ⊂ 풩푇1. — Let 0 < 푡 < 휀 and 푓0 ∈ (e푡퐴∗풩⊥ +푇1)⊥. For every 푔0 ∈ 풩⊥ +푇1, +we have +0 = ⟨e푡퐴∗푔0, 푓0⟩ = ⟨푔0, e푡퐴푓0⟩. +Thus, e푡퐴푓0 ∈ (풩⊥ +푇1)⊥. Since 풩푇1 is closed (step 3), e푡퐴푓0 ∈ 풩푇1. By definition of 풩푇1 and the +“extension by 0” property of 푈푇1, this proves that 푓0 ∈ 풩푇1+푡. According to the previous step, +풩푇1+푡 = 풩푇1, which proves the claim. +Step 6: 풩⊥ +푇1 is left-invariant by e푡퐴∗. — First, consider 0 < 푡 < 휀. According to the previous step, +풩⊥ +푇1 ⊂ ((e푡퐴∗풩⊥ +푇1)⊥)⊥. Since 풩⊥ +푇1 is finite dimensional hence closed, 풩⊥ +푇1 ⊂ e푡퐴∗풩⊥ +푇1. Moreover, +dim(e푡퐴∗풩⊥ +푇1) ≤ dim(풩⊥ +푇1). Thus, for 0 < 푡 < 휀, e푡퐴∗풩⊥ +푇1 = 풩⊥ +푇1. Thanks to the semigroup property, +this is true for all 푡 > 0. +Step 7: Unique continuation property associated to the control problem “steer every 푓0 ∈ 퐻 into 풩푇1 in +time 휀 with a control in 푈휀”. — The control problem is, in mathematical form, the following: +∀푓0 ∈ 퐻, ∃푢 ∈ 푈휀, 푓(푇) ∈ 풩푇1, where 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0. +(38) +Let Π∶ 퐻 → 퐻 the orthogonal projection on 풩⊥ +푇1. Set also 푅푇 ∶ 퐿2(0, 푇; 푈) → 퐻 the input-to- +output map defined by +푅푇푢 ∶= 푓(푇), where 푓′ = 퐴푓 + 퐵푢, 푓(0) = 0. +Then, the control problem (38) is equivalent to +∀푓0 ∈ 퐻, ∃푢 ∈ 푈휀, Πe휀퐴푓0 + Π푅휀푢 = 0. +27 + +We denote by 휄휀 the injection map 푈휀 → 퐿2(0, 푇; 푈). Then, the previous assertion is equivalent to +Range (Π◦e휀퐴) ⊂ Range (Π◦푅휀◦휄휀 +). +The observability inequality associated to this control problem is (see [14, Lemma 2.48]): +∀푔0 ∈ 퐻, ‖e휀퐴∗◦Π∗푔0‖ ≤ 퐶‖휄∗ +휀 ◦푅∗ +휀 ◦Π∗푔0‖. +Since Range(Π∗) = 풩⊥ +푇1 is finite-dimensional, and since ker(휄∗) = Range(휄)⊥ = {0}, this is equivalent +to +∀푔0 ∈ 풩⊥ +푇1, 푅∗ +휀 푔0 = 0 ⟹ e휀퐴∗푔0 = 0. +(39) +To conclude, since 풩⊥ +푇1 is finite dimensional and stable by e푡퐴∗, the semigroup e푡퐴∗ is in fact a group, +and in particular e휀퐴∗ is invertible on 풩⊥ +푇1. 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Control Lett., 93:21–29, 2016. +30 + diff --git a/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/load_file.txt b/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b836a64fb82d901cfea8fa523c073292b57c638 --- /dev/null +++ b/sNAyT4oBgHgl3EQfmfhl/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf,len=1084 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='00471v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='OC] 1 Jan 2023 Null-controllability of underactuated linear parabolic-transport systems with constant coefficients Armand Koenig, Pierre Lissy† January 3, 2023 Abstract The goal of the present article is to study controllability properties of mixed systems of lin- ear parabolic-transport equations, with possibly non-diagonalizable diffusion matrix, on the one- dimensional torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The equations are coupled by zero or first order coupling terms, with con- stant coupling matrices, without any structure assumptions on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The distributed control acts through a constant matrix operator on the system, so that there might be notably less controls than equations, encompassing the case of indirect and simultaneous controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' More precisely, we prove that in small time, such kind of systems are never controllable in appropriate Sobolev spaces, whereas in large time, null-controllability holds, for sufficiently regular initial data, if and and only if a spectral Kalman rank condition is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We also prove that initial data that are not regular enough are not controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Positive results are obtained by using the so-called fictitious control method together with an algebraic solvability argument, whereas the negative results are obtained by using an appropriate WKB construction of approximate solutions for the adjoint system asso- ciated to the control problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' As an application to our general results, we also investigate into details the case of 2 × 2 systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', one pure transport equation and one parabolic equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' MSC Classification 93B05, 93B07, 93C20, 35M30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Keywords Parabolic-transport systems, null-controllability, observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1 Context and state of the art Controllability properties of coupled systems of PDEs has attracted a lot of attention this last two decades, due to their link with real-life models and also the specific mathematical difficulties arising in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' An important part of the literature is devoted to systems where all components of the equations have the same qualitative behaviour (meaning that they are for instance all parabolic, or all hyperbolic, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' However, the case where different dynamics are mixed has been less studied, de- spite its mathematical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Indeed, in this context, the controllability propertiesof each equation taken separately might be totally different (for instance, the heat equation with distributed control is controllable in arbitrary small time from any open subset [29, 22], whereas the wave equation with IMT, Université de Toulouse, CNRS, Université Toulouse III-Paul Sabatier (UPS), Toulouse, France (armand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='koenig@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='univ-toulouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='fr) †CEREMADE, Université Paris-Dauphine & CNRS UMR 7534, Université PSL, 75016 Paris, France (lissy@ceremade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='dauphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1 distributed control is controllable in large time and under some geometric conditions [6]), so that the controllability properties of the final coupled system might be difficult to guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, when we are considering underactuated systems (in the sense that there are less controls than equations) as in the present article, additional mathematical difficulties are appearing, due notably to the algebraic and analytic effects of the coupling terms, that become predominant in the understanding of the con- trollability or observability properties of the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Here, in the present article, we aim to study the indirect controllability properties of a model of coupled parabolic-transport equations as introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us mention that many realistic models already studied in the literature can be reformulated in terms of coupled parabolic-transport equations, notably the wave equation with structural damping [37, 34, 10, 24], the heat equation with memory [26, 23], the 1D-Linearized compressible Navier- Stokes equations [20, 13, 12, 8], or the Benjamin-Bona-Mahony equation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For more details, we also refer to [7, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This justifies the interest of studying a general version of coupled parabolic- transport systems as in the present article, that can be seen as an attempt to find a unified framework in order to encompass many existing results of the literature and to generalize them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Other results of interest, related to the present work, are [2], where the authors study a one-dimensional system of one transport equation and one parabolic equation, for which they prove a non-controllability result in small time by a WKB approach, and [11], where the authors prove a controllability result in large time for a one-dimensional system of one transport equation and one elliptic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2 Presentation of the parabolic-transport system under study Let 푇 > 0 some final time , 핋 = ℝ∕(2휋ℤ) the one-dimensional torus,휔 an nonempty open subset of 핋, 푑 ∈ ℕ∗ (which represents the number of equations in our system) , 푚 ∈ {1, … , 푑} (which represents the number of controls in our system), 퐴, 퐵, 퐾 ∈ ℳ푑(ℝ) (that are some constant coupling matrices), and 푀 ∈ ℳ푑,푚(ℝ) (that is a constant control operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Our goal is to study the controllability properties of the following coupled system of parabolic-transport equations: { 휕푡푓 − 퐵휕2 푥푓 + 퐴휕푥푓 + 퐾푓 = 푀푢1휔 in (0, 푇) × 핋, 푓(0, ⋅) = 푓0 in 핋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (Sys) Here, the state is 푓∶ [0, 푇] × 핋 → ℝ푑, and the control is 푢∶ [0, 푇] × 핋 → ℝ푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The exact regularity chosen for 푓 and 푢 will be made more precise later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We assume that 푑 = 푑h + 푑p with 1 ≤ 푑h < 푑, 1 ≤ 푑p < 푑, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1) 퐵 = (0 0 0 퐷) , with 퐷 ∈ ℳ푑p(ℝ), (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2) ℜ(Sp(퐷)) ⊂ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3) 푑h represents the number of purely hyperbolic equations, whereas 푑p represents the number of parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Notice that (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3) is necessary to ensure that the matrix operator 휕푡−퐷∆ is parabolic is the sense of Petrovskii ([28, Chapter 7, Definition 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Introducing the similar block decomposition for the 푑 × 푑 matrix 퐴 = ( 퐴′ 퐴12 퐴21 퐴22 ), we make the following hypothesis on the matrix 퐴′ ∈ ℳ푑ℎ(ℝ) 퐴′ is diagonalizable with Sp(퐴′) ⊂ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4) Notice that it is well-known that (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4) is necessary (and sufficient, see [7, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2]) to ensure the well- posedness of (Sys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3 Main results To state our results, we need to introduce the following notations: 퓁(휔) ∶= sup{|퐼|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐼 connected component of 핋 ⧵ 휔}, (1) 휇∗ ∶= min{|휇|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 휇 ∈ Sp(퐴′)}, and 푇∗ = 푇∗(휔) ∶= { 퓁(휔) 휇∗ if 휇∗ > 0, +∞ if 휇∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (2) For 푛 ∈ ℤ, we also set 퐵푛 ∶= −푛2퐵 − i푛퐴 − 퐾 (3) and [퐵푛|푀] ∶= ( 푀 퐵푛푀 … 퐵푑−1 푛 푀 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (4) Our main result is the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that the hypotheses (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1)–(H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4) hold, that 푇 > 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the spectral Kalman rank condition rank([퐵푛|푀]) = 푑 holds for all 푛 ∈ ℤ if and only if for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔)푚 such that the solution 푓 of the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Recall that the Kalman rank condition is necessary for the control of ODE sys- tems [14, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Therefore, writing the parabolic-transport system in Fourier, we im- mediately find that for every 푇 > 0, the spectral Kalman-rank condition ∀푛 ∈ ℤ, rank([퐵푛|푀]) = 푑 is necessary for the null-controllability of every 퐻푘 initial conditions in time 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Actually, we prove two slightly stronger versions of this theorem, namely theorems 9 and 12, that are useful in order to obtain some controllability results under some constraints on Fourier coefficients of the hyperbolic part of the initial condition (see proposition 20, proposition 21, proposition 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' One can refine a little bit the regularity stated in theorem 1, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 푇 > 푇∗ and that for all 푛 ∈ ℤ, the spectral Kalman rank condition rank([퐵푛|푀]) = 푑 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑ℎ × 퐻4푑(푑−1)−1(핋)푑푝, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔)푚 such that the solution 푓 of the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' if 퐴12 = 0, for every 푓0 ∈ 퐻4푑(푑−1)(핋)푑ℎ × 퐻4푑(푑−1)−2(핋)푑푝, there exists a control 푢 ∈ 퐿2([0, 푇]×휔)푚 such that the solution 푓 of the parabolic-transportsystem (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Indeed, by letting evolve the system freely on a short interval of time, we can show using the method of lemma 23 that the parabolic component becomes 퐻4푑(푑−1)(핋)푑푝, so that theorem 1 can be applied, taking into account that the condition 푇 > 푇∗ is open and that the system is time-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The spectral Kalman rank condition rank([퐵푛|푀]) = 푑 was first introduced in [5] for coupled systems of heat equations with diagonalizable diffusions (see also [33] for non-diagonalizable diffusions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휇 ∈ Sp(퐴′), 푁 ∈ ℕ and 푇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that every initial condition 푓0 ∈ 퐿2(핋)푑 ∩ {∑ |푛|>푁 푋푛ei푛푥}issteerableto0intime푇 withcontrolin퐿2((0, 푇)×휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, thereexists푉0 ∈ ker(퐴′∗+ 휇) such that 푀∗( 푉0 0 ) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Theorems 1, 9 and 12 only ensures null-controllability of smooth enough initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Theorem 3 proves that such a regularity condition is needed in general: even if the time is large enough and if the Kalman rank condition is satisfied for every 푛, it might happen that some 퐿2 initial condition cannot be steered to 0 with a 퐿2 control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4 Precise scope and organization of the article This article can be seen as a continuation of [7], insofar as we generalize the results of the above- mentioned article, since we are able to treat any matrices 퐴, 퐵, 퐾, 푀 without any restrictions on their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Indeed, in [7], the authors treated the case where 푀 = 퐼푑 (where no Kalman rank condi- tion is needed), or particular cases where only the parabolic or the hyperbolic parts are controlled, under strong restrictions on the structure of the coupling matrices 퐴, 퐵 and 퐾 and also on the diffu- sion matrix 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us mention that our results are sharp in terms of the controllability conditions we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' However, it is very likely that the initial state space (whose choice is determined by technical reasons coming from the specific strategy we use, that is consuming in terms of regularity, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2) is almost never sharp and depends strongly on the structure of the coupling terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Finding the exact “good” state space remains an open problem that seems to be difficult to solve in all generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In section 2, we give some notations and we gather some existing results that will be used in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Section 3 is devoted to proving that the condition rank([퐵푛|푀]) = 푑 is sufficient in order to obtain our desired controllability result in large time The argument is based on a fictitious control argument detailed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1, where we first prove an auxiliary controllability result, in the case 푀 = 퐼푑, with regular enough controls for regular enough initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2, we explain how to obtain a control in the range on 푀 by performing algebraic manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Notice that the method of fictitious control plus algebraic solvability, that has been introduced in [16] in the context of the controllability of PDEs, has been successfully used for various problems [4, 17, 18, 32, 15, 39, 40, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' One of the main novelties here is that the algebraic solvability is not directly performed on the system (or its adjoint as in [19]) but on a projected version of the system on its Fourier components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Section 4 is devoted to proving some necessary conditions of controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1 is devoted to constructing WKB solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' These solutions are used to disprove controllability in small time in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2 and to prove theorem 3 in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Section 5 aims to give an application of our results to the particular case of 2 × 2 systems together with some considerationsabout the sharpness of our regularity assumptions in this precise setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To conclude, appendix A proves a general result about a “control up to a finite-dimensional space plus unique continuation” strategy that is used in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1, in the spirit of [30, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Acknowledgement Armand Koenig is supported by the ANR LabEx CIMI (under grant ANR- 11-LABX-0040) within the French State Programme “Investissements d’Avenir”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Pierre Lissy is supported by the Agence Nationale de la Recherche, Project TRECOS, under grant ANR-20-CE40-0009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 2 Some notations and preliminary results We will rely on some basic results on the parabolic-transport system (Sys) that are already known, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the reader convenience, we collect here the notations and results we will use most often, and we will recall some others along the way as they are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 4 Let ℒ be the unbounded operator on 퐿2(핋)푑 with domain 퐻1(핋)푑h × 퐻2(핋)푑p defined by ℒ푓 = −퐵휕2 푥푓 + 퐴휕푥푓 + 퐾푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The operator −ℒ generates a strongly continuous semigroup of bounded operators of 퐿2(핋)푑 [7, Proposition 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Every 퐻푘(핋)푑 is stable by e−푡ℒ, and the restriction of e−푡ℒ on 퐻푘(핋)푑 is a strongly continuous semigroup of bounded operators [7, Remark 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We denote by 푆(푇, 푓0, 푢) the solution at time 푇 of the parabolic-transport system (Sys) with control matrix 푀 = 퐼푑 (the identity matrix of size 푑, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', we control every component with a different control), initial condition 푓0 and control 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푛0 ∈ ℕ to be chosen large enough later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We denote by 푒푛 ∶ 푥 ∈ 핋 ↦ ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We also denote by 퐸 ∶ ℂ → ℳ푑(ℂ) the following function: 퐸(푧) = 퐵 + 푧퐴 − 푧2퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푟 > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For |푧| < 푟, let 푃h(푧) be the eigenprojection on the sum of eigenspaces of 퐸(푧) associated to the set of eigenvalues 휆(푧) ∈ Sp(퐸(푧)) such that |휆(푧)| < 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to [7, Proposition 5], 푃h(푧) satisfies: 푃h(0) = ( 퐼 0 0 0 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푧 ↦ 푃h(푧) is holomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푃h(푧) is a projection that commutes with 퐸(푧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푃h(푧)퐸(푧) = 푂(푧) as 푧 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We also set 푃p(푧) = 퐼 − 푃h(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This projection 푃p(푧) satisfies similar properties as 푃h(푧) ([7, Proposi- tions 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Following [7, Proposition 18], we denote by 퐹0 the space of frequencies less than 푛0 and by 퐹h (re- spectively 퐹p) the space of hyperbolic frequencies greater than 푛0 (respectively the space of parabolic frequencies greater than 푛0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐹0 = ⨁ |푛|≤푛0 Span(푒푛);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐹p = ⨁ |푛|>푛0 Range(푃p(i∕푛))푒푛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐹h = ⨁ |푛|>푛0 Range(푃h(i∕푛))푒푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' By [7, Proposition 18], we notably have 퐿2(핋)푑 = 퐹0 ⊕ 퐹p ⊕ 퐹h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The space 퐹p is stable by the semigroup e−푡ℒ (see the definition of 푃p [7, Proposition 5] and the definition of 퐹p [7, Proposition 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We denote by ℒp the restriction of ℒ to 퐹p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Similarly, the space 퐹h is stable by the semigroup e−푡ℒ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We denote by ℒh the restriction of ℒ to 퐹h, and −ℒh generates a strongly continuous group of bounded operators on 퐹h [7, Proposition 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let Π0, Πp, Πh and Π be the projections defined by 퐿2(핋)푑 = 퐹0 ⊕ 퐹p ⊕ 퐹h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Π0 = 퐼퐹0 + 0 + 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Πp = 0 + 퐼퐹p + 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Πh = 0 + 0 + 퐼퐹h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Π = 0 + 퐼퐹p + 퐼퐹h = Πp + Πh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' These projections are bounded operators on 퐿2(핋)푑 [7, Proposition 18] (and also on every 퐻푘(핋)푑, as one can readily convince by following the proof of [7, Proposition 18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 5 3 Null controllability of regular initial conditions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1 Regular controls for regular initial conditions As a technical preparation for the proof of theorem 1, we need some results regarding the regularity of controls, when the control matrix is 푀 = 퐼푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 푇 > 푇∗ (as defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (2)) and that 푀 = 퐼푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘, 퓁 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푓0 ∈ 퐻푘(핋)푑, there exists 푢 ∈ 퐻푘 0 ((0, 푇) × 휔)푑h × 퐻퓁 0 ((0, 푇) × 휔)푑p such that the solution of the parabolic-transport system (Sys) with initial condition 푓0 and control 푢 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We adapt the proof of the corresponding result when 푘 = 0 [7, Theorem 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' First, we prove the following adaptation of [7, Proposition 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇′ ∈ (푇∗, 푇) and 푘 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푛0 (in the definition of 퐹0, see [7, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (40–42)]) is large enough, there exists a continuous operator 풰h ∶ 퐻푘(핋)푑 × 퐻푘 0 ((푇′, 푇) × 휔)푑p→ 퐻푘 0((0, 푇′) × 휔)푑h (푓0, 푢p) ↦ 푢h, such that for every (푓0, 푢p) ∈ 퐻푘(핋)푑 × 퐻푘 0((푇′, 푇) × 휔)푑p (where 푢p is extended by 0 on (0, 푇′) and 푢h is extended by 0 on (푇′, 푇)), Πh푆(푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푓0, (풰h(푓0, 푢p), 푢p)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' As in [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1], the conclusion of proposition 6 is equivalent to the exact controllability of the system 휕푡푓 + ℒh푓 = Πh(푢, 0) at time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since −ℒh generates a strongly continuous group, the exact controllability at time 푇′ is equivalent to the null-controllability at time 푇′, which is what we are going to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' When 푘 = 0, [7, Proposition 23] is the claimed result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To extend this result to 푘 > 0, we use a general result of Ervedoza and Zuazua concerning the regularity of controls for regular initial data in the context of groups of operators [21, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let ˜휔 an open subset of 핋 such that ˜휔 ⊂ 휔 and 푇∗(˜휔) < 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휒 ∈ 퐶∞ 푐 (휔) such that 휒 = 1 on ˜휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휂 ∈ 퐶∞ 0 (0, 푇′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푧0 ∈ 퐻푘(핋)푑 be an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푌푇′ as defined by [21, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3] and define the control as 푉(푡) = 휂(푡)휒(푥)푀∗푌(푡), where 푌 is the solution to 휕푡푌 − 퐵∗휕2 푥푌 − 퐴∗휕푥푌 + 퐾∗푌 = 0 associated to the initial condition 푌(푇′) = 푌푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to [21, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3], 푉(푡) is a con- trol that steers 푧0 to 0 at time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to [21, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4], 푌푇′ ∈ 퐻푘(핋)푑 (hence 푉 ∈ 퐿2(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푘(휔)푑)) and 푉 ∈ 퐻푘(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐿2(휔)푑), with estimates of the form ‖푉‖2 퐿2(0,푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='퐻푘(휔)푑) + ‖푉‖2 퐻푘(0,푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='퐿2(휔)푑) ≤ 퐶푘‖푧0‖2 퐻푘(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We claim that 퐿2(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푘 0(휔))∩퐻푘 0(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐿2(휔)) ⊂ 퐻푘((0, 푇′)×휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Indeed, for every 휏 ∈ ℝ and 휉 ∈ ℝ, (1 + 휏2 + 휉2)푘 ≤ 퐶푘 ((1 + 휏2)푘 + (1 + 휉2)푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, integrating in Fourier space, ‖푓‖2 퐻푘(ℝ2) ≤ 퐶푘 (‖푓‖2 퐿2(ℝ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='퐻푘(ℝ)) + ‖푓‖2 퐻푘(ℝ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='퐿2(ℝ)) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 6 Recall that for Ω ⊂ ℝ푛 convex1, 퐻푘 0(Ω) is the set of functions whose extension by zero outside Ω are 퐻푘(ℝ푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, 퐿2(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푘 0 (휔)) ∩ 퐻푘 0(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐿2(휔)) ⊂ 퐻푘((0, 푇′) × 휔) as claimed, so that 푉 ∈ 퐻푘((0, 푇′) × 휔)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 휂 ∈ 퐶∞(0, 푇′) and 휒 ∈ 퐶∞ 0 (휔), we conclude that 푉 ∈ 퐻푘 0((0, 푇′) × 휔)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the proof of proposition 5, we will also use: Proposition 7 ([7], proposition 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇′ ∈ (푇∗, 푇) and 푘 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푛0 is large enough, there exists a continuous operator 풰p ∶ 퐿2(핋)푑 × 퐿2((0, 푇′) × 휔)푑h→ 퐶∞ 푐 ((푇′, 푇) × 휔)푑p (푓0, 푢h) ↦ 푢p, (in the sense that for any 푠 ∈ ℕ, 풰p ∶ 퐿2(핋)푑×퐿2((0, 푇′)×휔)푑h → 퐻푠 0(푇′, 푇)×휔)푑p is continuous for the natural topologies associated to these spaces) such that for every (푓0, 푢h) ∈ 퐿2(핋)푑 × 퐿2((0, 푇′) × 휔)푑h, Πp푆(푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푓0, (푢h, 풰p(푓0, 푢h)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We can now prove proposition 5 by mimicking the proof of the case 푘 = 0 [7, Proposition 20 & §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Control up to final dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — We claim that there exists a closed finite codimensional space 풢 of 퐻푘(핋)푑 and a continuous operator 풰 ∶ 풢 → 퐻푘 0((0, 푇′) × 휔)푑h × 퐶∞ 푐 ((푇′, 푇) × 휔)푑p (in the sense that for any 푠 ∈ ℕ, 풰 ∶ 풢 → 퐻푘 0((0, 푇′) × 휔)푑h × 퐻푠 0(푇′, 푇) × 휔)푑p is continuous for the natural topologies associated to these spaces) such that for every 푓0 ∈ 풢, Π푆(푇, 푓0, 풰푓0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The property Π푆(푇, 푓0, (푢h, 푢p)) = 0 holds if { 푢h = 풰h(푓0, 푢p) = 풰h 1 (푓0) + 풰h 2 (푢p), 푢p = 풰p(푓0, 푢h) = 풰p 1(푓0) + 풰p 2 (푢h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (5) Set 풞 = 풰p 1 + 풰2 p풰h 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the previous relations hold if 풞푓0 = (퐼 − 풰p 2 풰h 2 )푢p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (6) Since 풰p 2 is continuous from 퐻푘 0((푇′, 푇) × 휔)푑p into 퐶푐((푇′, 푇) × 휔)푑p, we deduce that the operator 풞∶ 퐻푘 0((푇′, 푇) × 휔) 푑p → 퐻푘 0((푇′, 푇) × 휔) 푑p is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, according to Fredholm’s alternative, the relation (6) holds on a closed finite codimensional space 풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Dealing with the finite (co)dimensional spaces 퐹0 and 풢 is a straightforward adaptation of [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' more specifically, we use proposition 25 proved in Appendix A with 퐻 = 푉 = 퐻푘(핋)푑, 푈푇 = 퐻푘 0 ((0, 푇) × 휔)푑h × 퐻퓁 0 ((0, 푇) × 휔), 퐴 = −ℒ, 퐵 = 1휔, 풢 = 풢 and ℱ = 퐹0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The control up to a finite dimensional space hypothesis is satisfied according to the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The unique continuation hypothesis is satisfied because every generalised eigenvector is a finite sum of elements of the form 푋푛ei푛푥 (푋푛 ∈ ℂ푑), and finite linear combinations of 푋푛ei푛푥 have the unique continuation property thanks to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', Jerison-Lebeau’s spectral inequality (see [30, Theorem 3], or [7, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (90)] for our specific case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For technical reasons, we will need the control to be in the form 푃(휕푥)푢, where 푃(휕푥) is a constant coefficients differential operator to be chosen later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1More generally, satisfiying the segment condition, see[1, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='21 & Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 7 Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 푇 > 푇∗ (as defined in (2)) and that 푀 = 퐼푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘, 퓁 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푃 be a nonzero polynomial with complex coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 퓁 ⩾ deg(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푓0 ∈ 퐻푘(핋)푑 be such that for every 푛 ∈ ℤ, 푃(i푛) = 0 ⟹ 푐푛(푓0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, there exists 푢 ∈ 퐻푘+deg(푃) 0 ((0, 푇) × 휔)푑h × 퐻퓁 0 ((0, 푇) × 휔)푑p such that the solution of the parabolic-transport system (Sys) with initial condition 푓0 and control 푃(휕푥)푢 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푘, 퓁 ∈ ℕ with 퓁 ⩾ deg(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='Let 푓0 ∈ 퐻푘(핋)푑 be such that for every 푛 ∈ ℤ, 푃(i푛) = 0 ⟹ 푐푛(푓0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We define ˜푓0 ∶= 푃(휕푥)−1푓0 by 푐푛( ˜푓0) ∶= 푃(i푛)−1푐푛(푓0) if 푃(i푛) ≠ 0 and 푐푛( ˜푓0) ∶= 0 if 푃(i푛) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Note that 푃(휕푥) ˜푓0 = 푓0 and that ˜푓0 ∈ 퐻푘+deg(푃) 0 (휔)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, applying proposition 5 to ˜푓0 leads to the fact that there exists ˜푢 ∈ 퐻푘+deg(푃) 0 ((0, 푇) × 휔)푑h × 퐻퓁 0 ((0, 푇) × 휔)푑p such that the solution ˜푓 of the parabolic-transport system (Sys) with initial condition ˜푓0 and control ˜푢 satisfies ˜푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, since ˜푓0 ∈ 퐻푘+deg(푃) 0 (휔)푑 and ˜푢 ∈ 퐻푘+deg(푃) 0 ((0, 푇) × 휔)푑h × 퐻퓁 0 ((0, 푇) × 휔)푑p with 퓁 ⩾ deg(푃), we notably have ˜푓 ∈ 퐿2((0, 푇);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푘+deg(푃)(핋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, setting 푓 = 푃(휕푥) ˜푓 and 푢 = 푃(휕푥) ˜푓, and using that 푃(휕푥) has constant coefficients (so that it commutes with the operator 휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾퐼푑)) ensures that 푓 verifies (Sys) with initial condition 푓0 and control 푃(휕푥)푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, since ˜푓(푇, ⋅) = 0, we also have푓(푇, ⋅) = 푃(휕푥) ˜푓(푇, ⋅) = 0, which leads to the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2 Algebraic solvability For 푘 ∈ ℕ, we define [퐵푛|푀]푘 ∶= ( 푀 퐵푛푀 … 퐵푘−1 푛 푀 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (7) We prove the following variant of theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that the hypotheses (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1)–(H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4) hold, and that 푇 > 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that for all |푛| ∈ ℕ large enough, the Kalman rank condition rank([퐵푛|푀]푘) = 푑 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Define the following space of functions 퐸 ∶= {푓 ∈ 퐿2(핋)푑 ∶ ∀푛 ∈ ℤ, 푐푛(푓) ∈ Range([퐵푛|푀])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set, when it is defined, [퐵푛|푀]+ 푘 ∶= [퐵푛|푀]∗ 푘 ( [퐵푛|푀]푘[퐵푛|푀]∗ 푘 )−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write [퐵푛|푀]+ 푘 by blocks as [퐵푛|푀]+ 푘 = ⎛ ⎜ ⎜ ⎝ 퐿h 푛,1 퐿p 푛,1 ⋮ ⋮ 퐿h 푛,푘 퐿p 푛,푘 ⎞ ⎟ ⎟ ⎠ , where the 퐿h 푛,푗 are of size 푚×푑h and the 퐿p 푛,푗 are of size 푚×푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Considering the 퐿h 푛,푗 as rational functions of 푛, and denoting their degree by deg(퐿h 푛,푗), set 푝 ∶= max 1≤푗≤푘 deg(푛푗−1퐿h 푛,푗) = max 1≤푗≤푘 (푗 − 1 + deg(퐿h 푛,푗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, for every 푓0 ∈ 퐻푝(핋)푑 ∩ 퐸, there exists a control 푢 ∈ 퐿2([0, 푇] × 휔) such that the solution 푓 of the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The idea of the proof is to first choose a “fictitious” control that acts on every components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, we look at the Fourier coefficients of 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This transforms the control system (Sys) into a family of 8 finite-dimensional control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' On each of these finite-dimensional system, we perform some algebraic manipulations, called algebraic solvability, that transform the fictitious control (that acted on every component) into an “actual” control (that acts only on Range(푀)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We begin with the algebraic solvability result we will use, which is essentially taken from [32, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘 ∈ ℕ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let ˜퐵 ∈ ℳ푑(ℂ) and ˜ 푀 ∈ ℳ푚,푑(ℝ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푋0 ∈ ℂ푑 and 푤 ∈ 퐻푘−1 0 (0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푚푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write 푤 by blocks as 푤 = ⎛ ⎜ ⎝ 푤1 ⋮ 푤푘 ⎞ ⎟ ⎠ , where 푤푗 ∈ 퐻푘−1 0 (핋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푚), and set 푢 = 푤1 + 푤′ 2 + ⋯ + 푤(푘−1) 푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푋, ˜푋 ∈ 퐶0(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푑) be the solutions of 푋′ = ˜퐵푋 + [˜퐵| ˜ 푀]푘푤, ˜푋′ = ˜퐵푋 + ˜ 푀푢, 푋(0) = ˜푋(0) = 푋0, where [˜퐵| ˜ 푀]푘 ∶= ( ˜ 푀 ˜퐵푀 … ˜퐵푘−1 ˜ 푀) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then 푋(푇) = ˜푋(푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Consider ˜ ℳ푘 the operator matrix with 푑 + 푚 rows and 푘푚 columns defined by blocks as ˜ ℳ푘 ∶= ( 0 − ˜ 푀 ⋯ − ∑푘−2 푗=0 휕푗 푡 ˜퐵푘−2−푗 ˜ 푀 −퐼 −휕푡 ⋯ −휕푘−1 푡 ) = ( ˜ ℳ푘,1 ˜ ℳ푘,2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set also 풫∶ (푋, 푊) ∈ 퐻1 0(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푑) × 퐿2(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푚) → 휕푡푋 − ˜퐵푋 − ˜ 푀푊 ∈ 퐿2(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We claim that 풫◦ ˜ ℳ푘 = [˜퐵| ˜ 푀]푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (8) Indeed, we have by blocks 풫◦ ˜ ℳ푘 = (퐶0 ⋯ 퐶푘−1 ) with 퐶퓁 = −(휕푡 − ˜퐵) 퓁−1 ∑ 푗=0 휕푗 푡 ˜퐵퓁−1−푗 ˜ 푀 + ˜ 푀휕퓁 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, remarking that this is a telescoping sum, 퐶퓁 = − 퓁∑ 푗=1 휕푗 푡 ˜퐵퓁−푗 ˜ 푀 + 퓁−1 ∑ 푗=0 휕푗 푡 ˜퐵퓁−푗 ˜ 푀 + ˜ 푀휕퓁 푡 = −휕퓁 푡 ˜ 푀 + ˜퐵퓁 ˜ 푀 − ˜ 푀휕퓁 푡 , which proves the claimed formula (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Now, plug eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (8) into the differential equation 푋′ = ˜퐵푋 + [˜퐵| ˜ 푀]푘푤, which gives 푋′ = ˜퐵푋 + (휕푡 − ˜퐵) ˜ ℳ푘,1푤 − ˜ 푀 ˜ ℳ푘,2푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' With 푌 ∶= 푋 − ˜ ℳ푘,1푤, and remarking that ˜ ℳ푘,2푤 = −푢, this can be written as 푌′ = ˜퐵푌 + ˜ 푀푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푤 ∈ 퐻푘−1 0 (0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ푚푘), ˜ ℳ푘,1푤(0) = ˜ ℳ푘,1푤(푇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence 푌(0) = 푋(0) = ˜푋(0) and 푌(푇) = 푋(푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus 푌 solves the same Cauchy problem as ˜푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This proves that 푌 = ˜푋, hence ˜푋(푇) = 푌(푇) = 푋(푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 9 We can now prove theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푓0 ∈ 퐻푝(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set 푋푛(푡) = 푐푛(푓(푡, ⋅)) and 푢푛(푡) = 푐푛(푢(푡, ⋅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The desired conclusion 푓(푇, ⋅) = 0 reads in Fourier as: ∀푛 ∈ ℤ, 푋푛(푇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, 푋푛 satisfies { 푋′ 푛(푡) = 퐵푛푋푛(푡) + 푀푢푛(푡), 푡 ∈ (0, 푇), 푋푛(0) = 푐푛(푓0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (9) First, let us give the idea of the proof: if 푣 steers 푓0 to 0 when 푀 = 퐼, we want to define 푤푛 by 푐푛(푣(푡, ⋅)) = [퐵푛|푀]푘푤푛 (this is possible for 푛 large enough) and choose 푢푛 ∶= 푤푛1+푤′ 푛2+⋯+푤(푘−1) 푛푘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, according to lemma 10, the function 푢푛 steers 푋푛 from 푐푛(푓0) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' There are two problems with this crude choice of 푢푛: this construction only works for 푛 large enough, and more importantly, we have no guarantee that the support of ∑ 푢푛ei푛푥 is included in [0, 푇] × 휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The control strategy is to first bring frequencies less than 푛0 to 0 in time 휀 for some 푛0 > 0 large enough to be chosen later and 휀 > 0 small enough so that 푇 > 푇∗ + 2휀, and second use a refined version of the construction outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Control of a finite number of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Recall that Π is the projection on frequencies larger than 푛0 and that 퐸 was defined in the statement of theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We claim that for any 푛0 ∈ ℕ∗, 휀 > 0 and 푓0 ∈ 퐸 there exists 푢 ∈ 퐿2(0, 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푝 0 (휔))푚 such that (1 − Π)푆(휀, 푓0, 푀푢) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This property is equivalent to the null-controllability of the system (Sys) projected on frequencies less or equal than 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The observability inequality associated with this problem [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='44] is: ∃퐶 > 0, ∀푔0 ∈ (1 − Π)퐸, ‖e−휀ℒ∗푔0‖2 퐻−푝(핋)푑 ≤ 퐶 ∫ 휀 0 ‖푀∗e−푡ℒ∗푔0‖2 퐿2(휔)푚 d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since (1 − Π)퐸 is finite dimensional, this is equivalent to the unique continuation property ∀푔0 ∈ (1 − Π)퐸, ( 푀∗e−푡ℒ∗푔0(푥) = 0 for (푡, 푥) ∈ (0, 휀) × 휔 ) ⟹ 푔0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us prove this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푔0 ∈ (1 − Π)퐸 such that 푀∗e−푡ℒ∗푔0(푥) = 0 for (푡, 푥) ∈ (0, 휀) × 휔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since finite sums of ei푛푥 have the unique continuation property, we have for every 0 < 푡 < 휀 and |푛| ≤ 푛0, 푐푛(푀∗e−푡ℒ∗푔0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We can rewrite this as 푀∗e−푡퐵∗ 푛푐푛(푔0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Differentiating 퓁 times in 푡 and evaluating at 푡 = 0, we get that for all 퓁 ∈ ℕ and |푛| ≤ 푛0, 푀∗(퐵∗ 푛)퓁푐푛(푔0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since we assumed that for |푛| > 푛0, 푐푛(푔0) = 0, this means that 푐푛(푔0) ∈ ker([퐵푛|푀]∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But, by definition of 퐸, 푐푛(푔0) ∈ Range([퐵푛|푀]) = ker([퐵푛|푀]∗)⟂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 푐푛(푔0) = 0 and 푔0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This proves the unique continuation property, and the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Construction of 푢푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — We set 푇′ = 푇∗ + 휀 = 푇 − 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us write [퐵푛|푀]+ 푘 = 푄(i푛)∕푃(i푛) where 푄 is a polynomial with matrix coefficients, 푃 is a polynomial (with scalar coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If we denote the adjugate matrix of a matrix 퐶 by Adj(퐶), note that we may take 푄(i푛) = [퐵푛|푀]∗ 푘 Adj([퐵푛|푀]푘[퐵푛|푀]∗ 푘);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푃(i푛) = det([퐵푛|푀]푘[퐵푛|푀]∗ 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 10 Increasing 푛0 if necessary, we may assume that for every |푛| > 푛0, 푃(i푛) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We first apply a control as in step 1: for any 푓0 ∈ 퐸, there exists 푢 ∈ 퐿2(0, 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푝 0 (휔))푚 such that (1−Π)푆(휀, 푓0, 푀푢) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the resulting solution 푓(휀, ⋅) is such that 푃(i푛) = 0 ⟹ 푐푛(푓(휀, ⋅)) = 0, since 푃(i푛) ≠ 0 for |푛| > 푛0 and 푐푛(푓(휀, ⋅)) = 0 for |푛| ⩽ 푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We consider this 푓(휀, ⋅) as our new initial condition, that we denote by 푓휀, and we have to steer it to 0 in time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Note that since 푓0 ∈ 퐻푝(핋) and 푢 ∈ 퐿2(0, 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐻푝 0 (휔))푑, according to Duahmel’s formula and the fact that the semigroup e−푡ℒ is strongly continuous on 퐻푝(핋)푑, the state 푓휀 also belongs to 퐻푝(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 퓁 ∈ ℕ large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to proposition 8, there exists 푣 ∈ 퐻푝+deg 푃 0 ((0, 푇′) × 휔)푑h × 퐻퓁 0 ((0, 푇′) × 휔)푑p such that 푆(푇′, 푓휀, 푃(휕푥)푣) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write 푄(i푛) by blocks as: 푄(i푛) = ⎛ ⎜ ⎝ 푄1(i푛) ⋮ 푄푘(i푛) ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ 푄h 1(i푛) 푄p 1(i푛) ⋮ ⋮ 푄h 푘(i푛) 푄p 푘(i푛) ⎞ ⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' where the 푄푗(i푛) are of size 푚 × 푑, the 푄h 푗 (i푛) are of size 푚 × 푑h and 푄p 푗(i푛) are of size 푚 × 푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Notice that the 퐿h 푛,푗 defined in the statement of theorem 9 are 퐿h 푛,푗 = 푄h 푗 (i푛)∕푃(i푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set also 푤푛(푡) ∶= 푄(i푛)푐푛(푣(푡, ⋅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write it by blocks as 푤푛(푡) = ⎛ ⎜ ⎝ 푤푛,1(푡) ⋮ 푤푛,푘(푡) ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ 푄1(i푛)푐푛(푣(푡, ⋅)) ⋮ 푄푘(i푛)푐푛(푣(푡, ⋅)) ⎞ ⎟ ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Finally, set 푢푛(푡) ∶= 푤푛,1(푡) + 푤′ 푛,2(푡) + ⋯ + 푤(푘−1) 푛,푘 (푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 3: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Remark that for every 푛 ∈ ℤ, [퐵푛|푀]푘푤푛(푡) = [퐵푛|푀]푘푄(i푛)푐푛(푣(푡, ⋅)) = [퐵푛|푀]푘[퐵푛|푀]∗ 푘 Adj([퐵푛|푀]푘[퐵푛|푀]∗ 푘)푐푛(푣(푡, ⋅)) = det([퐵푛|푀]푘[퐵푛|푀]∗ 푘)푐푛(푣(푡, ⋅)) = 푃(i푛)푐푛(푣(푡, ⋅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, since 푆(푇′, 푓휀, 푃(휕푥)푣) = 0, the control ˜푣푛(푡) ∶= 푃(i푛)푐푛(푣(푡, ⋅)) steers 푐푛(푓휀) to 0 for the system 푋′ 푛 = 퐵푛푋푛 + ˜푣푛 in time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' That is to say, 푤푛 steers 푐푛(푓휀) to 0 for the system 푋′ 푛 = 퐵푛푋푛 + [퐵푛|푀]푘푤푛 in time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, according to lemma 10, 푢푛 steers 푐푛(푓휀) to 0 for the system (9) in time 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, the control 푢 formally defined by 푢 ∶= ∑ 푛∈ℤ 푢푛푒푛 is such that 푆(푓휀, 푇′, 푀푢) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Notice that the previous sum is well-defined in 퐿2(0, 푇′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 퐿2(핋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark that, if we define 푢 in the sense of distributions, 푢 = (푄1(휕푥) + 휕푡푄2(휕푥) + ⋯ + 휕푘−1 푡 푄푘(휕푥))푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푣 is supportedon [0, 푇′]×휔, so is 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Considerthe differentialoperator풬 ∶= 푄1(휕푥)+휕푡푄2(휕푥)+ ⋯ + 휕푘−1 푡 푄푘(휕푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have 푢 = 풬푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write this operator by blocks as 풬 = (풬h 풬p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In other words, 풬h ∶= 푄h 1(휕푥) + 휕푡푄h 2(휕푥) + ⋯ + 휕푘−1 푡 푄h 푘(휕푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 11 The order of the differential operator 풬h is at most Order(풬h) ≤ max 1≤푗≤푘(푗 − 1 + deg(푄h 푗 )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 퐿h 푛,푗 = 푄h 푗 (i푛)∕푃(i푛), according to the definition of 푝 (see theorem 9), Order(풬h) ≤ 푝 + deg(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, recall that 푣 ∈ 퐻푝+deg(푃) 0 ((0, 푇′) × 휔)푑h × 퐻퓁 0 ((0, 푇′) × 휔)푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, if we choose 퓁 ≥ Order(풬p), 푢 ∈ 퐿2((0, 푇′) × 휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3 Upper bound on the loss of regularity Theorem 9 requires initial condition to be 퐻푝 for some 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In this section, we provide an elementary upper bound on 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that for |푛| large enough, the Kalman rank condition rank([퐵푛|푀]) = 푑 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘(푛) ∶= inf{푘∶ rank([퐵푛|푀]푘) = 푑} ∈ {−∞} ∩ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, thesequence(푘(푛))푛∈ℤ iseventuallyconstantwhen|푛| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Wewilldenote푘0 ∶= lim|푛|→+∞ 푘(푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The rank condition rank([퐵푛|푀]푘) = 푑 is equivalent to det([퐵푛|푀]푘[퐵푛|푀]∗ 푘) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푃푘(푛) = det([퐵푛|푀]푘[퐵푛|푀]∗ 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푃푘 is a polynomial in 푛, hence if 푃푘(푛0) ≠ 0 for some 푛0, then 푃푘(푛) ≠ 0 for every large enough |푛|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, for every 푛0, there exists 푛1 such that 푘(푛) ≤ 푘(푛0) whenever |푛| ≥ 푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푘(푛) is integer valued, it is eventually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, we have the following version of theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that the hypotheses (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1)–(H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4) hold, that 푇 > 푇∗ and that for all |푛| ∈ ℕ large enough, the Kalman rank condition rank([퐵푛|푀]) = 푑 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푘0 as in proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 퐸 as in theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, for every 푓0 ∈ 퐻4푑(푘0−1)(핋)푑∩퐸, there exists a control푢 ∈ 퐿2([0, 푇]×휔) such that the solution 푓 of the parabolic-transport system (Sys) with initial condition 푓0 satisfies 푓(푇, ⋅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The sufficient part of theorem 1, as stated in the introduction is a special case of this theorem, since we always have 푘0 ⩽ 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Here is the main lemma that allows us to bound the 푝 of theorem 9 (see also [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1] for similar considerations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 퐴 ∈ ℳ푑(ℂ)푝[푋] a polynomial of degree at most 푝 with 푑 × 푑 matrices coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that for some 푧0 ∈ ℂ, 퐴(푧0) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, 퐴−1 ∈ ℂ푑×푑 푝(푑−1)(푋), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', the coefficients of (퐴(푧))−1 are rational functions of 푧 of degree at most 푝(푑 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write 퐴(푧)−1 = 1 det(퐴(푧)) Adj(퐴(푧)), where Adj(퐴(푧)) is the adjugate matrix of 퐴(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' det(퐴(푧)) and Adj(퐴(푧)) are nonzero polynomials in 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, the coefficients of Adj(퐴(푧)) are sums of products on 푑 − 1 coefficients of 퐴(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, they are polynomials of degree at most (푑 − 1)푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The case we are interested in is: 12 Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' With 푘0 as in proposition 11, set, when it is defined [퐵푛|푀]+ 푘0 ∶= [퐵푛|푀]∗ 푘0 ([퐵푛|푀]푘0[퐵푛|푀]∗ 푘0 )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, as a function of 푛, [퐵푛|푀]+ 푘0 ∈ ℂ푑×푑 2(푘0−1)(2푑−1)(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have [퐵푛|푀]푘0 ∈ ℂ푑×푚푘0 2(푘0−1)[푋], hence [퐵푛|푀]푘0[퐵푛|푀]∗ 푘0 ∈ ℂ푑×푑 4(푘0−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the previous lemma, ([퐵푛|푀]푘0[퐵푛|푀]∗ 푘0 )−1 ∈ ℂ푑×푑 4(푘0−1)(푑−1)(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence [퐵푛|푀]+ 푘0 ∈ ℂ푑×푑 푘 (푋) with 푘 = 4(푘0 − 1)(푑 − 1) + 2(푘0 − 1) = 2(푘0 − 1)(2푑 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to theorem 9, every initial condition in 퐸∩퐻푝(핋)푑 can be steered to 0, where 푝 = deg([퐵푛|푀]+ 푘0) + 푘0 − 1 (degree as a rational function of 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But according to corollary 14, deg([퐵푛|푀]+ 푘0) ≤ 2(푘0 − 1)(2푑 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus 푝 ≤ 4푑(푘0 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, every initial condition in 퐸 ∩ 퐻4푑(푘0−1)(핋)푑 can be steered to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 4 Necessary conditions for null-controllability 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1 Construction of WKB solutions We will give other necessary conditions of null-controllability using so called WKB solutions, that we construct here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Using these kind of approximate solutions is standard for wave equation (see,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', [25, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 426–428] or [31, Appendix B] for a more elementary presentation) or Schrödinger equation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', [35, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 16–17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' WKB solutions were also used to disprove observability of some 2×2 parabolic- transport system with variable coefficients [2, §3] (see also [3, §3] for a Navier-Stokes system with Maxwell’s law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Our construction is a generalization of their construction for system of arbitrary size, which brings a few difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the sake of clarity, we construct WKB solutions only for systems with constant coefficients, which is enough for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But it is likely that such a construction could be adapted to a large class of variable-coefficients parabolic-transport systems of arbitrary sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To disprove the observability inequality, these WKB solutions ought to be constructed for the adjoint system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But the parabolic-transport system (Sys) and its adjoint have the same structure, so, in order to lighten the notations, we construct the WKB solutions for the system (Sys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휙 ∈ 퐶∞([0, 푇] × 핋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ℂ) such that ℑ(휙) ≥ 0 and 휕푥휙 never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We search approximate solutions 푔WKB ℎ (푡, 푥) of the parabolic-transport system (Sys) with the following ansatz, where ℎ > 0 is assumed to be small: ⎧ ⎨ ⎩ 푔WKB ℎ (푡, 푥) = 푋ℎ(푡, 푥)ei휙(푡,푥)∕ℎ, 푋ℎ(푡, 푥) ∼ ∑ 푗≥0 ℎ푗푌푗(푡, 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (10) 13 We have 휕푥푔WKB ℎ = (휕푥푋ℎ + i ℎ휕푥휙푋ℎ) ei휙∕ℎ, 휕푡푔WKB ℎ = (휕푡푋ℎ + i ℎ휕푡휙푋ℎ) ei휙∕ℎ, 휕2 푥푔WKB ℎ = (휕2 푥푋ℎ + 2i ℎ 휕푥휙휕푥푋ℎ − 1 ℎ2 (휕푥휙)2푋ℎ + i ℎ휕2 푥휙푋ℎ) ei휙∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assuming that this 푔WKB ℎ is solution of the parabolic-transport system (Sys), we get 0 = (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾) (푋ℎei휙∕ℎ) =[ (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾) 푋ℎ + 1 ℎ (i휕푡휙 + i퐴휕푥휙 − i퐵휕2 푥휙 − 2i퐵휕푥휙휕푥 ) 푋ℎ + 1 ℎ2 퐵(휕푥휙)2푋ℎ]ei휙∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Plugging in the asymptotic expansion of 푋ℎ, we get 0 ∼ ∑ 푗≥−2 [(휕푥휙)2퐵푌푗+2 + (i휕푡휙 + i퐴휕푥휙 − i퐵휕2 푥휙 − 2i퐵휕푥휙휕푥 ) 푌푗+1 + (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾) 푌푗]ℎ푗, where, by convention, 푌푗 = 0 for 푗 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We want to cancel each of the terms in this sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, we are looking for (푌푗)푗≥0 such that for all 푗 ≥ −2, (휕푥휙)2퐵푌푗+2 + (i휕푡휙 + i퐴휕푥휙 − i퐵휕2 푥휙 − 2i퐵휕푥휙휕푥 ) 푌푗+1 + (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾) 푌푗 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11) Solving this induction relation requires us to look at different projections of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' From now on, we will denote 푌푗 = ( 푌h 푗 푌p 푗 ) with 푌h 푗 ∈ ℂ푑h and 푌p 푗 ∈ ℂ푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, recalling that 퐵 = ( 0 0 0 퐷 ) and taking the parabolic components of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', the 푑p last components), we get (휕푥휙)2퐷푌p 푗 = − (0 퐼) [(i휕푡휙 + i퐴휕푥휙 − i퐵휕2 푥휙 − 2i퐵휕푥휙휕푥 ) 푌푗−1 + (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾) 푌푗−2 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12) Since 퐷 is invertible, this formula determines 푌p 푗 as a function of 푌푗−1 and 푌푗−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Before looking at the other projections of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11), let us recall that 퐴 = ( 퐴′ 퐴12 퐴21 퐴22 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We similarly write 퐾 in blocks as ( 퐾′ 퐾12 퐾21 퐾22 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, taking the transport (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', the first 푑h) components of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11), we get 0 = (i휕푡휙 + i휕푥휙퐴′)푌h 푗 + i휕푥휙퐴12푌p 푗 + (퐼 0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (13) From now on, we choose 휙 of the form2 휙(푡, 푥) = 휓(푥 − 휇푡), (14) 2Equations (12) and (13) with 푗 = 0 implies (휕푡휙 + 휕푥휙퐴′)푌h 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If we want a non-trivial 푌h 0 , this imposes 휙 to depend only on 푥 − 휇푡 for some 휇 ∈ Sp(퐴′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 14 where 휇 is an eigenvalue of 퐴′ an 휓′ never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' With this 휙, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (13) reads 0 = i휓′(푥 − 휇푡)(퐴′ − 휇)푌h 푗 + i휓′(푥 − 휇푡)퐴12푌p 푗 + (퐼 0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1 = i휓′(푥 − 휇푡)(퐴′ − 휇)푌h 푗 + i휓′(푥 − 휇푡)퐴12푌p 푗 + (휕푡 + 퐴′휕푥 + 퐾′)푌h 푗−1 + (퐴12휕푥 + 퐾12)푌p 푗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (15) Denote by 푃′ 휇 the projectionon the eigenspace of 퐴′ associatedwith 휇 along the other eigenspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We consider 푌h 푗,휇 ∈ Range(푃′ 휇) defined by 푌h 푗,휇 = 푃′ 휇푌h 푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Similarly, we set 푌h 푗,≠휇 ∈ ker(푃′ 휇) as 푌h 푗,≠휇 = (퐼 − 푃′ 휇)푌h 푗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Finally, we write in blocks 퐴′ and 퐾′ along the sum ℝ푑 = Range(푃′ 휇) ⊕ ker(푃′ 휇) as 퐴′ = (휇 0 0 퐴′ 22 ) , 퐾′ = (퐾′ 11 퐾′ 12 퐾′ 21 퐾′ 22 ) , where 퐴′ 22 ∈ ℒ(ker(푃′ 휇)), 퐾′ 11 = 푃′ 휇퐾′푃′ 휇 ∈ ℒ(Range(푃′ 휇)), 퐾′ 12 ∈ ℒ(ker(푃′ 휇), Range(푃′ 휇)), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, projecting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (15) on ker(푃′ 휇) along Range(푃′ 휇) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', multiplying by (퐼 − 푃′ 휇)), we get i휓′(푥 − 휇푡)(퐴′ 22 − 휇)푌h 푗,≠휇 = −(퐼 − 푃′ 휇) [ i휓′(푥 − 휇푡)퐴12푌p 푗 + (퐼 0) (휕푡 + 퐴휕푥 + 퐾)푌푗−1 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (16) Since 푃′ 휇 is the projection on the eigenspace of 퐴′ associated with the eigenvalue 휇, 퐴′−휇 is invertible on ker(푃′ 휇), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', 퐴′ 22 − 휇 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (16) determines 푌h 푗,≠휇 as a function of 푌p 푗 and 푌푗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Finally, we project eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (15) on Range(푃′ 휇), we get 0 = (휕푡 + 휇휕푥 + 퐾′ 11)푌h 푗,휇 + 퐾′ 12푌h 푗,≠휇 + 푃′ 휇(퐴12휕푥 + 퐾12)푌p 푗 + i휓′(푥 − 휇푡)푃′ 휇퐴12푌p 푗+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (17) We then use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12) to express 푌p 푗+1 as 푌p 푗+1 = 퐷1푌h 푗 + 퐷2푌p 푗 + 퐷3푌푗−1, with 퐷1 = − i 휓′(푥 − 휇푡)퐷−1퐴21, and where 퐷2 and 퐷3 are matrix first or second-order differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Their specific expressions do not matter for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Plugging this in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (17), we get (휕푡 + 휇휕푥 + 퐾′ 11 + 푃′ 휇퐴12퐷−1퐴21푃′ 휇)푌h 푗,휇 = −퐾′ 12푌h 푗,≠휇 − 푃′ 휇(퐴12휕푥 + 퐾12)푌p 푗 − i휓′(푥 − 휇푡)푃′ 휇퐴12(퐷1(퐼 − 푃′ 휇)푌h 푗,≠휇 + 퐷2푌p 푗 + 퐷3푌푗−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (18) If we chose an initial condition 푌h 푗,휇,0 for 푌h 푗,휇, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (18) determines 푌h 푗,휇 as a function of 푌h 푗,휇,0, 푌h 푗,≠휇, 푌p 푗 and 푌푗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have seenthat if 휙 is given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (14), the (푌푗)푗∈ℕ that solve the WKB recurrence equation (11) are given by eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12), (16) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To be rigorous, we have only proved that if (푌푗)푗푛ℕ solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11), then 푌p 푗 , 푌h 푗,≠휇 and 푌h 푗,휇 solves eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12), (16) and (18) respectively, but not the reciprocal (which is what we are actually interested in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' However, we easily rephrase the computations of this section as a sequence of equivalences: ∀푗 ≥ −2, 푌푗 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (11) if and only if;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ∀푗 ≥ 0, 푌p 푗 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12), 푌h 푗,≠휇 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (16) and 푌h 푗,휇 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (17) if and only if;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ∀푗 ≥ 0, 푌p 푗 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12), 푌h 푗,≠휇 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (16) and 푌h 푗,휇 solves eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 15 We summarize the computations of this section in the following proposition: Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휓 ∈ 퐶∞(핋) such that 휓′ never vanishes and ℑ(휓) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휇 ∈ Sp(퐴′) and set 휙 as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푗 ≥ 0, let 푌h 푗,휇,0 ∈ 퐶∞(핋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ker(퐴′ − 휇)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Define (푌p 푗 )푗≥−2, (푌h 푗,≠휇)푗≥−2 and (푌h 푗,휇)푗≥−2 with the following recursive procedure: set 푌p −2 = 푌p −1 = 0, 푌h −2,≠휇 = 푌h −1,≠휇 = 0, 푌h −2,휇 = 푌h −1,휇 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' if 푌p 푘, 푌h 푘,≠휇, 푌h 푘,휇 are defined for −2 ≤ 푘 ≤ 푗 − 1, define 푌p 푗 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (12), 푌h 푗,≠휇 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (16) and 푌h 푗,휇 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (18) with initial condition 푌h 푗,휇,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푗 ≥ 0, set 푌푗(푡, 푥) = ( 푌h 푗,휇+푌h 푗,≠휇 푌p 푗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푞 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let the function 푔WKB ℎ be defined by 푔WKB ℎ (푡, 푥) = 푞∑ 푗=0 ℎ푗푌푗ei휙(푡,푥)∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (19) Then, defining 푟ℎ by (휕푡 − 퐵휕2 푥 + 퐴휕푥 + 퐾)푔WKB ℎ (푡, 푥) = 푟ℎ(푡, 푥)ei휙(푡,푥)∕ℎ, for every 푘 ∈ ℕ, 퓁 ∈ ℕ, 푡 ∈ [0, 푇] and 푥 ∈ 핋, |휕푘 푡 휕퓁 푥푟ℎ(푡, 푥)| ≤ 퐶푘,퓁ℎ푞−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that ℎ−1 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, replacing 휙 by 휙 + 2푘휋 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (10) does not change the WKB solution 푔WKB ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, 휙 can be defined up to a factor 2푘휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' That way, 휙 can be non-periodic, as long as 휙 mod 2휋 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, we can choose 휙(푡, 푥) = i휑(푥 − 휇푡) + 푛0(푥 − 휇푡) with 휇 ∈ Sp(퐴′), 휑 ≥ 0, and 푛0 ∈ ℕ ⧵ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' These WKB solutions will be used to disproveobservability inequalities that often feature a projec- tion on high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To deal with these projection on high frequencies, we will use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푛 ∈ ℤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Under the assumptions of proposition 15, for every 퓁 ∈ ℕ, we have uniformly in 0 ≤ 푡 ≤ 푇, in the limit ℎ → 0+, (푔WKB ℎ (푡, ⋅), 푒푛)퐿2 = 푂(ℎ퓁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The scalar product (푔WKB ℎ (푡, ⋅), ei푛푥)퐿2 can be written as (푔WKB ℎ (푡, ⋅), 푒푛)퐿2 = ∫ 핋 푤푡,ℎ,푛(푥)ei휓(푥−휇푡)∕ℎ d푥, where 푤푡,ℎ,푛(푥) ∶= 푞∑ 푗=0 ℎ푗푌푗(푡, 푥)e−i푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 16 Note that 푤푡,ℎ,푛 and its derivative are uniformly bounded for 0 ≤ 푡 ≤ 푇 and ℎ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Consider the differential operator 퐿 ∶= (i휓′(푥 − 휇푡))−1휕푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Here, we use the fact that 휓′ never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This operator is such that ℎ퐿ei휓(푥−휇푡)∕ℎ = ei휓(푥−휇푡)∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, denoting 퐿∗ the adjoint of 퐿, by integration by parts, (푔WKB ℎ (푡, ⋅), 푒푛)퐿2 = ℎ푙 ∫ 핋 (퐿∗)퓁(푤푡,ℎ,푛)(푥)ei휓(푥−휇푡)∕ℎ d푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The operator 퐿∗ is a differential operator independent of ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, by definition of 푤푡,ℎ,푛 (푔WKB ℎ (푡, ⋅), 푒푛)퐿2 = 푂(ℎ퓁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2 The parabolic-transport system is not null controllable in small time We now prove that the time condition 푇 ⩾ 푇∗ is necessary (remark that the equality case 푇 = 푇∗ remains an open question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' It was already proved to be necessary for the null-controllability of every 퐿2 initial conditions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But this proof did not exclude the null-controllability of every 퐻푘 initial condition when 푇 < 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇 > 0 and assume that there exists 푁 ∈ ℕ∗ and 푘 ∈ ℕ such that every initial conditions in 퐻푘(핋)푑 ∩ {∑ |푛|>푁 푋푛ei푛푥} for the parabolic-transport system (Sys) can be steered to 0 in time 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then 푇 ≥ 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휇 ∈ Sp(퐴′) with maximum modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' By definition, 푇∗ = 퓁(휔)∕|휇|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇 < 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We aim to disprove that the observability inequality associated to the control problem of propo- sition 18 using the WKB solution constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We claim that this observability inequality is: there exists 퐶 > 0 such that for every 푔0 ∈ 퐿2(핋)푑, the solution 푔 of (휕푡 − 퐵∗휕푥 − 퐴∗휕푥 + 퐾∗휕푥)푔(푡, 푥) = 0, 푔(0, 푥) = 푔0(푥) (20) satisfies ‖휋푁푔(푇, ⋅)‖퐻−푘(핋) ≤ 퐶‖푀∗푔‖퐿2((0,푇)×휔), (21) where 휋푁 ∶ ∑ 푛∈ℤ 푋푛ei푛푥 ∈ 퐿2(핋) ↦ ∑ |푛|>푁 푋푛ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This is proved using a standard duality lemma, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='48] with 퐶2 = e−푡ℒ◦휋∗ 푁◦휄푘 and 퐶1 ∶ 푢 ∈ 퐿2((0, 푇) × 휔) ↦ ∫푇 0 e−(푇−푡)ℒ푀푢(푡) d푡, where 휄푘 is the injection 퐻푘(핋) → 퐿2(핋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Note that 휋∗ 푁 is the injection {∑ |푛|>푁 푋푛ei푛푥} → 퐿2(핋), and that 휄∗ 푘 is a bijective isometry 퐻−푘(핋) → 퐻푘(핋) ([7, Lemma 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Testing this observability inequality on initial conditions of the form 휕푘 푥푔0 instead of 푔0, we get ‖휋푁푔(푇, ⋅)‖퐿2(핋) ≤ 퐶‖휕푘 푥푀∗푔‖퐿2((0,푇)×휔), (22) Step 1: Construction of the counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Let 푇 < 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' There exists 푥0 ∉ 휔 such that 푥0 −휇푡 ∉ 휔 for every 0 ≤ 푡 ≤ 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Choose 휑 ∈ 퐶∞(핋) real-valued such that 휑(푥0) = 0, 휑′′(푥0) = 1 and 휑(푥) > 0 for every 푥 ≠ 푥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, choose 휙(푡, 푥) = i휑(푥 + 휇푡) + (푥 + 휇푡)푛0, as we did in remark 16 (the change from 휇 to −휇 is because we are considering −퐴∗ instead of 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This choice of 휙 ensures that whatever the choices of the 푌푗, the WKB solution 푔WKB ℎ defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (10) stays concentrated around 푥0 + 휇푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 17 Let 푌h 0,휇,0 ∈ 퐶∞(핋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' ker(퐴′∗ + 휇)) with 푌h 0,휇,0(푥0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푗 ≥ 1, set 푌h 푗,휇,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푞 > 푘 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Consider the function 푔WKB ℎ defined by proposition 15 (where 퐵, and 퐾 are replaced respectively by 퐵∗ and 퐾∗, and where 퐴 is replaced by −퐴∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set also 푔ℎ(푡, 푥) the solution of the adjoint system (20) with initial condition 푔WKB ℎ (푡 = 0, ⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Estimation of the difference between 푔WKB ℎ and 푔ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to proposition 15, (휕푡 − 퐵∗휕2 푥 − 퐴∗휕푥 + 퐾∗)푔WKB ℎ = 푂(ℎ푘+1)ei휙(푡,푥)∕ℎ, Hence, with 푟ℎ ∶= 푔WKB ℎ − 푔ℎ, we have 푟ℎ(0, 푥) = 0 and (휕푡 − 퐵∗휕2 푥 − 퐴∗휕푥 + 퐾∗)푟ℎ = 푂(ℎ푘+1)ei휙(푡,푥)∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' where the 푂 has to be understood in the 퐶∞-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since the parabolic-transport system is well- posed in 퐻푘(핋)푑, we get that for every 푗 ∈ ℕ, uniformly in 0 < 푡 < 푇, ‖휕푗 푥(푔WKB ℎ (푡, ⋅) − 푔ℎ(푡, ⋅))‖퐿2 ≤ 퐶푗ℎ푘−푗+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (23) Step 3: Upper bound on the right-hand side of the observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to the triangle inequality, ‖휕푘 푥푀∗푔ℎ‖퐿2((0,푇)×휔) ≤ ‖휕푘 푥푀∗푔WKB ℎ ‖퐿2((0,푇)×휔) + ‖휕푘 푥푀∗푟ℎ‖퐿2((0,푇)×휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to step 2, the second term of the right-hand side is 푂(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the first term of the right- hand side, we recall that 푔WKB ℎ = ∑푞 푗=0 ℎ푗푌푗ei휓(푥−휇푡), and that, thanks to our choice of 휓, ei휓(푥+휇푡) is exponentially small when 푥 + 휇푡 ≠ 푥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Therefore, since 푥0 − 휇푡 ∉ 휔 for every 0 ≤ 푡 ≤ 푇, for some 푐 > 0, ‖휕푘 푥푀∗푔WKB ℎ ‖퐿2((0,푇)×휔) = 푂(e−푐∕ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This proves that ‖휕푘 푥푀∗푔ℎ‖퐿2((0,푇)×휔) = 푂(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (24) Step 4: Lower bound on the left-hand side of the observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to lemma 17, for any 퓁 ≥ 0, ‖휋푁푔WKB ℎ (푇, ⋅)‖퐿2(핋) = ‖푔WKB ℎ (푇, ⋅)‖퐿2(핋) + 푂(ℎ퓁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (25) Thus, using the inverse triangle inequality, ‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ ‖휋푁푔WKB ℎ (푇, ⋅)‖퐿2(핋) − ‖휋푁푟ℎ(푇, ⋅)‖퐿2(핋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Using the error estimates of step 2, and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (25), we get ‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ ‖푔WKB ℎ (푇, ⋅)‖퐿2(핋) − 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (26) Thus, we only need to find a lower-bound for ‖푔WKB ℎ (푇, ⋅)‖퐿2(핋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have ‖푔WKB ℎ (푇, ⋅)‖2 퐿2(핋) = ∫ 핋 |||||||||| 푞∑ 푗=0 ℎ푗푌푗(푡, 푥) |||||||||| 2 e−2휑(푥+휇푇)∕ℎ d푥 = ∫ 핋 |푌0(푡, 푥)|2e−2휑(푥+휇푇)∕ℎ d푥 + 푂(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Recall that 휑(푥0) = 0, that for 푥 ≠ 푥0, 휑(푥) is strictly positive and that 휑′′(푥0) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, using Laplace’s method (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' [36, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2] and in particular [36, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='34)]), we get ‖푔WKB ℎ (푇, ⋅)‖2 퐿2(핋) = 푐 √ ℎ + 푂(ℎ3∕2) 18 for some 푐 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Plugging this into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (26), we get that for ℎ small enough, ‖휋푁푔ℎ(푇, ⋅)‖퐿2(핋) ≥ 푐 √ ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (27) Step 5: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Comparing the lower bound (27) and the upper bound (24) and taking ℎ small enough, we see that the observability inequality (21) cannot hold if 푇 < 푇∗, hence the parabolic- transport system (Sys) with initial conditions in 퐻푘 ∩ 휋푁(퐿2(핋)) is not null-controllable in time 푇 < 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3 Rough initial conditions are not null-controllable We now give necessary conditions for every 퐿2 initial condition to be steerable to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To do this, we only need the first term of the WKB expansion of proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' By analyzing higher-order terms of the WKB expansion, it is likely that we could get necessary conditions for the null-controllability of every 퐻푘 initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But doing this analysis in general seems hard, and we leave this for future work, or on a case-by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In fact, we will prove the following statement, which is a refined version of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 휇 ∈ Sp(퐴′), 푁 ∈ ℕ and 푇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푃′ 휇 be the projection on the eigenspace of 퐴′ associated to 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Write 퐾 in blocks as ( 퐾′ 퐾12 퐾21 퐾22 ), with 퐾′ ∈ ℳ푑ℎ(ℝ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set 퐾∗ 휇 ∶= (푃′ 휇)∗ ((퐾′)∗ + 퐴∗ 21(퐷∗)−1퐴∗ 12 ) (푃′ 휇)∗ Assume that every initial condition 푓0 ∈ 퐿2(핋)푑∩{∑ |푛|>푁 푋푛ei푛푥} is steerable to 0 in time 푇 with control in 퐿2((0, 푇) × 휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, for every 휇 ∈ Sp(퐴′) and for every non-zero subspace 푆 ⊂ Range((푃′ 휇)∗) that is stable by 퐾∗ 휇, there exists 푉0 ∈ 푆 such that 푀∗( 푉0 0 ) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Using a standard duality lemma [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='48], and as in the proof of proposition 18, we get an observability inequality that is equivalent to the null- controllability of the system (Sys) with initial conditions in 퐿2(핋)푑 ∩ {∑ |푛|>푁 푋푛ei푛푥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This observ- ability inequality is: there exists 퐶 > 0 such that for every 푔0 ∈ 퐿2(핋)푑, the solution 푔 of (휕푡 − 퐵∗휕2 푥 − 퐴∗휕푥 + 퐾∗)푔(푡, 푥) = 0, 푔(0, 푥) = 푔0(푥) (28) satisfies ‖휋푁푔(푇, ⋅)‖퐿2(핋) ≤ 퐶‖푀∗푔‖퐿2((0,푇)×휔), (29) where, as in the proof of proposition 18, 휋푁 ∶ ∑ 푛∈ℤ 푋푛ei푛푥 ∈ 퐿2(핋) ↦ ∑ |푛|>푁 푋푛ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Construction of the counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Let 푉0 ∈ 푆 ⧵ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set 휑 ∶== 0 and let 휙(푡, 푥) = 푛0(푥 − 휇푡) as in remark 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set 푌h 0,휇,0(푥) ∶= 푉0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푗 > 0, set 푌h 푗,휇,0 ∶= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푔WKB ℎ be defined by proposition 15 with 퐵 and 퐾 replaced respectively by 퐵∗ and 퐾∗ and 퐴 by −퐴∗, and with 푞 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푔ℎ be the solution of the parabolic-transport system (Sys) with initial condition 푔WKB ℎ (0, ⋅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark that according to proposition 15, and in particular eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (18), (휕푡 − 휇휕푥 + 퐾∗ 휇)푌h 0,휇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 푌h 0,휇(푡, 푥) = e−푡퐾∗ 휇푉0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, since 푆 is stable by 퐾∗ 휇, 푌h 0,휇(푡, 푥) ∈ 푆 for all 푡, 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 3: Error estimate between 푔WKB ℎ and 푔ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Set 푟ℎ ∶= 푔ℎ−푔WKB ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then 푟ℎ(0, 푥) = 0, and according to proposition 15, (휕푡 − 퐵∗휕2 푥 − 퐴∗휕푥 + 퐾∗)푟ℎ = 푂(ℎ)e푖휙(푡,푥)∕ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 19 Since the parabolic-transport system is well-posed in 퐿2(핋)푑, uniformly in 0 ≤ 푡 ≤ 푇, ‖푟ℎ(푡, ⋅)‖퐿2(핋) ≤ 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 4: Upper bound of the right-hand side of the observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Using the error estimate of the previous step, the right-hand side of the observability inequality (29) satisfies ‖푀∗푔ℎ‖2 퐿2((0,푇)×휔) ≤ ‖푀∗푔WKB ℎ ‖2 퐿2((0,푇)×휔) + 퐶ℎ ≤ ‖푀∗푌h 0ei휙∕ℎ‖2 퐿2((0,푇)×휔) + 퐶ℎ = ‖‖‖‖‖‖‖‖‖ 푀∗ (푌h 0,휇 0 ) ‖‖‖‖‖‖‖‖‖ 2 퐿2((0,푇)×휔) + 퐶ℎ = 2휋 ∫ 푇 0 ||||||||| 푀∗ (e−푡퐾∗ 휇푉0 0 ) ||||||||| 2 d푡 + 퐶ℎ, (30) where we used the definition of 푔WKB ℎ for the last three inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 5: Lower-bound of the left-hand side of the observability inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Using again the error estimate of step 3, the left-hand side of the observability inequality (29) satisfies ‖휋푁푔ℎ(푇, ⋅)‖2 퐿2 ≥ ‖휋푁푔WKB ℎ (푇, ⋅)‖2 퐿2 − 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, using the estimate on low frequencies of 푔WKB ℎ (lemma 17) ‖휋푁푔ℎ(푇, ⋅)‖2 퐿2 ≥ ‖푔WKB ℎ (푇, ⋅)‖2 퐿2 − 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Now, using the definition of 푔WKB ℎ , and the fact that |ei휙| = 1, ‖휋푁푔ℎ(푇, ⋅)‖2 퐿2 ≥ ‖푌h 0,휇(푇, ⋅)‖2 퐿2 − 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' = 2휋|e−푇퐾∗ 휇푉0|2 − 퐶ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (31) Step 6: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Comparing the upper bound on the right-hand side of the observability in- equality (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (30)) and the lower bound on the left-hand side (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (31)), we see that 푀∗e−푡퐾∗ 휇푉0 cannot vanish for every 0 ≤ 푡 ≤ 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since e−푡퐾∗ 휇푉0 ∈ 푆 for every 푡, this proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 5 Systems of two equations We apply the general theorems of the previous sections on 2 × 2 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Some of these results are not new (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Our goal here is only to check whether our results are optimal, at least in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1 Control properties of 2 × 2 systems: statements Here, we consider the parabolic transport-system (Sys) with 퐵 = (0 0 0 푑) , 퐴 = ( 푎′ 푎12 푎21 푎22) , 퐾 = (푘11 푘12 푘21 푘22) , 푀 = (푚1 푚2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (32) 20 where all lower-case letters are real numbers, with 푑 > 0 and 푎′ ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Here, we assume that 푀 has rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We do not need to treat the case where rank(푀) = 2, because it is already covered with the general theorem where there is a control on every component (see [7, Theorem 2] or theorem 12 with 푘 = 1): every initial condition in 퐿2(핋)푑 is null-controllable in time 푇 > 푇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In the following three propositions, we detail the applications of our general theorem to eleven cases, showcasing the variety of phenomena that can appear depending on the values of every coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The proofs are given in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 퐵, 퐴, 퐾, 푀 are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that (푚1, 푚2) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If (푎21, 푘21) = (0, 0), the parabolic-transport system (Sys) is not null-controllable, whatever the time 푇 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇 > 퓁(휔)∕|푎′| (where 퓁(휔) is defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푘21 ≠ 0, every initial condition in 퐿2(핋)2 for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎21 ≠ 0 and 푘21 = 0, every initial condition 푓0 = (푓h 0, 푓p 0) in 퐿2(핋)2 such that ∫핋 푓p 0 = 0 for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 퐵, 퐴, 퐾, 푀 are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that (푚1, 푚2) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If (푎12, 푘12) = (0, 0), the parabolic-transport system (Sys) is not null-controllable, whatever the time 푇 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎12 ≠ 0 and 푘12 ≠ 0, every initial condition in 퐻1(핋)×퐿2(핋) for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎12 ≠ 0 and 푘12 = 0, every initial condition 푓0 = (푓h 0, 푓p 0) in 퐻1(핋)×퐿2(핋) such that ∫핋 푓h 0 = 0 for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎12 = 0 and 푘12 ≠ 0, every initial condition in 퐻2(핋)×퐿2(핋) for the system (Sys) can be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In every cases, there exists an initial condition 푓0 in 퐿2(핋) such that ∫핋 푓0 = 0 that cannot be steered to 0 in time 푇 with 퐿2 controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In the case where 푎21 = 0 and 푘21 ≠ 0, there is a gap in the regularity condition that is sufficient for the null controllability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=', 퐻2 × 퐿2), and the lack of null-controllability of 퐿2 × 퐿2 initial condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Are every 퐻1 × 퐿2 initial conditions steerable to 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We conjecture that this is not the case, but theorem 3 is not enough to prove so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We would need to look at the second term in the WKB expan- sion to find out, or use another method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' maybe using a refined version of regularization properties of lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We do not detail in general the case where 푚1 ≠ 0 and 푚2 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us just mention that there is no regularity condition for null-controllability to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' But depending on whether the solution of det([퐵푛, 푀]) = 0 (which is a quadratic equation in 푛) are integer, there might be a condition on at most two fourier components for an initial condition to be steerable to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We only detail the following case that is about the simultaneous control of a transport and a parabolic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 퐵, 푀 are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that 퐴 = ( 푎′ 0 0 푎22 ) and 퐾 = ( 푘11 0 0 푘22 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that (푚1, 푚2) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎′ ≠ 푎22 and 푘11 = 푘22, every initial condition 푓0 = (푓h 0, 푓p 0) ∈ 퐿2(핋)2 such that ∫푇 푓h 0 = ∫핋 푓p 0 can be steered to zero with controls in 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎′ ≠ 푎22 and 푘11 ≠ 푘22, every initial condition in 퐿2(핋)2 can be steered to zero with controls in 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 21 If 푎′ = 푎22 and √ (푘22 − 푘11)∕푑 ∉ ℕ, every initial condition in 퐿2(핋)2 can be steered to zero with controls in 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푎′ = 푎22 and 푛0 ∶= √ (푘22 − 푘11)∕푑 ∈ ℕ, every initial condition 푓0 = (푓h 0, 푓p 0) ∈ 퐿2(핋)2 such that 푐±푛0(푓h 0) = 푐±푛0(푓p 0) can be steered to zero with controls in 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The case 푎′ ≠ 푎22 and 푘11 = 푘22 is not new, at least in spirit: the simultaneous controllability (equivalently, additive observability) of a heat equation and a wave equation has been studied by Zuazua [41, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='2 Regularity of the free equation We will use some basic regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푓0 ∈ 퐻1(핋)푑h × 퐿2(핋)푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푡 > 0, e−푡ℒ푓0 ∈ 퐻1(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume in addition that 퐴12 = 0, and that 푓0 ∈ 퐻2(핋)푑h × 퐿2(핋)푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푡 > 0, e−푡ℒ푓0 ∈ 퐻2(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' To prove it, we will use the following (sub)lemma: Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Consider ℒp and 퐹p as defined in section 2 (or [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푡 > 0, 푘 ∈ ℕ and 푓0 ∈ 퐹p, e−푡ℒp푓0 ∈ 퐻푘(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set 푓(푡) = e−푡ℒp푓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Denote the first 푑h components of 푓(푡) by 푓h(푡) and the last 푑p compo- nents of 푓(푡) by 푓p(푡) (and similarly for 푓0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We will use some simple tools from [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the sake of readability, we redo the proof in full here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Computing 푓h(푡) as a function of 푓p(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Since 푓(푡) ∈ 퐹p, by definition of 퐹p (section 2), for every |푛| > 푛0, 푃p(i∕푛)푐푛(푓(푡)) = 푐푛(푓(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Writing 푃p(푧) by blocks as ( 푝11(푧) 푝12(푧) 푝21(푧) 푝22(푧) ), and taking the first 푑h components, 푝11(i∕푛)푐푛(푓h(푡)) + 푝12(i∕푛)푐푛(푓p(푡)) = 푐푛(푓h(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푃p(0) = ( 0 0 0 퐼 ), 푝11(0) = 0 and for 푧 small enough, |푝11(푧)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, increasing푛0 if necessary, for |푛| > 푛0, 푐푛(푓h(푡)) = (퐼 − 푝11(i∕푛))−1푝12(i∕푛)푐푛(푓p(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푧 ∈ ℂ small enough, let 퐺(푧) = (퐼 − 푝11(푧))−1푝12(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, 퐺 depends holomorphically in 푧 small enough, and for |푛| > 푛0 푐푛(푓h(푡)) = 퐺(i∕푛)푐푛(푓p(푡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Define 풟 the unbounded operator on 퐿2(핋)푑p with domain 퐻2(핋)푑p by 풟( ∑ 푛 푋푛ei푛푥) ∶= ∑ 푛 (푛2퐷 − i푛퐴22 − 퐾22 − 퐺(i∕푛)(i푛퐴21 + 퐾21))푋푛ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Recall that (휕푡 − 퐷휕2 푥 + 퐴22휕푥 + 퐾22)푓p(푡) + (퐴21휕푥 + 퐾21)푓h(푡) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푐푛(푓h(푡)) = 퐺(i∕푛)푐푛(푓p(푡)), this can be written as (휕푡 + 풟)푓p(푡) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, 푓p(푡) = e−푡풟푓p 0 = ∑ |푛|>푛0 e−푡( 푛2퐷+i푛퐴22+퐾22+퐺(i∕푛)(i푛퐴21+퐾21)) 푐푛(푓p 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 22 Since ℜ(Sp(퐷)) ⊂ (0, +∞), 푓p(푡) is in every 퐻푘(핋)푑p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since the first 푑h components of 푓(푡) are 푓h(푡) = ∑ |푛|>푛0 퐺(i∕푛)푐푛(푓p(푡))푒푛, and since 퐺(i∕푛) is bounded as |푛| → +∞, 푓h(푡) also belongs in every 퐻푘(핋)푑h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The proof consists in looking at the projection on hyperbolic (respectively parabolic) components of e−푡ℒ푓0, using the asymptotics for the hyperbolic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' As in the previous proof, we denote the first 푑h components of 푓0 by 푓h 0 and the last 푑p components by 푓p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let us also recall that according to [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='1], e−푡ℒ푓0 = e−푡ℒ0Π0푓0 + e−푡ℒhΠh푓0 + e−푡ℒpΠ0푓p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (33) Step 1: Asymptotics for the hyperbolic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — We use the notations 푃p(푧), 푃h(푧) defined in [7, Proposition 5–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Using the series for the perturbation of the total eigenprojections [27, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' II, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='14)], we get 푃h(푧) = (퐼 0 0 0) − 푧 ((퐼 0 0 0) 퐴 (0 0 0 퐷−1) + (0 0 0 퐷−1) 퐴 (퐼 0 0 0)) + 푂(푧2) = (퐼 0 0 0) − 푧 ( 0 퐴12퐷−1 퐷−1퐴21 0 ) + 푂(푧2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, Πh푓0 = ∑ |푛|>푛0 [(푐푛(푓h 0) 0 ) − i 푛 (퐴12퐷−1푐푛(푓p 0) 퐷−1퐴21푐푛(푓h 0)) + 푂(푛−2푐푛(푓0))] ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (34) Step 2: Case where 푓0 ∈ 퐻1 × 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Since Π0푓0 is a finite sum of ei푛푥, it is in every 퐻푘, and so is e−푡ℒ0Π0푓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the regularity of the parabolic frequencies (lemma 24), e−푡ℒpΠp푓0 is in every 퐻푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푓h 0 ∈ 퐻1(핋)푑h, (푐푛(푓h 0))푛 ∈ 퓁2(ℤ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1 + 푛2) (the 퓁2 space with weight 1 + 푛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 푓p 0 ∈ 퐿2(핋)푑p, (푐푛(푓p 0))푛 ∈ 퓁2(ℤ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, (푐푛(푓h 0) − i 푛퐴12퐷−1푐푛(푓p 0)) |푛|>푛0 ∈ 퓁2(|푛| > 푛0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1 + 푛2), and (퐷−1퐴21푐푛(푓h 0)) |푛|>푛0 ∈ 퓁2(|푛| > 푛0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 1 + 푛2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to the asymptotics for Πh of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (34), Πh푓0 ∈ 퐻1(핋)푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since e−푡ℒh is continuous on every 퐻푘, e−푡ℒhΠh푓0 ∈ 퐻1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 3: Case where 푓0 ∈ 퐻2 × 퐿2 and 퐴12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — The asymptotics (34) reads Πh푓0 = ∑ |푛|>푛0 [(푐푛(푓h 0) 0 ) − i 푛 ( 0 퐷−1퐴21푐푛(푓h 0)) + 푂(푛−2푐푛(푓0))] ei푛푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (35) The rest of the proof is very similar to the previous case: e−푡ℒ0Π0푓0 and e−푡ℒpΠp푓0 are in every 퐻푘, while the asymptotics (35) proves that Πℎ푓0 “gains” two derivatives compared to 푓p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3 Control properties of 2 × 2 systems: proofs Proof of proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In this case, [퐵푛|푀] = (1 i푛푎′ + 푘22 0 i푛푎21 + 푘21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, det([퐵푛|푀]) = i푛푎21 +푘21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We see that if (푎21, 푘21) = (0, 0), the Kalman rank condition never holds, whatever 푛 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to remark 2, item 1, null-controllability does not hold, whatever 푇 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Note that in our case, [퐵푛|푀]+ = [퐵푛|푀]−1 (when the right-hand side exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, [퐵푛|푀]−1 = 1 i푛푎21 + 푘21 (i푛푎21 + 푘21 −i푛푎′ − 푘22 0 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, with the notations of theorem 9 with 푘 = 2, 퐿h 푛,1 = 1 and 퐿h 푛,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 푝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푘21 ≠ 0, det([퐵푛|푀]) = i푛푎21 + 푘21 never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In this case, 퐸 (as defined in theorem 9) is 퐸 = 퐿2(핋)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to theorem 9, every 퐿2(핋)2 can be steered to 0 with 퐿2 controls in time 푇 > 퓁(휔)∕|푎′| If 푎21 ≠ 0 and 푘21 = 0, the Kalman rank condition holds for every 푛 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푛 = 0, according to the formula for [퐵푛|푀], rank([퐵0|푀]) = ℂ × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 퐸 = {(푓h 0, 푓p 0) ∈ 퐿2(핋)2, ∫핋 푓p 0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Therefore, according to theorem 9, every initial condition (푓h 0, 푓p 0) ∈ 퐿2(핋)2 such that ∫핋 푓p 0 = 0 can be steered to 0 with controls in 퐿2 in time 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In this case, [퐵푛|푀] = (0 i푛푎12 + 푘12 1 −푛2푑 + i푛푎22 + 푘22) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, det([퐵푛|푀]) = −i푛푎12 − 푘12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We see that if (푎12, 푘12) = (0, 0), the Kalman rank condi- tion never holds, whatever 푛 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to remark 2 item 1, null-controllability does not hold, whatever 푇 is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' As in the previous proof, [퐵푛|푀]+ = [퐵푛|푀]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, [퐵푛|푀]−1 = 1 −i푛푎12 − 푘12 (−푛2푑 + i푛푎22 + 푘22 −i푛푎12 − 푘12 −1 0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, with the notations of theorem 9 with 푘 = 2, 퐿h 푛,1 = −(−푛2푑 +i푛푎22 +푘22)∕(i푛푎12 +푘12) and 퐿h 푛,2 = 1∕(i푛푎12 + 푘12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, if 푎12 ≠ 0, 푝 = max(1, 1 − 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' And if 푎12 = 0 and 푘12 ≠ 0, 푝 = max(2, 1 + 0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Case 푎12 ≠ 0 and 푘12 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — The Kalman rank condition holds for every 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, with the notations of theorem 9, 푝 = 1 and 퐸 = 퐿2(핋)2, and every initial condition in 퐻1(핋)2 can be steered to 0 with controls in 퐿2 in time 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The strategy to control initial conditions in 퐻1 × 퐿2 is first to let the solution evolve freely during an arbitrarily small time, which gives a 퐻1(핋)2 state (lemma 23), that we can steer to 0 according to the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Case 푎12 ≠ 0 and 푘12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — The case is almost the same as the previous one, except that the Kalman rank condition is not satisfied for 푛 = 0 (and only for 푛 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have rank([퐵0|푀]) = 24 {0} × ℂ and 퐸 = {(푓h 0, 푓p 0) ∈ 퐿2(핋)2, ∫핋 푓h 0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We still have 푝 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, we can steer every initial condition (푓h 0, 푓p 0) ∈ 퐻1(핋)2 such that ∫핋 푓h 0 = 0 an be steered to 0 with controls in time 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' As in the previous case, to control initial conditions in 퐻1 × 퐿2, we let the solution evolve freely, which gives a 퐻1(핋)2 state, and preserves the property ∫핋 푓h 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, we can steer this state in time 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 3: Case 푎12 = 0 and 푘12 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — In this case, the Kalman rank condition is satisfied for every 푛, and 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to theorem 9, we can steer every 퐻2(핋)2 initial condition to 0 in time 푇 > 퓁(휔)∕|푎′| with controls in 퐿2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Again, to control an initial condition in 퐻2 × 퐿2, we let the solution evolve freely for a small time, which gives a 퐻2(핋)2 state (lemma 23), that we can steer to 0 in time 푇 > 퓁(휔)∕|푎′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 4: Lack of null-controllability of 퐿2 initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — We have 푀∗( 1 0 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, according to theorem 3, (recall that 퐴′ has size 1 × 1), there exists a 퐿2(핋)2 initial condition with zero average that cannot be steered to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof of proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We have [퐵푛|푀] = (1 i푛푎′ + 푘11 1 −푑푛2 + i푛푎22 + 푘22) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, det([퐵푛|푀]) = −푑푛2 + i푛(푎22 − 푎′) + 푘22 − 푘11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We see that for 푛 large enough, this determinant is non zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In fact, taking the real and imaginary parts, det([퐵푛|푀]) = 0 ⇔ { −푑푛2 + 푘22 − 푘11 = 0 푛(푎22 − 푎′) = 0 (36) Moreover, [퐵푛|푀]+ = [퐵푛|푀]−1 = 1 det([퐵푛|푀]) (−푑푛2 + i푛푎22 + 푘22 −i푛푎′ − 푘11 −1 1 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 퐿h 푛,1 = −푑푛2 + 푂(푛) −푑푛2 + 푂(푛), and 퐿h 푛,2 = −1 −푑푛2 + 푂(푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, 푝 = max(0, 1 − 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: Case 푎′ ≠ 푎22 and 푘11 = 푘22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (36), the Kalman condition is satisfied for 푛 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, for 푛 = 0, Range([퐵0|푀]) = ℂ푀, thus 퐸 = {(푓h 0, 푓p 0) ∈ 퐿2(핋)2, ∫핋 푓h 0 = ∫핋 푓p 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The theorem 9 gives the claimed controllability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: Case 푎′ ≠ 푎22 and 푘11 ≠ 푘22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (36), the Kalman condition is satisfied for every 푛 ∈ ℤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The theorem 9 gives the claimed controllability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 3: Case 푎′ = 푎22 and √ (푘22 − 푘11)∕푑 ∉ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — As in the previous case, according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (36), the Kalman condition is satisfied for every 푛 ∈ ℤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The theorem 9 gives the claimed controllability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 4: Case 푎′ = 푎22 and 푛0 ∶= √ (푘22 − 푘11)∕푑 ∈ ℕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — According to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (36), the Kalman condition is satisfied for 푛 ≠ ±푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푛 = ±푛0, Range([퐵±푛0|푀]) = ℂ푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The theorem 9 gives the claimed controllability result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 25 A A finite dimension-uniqueness principle for the null-controllability In the null controllability of parabolic-transport systems, we sometimes prove null-controllability “up to a finite dimensional space”, and then use functional analysis arguments to deal with the finite- dimensional spaces that are left [30, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In the previous articles, this was not stated as a general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' This is the purpose of this appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푇0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 퐻 be a complex Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 퐴 be an unbounded operator on 퐻 that generates a strongly continuous semigroup of bounded operator on 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푈 be another Hilbert space and let 퐵∶ 푈 → 퐻 a bounded control operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푇 > 0, let 푈푇 be a Hilbert space that is a subspace of 퐿2(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푈) with continuous and dense injection that satisfies the following “extension by 0 property”:3 if 푢 ∈ 푈푇, 푎, 푏 > 0, then the function ˜푢 defined by ˜푢(푡) = 0 for 0 < 푡 < 푎, ˜푢(푡) = 푢(푡 − 푎) for 푎 < 푡 < 푇 + 푎, and ˜푢(푡) = 0 for 푇 + 푎 < 푡 < 푇 + 푎 + 푏 is in 푈푇+푎+푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Assume that there exists a finite dimensional space ℱ of 퐻 that is stable by the semigroup e푡퐴 and a closed finite codimensional space4 풢 of 퐻 such that: (control up to finite dimension) for every 푓0 ∈ 풢, there exists 푢 ∈ 푈푇0 such that the solution 푓 of 푓′ = 퐴푓 + 퐵푢 satisfies 푓(푇0) ∈ ℱ, (unique continuation) for every 휀 > 0 and for every finite linear combination of generalized eigen- functions 푔0 ∈ 퐻 of 퐴∗, we have 퐵∗(e푡퐴∗푔0) = 0 on 푡 ∈ (0, 휀) ⟹ 푔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, for every 푇 > 푇0 and every 푓0 ∈ 퐻, there exists 푢 ∈ 푈푇 such that the solution 푓 of 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0 satisfies 푓(푇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Remark 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In this proposition, we can weaken the hypothesis “퐵 bounded” into “퐵 admissi- ble” (see [14, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='3]), but in this article, 퐵 is always bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If the assertion “(푔0 ∈ 퐻 is a finite linear combination of generalized eigenfunctions of 퐴∗ and 퐵∗푔0 = 0) ⟹ 푔0 = 0” holds, the unique continuation hypothesis is satisfied by well- posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 1: We may assume that ℱ ⊂ 풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — We prove that if we replace 풢 by ℱ +풢, the hypotheses are still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Let 푓0 ∈ ℱ + 풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We write 푓0 = 푓ℱ + 푓풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the hypotheses, there exists 푢 ∈ 푈푇0 such that the solution 푓 of 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓풢 is such that 푓(푇0) ∈ ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the solution ˜푓 of ˜푓′ = 퐴 ˜푓 + 퐵푢, ˜푓(0) = 푓0 is such that ˜푓(푇0) = e푇0퐴푓ℱ ⏟ ⏟ ⏟ ∈ℱ + 푓(푇0) ⏟⏟⏟ ∈ℱ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Note that if we replace 푇0 by any 푇1 > 푇0, the hypotheses are still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 2: For 푇 > 푇0, the control 푢 ∈ 푈푇 such that 푓(푇) ∈ ℱ may be chosen linearly and continuously in 푓0 ∈ 풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — This is a standard proof of control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푓0 ∈ 풢, set 푉(푓0) ∶= {푢 ∈ 푈푇 ∶ 푓(푇) ∈ ℱ, 푓 solves 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 퐴 generates a strongly continuous semigroup, 푉(푓0) is a closed affine subspace of 푈푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, we can define 풰(푓0) as the orthogonal projection of 0 onto 푉(푓0) for the 푈푇-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Using the charac- terization of orthogonal projection on closed convex set, we see that 풰 is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Using the fact that 퐴 3In the application we use here, 푈 = 퐿2(휔) and 푈푇 = 퐻푘 0 ((0, 푇) × 휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The hypotheses of proposition 25 are tailored to allow this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 4We do not require 풢 to be stable by e푡퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 26 generates a strongly continuous semigroup, the characterization of the projection on closed convex subsets and the closed graph theorem, we see that 풰 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For the rest of the proof we set 풰푇 ∶ 풢 → 푈푇 such a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' We also set 풩푇 ∶= {푓0 ∈ 퐻 ∶ ∃푢 ∈ 푈푇, 푓(푇) = 0, 푓 solves 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (37) Step 3: For 푇 ≥ 푇0, 풩푇 is a closed finite codimensional subspace of 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Set 푆0(푡) the semigroup e푡퐴 restricted to ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since ℱ is finite dimensional, 푆0(푡) can be written as e푡퐴0, where 퐴0 is a bounded operator of ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, 퐴0 = 퐴|ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' In particular, 푆0 is actually a group of bounded operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For 푓0 ∈ 풢, and 푓′ = 퐴푓 + 퐵풰푇푓0, 푓(0) = 푓0, we have 푓(푇) ∈ ℱ, which allows us to define 풦 ∶ 푓0 ∈ 풢 ↦ −푆0(−푇)푓(푇) ∈ ℱ The range of this operator 풦 satisfies Range(풦) ⊂ ℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, 풦 has finite rank and is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, according to Fredholm’s alternative, (퐼 + 풦)풢 is a closed subspace of 풢 of finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, for every 푓0 ∈ 풢, the solution ˜푓 of ˜푓′ = 퐴 ˜푓 + 퐵풰푇푓0, ˜푓(0) = 푓0 + 풦푓0 satisfies ˜푓(푇) = 푓(푇) + e푇퐴풦푓0 = 푓(푇) − 푆0(푇)푆0(−푇)푓(푇) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, (퐼 + 풦)풢 ⊂ ℱ푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to [9, Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='5], this proves that 풩푇 is closed and has finite codimension in 퐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 4: There exists 훿 > 0 such that for every 푇, 푇′ ∈ (푇0, 푇0 + 훿), 풩푇 = 풩푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Assume 푇0 < 푇 < 푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' If 푢 ∈ 풩푇, and if we extend푢 by 0 on (푇, 푇′), we have have 푢 ∈ 풩푇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus codim(풩푇′) ≤ codim(풩푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since codim(풩푇) is an integer, the discontinuities of 푇 ↦ codim(풩푇) are isolated, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' From now on, we choose 휀 ∈ (0, 훿∕2) arbitrarily small and we set 푇1 = 푇0 + 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 5: For 푡 ∈ (0, 휀), (e푡퐴∗풩⊥ 푇1)⊥ ⊂ 풩푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — Let 0 < 푡 < 휀 and 푓0 ∈ (e푡퐴∗풩⊥ 푇1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' For every 푔0 ∈ 풩⊥ 푇1, we have 0 = ⟨e푡퐴∗푔0, 푓0⟩ = ⟨푔0, e푡퐴푓0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, e푡퐴푓0 ∈ (풩⊥ 푇1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 풩푇1 is closed (step 3), e푡퐴푓0 ∈ 풩푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' By definition of 풩푇1 and the “extension by 0” property of 푈푇1, this proves that 푓0 ∈ 풩푇1+푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the previous step, 풩푇1+푡 = 풩푇1, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 6: 풩⊥ 푇1 is left-invariant by e푡퐴∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — First, consider 0 < 푡 < 휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the previous step, 풩⊥ 푇1 ⊂ ((e푡퐴∗풩⊥ 푇1)⊥)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 풩⊥ 푇1 is finite dimensional hence closed, 풩⊥ 푇1 ⊂ e푡퐴∗풩⊥ 푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, dim(e푡퐴∗풩⊥ 푇1) ≤ dim(풩⊥ 푇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, for 0 < 푡 < 휀, e푡퐴∗풩⊥ 푇1 = 풩⊥ 푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thanks to the semigroup property, this is true for all 푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Step 7: Unique continuation property associated to the control problem “steer every 푓0 ∈ 퐻 into 풩푇1 in time 휀 with a control in 푈휀”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — The control problem is, in mathematical form, the following: ∀푓0 ∈ 퐻, ∃푢 ∈ 푈휀, 푓(푇) ∈ 풩푇1, where 푓′ = 퐴푓 + 퐵푢, 푓(0) = 푓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (38) Let Π∶ 퐻 → 퐻 the orthogonal projection on 풩⊥ 푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Set also 푅푇 ∶ 퐿2(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푈) → 퐻 the input-to- output map defined by 푅푇푢 ∶= 푓(푇), where 푓′ = 퐴푓 + 퐵푢, 푓(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the control problem (38) is equivalent to ∀푓0 ∈ 퐻, ∃푢 ∈ 푈휀, Πe휀퐴푓0 + Π푅휀푢 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 27 We denote by 휄휀 the injection map 푈휀 → 퐿2(0, 푇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' 푈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Then, the previous assertion is equivalent to Range (Π◦e휀퐴) ⊂ Range (Π◦푅휀◦휄휀 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' The observability inequality associated to this control problem is (see [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='48]): ∀푔0 ∈ 퐻, ‖e휀퐴∗◦Π∗푔0‖ ≤ 퐶‖휄∗ 휀 ◦푅∗ 휀 ◦Π∗푔0‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since Range(Π∗) = 풩⊥ 푇1 is finite-dimensional, and since ker(휄∗) = Range(휄)⊥ = {0}, this is equivalent to ∀푔0 ∈ 풩⊥ 푇1, 푅∗ 휀 푔0 = 0 ⟹ e휀퐴∗푔0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (39) To conclude, since 풩⊥ 푇1 is finite dimensional and stable by e푡퐴∗, the semigroup e푡퐴∗ is in fact a group, and in particular e휀퐴∗ is invertible on 풩⊥ 푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Moreover, 푅∗ 휀 푔0(푡) = 퐵∗e(휀−푡)퐴∗푔0 (see [14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content='47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, the assertion (39) is equivalent to ∀푔0 ∈ 풩⊥ 푇1, ( 퐵∗e푡퐴∗푔0 = 0 for 0 < 푡 < 휀 ) ⟹ 푔0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' (40) Step 8: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' — The unique continuation property (40) of the previous step is exactly the unique continuation property we assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Thus, according to the previous step, we can steer every 푓0 ∈ 퐻 into 풩푇1 in time 휀 with a control in 푈휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' According to the definition of 풩푇1, we can steer every 푓0 ∈ 풩푇1 to 0 in time 푇1 = 푇 + 휀 with a control in 푈푇1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Hence, we can steer every 푓0 ∈ 퐻 to 0 in time 푇1 + 휀 = 푇 + 2휀 with a control in 푈푇+2휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Since 휀 can be chosen arbitrarily small, this proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' References [1] Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Adams and John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNAyT4oBgHgl3EQfmfhl/content/2301.00471v1.pdf'} +page_content=' Fournier.' metadata={'source': 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[clue-ai/ChatYuan](https://github.com/clue-ai/ChatYuan) 项目的模型 [ClueAI/ChatYuan-large-v2](https://huggingface.co/ClueAI/ChatYuan-large-v2) 的支持。 + +💡 受 [GanymedeNil](https://github.com/GanymedeNil) 的项目 [document.ai](https://github.com/GanymedeNil/document.ai) 和 [AlexZhangji](https://github.com/AlexZhangji) 创建的 [ChatGLM-6B Pull Request](https://github.com/THUDM/ChatGLM-6B/pull/216) 启发，建立了全部基于开源模型实现的本地知识问答应用。 + +✅ 本项目中 Embedding 默认选用的是 [GanymedeNil/text2vec-large-chinese](https://huggingface.co/GanymedeNil/text2vec-large-chinese/tree/main)，LLM 默认选用的是 [ChatGLM-6B](https://github.com/THUDM/ChatGLM-6B)。依托上述模型，本项目可实现全部使用**开源**模型**离线私有部署**。 + +⛓️ 本项目实现原理如下图所示，过程包括加载文件 -> 读取文本 -> 文本分割 -> 文本向量化 -> 问句向量化 -> 在文本向量中匹配出与问句向量最相似的`top k`个 -> 匹配出的文本作为上下文和问题一起添加到`prompt`中 -> 提交给`LLM`生成回答。 + +![实现原理图](img/langchain+chatglm.png) + +从文档处理角度来看，实现流程如下： + +![实现原理图2](img/langchain+chatglm2.png) + +🚩 本项目未涉及微调、训练过程，但可利用微调或训练对本项目效果进行优化。 + +🌐 [AutoDL 镜像](https://www.codewithgpu.com/i/imClumsyPanda/langchain-ChatGLM/langchain-ChatGLM) + +📓 [ModelWhale 在线运行项目](https://www.heywhale.com/mw/project/643977aa446c45f4592a1e59) + +## 变更日志 + +参见 [变更日志](docs/CHANGELOG.md)。 + +## 硬件需求 + +- ChatGLM-6B 模型硬件需求 + + 注：如未将模型下载至本地，请执行前检查`$HOME/.cache/huggingface/`文件夹剩余空间，模型文件下载至本地需要 15 GB 存储空间。 + + 模型下载方法可参考 [常见问题](docs/FAQ.md) 中 Q8。 + + | **量化等级** | **最低 GPU 显存**（推理） | **最低 GPU 显存**（高效参数微调） | + | -------------- | ------------------------- | --------------------------------- | + | FP16（无量化） | 13 GB | 14 GB | + | INT8 | 8 GB | 9 GB | + | INT4 | 6 GB | 7 GB | + +- MOSS 模型硬件需求 + + 注：如未将模型下载至本地，请执行前检查`$HOME/.cache/huggingface/`文件夹剩余空间，模型文件下载至本地需要 70 GB 存储空间 + + 模型下载方法可参考 [常见问题](docs/FAQ.md) 中 Q8。 + + | **量化等级** | **最低 GPU 显存**（推理） | **最低 GPU 显存**（高效参数微调） | + |-------------------|-----------------------| --------------------------------- | + | FP16（无量化） | 68 GB | - | + | INT8 | 20 GB | - | + +- Embedding 模型硬件需求 + + 本项目中默认选用的 Embedding 模型 [GanymedeNil/text2vec-large-chinese](https://huggingface.co/GanymedeNil/text2vec-large-chinese/tree/main) 约占用显存 3GB，也可修改为在 CPU 中运行。 + +## Docker 部署 +为了能让容器使用主机GPU资源，需要在主机上安装 [NVIDIA Container Toolkit](https://github.com/NVIDIA/nvidia-container-toolkit)。具体安装步骤如下： +```shell +sudo apt-get update +sudo apt-get install -y nvidia-container-toolkit-base +sudo systemctl daemon-reload +sudo systemctl restart docker +``` +安装完成后，可以使用以下命令编译镜像和启动容器： +``` +docker build -f Dockerfile-cuda -t chatglm-cuda:latest . +docker run --gpus all -d --name chatglm -p 7860:7860 chatglm-cuda:latest + +#若要使用离线模型，请配置好模型路径，然后此repo挂载到Container +docker run --gpus all -d --name chatglm -p 7860:7860 -v ~/github/langchain-ChatGLM:/chatGLM chatglm-cuda:latest +``` + + +## 开发部署 + +### 软件需求 + +本项目已在 Python 3.8 - 3.10，CUDA 11.7 环境下完成测试。已在 Windows、ARM 架构的 macOS、Linux 系统中完成测试。 + +vue前端需要node18环境 +### 从本地加载模型 + +请参考 [THUDM/ChatGLM-6B#从本地加载模型](https://github.com/THUDM/ChatGLM-6B#从本地加载模型) + +### 1. 安装环境 + +参见 [安装指南](docs/INSTALL.md)。 + +### 2. 设置模型默认参数 + +在开始执行 Web UI 或命令行交互前，请先检查 [configs/model_config.py](configs/model_config.py) 中的各项模型参数设计是否符合需求。 + +### 3. 执行脚本体验 Web UI 或命令行交互 + +> 注：鉴于环境部署过程中可能遇到问题，建议首先测试命令行脚本。建议命令行脚本测试可正常运行后再运行 Web UI。 + +执行 [cli_demo.py](cli_demo.py) 脚本体验**命令行交互**： +```shell +$ python cli_demo.py +``` + +或执行 [webui.py](webui.py) 脚本体验 **Web 交互** + +```shell +$ python webui.py +``` + +或执行 [api.py](api.py) 利用 fastapi 部署 API +```shell +$ python api.py +``` +或成功部署 API 后，执行以下脚本体验基于 VUE 的前端页面 +```shell +$ cd views + +$ pnpm i + +$ npm run dev +``` + +执行后效果如下图所示： +1. `对话` Tab 界面 +![](img/webui_0510_0.png) +2. `知识库测试 Beta` Tab 界面 +![](img/webui_0510_1.png) +3. `模型配置` Tab 界面 +![](img/webui_0510_2.png) + +Web UI 可以实现如下功能： + +1. 运行前自动读取`configs/model_config.py`中`LLM`及`Embedding`模型枚举及默认模型设置运行模型，如需重新加载模型，可在 `模型配置` Tab 重新选择后点击 `重新加载模型` 进行模型加载； +2. 可手动调节保留对话历史长度、匹配知识库文段数量，可根据显存大小自行调节； +3. `对话` Tab 具备模式选择功能，可选择 `LLM对话` 与 `知识库问答` 模式进行对话，支持流式对话； +4. 添加 `配置知识库` 功能，支持选择已有知识库或新建知识库，并可向知识库中**新增**上传文件/文件夹，使用文件上传组件选择好文件后点击 `上传文件并加载知识库`，会将所选上传文档数据加载至知识库中，并基于更新后知识库进行问答； +5. 新增 `知识库测试 Beta` Tab，可用于测试不同文本切分方法与检索相关度阈值设置，暂不支持将测试参数作为 `对话` Tab 设置参数。 +6. 后续版本中将会增加对知识库的修改或删除，及知识库中已导入文件的查看。 + +### 常见问题 + +参见 [常见问题](docs/FAQ.md)。 + +## Demo + +以问题`chatglm-6b 的局限性具体体现在哪里，如何实现改进`为例： + +未使用 langchain 接入本地文档时： + +>ChatGLM-6B 是一个基于 GPT-3 的聊天机器人,它的局限性主要体现在以下几个方面: +>1. 语言模型的精度:GPT-3 是一种大型的语言模型,它的精度比较高,但是仍然有一些局限性。例如,它可能无法准确地理解某些方言、口音或者复杂的句子结构。 +>2. 知识图谱的完整性:GPT-3 可以生成一些知识图谱,但是其完整性和准确性可能不够。例如,它可能无法准确地理解某些领域的专业知识和术语。 +>3. 对话生成的连贯性:GPT-3 可以生成一些连贯的对话,但是其生成的对话可能不够自然。例如,它可能会生成一些虚假的、不连贯的对话或者误解用户的意图。 +> +>为了改进 ChatGLM-6B,可以考虑以下几个方面: +>1. 提高语言模型的精度:可以通过增加训练数据、使用更好的语言模型架构、优化模型训练过程等方式来提高语言模型的精度。 +>2. 增强知识图谱的完整性:可以通过增加知识图谱的维度、使用更好的知识图谱构建方法、优化知识图谱的匹配算法等方式来增强知识图谱的完整性。 +>3. 提高对话生成的连贯性:可以通过增加对话生成模型的上下文信息、使用更好的上下文信息提取方法、优化对话生成模型的算法等方式来提高对话生成的连贯性。 +>4. 引入更多的人类反馈:人类反馈可以帮助 ChatGLM-6B 更好地理解用户的意图和需求,从而提高对话生成的质量和准确性。可以通过引入更多的人类客服、使用人类反馈收集器等方式来引入更多的人类反馈。 + +使用 langchain 接入 ChatGLM-6B 项目的 README.md 文件后 ChatGLM 的回答： +>ChatGLM-6B 是一个基于深度学习的自然语言处理模型,它在回答问题方面表现出色。但是,它也有一些局限性,具体如下: +>1. 语言理解有限:ChatGLM-6B 只学习了中文自然语言,对于其他语言可能会存在理解上的局限性。 +>2. 知识储备不足:ChatGLM-6B 的训练数据集只包含了中文语料,因此它可能无法回答一些非中文的问题或者某些特定领域的问题。 +>3. 数据量有限:ChatGLM-6B 的训练数据集只有几百万条记录,因此它可能无法回答一些非常具体或者复杂的问题。 +> +>为了改进 ChatGLM-6B,可以考虑以下几个方面: +>1. 学习更多的语言知识:可以学习其他语言的自然语言处理技术,扩大语言理解的范围。 +>2. 扩大知识储备:可以收集更多的中文语料,或者使用其他语言的数据集来扩充知识储备。 +>3. 增加数据量:可以使用更大的数据集来训练 ChatGLM-6B,提高模型的表现。 +>4. 引入更多的评估指标:可以引入更多的评估指标来评估模型的表现,从而发现 ChatGLM-6B 存在的不足和局限性。 +>5. 改进模型架构:可以改进 ChatGLM-6B 的模型架构,提高模型的性能和表现。例如,可以使用更大的神经网络或者改进的卷积神经网络结构。 + +## 路线图 + +- [ ] Langchain 应用 + - [x] 接入非结构化文档（已支持 md、pdf、docx、txt 文件格式） + - [ ] 搜索引擎与本地网页接入 + - [ ] 结构化数据接入（如 csv、Excel、SQL 等） + - [ ] 知识图谱/图数据库接入 + - [ ] Agent 实现 +- [ ] 增加更多 LLM 模型支持 + - [x] [THUDM/chatglm-6b](https://huggingface.co/THUDM/chatglm-6b) + - [x] [THUDM/chatglm-6b-int8](https://huggingface.co/THUDM/chatglm-6b-int8) + - [x] [THUDM/chatglm-6b-int4](https://huggingface.co/THUDM/chatglm-6b-int4) + - [x] [THUDM/chatglm-6b-int4-qe](https://huggingface.co/THUDM/chatglm-6b-int4-qe) + - [x] [ClueAI/ChatYuan-large-v2](https://huggingface.co/ClueAI/ChatYuan-large-v2) + - [x] [fnlp/moss-moon-003-sft](https://huggingface.co/fnlp/moss-moon-003-sft) +- [ ] 增加更多 Embedding 模型支持 + - [x] [nghuyong/ernie-3.0-nano-zh](https://huggingface.co/nghuyong/ernie-3.0-nano-zh) + - [x] [nghuyong/ernie-3.0-base-zh](https://huggingface.co/nghuyong/ernie-3.0-base-zh) + - [x] [shibing624/text2vec-base-chinese](https://huggingface.co/shibing624/text2vec-base-chinese) + - [x] [GanymedeNil/text2vec-large-chinese](https://huggingface.co/GanymedeNil/text2vec-large-chinese) +- [ ] Web UI + - [x] 利用 gradio 实现 Web UI DEMO + - [x] 添加输出内容及错误提示 + - [x] 引用标注 + - [ ] 增加知识库管理 + - [x] 选择知识库开始问答 + - [x] 上传文件/文件夹至知识库 + - [ ] 删除知识库中文件 + - [ ] 利用 streamlit 实现 Web UI Demo +- [ ] 增加 API 支持 + - [x] 利用 fastapi 实现 API 部署方式 + - [ ] 实现调用 API 的 Web UI Demo + +## 项目交流群 +![二维码](img/qr_code_17.jpg) + +🎉 langchain-ChatGLM 项目交流群，如果你也对本项目感兴趣，欢迎加入群聊参与讨论交流。 diff --git a/samples/content/test.txt b/samples/content/test.txt new file mode 100644 index 0000000000000000000000000000000000000000..34f4f179e09a9fb742631e3b16c67fc87ddbca39 --- /dev/null +++ b/samples/content/test.txt @@ -0,0 +1,835 @@ +ChatGPT是OpenAI开发的一个大型语言模型，可以提供各种主题的信息， + +# 如何向 ChatGPT 提问以获得高质量答案：提示技巧工程完全指南 + +## 介绍 + +我很高兴欢迎您阅读我的最新书籍《The Art of Asking ChatGPT for High-Quality Answers: A complete Guide to Prompt Engineering Techniques》。本书是一本全面指南，介绍了各种提示技术，用于从ChatGPT中生成高质量的答案。 + +我们将探讨如何使用不同的提示工程技术来实现不同的目标。ChatGPT是一款最先进的语言模型，能够生成类似人类的文本。然而，理解如何正确地向ChatGPT提问以获得我们所需的高质量输出非常重要。而这正是本书的目的。 + +无论您是普通人、研究人员、开发人员，还是只是想在自己的领域中将ChatGPT作为个人助手的人，本书都是为您编写的。我使用简单易懂的语言，提供实用的解释，并在每个提示技术中提供了示例和提示公式。通过本书，您将学习如何使用提示工程技术来控制ChatGPT的输出，并生成符合您特定需求的文本。 + +在整本书中，我们还提供了如何结合不同的提示技术以实现更具体结果的示例。我希望您能像我写作时一样，享受阅读本书并从中获得知识。 + +
+ +## 第一章：Prompt 工程技术简介 + +什么是 Prompt 工程？ + +Prompt 工程是创建提示或指导像 ChatGPT 这样的语言模型输出的过程。它允许用户控制模型的输出并生成符合其特定需求的文本。 + +ChatGPT 是一种先进的语言模型，能够生成类似于人类的文本。它建立在 Transformer 架构上，可以处理大量数据并生成高质量的文本。 + +然而，为了从 ChatGPT 中获得最佳结果，重要的是要了解如何正确地提示模型。 提示可以让用户控制模型的输出并生成相关、准确和高质量的文本。 在使用 ChatGPT 时，了解它的能力和限制非常重要。 + +该模型能够生成类似于人类的文本，但如果没有适当的指导，它可能无法始终产生期望的输出。 + +这就是 Prompt 工程的作用，通过提供清晰而具体的指令，您可以引导模型的输出并确保其相关。 + +**Prompt 公式是提示的特定格式，通常由三个主要元素组成：** + +- 任务：对提示要求模型生成的内容进行清晰而简洁的陈述。 + +- 指令：在生成文本时模型应遵循的指令。 + +- 角色：模型在生成文本时应扮演的角色。 + +在本书中，我们将探讨可用于 ChatGPT 的各种 Prompt 工程技术。我们将讨论不同类型的提示，以及如何使用它们实现您想要的特定目标。 + +
+ +## 第二章：指令提示技术 + +现在，让我们开始探索“指令提示技术”，以及如何使用它从ChatGPT中生成高质量的文本。 + +指令提示技术是通过为模型提供具体指令来引导ChatGPT的输出的一种方法。这种技术对于确保输出相关和高质量非常有用。 + +要使用指令提示技术，您需要为模型提供清晰简洁的任务，以及具体的指令以供模型遵循。 + +例如，如果您正在生成客户服务响应，您将提供任务，例如“生成响应客户查询”的指令，例如“响应应该专业且提供准确的信息”。 + +提示公式：“按照以下指示生成[任务]：[指令]” + +示例： + +**生成客户服务响应：** + +- 任务：生成响应客户查询 +- 指令：响应应该专业且提供准确的信息 +- 提示公式：“按照以下指示生成专业且准确的客户查询响应：响应应该专业且提供准确的信息。” + +**生成法律文件：** + +- 任务：生成法律文件 +- 指令：文件应符合相关法律法规 +- 提示公式：“按照以下指示生成符合相关法律法规的法律文件：文件应符合相关法律法规。” + +使用指令提示技术时，重要的是要记住指令应该清晰具体。这将有助于确保输出相关和高质量。可以将指令提示技术与下一章节中解释的“角色提示”和“种子词提示”相结合，以增强ChatGPT的输出。 + +
+ +## 第三章：角色提示 + +角色提示技术是通过为ChatGPT指定一个特定的角色来引导其输出的一种方式。这种技术对于生成针对特定上下文或受众的文本非常有用。 + +要使用角色提示技术，您需要为模型提供一个清晰具体的角色。 + +例如，如果您正在生成客户服务回复，您可以提供一个角色，如“客户服务代表”。 + +提示公式：“作为[角色]生成[任务]” + +示例： + +**生成客户服务回复：** + +- 任务：生成对客户查询的回复 +- 角色：客户服务代表 +- 提示公式：“作为客户服务代表，生成对客户查询的回复。” + +**生成法律文件：** + +- 任务：生成法律文件 +- 角色：律师 +- 提示公式：“作为律师，生成法律文件。” + +将角色提示技术与指令提示和种子词提示结合使用可以增强ChatGPT的输出。 + +**下面是一个示例，展示了如何将指令提示、角色提示和种子词提示技术结合使用：** + +- 任务：为新智能手机生成产品描述 +- 指令：描述应该是有信息量的，具有说服力，并突出智能手机的独特功能 +- 角色：市场代表 种子词：“创新的” +- 提示公式：“作为市场代表，生成一个有信息量的、有说服力的产品描述，突出新智能手机的创新功能。该智能手机具有以下功能[插入您的功能]” + +在这个示例中，指令提示用于确保产品描述具有信息量和说服力。角色提示用于确保描述是从市场代表的角度书写的。而种子词提示则用于确保描述侧重于智能手机的创新功能。 + +
+ +## 第四章：标准提示 + +标准提示是一种简单的方法，通过为模型提供一个特定的任务来引导ChatGPT的输出。例如，如果您想生成一篇新闻文章的摘要，您可以提供一个任务，如“总结这篇新闻文章”。 + +提示公式：“生成一个[任务]” + +例如： + +**生成新闻文章的摘要：** + +- 任务：总结这篇新闻文章 +- 提示公式：“生成这篇新闻文章的摘要” + +**生成一篇产品评论：** + +- 任务：为一款新智能手机撰写评论 +- 提示公式：“生成这款新智能手机的评论” + +此外，标准提示可以与其他技术（如角色提示和种子词提示）结合使用，以增强ChatGPT的输出。 + +**以下是如何将标准提示、角色提示和种子词提示技术结合使用的示例：** + +- 任务：为一台新笔记本电脑撰写产品评论 +- 说明：评论应客观、信息丰富，强调笔记本电脑的独特特点 +- 角色：技术专家 +- 种子词：“强大的” +- 提示公式：“作为一名技术专家，生成一个客观而且信息丰富的产品评论，强调新笔记本电脑的强大特点。” + +在这个示例中，标准提示技术用于确保模型生成产品评论。角色提示用于确保评论是从技术专家的角度写的。而种子词提示用于确保评论侧重于笔记本电脑的强大特点。 + +
+ +## 第五章：零、一和少样本提示 + +零样本、一样本和少样本提示是用于从ChatGPT生成文本的技术，最少或没有任何示例。当特定任务的数据有限或任务是新的且未定义时，这些技术非常有用。 + +当任务没有可用的示例时，使用零样本提示技术。模型提供一个通用任务，根据对任务的理解生成文本。 + +当任务只有一个示例可用时，使用一样本提示技术。模型提供示例，并根据对示例的理解生成文本。 + +当任务只有有限数量的示例可用时，使用少样本提示技术。模型提供示例，并根据对示例的理解生成文本。 + +提示公式：“基于[数量]个示例生成文本” + +例如： + +**为没有可用示例的新产品编写产品描述：** + +- 任务：为新的智能手表编写产品描述 + +- 提示公式：“基于零个示例为这款新智能手表生成产品描述” + +**使用一个示例生成产品比较：** + +- 任务：将新款智能手机与最新的iPhone进行比较 + +- 提示公式：“使用一个示例（最新的iPhone）为这款新智能手机生成产品比较” + +**使用少量示例生成产品评论：** + +- 任务：为新的电子阅读器撰写评论 + +- 提示公式：“使用少量示例（3个其他电子阅读器）为这款新电子阅读器生成评论” + + +这些技术可用于根据模型对任务或提供的示例的理解生成文本。 + +
+ +## 第六章：“让我们思考一下”提示 + +“让我们思考一下”提示是一种技巧，可鼓励ChatGPT生成反思和思考性的文本。这种技术适用于撰写论文、诗歌或创意写作等任务。 + +“让我们思考一下”提示的公式非常简单，即“让我们思考一下”后跟一个主题或问题。 + +例如： + +**生成一篇反思性论文：** + +- 任务：就个人成长主题写一篇反思性论文 + +- 提示公式：“让我们思考一下：个人成长” + +**生成一首诗：** + +- 任务：写一首关于季节变化的诗 + +- 提示公式：“让我们思考一下：季节变化” + + +这个提示要求对特定主题或想法展开对话或讨论。发言者邀请ChatGPT参与讨论相关主题。 + +模型提供了一个提示，作为对话或文本生成的起点。 + +然后，模型使用其训练数据和算法生成与提示相关的响应。这种技术允许ChatGPT根据提供的提示生成上下文适当且连贯的文本。 + +**要使用“让我们思考一下提示”技术与ChatGPT，您可以遵循以下步骤：** + +1. 确定您要讨论的主题或想法。 + +2. 制定一个明确表达主题或想法的提示，并开始对话或文本生成。 + +3. 用“让我们思考”或“让我们讨论”开头的提示，表明您正在启动对话或讨论。 + +**以下是使用此技术的一些提示示例：** + +- 提示：“让我们思考气候变化对农业的影响” + +- 提示：“让我们讨论人工智能的当前状态” + +- 提示：“让我们谈谈远程工作的好处和缺点” 您还可以添加开放式问题、陈述或一段您希望模型继续或扩展的文本。 + + +提供提示后，模型将使用其训练数据和算法生成与提示相关的响应，并以连贯的方式继续对话。 + +这种独特的提示有助于ChatGPT以不同的视角和角度给出答案，从而产生更具动态性和信息性的段落。 + +使用提示的步骤简单易行，可以真正提高您的写作水平。尝试一下，看看效果如何吧。 + +
+ +## 第七章：自洽提示 + +自洽提示是一种技术，用于确保ChatGPT的输出与提供的输入一致。这种技术对于事实核查、数据验证或文本生成中的一致性检查等任务非常有用。 + +自洽提示的提示公式是输入文本后跟着指令“请确保以下文本是自洽的”。 + +或者，可以提示模型生成与提供的输入一致的文本。 + +提示示例及其公式： + +**示例1：文本生成** + +- 任务：生成产品评论 + +- 指令：评论应与输入中提供的产品信息一致 + +- 提示公式：“生成与以下产品信息一致的产品评论[插入产品信息]” + +**示例2：文本摘要** + +- 任务：概括一篇新闻文章 + +- 指令：摘要应与文章中提供的信息一致 + +- 提示公式：“用与提供的信息一致的方式概括以下新闻文章[插入新闻文章]” + +**示例3：文本完成** + +- 任务：完成一个句子 + +- 指令：完成应与输入中提供的上下文一致 + +- 提示公式：“以与提供的上下文一致的方式完成以下句子[插入句子]” + +**示例4：** + +1. **事实核查：** + + 任务：检查给定新闻文章的一致性 + + 输入文本：“文章中陈述该城市的人口为500万，但后来又说该城市的人口为700万。” + + 提示公式：“请确保以下文本是自洽的：文章中陈述该城市的人口为500万，但后来又说该城市的人口为700万。” + +2. **数据验证：** + + 任务：检查给定数据集的一致性 + + 输入文本：“数据显示7月份的平均温度为30度，但最低温度记录为20度。” + + 提示公式：“请确保以下文本是自洽的：数据显示7月份的平均温度为30度，但最低温度记录为20度。” + +
+ +## 第八章：种子词提示 + +种子词提示是一种通过提供特定的种子词或短语来控制ChatGPT输出的技术。种子词提示的提示公式是种子词或短语，后跟指令“请根据以下种子词生成文本”。 + +示例： + +**文本生成：** + +- 任务：编写一篇有关龙的故事 +- 种子词：“龙” +- 提示公式：“请根据以下种子词生成文本：龙” + +**语言翻译：** + +- 任务：将一句话从英语翻译成西班牙语 +- 种子词：“你好” +- 提示公式：“请根据以下种子词生成文本：你好” + +这种技术允许模型生成与种子词相关的文本并对其进行扩展。这是一种控制模型生成文本与某个特定主题或背景相关的方式。 + +种子词提示可以与角色提示和指令提示相结合，以创建更具体和有针对性的生成文本。通过提供种子词或短语，模型可以生成与该种子词或短语相关的文本，并通过提供有关期望输出和角色的信息，模型可以以特定于角色或指令的风格或语气生成文本。这样可以更好地控制生成的文本，并可用于各种应用程序。 + +以下是提示示例及其公式： + +**示例1：文本生成** + +- 任务：编写一首诗 +- 指令：诗应与种子词“爱”相关，并以十四行诗的形式书写。 +- 角色：诗人 +- 提示公式：“作为诗人，根据以下种子词生成与“爱”相关的十四行诗：” + +**示例2：文本完成** + +- 任务：完成一句话 +- 指令：完成应与种子词“科学”相关，并以研究论文的形式书写。 +- 角色：研究员 +- 提示公式：“作为研究员，请在与种子词“科学”相关且以研究论文的形式书写的情况下完成以下句子：[插入句子]” + +**示例3：文本摘要** + +- 任务：摘要一篇新闻文章 +- 指令：摘要应与种子词“政治”相关，并以中立和公正的语气书写。 +- 角色：记者 +- 提示公式：“作为记者，请以中立和公正的语气摘要以下新闻文章，与种子词“政治”相关：[插入新闻文章]” + +
+ +## 第九章：知识生成提示 + +知识生成提示是一种从ChatGPT中引出新的、原创的信息的技术。 + +知识生成提示的公式是“请生成关于X的新的和原创的信息”，其中X是感兴趣的主题。 + +这是一种利用模型预先存在的知识来生成新的信息或回答问题的技术。 + +要将此提示与ChatGPT一起使用，需要将问题或主题作为输入提供给模型，以及指定所生成文本的任务或目标的提示。 + +提示应包括有关所需输出的信息，例如要生成的文本类型以及任何特定的要求或限制。 + +以下是提示示例及其公式： + +**示例1：知识生成** + +- 任务：生成有关特定主题的新信息 +- 说明：生成的信息应准确且与主题相关 +- 提示公式：“生成有关[特定主题]的新的准确信息” + +**示例2：问答** + +- 任务：回答问题 +- 说明：答案应准确且与问题相关 +- 提示公式：“回答以下问题：[插入问题]” + +**示例3：知识整合** + +- 任务：将新信息与现有知识整合 +- 说明：整合应准确且与主题相关 +- 提示公式：“将以下信息与有关[特定主题]的现有知识整合：[插入新信息]” + +**示例4：数据分析** + +- 任务：从给定的数据集中生成有关客户行为的见解 +- 提示公式：“请从这个数据集中生成有关客户行为的新的和原创的信息” + +
+ +## 第十章：知识整合提示 + +这种技术利用模型的现有知识来整合新信息或连接不同的信息片段。 + +这种技术对于将现有知识与新信息相结合，以生成更全面的特定主题的理解非常有用。 + +**如何与ChatGPT一起使用：** + +- 模型应该提供新信息和现有知识作为输入，以及指定生成文本的任务或目标的提示。 +- 提示应包括有关所需输出的信息，例如要生成的文本类型以及任何特定的要求或限制。 + +提示示例及其公式： + +**示例1：知识整合** + +- 任务：将新信息与现有知识整合 +- 说明：整合应准确且与主题相关 +- 提示公式：“将以下信息与关于[具体主题]的现有知识整合：[插入新信息]” + +**示例2：连接信息片段** + +- 任务：连接不同的信息片段 +- 说明：连接应相关且逻辑清晰 +- 提示公式：“以相关且逻辑清晰的方式连接以下信息片段：[插入信息1] [插入信息2]” + +**示例3：更新现有知识** + +- 任务：使用新信息更新现有知识 +- 说明：更新的信息应准确且相关 +- 提示公式：“使用以下信息更新[具体主题]的现有知识：[插入新信息]” + +
+ +## 第十一章：多项选择提示 + +这种技术向模型提供一个问题或任务以及一组预定义的选项作为潜在答案。 + +该技术对于生成仅限于特定选项集的文本非常有用，可用于问答、文本完成和其他任务。模型可以生成仅限于预定义选项的文本。 + +要使用ChatGPT的多项选择提示，需要向模型提供一个问题或任务作为输入，以及一组预定义的选项作为潜在答案。提示还应包括有关所需输出的信息，例如要生成的文本类型以及任何特定要求或限制。 + +提示示例及其公式： + +**示例1：问答** + +- 任务：回答一个多项选择题 +- 说明：答案应该是预定义的选项之一 +- 提示公式：“通过选择以下选项之一回答以下问题：[插入问题] [插入选项1] [插入选项2] [插入选项3]” + +**示例2：文本完成** + +- 任务：使用预定义选项之一完成句子 +- 说明：完成应该是预定义的选项之一 +- 提示公式：“通过选择以下选项之一完成以下句子：[插入句子] [插入选项1] [插入选项2] [插入选项3]” + +**示例3：情感分析** + +- 任务：将文本分类为积极、中立或消极 +- 说明：分类应该是预定义的选项之一 +- 提示公式：“通过选择以下选项之一，将以下文本分类为积极、中立或消极：[插入文本] [积极] [中立] [消极]” + +
+ +## 第十二章：可解释的软提示 + +可解释的软提示是一种技术，可以在提供一定的灵活性的同时控制模型生成的文本。它通过提供一组受控输入和关于所需输出的附加信息来实现。这种技术可以生成更具解释性和可控性的生成文本。 + +提示示例及其公式： + +**示例1：文本生成** + +- 任务：生成一个故事 +- 指令：故事应基于一组给定的角色和特定的主题 +- 提示公式：“基于以下角色生成故事：[插入角色]和主题：[插入主题]” + +**示例2：文本完成** + +- 任务：完成一句话 +- 指令：完成应以特定作者的风格为基础 +- 提示公式：“以[特定作者]的风格完成以下句子：[插入句子]” + +**示例3：语言建模** + +- 任务：以特定风格生成文本 +- 指令：文本应以特定时期的风格为基础 +- 提示公式：“以[特定时期]的风格生成文本：[插入上下文]” + +
+ +## 第十三章：控制生成提示 + +控制生成提示是一种技术，可让模型在生成文本时对输出进行高度控制。 + +这可以通过提供一组特定的输入来实现，例如模板、特定词汇或一组约束条件，这些输入可用于指导生成过程。 + +以下是一些示例和它们的公式： + +**示例1：文本生成** + +- 任务：生成一个故事 +- 说明：该故事应基于特定的模板 +- 提示公式：“根据以下模板生成故事：[插入模板]” + +**示例2：文本补全** + +- 任务：完成一句话 +- 说明：完成应使用特定的词汇 +- 提示公式：“使用以下词汇完成以下句子：[插入词汇]：[插入句子]” + +**示例3：语言建模** + +- 任务：以特定风格生成文本 +- 说明：文本应遵循一组特定的语法规则 +- 提示公式：“生成遵循以下语法规则的文本：[插入规则]：[插入上下文]” + +通过提供一组特定的输入来指导生成过程，控制生成提示使得生成的文本更具可控性和可预测性。 + +
+ +## 第十四章：问答提示 + +问答提示是一种技术，可以让模型生成回答特定问题或任务的文本。通过将问题或任务与可能与问题或任务相关的任何其他信息一起作为输入提供给模型来实现此目的。 + +一些提示示例及其公式如下： + +**示例1：事实问题回答** + +- 任务：回答一个事实性问题 +- 说明：答案应准确且相关 +- 提示公式：“回答以下事实问题：[插入问题]” + +**示例2：定义** + +- 任务：提供一个词的定义 +- 说明：定义应准确 +- 提示公式：“定义以下词汇：[插入单词]” + +**示例3：信息检索** + +- 任务：从特定来源检索信息 +- 说明：检索到的信息应相关 +- 提示公式：“从以下来源检索有关[特定主题]的信息：[插入来源]” 这对于问答和信息检索等任务非常有用。 + +
+ +## 第十五章：概述提示 + +概述提示是一种技术，允许模型在保留其主要思想和信息的同时生成给定文本的较短版本。 + +这可以通过将较长的文本作为输入提供给模型并要求其生成该文本的摘要来实现。 + +这种技术对于文本概述和信息压缩等任务非常有用。 + +**如何在ChatGPT中使用：** + +- 应该向模型提供较长的文本作为输入，并要求其生成该文本的摘要。 +- 提示还应包括有关所需输出的信息，例如摘要的所需长度和任何特定要求或限制。 + +提示示例及其公式： + +**示例1：文章概述** + +- 任务：概述新闻文章 +- 说明：摘要应是文章主要观点的简要概述 +- 提示公式：“用一句简短的话概括以下新闻文章：[插入文章]” + +**示例2：会议记录** + +- 任务：概括会议记录 +- 说明：摘要应突出会议的主要决策和行动 +- 提示公式：“通过列出主要决策和行动来总结以下会议记录：[插入记录]” + +**示例3：书籍摘要** + +- 任务：总结一本书 +- 说明：摘要应是书的主要观点的简要概述 +- 提示公式：“用一段简短的段落总结以下书籍：[插入书名]” + +
+ +## 第十六章：对话提示 + +对话提示是一种技术，允许模型生成模拟两个或更多实体之间对话的文本。通过为模型提供一个上下文和一组角色或实体，以及它们的角色和背景，并要求模型在它们之间生成对话。 + +因此，应为模型提供上下文和一组角色或实体，以及它们的角色和背景。还应向模型提供有关所需输出的信息，例如对话或交谈的类型以及任何特定的要求或限制。 + +提示示例及其公式： + +**示例1：对话生成** + +- 任务：生成两个角色之间的对话 +- 说明：对话应自然且与给定上下文相关 +- 提示公式：“在以下情境中生成以下角色之间的对话[插入角色]” + +**示例2：故事写作** + +- 任务：在故事中生成对话 +- 说明：对话应与故事的角色和事件一致 +- 提示公式：“在以下故事中生成以下角色之间的对话[插入故事]” + +**示例3：聊天机器人开发** + +- 任务：为客服聊天机器人生成对话 +- 说明：对话应专业且提供准确的信息 +- 提示公式：“在客户询问[插入主题]时，为客服聊天机器人生成专业和准确的对话” + +因此，这种技术对于对话生成、故事写作和聊天机器人开发等任务非常有用。 + +
+ +## 第十七章：对抗性提示 + +对抗性提示是一种技术，它允许模型生成抵抗某些类型的攻击或偏见的文本。这种技术可用于训练更为稳健和抵抗某些类型攻击或偏见的模型。 + +要在ChatGPT中使用对抗性提示，需要为模型提供一个提示，该提示旨在使模型难以生成符合期望输出的文本。提示还应包括有关所需输出的信息，例如要生成的文本类型和任何特定要求或约束。 + +提示示例及其公式： + +**示例1：用于文本分类的对抗性提示** + +- 任务：生成被分类为特定标签的文本 +- 说明：生成的文本应难以分类为特定标签 +- 提示公式：“生成难以分类为[插入标签]的文本” + +**示例2：用于情感分析的对抗性提示** + +- 任务：生成难以分类为特定情感的文本 +- 说明：生成的文本应难以分类为特定情感 +- 提示公式：“生成难以分类为具有[插入情感]情感的文本” + +**示例3：用于语言翻译的对抗性提示** + +- 任务：生成难以翻译的文本 +- 说明：生成的文本应难以翻译为目标语言 +- 提示公式：“生成难以翻译为[插入目标语言]的文本” + +
+ +## 第十八章：聚类提示 + +聚类提示是一种技术，它可以让模型根据某些特征或特点将相似的数据点分组在一起。 + +通过提供一组数据点并要求模型根据某些特征或特点将它们分组成簇，可以实现这一目标。 + +这种技术在数据分析、机器学习和自然语言处理等任务中非常有用。 + +**如何在ChatGPT中使用：** + +应该向模型提供一组数据点，并要求它根据某些特征或特点将它们分组成簇。提示还应包括有关所需输出的信息，例如要生成的簇数和任何特定的要求或约束。 + +提示示例及其公式： + +**示例1：客户评论的聚类** + +- 任务：将相似的客户评论分组在一起 +- 说明：应根据情感将评论分组 +- 提示公式：“将以下客户评论根据情感分组成簇：[插入评论]” + +**示例2：新闻文章的聚类** + +- 任务：将相似的新闻文章分组在一起 +- 说明：应根据主题将文章分组 +- 提示公式：“将以下新闻文章根据主题分组成簇：[插入文章]” + +**示例3：科学论文的聚类** + +- 任务：将相似的科学论文分组在一起 +- 说明：应根据研究领域将论文分组 +- 提示公式：“将以下科学论文根据研究领域分组成簇：[插入论文]” + +
+ +## 第十九章：强化学习提示 + +强化学习提示是一种技术，可以使模型从过去的行动中学习，并随着时间的推移提高其性能。要在ChatGPT中使用强化学习提示，需要为模型提供一组输入和奖励，并允许其根据接收到的奖励调整其行为。提示还应包括有关期望输出的信息，例如要完成的任务以及任何特定要求或限制。这种技术对于决策制定、游戏玩法和自然语言生成等任务非常有用。 + +提示示例及其公式： + +**示例1：用于文本生成的强化学习** + +- 任务：生成与特定风格一致的文本 +- 说明：模型应根据为生成与特定风格一致的文本而接收到的奖励来调整其行为 +- 提示公式：“使用强化学习来生成与以下风格一致的文本[插入风格]” + +**示例2：用于语言翻译的强化学习** + +- 任务：将文本从一种语言翻译成另一种语言 +- 说明：模型应根据为生成准确翻译而接收到的奖励来调整其行为 +- 提示公式：“使用强化学习将以下文本[插入文本]从[插入语言]翻译成[插入语言]” + +**示例3：用于问答的强化学习** + +- 任务：回答问题 +- 说明：模型应根据为生成准确答案而接收到的奖励来调整其行为 +- 提示公式：“使用强化学习来回答以下问题[插入问题]” + +
+ +## 第二十章：课程学习提示 + +课程学习是一种技术，允许模型通过先训练简单任务，逐渐增加难度来学习复杂任务。 + +要在ChatGPT中使用课程学习提示，模型应该提供一系列任务，这些任务逐渐增加难度。 + +提示还应包括有关期望输出的信息，例如要完成的最终任务以及任何特定要求或约束条件。 + +此技术对自然语言处理、图像识别和机器学习等任务非常有用。 + +提示示例及其公式： + +**示例1：用于文本生成的课程学习** + +- 任务：生成与特定风格一致的文本 +- 说明：模型应该在移动到更复杂的风格之前先在简单的风格上进行训练。 +- 提示公式：“使用课程学习来生成与以下风格[插入风格]一致的文本，按照以下顺序[插入顺序]。” + +**示例2：用于语言翻译的课程学习** + +- 任务：将文本从一种语言翻译成另一种语言 +- 说明：模型应该在移动到更复杂的语言之前先在简单的语言上进行训练。 +- 提示公式：“使用课程学习将以下语言[插入语言]的文本翻译成以下顺序[插入顺序]。” + +**示例3：用于问题回答的课程学习** + +- 任务：回答问题 +- 说明：模型应该在移动到更复杂的问题之前先在简单的问题上进行训练。 +- 提示公式：“使用课程学习来回答以下问题[插入问题]，按照以下顺序[插入顺序]生成答案。” + +
+ +## 第二十一章：情感分析提示 + +情感分析是一种技术，允许模型确定文本的情绪色彩或态度，例如它是积极的、消极的还是中立的。 + +要在ChatGPT中使用情感分析提示，模型应该提供一段文本并要求根据其情感分类。 + +提示还应包括关于所需输出的信息，例如要检测的情感类型（例如积极的、消极的、中立的）和任何特定要求或约束条件。 + +提示示例及其公式： + +**示例1：客户评论的情感分析** + +- 任务：确定客户评论的情感 +- 说明：模型应该将评论分类为积极的、消极的或中立的 +- 提示公式：“对以下客户评论进行情感分析[插入评论]，并将它们分类为积极的、消极的或中立的。” + +**示例2：推文的情感分析** + +- 任务：确定推文的情感 +- 说明：模型应该将推文分类为积极的、消极的或中立的 +- 提示公式：“对以下推文进行情感分析[插入推文]，并将它们分类为积极的、消极的或中立的。” + +**示例3：产品评论的情感分析** + +- 任务：确定产品评论的情感 +- 说明：模型应该将评论分类为积极的、消极的或中立的 +- 提示公式：“对以下产品评论进行情感分析[插入评论]，并将它们分类为积极的、消极的或中立的。” + +这种技术对自然语言处理、客户服务和市场研究等任务非常有用。 + +
+ +## 第二十二章：命名实体识别提示 + +命名实体识别（NER）是一种技术，它可以使模型识别和分类文本中的命名实体，例如人名、组织机构、地点和日期等。 + +要在ChatGPT中使用命名实体识别提示，需要向模型提供一段文本，并要求它识别和分类文本中的命名实体。 + +提示还应包括有关所需输出的信息，例如要识别的命名实体类型（例如人名、组织机构、地点、日期）以及任何特定要求或约束条件。 + +提示示例及其公式： + +**示例1：新闻文章中的命名实体识别** + +- 任务：在新闻文章中识别和分类命名实体 +- 说明：模型应识别和分类人名、组织机构、地点和日期 +- 提示公式：“在以下新闻文章[插入文章]上执行命名实体识别，并识别和分类人名、组织机构、地点和日期。” + +**示例2：法律文件中的命名实体识别** + +- 任务：在法律文件中识别和分类命名实体 +- 说明：模型应识别和分类人名、组织机构、地点和日期 +- 提示公式：“在以下法律文件[插入文件]上执行命名实体识别，并识别和分类人名、组织机构、地点和日期。” + +**示例3：研究论文中的命名实体识别** + +- 任务：在研究论文中识别和分类命名实体 +- 说明：模型应识别和分类人名、组织机构、地点和日期 +- 提示公式：“在以下研究论文[插入论文]上执行命名实体识别，并识别和分类人名、组织机构、地点和日期。” + +
+ +## 第二十三章：文本分类提示 + +文本分类是一种技术，它可以让模型将文本分成不同的类别。这种技术对于自然语言处理、文本分析和情感分析等任务非常有用。 + +需要注意的是，文本分类和情感分析是不同的。情感分析特别关注于确定文本中表达的情感或情绪。这可能包括确定文本表达了积极、消极还是中性的情感。情感分析通常用于客户评论、社交媒体帖子和其他需要表达情感的文本。 + +要在ChatGPT中使用文本分类提示，模型需要提供一段文本，并要求它根据预定义的类别或标签进行分类。提示还应包括有关所需输出的信息，例如类别或标签的数量以及任何特定的要求或约束。 + +提示示例及其公式： + +**示例1：对客户评论进行文本分类** + +- 任务：将客户评论分类为不同的类别，例如电子产品、服装和家具 +- 说明：模型应根据评论的内容对其进行分类 +- 提示公式：“对以下客户评论 [插入评论] 进行文本分类，并根据其内容将其分类为不同的类别，例如电子产品、服装和家具。” + +**示例2：对新闻文章进行文本分类** + +- 任务：将新闻文章分类为不同的类别，例如体育、政治和娱乐 +- 说明：模型应根据文章的内容对其进行分类 +- 提示公式：“对以下新闻文章 [插入文章] 进行文本分类，并根据其内容将其分类为不同的类别，例如体育、政治和娱乐。” + +**示例3：对电子邮件进行文本分类** + +- 任务：将电子邮件分类为不同的类别，例如垃圾邮件、重要邮件或紧急邮件 +- 说明：模型应根据电子邮件的内容和发件人对其进行分类 +- 提示公式：“对以下电子邮件 [插入电子邮件] 进行文本分类，并根据其内容和发件人将其分类为不同的类别，例如垃圾邮件、重要邮件或紧急邮件。” + +
+ +## 第二十四章：文本生成提示 + +文本生成提示与本书中提到的其他提示技术相关，例如：零、一、几次提示，受控生成提示，翻译提示，语言建模提示，句子补全提示等。这些提示都与生成文本有关，但它们在生成文本的方式和放置在生成文本上的特定要求或限制方面有所不同。文本生成提示可用于微调预训练模型或训练新模型以执行特定任务。 + +提示示例及其公式： + +**示例1：故事创作的文本生成** + +- 任务：根据给定的提示生成故事 +- 说明：故事应至少包含1000个单词，并包括一组特定的角色和情节。 +- 提示公式：“根据以下提示[插入提示]生成一个至少包含1000个单词，包括角色[插入角色]和情节[插入情节]的故事。” + +**示例2：语言翻译的文本生成** + +- 任务：将给定的文本翻译成另一种语言 +- 说明：翻译应准确并符合习惯用语。 +- 提示公式：“将以下文本[插入文本]翻译成[插入目标语言]，并确保其准确且符合习惯用语。” + +**示例3：文本完成的文本生成** + +- 任务：完成给定的文本 +- 说明：生成的文本应与输入文本连贯一致。 +- 提示公式：“完成以下文本[插入文本]，并确保其连贯一致且符合输入文本。” + +
+ +## 结语 + +正如本书中所探讨的那样，快速工程是一种利用像ChatGPT这样的语言模型获得高质量答案的强大工具。通过精心设计各种技巧的提示，我们可以引导模型生成符合我们特定需求和要求的文本。 + +在第二章中，我们讨论了如何使用指令提示向模型提供清晰明确的指导。在第三章中，我们探讨了如何使用角色提示生成特定的语音或风格的文本。在第四章中，我们研究了如何使用标准提示作为微调模型性能的起点。我们还研究了几种高级提示技术，例如Zero、One和Few Shot Prompting、Self-Consistency、Seed-word Prompt、Knowledge Generation Prompt、Knowledge Integration prompts、Multiple Choice prompts、Interpretable Soft Prompts、Controlled generation prompts、Question-answering prompts、Summarization prompts、Dialogue prompts、Adversarial prompts、Clustering prompts、Reinforcement learning prompts、Curriculum learning prompts、Sentiment analysis prompts、Named entity recognition prompts和Text classification prompts（对应章节的名字）。 + +这些技术中的每一种都可以以不同的方式使用，以实现各种不同的结果。随着您继续使用ChatGPT和其他语言模型，值得尝试不同的技巧组合，以找到最适合您特定用例的方法。 + +最后，您可以查看我写的其他主题的书籍。 + +感谢您阅读整本书。期待在我的其他书中与您见面。 + +(本文翻译自《The Art of Asking ChatGPT for High-Quality Answers A Complete Guide to Prompt Engineering Techniques》这本书，本文的翻译全部由ChatGpt完成，我只是把翻译内容给稍微排版了一下。做完了才发现这个工作早就有人做过了...下面是我以此事件让New Bing编写的一个小故事，希望大家喜欢) + +> 他终于画完了他的画，心满意足地把它挂在了墙上。他觉得这是他一生中最伟大的作品，无人能及。他邀请了所有的朋友来欣赏，期待着他们的赞美和惊叹。 可是当他们看到画时，却没有一个人说话。他们只是互相对视，然后低头咳嗽，或者假装看手机。他感到很奇怪，难道他们都不懂艺术吗？难道他们都没有眼光吗？ 他忍不住问其中一个朋友：“你觉得我的画怎么样？” 朋友犹豫了一下，说：“嗯……其实……这个画……我以前在哪里见过。” “见过？你在哪里见过？”他惊讶地问。 “就在……就在那边啊。”朋友指了指墙角的一个小框架，“那不就是你上个月买回来的那幅名画吗？你怎么把它照抄了一遍？ ——New Bing + +[这就是那幅名画]: http://yesaiwen.com/art-of-asking-chatgpt-for-high-quality-answ-engineering-techniques/#i-3 "《如何向ChatGPT提问并获得高质量的答案》" \ No newline at end of file diff --git a/samples/isssues_merge/langchain-ChatGLM_closed.xlsx b/samples/isssues_merge/langchain-ChatGLM_closed.xlsx new file mode 100644 index 0000000000000000000000000000000000000000..24119cc849e97ddbae577dfb1cc595fb1b512d37 Binary files /dev/null and b/samples/isssues_merge/langchain-ChatGLM_closed.xlsx differ diff --git a/samples/isssues_merge/langchain-ChatGLM_open.jsonl b/samples/isssues_merge/langchain-ChatGLM_open.jsonl new file mode 100644 index 0000000000000000000000000000000000000000..a34816e89f4e085b305df0bcf99971a5dcbd490f --- /dev/null +++ b/samples/isssues_merge/langchain-ChatGLM_open.jsonl @@ -0,0 +1,323 @@ +{"title": "效果如何优化", "file": "2023-04-04.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/14", "detail": "如图所示，将该项目的README.md和该项目结合后，回答效果并不理想，请问可以从哪些方面进行优化", "id": 0} +{"title": "怎么让模型严格根据检索的数据进行回答，减少胡说八道的回答呢", "file": "2023-04-04.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/15", "detail": "举个例子：", "id": 1} +{"title": "When I try to run the `python knowledge_based_chatglm.py`, I got this error in macOS(M1 Max, OS 13.2)", "file": "2023-04-07.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/32", "detail": "```python", "id": 2} +{"title": "萌新求教大佬怎么改成AMD显卡或者CPU？", "file": "2023-04-10.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/48", "detail": "把.cuda()去掉就行", "id": 3} +{"title": "输出answer的时间很长，是否可以把文本向量化的部分提前做好存储起来？", "file": "2023-04-10.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/50", "detail": "GPU：4090 24G显存", "id": 4} +{"title": "报错Use `repo_type` argument if needed.", "file": "2023-04-11.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/57", "detail": "Traceback (most recent call last):", "id": 5} +{"title": "无法打开gradio的页面", "file": "2023-04-11.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/58", "detail": "$ python webui.py", "id": 6} +{"title": "支持word，那word里面的图片正常显示吗？", "file": "2023-04-12.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/60", "detail": "如题，刚刚从隔壁转过来的，想先了解下", "id": 7} +{"title": "detectron2 is not installed. Cannot use the hi_res partitioning strategy. Falling back to partitioning with the fast strategy.", "file": "2023-04-12.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/63", "detail": "能够正常的跑起来，在加载content文件夹中的文件时，每加载一个文件都会提示：", "id": 8} +{"title": "cpu上运行webui，step3 asking时报错", "file": "2023-04-12.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/66", "detail": "web运行，文件加载都正常，asking时报错", "id": 9} +{"title": "建议弄一个插件系统", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/67", "detail": "如题弄成stable-diffusion-webui那种能装插件，再开一个存储库给使用者或插件开发，存储或下载插件。", "id": 10} +{"title": "请教加载模型出错！？", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/75", "detail": "AttributeError: module 'transformers_modules.chatglm-6b.configuration_chatglm' has no attribute 'ChatGLMConfig 怎么解决呀", "id": 11} +{"title": "从本地知识检索内容的时候，是否可以设置相似度阈值，小于这个阈值的内容不返回，即使会小于设置的VECTOR_SEARCH_TOP_K参数呢？谢谢大佬", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/76", "detail": "比如 问一些 你好/你是谁 等一些跟本地知识库无关的问题", "id": 12} +{"title": "如何改成多卡推理？", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/77", "detail": "+1", "id": 13} +{"title": "能否弄个懒人包，可以一键体验？", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/78", "detail": "能否弄个懒人包，可以一键体验？", "id": 14} +{"title": "连续问问题会导致崩溃", "file": "2023-04-13.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/79", "detail": "看上去不是爆内存的问题，连续问问题后，会出现如下报错", "id": 15} +{"title": "AttributeError: 'NoneType' object has no attribute 'as_retriever'", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/86", "detail": "环境：windows 11, anaconda/python 3.8", "id": 16} +{"title": "FileNotFoundError: Could not find module 'nvcuda.dll' (or one of its dependencies). Try using the full path with constructor syntax.", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/87", "detail": "请检查一下cuda或cudnn是否存在安装问题", "id": 17} +{"title": "加载txt文件失败？", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/89", "detail": "![JppHrGOWFa](https://user-images.githubusercontent.com/109277248/232009383-bf7c46d1-a01e-4e0a-9de6-5b5ed3e36158.jpg)", "id": 18} +{"title": "NameError: name 'chatglm' is not defined", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/90", "detail": "This share link expires in 72 hours. For free permanent hosting and GPU upgrades (NEW!), check out Spaces: https://huggingface.co/spaces", "id": 19} +{"title": "打不开地址？", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/91", "detail": "报错数据如下：", "id": 20} +{"title": "加载md文件出错", "file": "2023-04-14.00", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/98", "detail": "运行 webui.py后能访问页面，上传一个md文件后，日志中有错误。等待后能加载完成，提示可以提问了，但提问没反应，日志中有错误。 具体日志如下。", "id": 21} +{"title": "建议增加获取在线知识的能力", "file": "2023-04-15.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/101", "detail": "建议增加获取在线知识的能力", "id": 22} +{"title": "txt 未能成功加载", "file": "2023-04-15.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/103", "detail": "hinese. Creating a new one with MEAN pooling.", "id": 23} +{"title": "pdf加载失败", "file": "2023-04-15.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/105", "detail": "e:\\a.txt加载成功了，e:\\a.pdf加载就失败，pdf文件里面前面几页是图片，后面都是文字，加载失败没有报更多错误，请问该怎么排查？", "id": 24} +{"title": "一直停在文本加载处", "file": "2023-04-15.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/108", "detail": "一直停在文本加载处", "id": 25} +{"title": " File \"/root/.cache/huggingface/modules/transformers_modules/chatglm-6b/modeling_chatglm.py\", line 440, in forward new_tensor_shape = mixed_raw_layer.size()[:-1] + ( TypeError: torch.Size() takes an iterable of 'int' (item 2 is 'float')", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/113", "detail": "按照最新的代码，发现", "id": 26} +{"title": "后续会提供前后端分离的功能吗？", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/114", "detail": "类似这种https://github.com/lm-sys/FastChat/tree/main/fastchat/serve", "id": 27} +{"title": "安装依赖报错", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/115", "detail": "(test) C:\\Users\\linh\\Desktop\\langchain-ChatGLM-master>pip install -r requirements.txt", "id": 28} +{"title": "问特定问题会出现爆显存", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/117", "detail": "正常提问没问题。", "id": 29} +{"title": "Expecting value: line 1 column 1 (char 0)", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/118", "detail": "运行后 第一步加载配置一直报错：", "id": 30} +{"title": "embedding https://huggingface.co/GanymedeNil/text2vec-large-chinese/tree/main是免费的，效果比对openai的如何？", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/119", "detail": "-------------------------------------------------------------------------------", "id": 31} +{"title": "这是什么错误，在Colab上运行的。", "file": "2023-04-17.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/120", "detail": "libcuda.so.1: cannot open shared object file: No such file or directory", "id": 32} +{"title": "只想用自己的lora微调后的模型进行对话，不想加载任何本地文档，该如何调整？", "file": "2023-04-18.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/121", "detail": "能出一个单独的教程吗", "id": 33} +{"title": "租的gpu,Running on local URL: http://0.0.0.0:7860 To create a public link, set `share=True` in `launch()`. 浏览器上访问不了？？？", "file": "2023-04-18.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/122", "detail": "(chatglm20230401) root@autodl-container-e82d11963c-10ece0d7:~/autodl-tmp/chatglm/langchain-ChatGLM-20230418# python3.9 webui.py", "id": 34} +{"title": "本地部署中的报错请教", "file": "2023-04-18.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/124", "detail": "您好，在本地运行langchain-ChatGLM过程中，环境及依赖的包都已经满足条件，但是运行webui.py,报错如下（运行cli_demo.py报错类似），请问是哪里出了错呢？盼望您的回复，谢谢！", "id": 35} +{"title": "报错。The dtype of attention mask (torch.int64) is not bool", "file": "2023-04-18.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/131", "detail": "The dtype of attention mask (torch.int64) is not bool", "id": 36} +{"title": "[求助] pip install -r requirements.txt 的时候出现以下报错。。。有大佬帮忙看看怎么搞么，下的release里面的包", "file": "2023-04-18.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/134", "detail": "$ pip install -r requirements.txt", "id": 37} +{"title": "如何提升根据问题搜索到对应知识的准确率", "file": "2023-04-19.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/136", "detail": "外链知识库最大的问题在于问题是短文本，知识是中长文本。如何根据问题精准的搜索到对应的知识是个最大的问题。这类本地化项目不像百度，由无数的网页，基本上每个问题都可以找到对应的页面。", "id": 38} +{"title": "是否可以增加向量召回的阈值设定，有些召回内容相关性太低，导致模型胡言乱语", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/140", "detail": "如题", "id": 39} +{"title": "输入长度问题", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/141", "detail": "感谢作者支持ptuning微调模型。", "id": 40} +{"title": "已有部署好的chatGLM-6b，如何通过接口接入？", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/144", "detail": "已有部署好的chatGLM-6b，如何通过接口接入，而不是重新加载一个模型；", "id": 41} +{"title": "执行web_demo.py后，显示Killed，就退出了，是不是配置不足呢？", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/146", "detail": "![图片](https://user-images.githubusercontent.com/26102866/233256425-c7aab999-11d7-4de9-867b-23ef18d519e4.png)", "id": 42} +{"title": "执行python cli_demo1.py", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/147", "detail": "Traceback (most recent call last):", "id": 43} +{"title": "报错：ImportError: cannot import name 'GENERATION_CONFIG_NAME' from 'transformers.utils'", "file": "2023-04-20.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/149", "detail": "(mychatGLM) PS D:\\Users\\admin3\\zrh\\langchain-ChatGLM> python cli_demo.py", "id": 44} +{"title": "上传文件并加载知识库时，会不停地出现临时文件", "file": "2023-04-21.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/153", "detail": "环境：ubuntu 18.04", "id": 45} +{"title": "向知识库中添加文件后点击”上传文件并加载知识库“后Segmentation fault报错。", "file": "2023-04-23.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/161", "detail": "运行服务后的提示如下：", "id": 46} +{"title": "langchain-serve 集成", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/162", "detail": "Hey 我是来自 [langchain-serve](https://github.com/jina-ai/langchain-serve) 的dev！", "id": 47} +{"title": "大佬们，wsl的ubuntu怎么配置用cuda加速，装了运行后发现是cpu在跑", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/164", "detail": "大佬们，wsl的ubuntu怎么配置用cuda加速，装了运行后发现是cpu在跑", "id": 48} +{"title": "在github codespaces docker运行出错", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/165", "detail": "docker run -d --restart=always --name chatglm -p 7860:7860 -v /www/wwwroot/code/langchain-ChatGLM:/chatGLM chatglm", "id": 49} +{"title": "有计划接入Moss模型嘛", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/166", "detail": "后续会开展测试，目前主要在优化langchain部分效果，如果有兴趣也欢迎提PR", "id": 50} +{"title": "怎么实现 API 部署？", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/168", "detail": "利用 fastapi 实现 API 部署方式，具体怎么实现，有方法说明吗？", "id": 51} +{"title": " 'NoneType' object has no attribute 'message_types_by_name'报错", "file": "2023-04-24.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/169", "detail": "_HISTOGRAMPROTO = DESCRIPTOR.message_types_by_name['HistogramProto']", "id": 52} +{"title": "能否指定自己训练的text2vector模型？", "file": "2023-04-25.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/172", "detail": "请问大佬：", "id": 53} +{"title": "关于项目支持的模型以及quantization_bit潜在的影响的问题", "file": "2023-04-26.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/176", "detail": "作者您好～", "id": 54} +{"title": "运行python3.9 api.py WARNING: You must pass the application as an import string to enable 'reload' or 'workers'.", "file": "2023-04-26.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/179", "detail": "api.py文件最下面改成这样试试：", "id": 55} +{"title": "ValidationError: 1 validation error for HuggingFaceEmbeddings model_kwargs extra fields not permitted (type=value_error.extra)", "file": "2023-04-26.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/180", "detail": "ValidationError: 1 validation error for HuggingFaceEmbeddings", "id": 56} +{"title": "如果没有检索到相关性比较高的，回答“我不知道”", "file": "2023-04-26.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/181", "detail": "如果通过设计system_template，让模型在搜索到的文档都不太相关的情况下回答“我不知道”", "id": 57} +{"title": "请问如果不能联网，6B之类的文件从本地上传需要放到哪里", "file": "2023-04-26.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/182", "detail": "感谢大佬的项目，很有启发~", "id": 58} +{"title": "知识库问答--输入新的知识库名称是中文的话，会报error", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/184", "detail": "知识库问答--输入新的知识库名称是中文的话，会报error，选择要加载的知识库那里也不显示之前添加的知识库", "id": 59} +{"title": "现在能通过问题匹配的相似度值，来直接返回文档中的文段，而不经过模型吗？因为有些答案在文档中，模型自己回答，不能回答文档中的答案", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/186", "detail": "现在能通过问题匹配的相似度值，来直接返回文档中的文段，而不经过模型吗？因为有些答案在文档中，模型自己回答，不能回答文档中的答案。也就是说，提供向量检索回答+模型回答相结合的策略。如果相似度值高于一定数值，直接返回文档中的文本，没有高于就返回模型的回答或者不知道", "id": 60} +{"title": "TypeError: The type of ChatGLM.callback_manager differs from the new default value; if you wish to change the type of this field, please use a type annotation", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/188", "detail": "Mac 运行 python3 ./webui.py 报 TypeError: The type of ChatGLM.callback_manager differs from the new default value; if you wish to change the type of this field, please use a type annotation", "id": 61} +{"title": "Not Enough Memory", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/190", "detail": "运行命令行程序python cli_demo.py， 已经成功加载pdf文件, 报“DefaultCPUAllocator: not enough memory: you tried to allocate 458288380900 bytes”错误，请问哪里可以配置default memory", "id": 62} +{"title": "参与开发问题", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/191", "detail": "1.是否需要进专门的开发群", "id": 63} +{"title": "对话框中代码片段格式需改进", "file": "2023-04-27.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/192", "detail": "最好能改进下输出代码片段的格式，目前输出的格式还不友好。", "id": 64} +{"title": "请问未来有可能支持belle吗", "file": "2023-04-28.01", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/195", "detail": "如题，谢谢大佬", "id": 65} +{"title": "TypeError: cannot unpack non-iterable NoneType object", "file": "2023-04-28.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/200", "detail": "When i tried to change the knowledge vector store through `init_knowledge_vector_store`, the error `TypeError: cannot unpack non-iterable NoneType object` came out.", "id": 66} +{"title": "生成结果", "file": "2023-04-28.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/202", "detail": "你好，想问一下langchain+chatglm-6B，找到相似匹配的prompt，是直接返回prompt对应的答案信息，还是chatglm-6B在此基础上自己优化答案？", "id": 67} +{"title": "在win、ubuntu下都出现这个错误：attributeerror: 't5forconditionalgeneration' object has no attribute 'stream_chat'", "file": "2023-04-29.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/207", "detail": "在win、ubuntu。下载完模型后，没办法修改代码以执行本地模型，每次都要重新输入路径； LLM 模型、Embedding 模型支持也都在官网下的，在其他项目（wenda）下可以使用", "id": 68} +{"title": "[FEATURE] knowledge_based_chatglm.py: renamed or missing?", "file": "2023-04-30.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/210", "detail": "Not found. Was it renamed? Or, is it missing? How can I get it?", "id": 69} +{"title": "sudo apt-get install -y nvidia-container-toolkit-base执行报错", "file": "2023-05-01.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/211", "detail": "**问题描述 / Problem Description**", "id": 70} +{"title": "效果不佳几乎答不上来", "file": "2023-05-01.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/212", "detail": "提供了50条问答的docx文件", "id": 71} +{"title": "有没有可能新增一个基于chatglm api调用的方式构建langchain", "file": "2023-05-02.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/218", "detail": "我有两台8G GPU/40G内存的服务器，一个台做成了chatglm的api ；想基于另外一台服务器部署langchain；网上好像没有类似的代码。", "id": 72} +{"title": "电脑是intel的集成显卡； 运行时告知我找不到nvcuda.dll，模型无法运行", "file": "2023-05-02.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/219", "detail": "您好，我的电脑是intel的集成显卡，不过CPU是i5-11400 @ 2.60GHz ，内存64G；", "id": 73} +{"title": "根据langchain官方的文档和使用模式，是否可以改Faiss为Elasticsearch？会需要做哪些额外调整？求解", "file": "2023-05-03.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/221", "detail": "本人新手小白，由于业务模式的原因（有一些自己的场景和优化），希望利用Elasticsearch做这个体系内部的检索机制，不知道是否可以替换，同时，还会涉及到哪些地方的改动？或者说可能会有哪些其他影响，希望作者和大佬们不吝赐教！", "id": 74} +{"title": "请问未来有可能支持t5吗", "file": "2023-05-04.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/224", "detail": "请问可能支持基於t5的模型吗?", "id": 75} +{"title": "[BUG] 内存溢出 / torch.cuda.OutOfMemoryError:", "file": "2023-05-04.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/229", "detail": "**问题描述 / Problem Description**", "id": 76} +{"title": "报错 No module named 'chatglm_llm'", "file": "2023-05-04.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/230", "detail": "明明已经安装了包，却在python里吊不出来", "id": 77} +{"title": "能出一个api部署的描述文档吗", "file": "2023-05-04.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/233", "detail": "**功能描述 / Feature Description**", "id": 78} +{"title": "使用docs/API.md 出错", "file": "2023-05-04.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/234", "detail": "使用API.md文档2种方法，出错", "id": 79} +{"title": "加载pdf文档报错？", "file": "2023-05-05.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/238", "detail": "ew one with MEAN pooling.", "id": 80} +{"title": "上传的本地知识文件后再次上传不能显示，只显示成功了一个，别的上传成功后再次刷新就没了", "file": "2023-05-05.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/239", "detail": "您好，项目有很大启发，感谢~", "id": 81} +{"title": "创建了新的虚拟环境，安装了相关包，并且自动下载了相关的模型，但是仍旧出现：OSError: Unable to load weights from pytorch checkpoint file for", "file": "2023-05-05.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/240", "detail": "![78ac8e663fdc312d0e9d78da95925c4](https://user-images.githubusercontent.com/34124260/236378728-9ea4424f-0f7f-4013-9d33-820b723de321.png)", "id": 82} +{"title": "[BUG] 数据加载不进来", "file": "2023-05-05.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/243", "detail": "使用的.txt格式，utf-8编码，报以下错误", "id": 83} +{"title": "不能读取pdf", "file": "2023-05-05.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/244", "detail": "请问是webui还是cli_demo", "id": 84} +{"title": "本地txt文件有500M，加载的时候很慢，如何提高速度？", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/251", "detail": "![yayRzxSYHP](https://user-images.githubusercontent.com/109277248/236592902-f5ab338d-c1e9-43dc-ae16-9df2cd3c1378.jpg)", "id": 85} +{"title": "[BUG] gradio上传知识库后刷新之后 知识库就不见了 只有重启才能看到之前的上传的知识库", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/253", "detail": "gradio上传知识库后刷新之后 知识库就不见了 只有重启才能看到之前的上传的知识库", "id": 86} +{"title": "[FEATURE] 可以支持 OpenAI 的模型嘛？比如 GPT-3、GPT-3.5、GPT-4；embedding 增加 text-embedding-ada-002", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/254", "detail": "**功能描述 / Feature Description**", "id": 87} +{"title": "[FEATURE] 能否增加对于milvus向量数据库的支持 / Concise description of the feature", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/256", "detail": "**功能描述 / Feature Description**", "id": 88} +{"title": "CPU和GPU上跑，除了速度有区别，准确率效果回答上有区别吗？", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/259", "detail": "理论上没有区别", "id": 89} +{"title": "m1，请问在生成回答时怎么看是否使用了mps or cpu？", "file": "2023-05-06.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/260", "detail": "m1，请问在生成回答时怎么看是否使用了mps or cpu？", "id": 90} +{"title": "知识库一刷新就没了", "file": "2023-05-07.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/263", "detail": "知识库上传后刷新就没了", "id": 91} +{"title": "本地部署报没有模型", "file": "2023-05-07.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/267", "detail": "建议在下载llm和embedding模型至本地后在configs/model_config中写入模型本地存储路径后再运行", "id": 92} +{"title": "[BUG] python3: can't open file 'webui.py': [Errno 2] No such file or directory", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/269", "detail": "**问题描述 / Problem Description**", "id": 93} +{"title": "模块缺失提示", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/271", "detail": "因为已有自己使用的docker环境，直接启动webui.py，提示", "id": 94} +{"title": "运行api.py后，执行curl -X POST \"http://127.0.0.1:7861\" 报错？", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/272", "detail": "执行curl -X POST \"http://127.0.0.1:7861\" \\ -H 'Content-Type: application/json' \\ -d '{\"prompt\": \"你好\", \"history\": []}'，报错怎么解决", "id": 95} +{"title": "[BUG] colab安装requirements提示protobuf版本问题？", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/273", "detail": "pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.", "id": 96} +{"title": "请问项目里面向量相似度使用了什么方法计算呀？", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/275", "detail": "基本按照langchain里的FAISS.similarity_search_with_score_by_vector实现", "id": 97} +{"title": "[BUG] 安装detectron2后，pdf无法加载", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/276", "detail": "**问题描述 / Problem Description**", "id": 98} +{"title": "[BUG] 使用ChatYuan-V2模型无法流式输出，会报错", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/277", "detail": "一方面好像是ChatYuan本身不支持stream_chat，有人在clueai那边提了issue他们说还没开发，所以估计这个attribute调不起来；但是另一方面看报错好像是T5模型本身就不是decoder-only模型，所以不能流式输出吧（个人理解）", "id": 99} +{"title": "[BUG] 无法加载text2vec模型", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/278", "detail": "**问题描述 / Problem Description**", "id": 100} +{"title": "请问能否增加网络搜索功能", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/281", "detail": "请问能否增加网络搜索功能", "id": 101} +{"title": "[FEATURE] 结构化数据sql、excel、csv啥时会支持呐。", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/283", "detail": "**功能描述 / Feature Description**", "id": 102} +{"title": "TypeError: ChatGLM._call() got an unexpected keyword argument 'stop'", "file": "2023-05-08.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/284", "detail": "No sentence-transformers model found with name D:\\DevProject\\langchain-ChatGLM\\GanymedeNil\\text2vec-large-chinese. Creating a new one with MEAN pooling.", "id": 103} +{"title": "关于api.py的一些bug和设计逻辑问题？", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/285", "detail": "首先冒昧的问一下，这个api.py，开发者大佬们是在自己电脑上测试后确实没问题吗？", "id": 104} +{"title": "有没有租用的算力平台上，运行api.py后，浏览器http://localhost:7861/报错", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/287", "detail": "是不是租用的gpu平台上都会出现这个问题？？？", "id": 105} +{"title": "请问一下项目中有用到文档段落切割方法吗？", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/288", "detail": "text_load中的文档切割方法用上了吗？在代码中看好像没有用到？", "id": 106} +{"title": "报错 raise ValueError(f\"Knowledge base {knowledge_base_id} not found\") ValueError: Knowledge base ./vector_store not found", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/289", "detail": "File \"/root/autodl-tmp/chatglm/langchain-ChatGLM-master/api.py\", line 183, in chat", "id": 107} +{"title": "能接入vicuna模型吗", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/290", "detail": "目前本地已经有了vicuna模型能直接接入吗？", "id": 108} +{"title": "[BUG] 提问公式相关问题大概率爆显存", "file": "2023-05-09.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/291", "detail": "**问题描述 / Problem Description**", "id": 109} +{"title": "安装pycocotools失败，找了好多方法都不能解决。", "file": "2023-05-10.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/292", "detail": "**问题描述 / Problem Description**", "id": 110} +{"title": "使用requirements安装，PyTorch安装的是CPU版本", "file": "2023-05-10.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/294", "detail": "如题目，使用requirements安装，PyTorch安装的是CPU版本，运行程序的时候，也是使用CPU在工作。", "id": 111} +{"title": "能不能给一个毛坯服务器的部署教程", "file": "2023-05-10.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/298", "detail": "“开发部署”你当成服务器的部署教程用就行了。", "id": 112} +{"title": " Error(s) in loading state_dict for ChatGLMForConditionalGeneration:", "file": "2023-05-10.02", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/299", "detail": "运行中出现的问题，7860的端口页面显示不出来，求助。", "id": 113} +{"title": "ChatYuan-large-v2模型加载失败", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/300", "detail": "**实际结果 / Actual Result**", "id": 114} +{"title": "新增摘要功能", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/303", "detail": "你好，后续会考虑新增对长文本信息进行推理和语音理解功能吗？比如生成摘要", "id": 115} +{"title": "[BUG] pip install -r requirements.txt 出错", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/304", "detail": "pip install langchain -i https://pypi.org/simple", "id": 116} +{"title": "[BUG] 上传知识库文件报错", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/305", "detail": "![19621e29eaa547d01213bee53d81e6a](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/84606552/7f6ceb46-e494-4b0e-939c-23b585a6d9d8)", "id": 117} +{"title": "[BUG] AssertionError: Component with id 41 not a valid input component.", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/306", "detail": "**问题描述 / Problem Description**", "id": 118} +{"title": "[BUG] CUDA out of memory with container deployment", "file": "2023-05-10.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/310", "detail": "**问题描述 / Problem Description**", "id": 119} +{"title": "[FEATURE] 增加微调训练功能", "file": "2023-05-11.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/311", "detail": "**功能描述 / Feature Description**", "id": 120} +{"title": "如何使用多卡部署，多个gpu", "file": "2023-05-11.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/315", "detail": "机器上有多个gpu,如何全使用了", "id": 121} +{"title": "请问这个知识库问答，和chatglm的关系是什么", "file": "2023-05-11.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/319", "detail": "这个知识库问答，哪部分关联到了chatglm，是不是没有这个chatglm，知识库问答也可单单拎出来", "id": 122} +{"title": "[BUG] 运行的时候报错ImportError: libcudnn.so.8: cannot open shared object file: No such file or directory", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/324", "detail": "**问题描述 / Problem Description**raceback (most recent call last):", "id": 123} +{"title": "webui启动成功，但有报错", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/325", "detail": "**问题描述 / Problem Description**", "id": 124} +{"title": "切换MOSS的时候报错", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/327", "detail": "danshi但是发布的源码中，", "id": 125} +{"title": "vicuna模型是否能接入？", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/328", "detail": "您好！关于MOSS模型和vicuna模型，都是AutoModelForCausalLM来加载模型的，但是稍作更改（模型路径这些）会报这个错误。这个错误的造成是什么", "id": 126} +{"title": "你好，请问一下在阿里云CPU服务器上跑可以吗？可以的话比较理想的cpu配置是什么？", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/330", "detail": "你好，请问一下在阿里云CPU服务器上跑可以吗？可以的话比较理想的cpu配置是什么？", "id": 127} +{"title": "你好，请问8核32g的CPU可以跑多轮对话吗？", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/331", "detail": "什么样的cpu配置比较好呢？我目前想部署CPU下的多轮对话？", "id": 128} +{"title": "[BUG] 聊天内容输入超过10000个字符系统出现错误", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/332", "detail": "聊天内容输入超过10000个字符系统出现错误，如下图所示：", "id": 129} +{"title": "能增加API的多用户访问接口部署吗？", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/333", "detail": "默认部署程序仅支持单用户访问，多用户则需要排队访问。测试过相关的几个Github多用户工程，但是其中一些仍然不满足要求。本节将系统介绍如何实现多用户同时访问ChatGLM的部署接口，包括http、websocket（流式输出，stream）和web页面等方式，主要目录如下所示。", "id": 130} +{"title": "多卡部署", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/334", "detail": "用单机多卡或多机多卡，fastapi部署模型，怎样提高并发", "id": 131} +{"title": "WEBUI能否指定知识库目录？", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/335", "detail": "**功能描述 / Feature Description**", "id": 132} +{"title": "[BUG] Cannot read properties of undefined (reading 'error')", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/336", "detail": "**问题描述 / Problem Description**", "id": 133} +{"title": "[BUG] 1 validation error for HuggingFaceEmbeddings model_kwargs extra fields not permitted.", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/337", "detail": "模型加载到 100% 后出现问题：", "id": 134} +{"title": "上传知识库需要重启能不能修复一下", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/338", "detail": "挺严重的这个问题", "id": 135} +{"title": "[BUG] 4块v100卡爆显存，在LLM会话模式也一样", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/339", "detail": "**问题描述 / Problem Description**", "id": 136} +{"title": "针对上传的文件配置不同的TextSpliter", "file": "2023-05-12.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/341", "detail": "1. 目前的ChineseTextSpliter切分对英文尤其是代码文件不友好，而且限制固定长度；导致对话结果不如人意", "id": 137} +{"title": "[FEATURE] 未来可增加Bloom系列模型吗？根据甲骨易的测试，这系列中文评测效果不错", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/346", "detail": "**功能描述 / Feature Description**", "id": 138} +{"title": "[BUG] v0.1.12打包镜像后启动webui.py失败 / Concise description of the issue", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/347", "detail": "**问题描述 / Problem Description**", "id": 139} +{"title": "切换MOSS模型时报错", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/349", "detail": "昨天问了下，说是transformers版本不对，需要4.30.0，发现没有这个版本，今天更新到4.29.1，依旧报错，错误如下", "id": 140} +{"title": "[BUG] pdf文档加载失败", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/350", "detail": "**问题描述 / Problem Description**", "id": 141} +{"title": "建议可以在后期增强一波注释，这样也有助于更多人跟进提PR", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/351", "detail": "知道作者和团队在疯狂更新审查代码，只是建议后续稳定后可以把核心代码进行一些注释的补充，从而能帮助更多人了解各个模块作者的思路从而提出更好的优化。", "id": 142} +{"title": "[FEATURE] MOSS 量化版支援", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/353", "detail": "**功能描述 / Feature Description**", "id": 143} +{"title": "[BUG] moss模型无法加载", "file": "2023-05-13.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/356", "detail": "**问题描述 / Problem Description**", "id": 144} +{"title": "[BUG] load_doc_qa.py 中的 load_file 函数有bug", "file": "2023-05-14.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/358", "detail": "原函数为：", "id": 145} +{"title": "[FEATURE] API模式，知识库加载优化", "file": "2023-05-14.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/359", "detail": "如题，当前版本，每次调用本地知识库接口，都将加载一次知识库，是否有更好的方式？", "id": 146} +{"title": "运行Python api.py脚本后端部署后，怎么使用curl命令调用？", "file": "2023-05-15.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/361", "detail": "也就是说，我现在想做个对话机器人，想和公司的前后端联调？怎么与前后端相互调用呢？可私信，有偿解答！！！", "id": 147} +{"title": "上传知识库需要重启能不能修复一下", "file": "2023-05-15.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/363", "detail": "上传知识库需要重启能不能修复一下", "id": 148} +{"title": "[BUG] pip install -r requirements.txt -i https://pypi.tuna.tsinghua.edu.cn/simple", "file": "2023-05-15.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/364", "detail": "我的python是3.8.5的", "id": 149} +{"title": "pip install gradio 报错", "file": "2023-05-15.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/367", "detail": "大佬帮我一下", "id": 150} +{"title": "[BUG] pip install gradio 一直卡不动", "file": "2023-05-15.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/369", "detail": "![aba82742dd9d4d242181662eb5027a7](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/84606552/cd9600d9-f6e7-46b7-b1be-30ed8b99f76b)", "id": 151} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/370", "detail": "初次加载本地知识库成功，但提问后，就无法重写加载本地知识库", "id": 152} +{"title": "[FEATURE] 简洁阐述功能 / Concise description of the feature", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/371", "detail": "**功能描述 / Feature Description**", "id": 153} +{"title": "在windows上，模型文件默认会安装到哪", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/372", "detail": "-------------------------------------------------------------------------------", "id": 154} +{"title": "[FEATURE] 兼顾对话管理", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/374", "detail": "如何在知识库检索的情况下，兼顾对话管理？", "id": 155} +{"title": "llm device: cpu embedding device: cpu", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/376", "detail": "**问题描述 / Problem Description**", "id": 156} +{"title": "[FEATURE] 简洁阐述功能 /文本文件的知识点之间使用什么分隔符可以分割？", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/377", "detail": "**功能描述 / Feature Description**", "id": 157} +{"title": "[BUG] 上传文件失败：PermissionError: [WinError 32] 另一个程序正在使用此文件，进程无法访问。", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/379", "detail": "**问题描述 / Problem Description**", "id": 158} +{"title": "[BUG] 执行python api.py 报错", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/383", "detail": "错误信息", "id": 159} +{"title": "model_kwargs extra fields not permitted (type=value_error.extra)", "file": "2023-05-16.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/384", "detail": "大家好，请问这个有遇到的么,？", "id": 160} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-05-17.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/385", "detail": "执行的时候出现了ls1 = [ls[0]]", "id": 161} +{"title": "[FEATURE] 性能优化", "file": "2023-05-17.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/388", "detail": "**功能描述 / Feature Description**", "id": 162} +{"title": "[BUG] Moss模型问答，RuntimeError: probability tensor contains either inf, nan or element < 0", "file": "2023-05-17.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/390", "detail": "**问题描述 / Problem Description**", "id": 163} +{"title": "有没有人知道v100GPU的32G显存，会报错吗？支持V100GPU吗？", "file": "2023-05-17.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/392", "detail": "**问题描述 / Problem Description**", "id": 164} +{"title": "针对于编码问题比如'gbk' codec can't encode character '\\xab' in position 14: illegal multibyte sequence粗浅的解决方法", "file": "2023-05-17.03", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/397", "detail": "**功能描述 / Feature Description**", "id": 165} +{"title": "Could not import sentence_transformers python package. Please install it with `pip install sentence_transformers`.", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/400", "detail": "**问题描述 / Problem Description**", "id": 166} +{"title": "支持模型问答与检索问答", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/401", "detail": "不同的query，根据意图不一致，回答也应该不一样。", "id": 167} +{"title": "文本分割的时候，能不能按照txt文件的每行进行分割，也就是按照换行符号\\n进行分割？？？", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/403", "detail": "下面的代码应该怎么修改？", "id": 168} +{"title": "local_doc_qa/local_doc_chat 接口响应是串行", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/405", "detail": "**问题描述 / Problem Description**", "id": 169} +{"title": "为什么找到出处了,但是还是无法回答该问题?", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/406", "detail": "![图片](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/3349611/1fc81d61-2409-4330-9065-fdda1a27c86a)", "id": 170} +{"title": "请问下:知识库测试中的:添加单条内容,如果换成文本导入是是怎样的格式?我发现添加单条内容测试效果很好.", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/412", "detail": "我发现在知识库测试中`添加单条内容`,并且勾选`禁止内容分句入库`,即使 `不开启上下文关联`的测试效果都非常好.", "id": 171} +{"title": "[BUG] 无法配置知识库", "file": "2023-05-18.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/413", "detail": "**问题描述 / Problem Description**", "id": 172} +{"title": "[BUG] 部署在阿里PAI平台的EAS上访问页面是白屏", "file": "2023-05-19.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/414", "detail": "**问题描述 / Problem Description**", "id": 173} +{"title": "API部署后调用/local_doc_qa/local_doc_chat 返回Knowledge base samples not found", "file": "2023-05-19.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/416", "detail": "入参", "id": 174} +{"title": "[FEATURE] 上传word另存为的txt文件报 'ascii' codec can't decode byte 0xb9 in position 6: ordinal not in range(128)", "file": "2023-05-20.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/421", "detail": "上传word另存为的txt文件报", "id": 175} +{"title": "创建保存的知识库刷新后没有出来，这个知识库是永久保存的吗？可以连外部的 向量知识库吗？", "file": "2023-05-21.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/422", "detail": "创建保存的知识库刷新后没有出来，这个知识库是永久保存的吗？可以连外部的 向量知识库吗？", "id": 176} +{"title": "[BUG] 用colab运行，无法加载模型，报错：'NoneType' object has no attribute 'message_types_by_name'", "file": "2023-05-21.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/423", "detail": "**问题描述 / Problem Description**", "id": 177} +{"title": "请问是否需要用到向量数据库？以及什么时候需要用到向量数据库？", "file": "2023-05-21.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/424", "detail": "目前用的是 text2vec ， 请问是否需要用到向量数据库？以及什么时候需要用到向量数据库？", "id": 178} +{"title": "huggingface模型引用问题", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/427", "detail": "它最近似乎变成了一个Error？", "id": 179} +{"title": "你好，加载本地txt文件出现这个killed错误，TXT文件有100M左右大小。原因是？谢谢。", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/429", "detail": "\"929aca3b22b8cd74e997a87b61d241b\"", "id": 180} +{"title": "想请问一下，关于对本地知识的管理是如何管理？例如：通过http API接口添加数据 或者 删除某条数据", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/430", "detail": "例如：通过http API接口添加、删除、修改 某条数据。", "id": 181} +{"title": "[FEATURE] 双栏pdf识别问题", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/432", "detail": "试了一下模型，感觉对单栏pdf识别的准确性较高，但是由于使用的基本是ocr的技术，对一些双栏pdf论文识别出来有很多问题，请问有什么办法改善吗？", "id": 182} +{"title": "部署启动小问题，小弟初学求大佬解答", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/433", "detail": "1.python loader/image_loader.py时，提示ModuleNotFoundError: No module named 'configs'，但是跑python webui.py还是还能跑", "id": 183} +{"title": "能否支持检测到目录下文档有增加而去增量加载文档，不影响前台对话，其实就是支持读写分离。如果能支持查询哪些文档向量化了，删除过时文档等就更好了，谢谢。", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/434", "detail": "**功能描述 / Feature Description**", "id": 184} +{"title": "[BUG] 简洁阐述问题 / windows 下cuda错误，请用https://github.com/Keith-Hon/bitsandbytes-windows.git", "file": "2023-05-22.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/435", "detail": "pip install git+https://github.com/Keith-Hon/bitsandbytes-windows.git", "id": 185} +{"title": "[BUG] from commit 33bbb47, Required library version not found: libbitsandbytes_cuda121_nocublaslt.so. Maybe you need to compile it from source?", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/438", "detail": "**问题描述 / Problem Description**", "id": 186} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue上传60m的txt文件报错，显示超时，请问这个能上传的文件大小有限制吗", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/439", "detail": "ERROR 2023-05-23 11:13:09,627-1d: Timeout reached while detecting encoding for ./docs/GLM模型格式数据.txt", "id": 187} +{"title": "[BUG] TypeError: issubclass() arg 1 must be a class", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/440", "detail": "**问题描述**", "id": 188} +{"title": "执行python3 webui.py后，一直提示”模型未成功加载，请到页面左上角\"模型配置\"选项卡中重新选择后点击\"加载模型\"按钮“", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/441", "detail": "**问题描述 / Problem Description**", "id": 189} +{"title": "是否能提供网页文档得导入支持", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/444", "detail": "现在很多都是在线文档作为协作得工具，所以通过URL导入在线文档需求更大", "id": 190} +{"title": "[BUG] history 索引问题", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/445", "detail": "在比较对话框的history和模型chat function 中的history时， 发现并不匹配，在传入 llm._call 时，history用的索引是不是有点问题，导致上一轮对话的内容并不输入给模型。", "id": 191} +{"title": "[BUG] moss_llm没有实现", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/447", "detail": "有些方法没支持，如history_len", "id": 192} +{"title": "请问langchain-ChatGLM如何删除一条本地知识库的数据？", "file": "2023-05-23.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/448", "detail": "例如：用户刚刚提交了一条错误的数据到本地知识库中了，现在如何在本地知识库从找到，并且对此删除。", "id": 193} +{"title": "[BUG] 简洁阐述问题 / UnboundLocalError: local variable 'resp' referenced before assignment", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/450", "detail": "在最新一版的代码中， 运行api.py 出现了以上错误（UnboundLocalError: local variable 'resp' referenced before assignment）， 通过debug的方式观察到local_doc_qa.llm.generatorAnswer(prompt=question, history=history,streaming=True)可能不返回任何值。", "id": 194} +{"title": "请问有没有 PROMPT_TEMPLATE 能让模型不回答敏感问题", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/452", "detail": "## PROMPT_TEMPLATE问题", "id": 195} +{"title": "[BUG] 测试环境 Python 版本有误", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/456", "detail": "**问题描述 / Problem Description**", "id": 196} +{"title": "[BUG] webui 部署后样式不正确", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/458", "detail": "**问题描述 / Problem Description**", "id": 197} +{"title": "配置默认LLM模型的问题", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/459", "detail": "**问题描述 / Problem Description**", "id": 198} +{"title": "[FEATURE]是时候更新一下autoDL的镜像了", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/460", "detail": "如题，跑了下autoDL的镜像，发现是4.27号的，git pull新版本的代码功能+老的依赖环境，各种奇奇怪怪的问题。", "id": 199} +{"title": "[BUG] tag:0.1.13 以cpu模式下，想使用本地模型无法跑起来，各种路径参数问题", "file": "2023-05-24.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/462", "detail": "-------------------------------------------------------------------------------", "id": 200} +{"title": "[BUG] 有没有同学遇到过这个错！！！加载本地txt文件出现这个killed错误，TXT文件有100M左右大小。", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/463", "detail": "运行cli_demo.py。是本地的txt文件太大了吗？100M左右。", "id": 201} +{"title": "API版本能否提供WEBSOCKET的流式接口", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/464", "detail": "webui 版本中，采用了WS的流式输出，整体感知反应很快", "id": 202} +{"title": "[BUG] 安装bug记录", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/465", "detail": "按照[install文档](https://github.com/imClumsyPanda/langchain-ChatGLM/blob/master/docs/INSTALL.md)安装的，", "id": 203} +{"title": "VUE的pnmp i执行失败的修复-用npm i命令即可", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/466", "detail": "感谢作者！非常棒的应用，用的很开心。", "id": 204} +{"title": "请教个问题，有没有人知道cuda11.4是否支持？？？", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/467", "detail": "请教个问题，有没有人知道cuda11.4是否支持？？？", "id": 205} +{"title": "请问有实现多轮问答中基于问题的搜索上下文关联么", "file": "2023-05-25.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/468", "detail": "在基于知识库的多轮问答中，第一个问题讲述了一个主题，后续的问题描述没有包含这个主题的关键词，但又存在上下文的关联。如果用后续问题去搜索知识库有可能会搜索出无关的信息，从而导致大模型无法正确回答问题。请问这个项目要考虑这种情况吗？", "id": 206} +{"title": "[BUG] 内存不足的问题", "file": "2023-05-26.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/470", "detail": "我用了本地的chatglm-6b-int4模型，然后显示了内存不足（win10+32G内存+1080ti11G），一般需要多少内存才足够？这个bug应该如何解决？", "id": 207} +{"title": "[BUG] 纯内网环境安装pycocotools失败", "file": "2023-05-26.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/472", "detail": "**问题描述 / Problem Description**", "id": 208} +{"title": "[BUG] webui.py 重新加载模型会导致 KeyError", "file": "2023-05-26.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/473", "detail": "**问题描述 / Problem Description**", "id": 209} +{"title": "chatyuan无法使用", "file": "2023-05-26.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/475", "detail": "**问题描述 / Problem Description**", "id": 210} +{"title": "[BUG] 文本分割模型AliTextSplitter存在bug，会把“.”作为分割符", "file": "2023-05-26.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/476", "detail": "阿里达摩院的语义分割模型存在bug，默认会把\".”作为分割符进行分割而不管上下文语义。是否还有其他分割符则未知。建议的修改方案：把“.”统一替换为其他字符，分割后再替换回来。或者添加其他分割模型。", "id": 211} +{"title": "[BUG] RuntimeError: Error in faiss::FileIOReader::FileIOReader(const char*) a", "file": "2023-05-27.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/479", "detail": "**问题描述 / Problem Description**", "id": 212} +{"title": "[FEATURE] 安装，为什么conda create要额外指定路径 用-p ，而不是默认的/envs下面", "file": "2023-05-28.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/481", "detail": "##**功能描述 / Feature Description**", "id": 213} +{"title": "[小白求助] 通过Anaconda执行webui.py后，无法打开web链接", "file": "2023-05-28.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/485", "detail": "在执行webui.py命令后，http://0.0.0.0:7860复制到浏览器后无法打开，显示“无法访问此网站”。", "id": 214} +{"title": "[BUG] 使用 p-tuningv2后的模型，重新加载报错", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/486", "detail": "把p-tunningv2训练完后的相关文件放到了p-tunningv2文件夹下，勾选使用p-tuningv2点重新加载模型，控制台输错错误信息：", "id": 215} +{"title": "[小白求助] 服务器上执行webui.py后，在本地无法打开web链接", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/487", "detail": "此项目执行在xxx.xx.xxx.xxx服务器上，我在webui.py上的代码为 (demo", "id": 216} +{"title": "[FEATURE] 能不能支持VisualGLM-6B", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/488", "detail": "**功能描述 / Feature Description**", "id": 217} +{"title": "你好，问一下各位，后端api部署的时候，支持多用户同时问答吗？？？", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/489", "detail": "支持多用户的话，最多支持多少用户问答？根据硬件而定吧？", "id": 218} +{"title": "V100GPU显存占满，而利用率却为0，这是为什么？", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/491", "detail": "\"de45fe2b6cb76fa091b6e8f76a3de60\"", "id": 219} +{"title": "[求助] 如果在公司内部搭建产品知识库，使用INT-4模型，200人规模需要配置多少显存的服务器？", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/492", "detail": "如题，计划给公司搭一个在线知识库。", "id": 220} +{"title": "你好，请教个问题，目前问答回复需要20秒左右，如何提高速度？V10032G服务器。", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/493", "detail": "**问题描述 / Problem Description**", "id": 221} +{"title": "[FEATURE] 如何实现只匹配下文，而不要上文的结果", "file": "2023-05-29.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/494", "detail": "在构建自己的知识库时，主要采用问答对的形式，那么也就是我需要的回答是在我的问题下面的内容，但是目前设置了chunk_size的值以后匹配的是上下文的内容，但我实际并不需要上文的。为了实现更完整的展示下面的答案，我只能调大chunk_size的值，但实际上上文的一半内容都是我不需要的。也就是扔了一半没用的东西给prompt，在faiss.py中我也没找到这块的一些描述，请问该如何进行修改呢？", "id": 222} +{"title": "你好，问一下，我调用api.py部署，为什么用ip加端口可以使用postman调用，而改为域名使用postman无法调用？", "file": "2023-05-30.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/497", "detail": "![5ufBSWxLyF](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/109277248/70e2fbac-5699-48d0-b0d1-3dc84fd042c2)", "id": 223} +{"title": "调用api.py中的stream_chat，返回source_documents中出现中文乱码。", "file": "2023-05-30.04", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/498", "detail": "-------------------------------------------------------------------------------", "id": 224} +{"title": "[BUG] 捉个虫，api.py中的stream_chat解析json问题", "file": "2023-05-30.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/501", "detail": "**问题描述 / Problem Description**", "id": 225} +{"title": "windows本地部署遇到了omp错误", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/502", "detail": "**问题描述 / Problem Description**", "id": 226} +{"title": "[BUG] bug14 ,\"POST /local_doc_qa/upload_file HTTP/1.1\" 422 Unprocessable Entity", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/503", "detail": "上传的文件报错，返回错误，api.py", "id": 227} +{"title": "你好，请教个问题，api.py部署的时候，如何改为多线程调用？谢谢", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/505", "detail": "目前的api.py脚本不支持多线程", "id": 228} +{"title": "你好，请教一下。api.py部署的时候，能不能提供给后端流失返回结果。", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/507", "detail": "curl -X 'POST' \\", "id": 229} +{"title": "流式输出，流式接口，使用server-sent events技术。", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/508", "detail": "想这样一样，https://blog.csdn.net/weixin_43228814/article/details/130063010", "id": 230} +{"title": "计划增加流式输出功能吗？ChatGLM模型通过api方式调用响应时间慢怎么破，Fastapi流式接口来解惑，能快速提升响应速度", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/509", "detail": "**问题描述 / Problem Description**", "id": 231} +{"title": "[BUG] 知识库上传时发生ERROR (could not open xxx for reading: No such file or directory)", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/510", "detail": "**问题描述 / Problem Description**", "id": 232} +{"title": "api.py脚本打算增加SSE流式输出吗？", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/511", "detail": "curl调用的时候可以检测第一个字，从而提升回复的体验", "id": 233} +{"title": "[BUG] 使用tornado实现webSocket，可以多个客户端同时连接，并且实现流式回复，但是多个客户端同时使用，答案就很乱，是模型不支持多线程吗", "file": "2023-05-31.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/512", "detail": "import asyncio", "id": 234} +{"title": "支持 chinese_alpaca_plus_lora 吗 基于llama的", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/514", "detail": "支持 chinese_alpaca_plus_lora 吗 基于llama的，https://github.com/ymcui/Chinese-LLaMA-Alpaca这个项目的", "id": 235} +{"title": "[BUG] 现在能读图片的pdf了，但是文字的pdf反而读不了了，什么情况？？？", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/515", "detail": "**问题描述 / Problem Description**", "id": 236} +{"title": "在推理的过程中卡住不动，进程无法正常结束", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/516", "detail": "**问题描述 / Problem Description**", "id": 237} +{"title": "curl调用的时候，从第二轮开始，curl如何传参可以实现多轮对话？", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/517", "detail": "第一轮调用：", "id": 238} +{"title": "建议添加api.py部署后的日志管理功能？", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/518", "detail": "-------------------------------------------------------------------------------", "id": 239} +{"title": "有大佬知道，怎么多线程部署api.py脚本吗？", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/519", "detail": "api.py部署后，使用下面的请求，时间较慢，好像是单线程，如何改为多线程部署api.py：", "id": 240} +{"title": "[BUG] 上传文件到知识库 任何格式与内容都永远失败", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/520", "detail": "上传知识库的时候，传txt无法解析，就算是穿content/sample里的样例txt也无法解析，上传md、pdf等都无法加载，会持续性等待，等到了超过30分钟也不行。", "id": 241} +{"title": "关于prompt_template的问题", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/521", "detail": "请问这段prompt_template是什么意思，要怎么使用？可以给一个具体模板参考下吗？", "id": 242} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-06-01.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/522", "detail": "**问题描述 / Problem Description**", "id": 243} +{"title": "中文分词句号处理（关于表达金额之间的\".\"）", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/523", "detail": "建议处理12.6亿元的这样的分词，最好别分成12 和6亿这样的，需要放到一起", "id": 244} +{"title": "ImportError: cannot import name 'inference' from 'paddle'", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/526", "detail": "在网上找了一圈，有说升级paddle的，我做了还是没有用，有说安装paddlepaddle的，我找了豆瓣的镜像源，但安装报错cannot detect archive format", "id": 245} +{"title": "[BUG] webscoket 接口串行问题（/local_doc_qa/stream-chat/{knowledge_base_id}）", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/527", "detail": "**问题描述 / Problem Description**", "id": 246} +{"title": "[FEATURE] 刷新页面更新知识库列表", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/528", "detail": "**功能描述以及改进方案**", "id": 247} +{"title": "[BUG] 使用ptuning微调模型后，问答效果并不好", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/530", "detail": "### 未调用ptuning", "id": 248} +{"title": "[BUG] 多轮对话效果不佳", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/532", "detail": "在进行多轮对话的时候，无论设置的history_len是多少，效果都不好。事实上我将其设置成了最大值10，但在对话中，仍然无法实现多轮对话：", "id": 249} +{"title": "RuntimeError: MPS backend out of memory (MPS allocated: 18.00 GB, other allocations: 4.87 MB, max allowed: 18.13 GB)", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/533", "detail": "**问题描述**", "id": 250} +{"title": " 请大家重视这个issue！真正使用肯定是多用户并发问答，希望增加此功能！！！", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/534", "detail": "这得看你有多少显卡", "id": 251} +{"title": "在启动项目的时候如何使用到多张gpu啊？", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/535", "detail": "**在启动项目的时候如何使用到多张gpu啊？**", "id": 252} +{"title": " 使用流式输出的时候，curl调用的格式是什么？", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/536", "detail": "app.websocket(\"/local_doc_qa/stream-chat/{knowledge_base_id}\")(stream_chat)中的knowledge_base_id应该填什么？？？", "id": 253} +{"title": "使用本地 vicuna-7b模型启动错误", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/538", "detail": "环境: ubuntu 22.04 cuda 12.1 没有安装nccl，使用rtx2080与m60显卡并行计算", "id": 254} +{"title": "为什么会不调用GPU直接调用CPU呢", "file": "2023-06-02.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/539", "detail": "我的阿里云配置是16G显存，用默认代码跑webui.py时提示", "id": 255} +{"title": "上传多个文件时会互相覆盖", "file": "2023-06-03.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/541", "detail": "1、在同一个知识库中上传多个文件时会互相覆盖，无法结合多个文档的知识，有大佬知道怎么解决吗？", "id": 256} +{"title": "[BUG] ‘gcc’不是内部或外部命令/LLM对话只能持续一轮", "file": "2023-06-03.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/542", "detail": "No compiled kernel found.", "id": 257} +{"title": "以API模式启动项目却没有知识库的接口列表？", "file": "2023-06-04.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/544", "detail": "请问如何获取知识库的接口列表？如果没有需要自行编写的话，可不可以提供相关的获取方式，感谢", "id": 258} +{"title": "程序以API模式启动的时候，如何才能让接口以stream模式被调用呢？", "file": "2023-06-05.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/546", "detail": "作者您好，我在以API模式进行程序启动后，我发现接口响应时间很长，怎么样才能让接口以stream模式被调用呢？我想实现像webui模式的回答那样", "id": 259} +{"title": "关于原文中表格转为文本后数据相关度问题。", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/547", "detail": "原文中表格数据转换为文本，以 （X-Y：值；...） 的格式每一行组织成一句话，但这样做后发现相关度较低，效果很差，有何好的方案吗？", "id": 260} +{"title": "启动后LLM和知识库问答模式均只有最后一轮记录", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/548", "detail": "拉取最新代码，问答时，每次页面只显示最后一次问答记录，需要修改什么参数才可以保留历史记录？", "id": 261} +{"title": "提供system message配置，以便于让回答不要超出知识库范围", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/549", "detail": "**功能描述 / Feature Description**", "id": 262} +{"title": "[BUG] 使用p-tunningv2报错", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/551", "detail": "按照readme的指示把p-tunningv2训练完后的文件放到了p-tunningv2文件夹下，勾选使用p-tuningv2点重新加载模型，控制台提示错误信息：", "id": 263} +{"title": "[BUG] 智障，这么多问题，也好意思放出来，浪费时间", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/553", "detail": "。。。", "id": 264} +{"title": "[FEATURE] 我看代码文件中有一个ali_text_splitter.py，为什么不用他这个文本分割器了？", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/554", "detail": "我看代码文件中有一个ali_text_splitter.py，为什么不用他这个文本分割器了？", "id": 265} +{"title": "加载文档函数报错", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/557", "detail": "def load_file(filepath, sentence_size=SENTENCE_SIZE):", "id": 266} +{"title": "参考指引安装docker后，运行cli_demo.py，提示killed", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/558", "detail": "root@b3d1bd08095c:/chatGLM# python3 cli_demo.py", "id": 267} +{"title": "注意：如果安装错误，注意这两个包的版本 wandb==0.11.0 protobuf==3.18.3", "file": "2023-06-06.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/559", "detail": "Error1: 如果启动异常报错 `protobuf` 需要更新到 `protobuf==3.18.3 `", "id": 268} +{"title": "知识库对长文的知识相关度匹配不太理想有何优化方向", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/563", "detail": "我们可能录入一个文章有 1W 字，里面涉及这个文章主题的很多角度问题，我们针对他提问，他相关度匹配的内容和实际我们需要的答案相差很大怎么办。", "id": 269} +{"title": "使用stream-chat函数进行流式输出的时候，能使用curl调用吗？", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/565", "detail": "为什么下面这样调用会报错？？？", "id": 270} +{"title": "有大佬实践过 并行 或者 多线程 的部署方案吗？", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/566", "detail": "+1", "id": 271} +{"title": "多线程部署遇到问题？", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/567", "detail": "\"3d87bf74f0cf1a4820cc9e46b245859\"", "id": 272} +{"title": "[BUG] 用fastchat加载vicuna-13b模型进行知识库的问答有token的限制错误", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/569", "detail": "当我开启fastchat的vicuna-13b的api服务，然后config那里配置好(api本地测试过可以返回结果)，然后知识库加载好之后(知识库大概有1000多个文档，用chatGLM可以正常推理)，进行问答时出现token超过限制，就问了一句hello；", "id": 273} +{"title": "现在的添加知识库，文件多了总是报错，也不知道自己加载了哪些文件，报错后也不知道是全部失败还是一部分成功；希望能有个加载指定文件夹作为知识库的功能", "file": "2023-06-07.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/574", "detail": "**功能描述 / Feature Description**", "id": 274} +{"title": "[BUG] moss模型本地加载报错", "file": "2023-06-08.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/577", "detail": "moss模型本地加载报错：", "id": 275} +{"title": "加载本地moss模型报错Can't instantiate abstract class MOSSLLM with abstract methods _history_len", "file": "2023-06-08.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/578", "detail": "(vicuna) ps@ps[13:56:20]:/data/chat/langchain-ChatGLM2/langchain-ChatGLM-0.1.13$ python webui.py --model-dir local_models --model moss --no-remote-model", "id": 276} +{"title": "[FEATURE] 能增加在前端页面控制prompt_template吗？或是能支持前端页面选择使用哪个prompt？", "file": "2023-06-08.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/579", "detail": "目前只能在config里修改一个prompt，想在多个不同场景切换比较麻烦", "id": 277} +{"title": "[BUG] streamlit ui的bug，在增加知识库时会报错", "file": "2023-06-08.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/580", "detail": "**问题描述 / Problem Description**", "id": 278} +{"title": "[FEATURE] webui/webui_st可以支持history吗？目前仅能一次对话", "file": "2023-06-08.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/581", "detail": "试了下webui和webui_st都不支持历史对话啊，只能对话一次，不能默认开启所有history吗？", "id": 279} +{"title": "启动python cli_demo.py --model chatglm-6b-int4-qe报错", "file": "2023-06-09.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/585", "detail": "下载好模型，和相关依赖环境，之间运行`python cli_demo.py --model chatglm-6b-int4-qe`报错了：", "id": 280} +{"title": "重新构建知识库报错", "file": "2023-06-09.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/586", "detail": "**问题描述 / Problem Description**", "id": 281} +{"title": "[FEATURE] 能否屏蔽paddle，我不需要OCR，效果差依赖环境还很复杂", "file": "2023-06-09.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/587", "detail": "希望能不依赖paddle", "id": 282} +{"title": "question ：文档向量化这个可以自己手动实现么？", "file": "2023-06-09.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/589", "detail": "现有公司级数据500G+，需要使用这个功能，请问如何手动实现这个向量化，然后并加载", "id": 283} +{"title": "view前端能进行流式的返回吗？？", "file": "2023-06-09.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/590", "detail": "view前端能进行流式的返回吗？？", "id": 284} +{"title": "[BUG] Load parallel cpu kernel failed, using default cpu kernel code", "file": "2023-06-11.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/594", "detail": "**问题描述 / Problem Description**", "id": 285} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-06-11.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/595", "detail": "**问题描述 / Problem Description**", "id": 286} +{"title": "我在上传本地知识库时提示KeyError: 'name'错误，本地知识库都是.txt文件，文件数量大约是2000+。", "file": "2023-06-12.05", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/597", "detail": "\"KError\"", "id": 287} +{"title": "model_config.py中有vicuna-13b-hf模型的配置信息，但是好像还是不可用？", "file": "2023-06-12.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/600", "detail": "@dongyihua543", "id": 288} +{"title": "ImportError: Using SOCKS proxy, but the 'socksio' package is not installed. Make sure to install httpx using `pip install httpx[socks]`.", "file": "2023-06-12.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/605", "detail": "应该代理问题，但是尝试了好多方法都解决不了，", "id": 289} +{"title": "[BUG] similarity_search_with_score_by_vector在找不到匹配的情况下出错", "file": "2023-06-12.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/607", "detail": "在设置匹配阈值 VECTOR_SEARCH_SCORE_THRESHOLD 的情况下，vectorstore会返回空，此时上述处理函数会出错", "id": 290} +{"title": "[FEATURE] 请问如何搭建英文知识库呢", "file": "2023-06-12.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/609", "detail": "**功能描述 / Feature Description**", "id": 291} +{"title": "谁有vicuna权重？llama转换之后的", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/611", "detail": "**问题描述 / Problem Description**", "id": 292} +{"title": "[FEATURE] API能实现上传文件夹的功能么？", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/612", "detail": "用户懒得全选所有的文件，就想上传个文件夹，请问下API能实现这个功能么？", "id": 293} +{"title": "请问在多卡部署后，上传单个文件作为知识库，用的是单卡在生成向量还是多卡？", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/614", "detail": "目前我检测我本地多卡部署的，好像生成知识库向量的时候用的还是单卡", "id": 294} +{"title": "[BUG] python webui.py提示非法指令", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/615", "detail": "(/data/conda-langchain [root@chatglm langchain-ChatGLM]# python webui.py", "id": 295} +{"title": "知识库文件跨行切分问题", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/616", "detail": "我的知识库文件txt文件，是一行一条知识，用\\n分行。", "id": 296} +{"title": "[FEATURE] bing搜索问答有流式的API么？", "file": "2023-06-13.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/617", "detail": "web端是有这个bing搜索回答，但api接口没有发现，大佬能给个提示么？", "id": 297} +{"title": "希望出一个macos m2的安装教程", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/620", "detail": "mac m2安装，模型加载成功了，知识库文件也上传成功了，但是一问答就会报错，报错内容如下", "id": 298} +{"title": "为【出处】提供高亮显示", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/621", "detail": "具体出处里面，对相关的内容高亮显示，不包含前后文。", "id": 299} +{"title": "[BUG] CPU运行cli_demo.py，不回答，hang住", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/622", "detail": "没有GPU；32G内存的ubuntu机器。", "id": 300} +{"title": "关于删除知识库里面的文档后，LLM知识库对话的时候还是会返回该被删除文档的内容", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/623", "detail": "如题，在vue前端成功执行删除知识库里面文档A.txt后，未能也在faiss索引中也删除该文档，LLM还是会返回这个A.txt的内容，并且以A.txt为出处，未能达到删除的效果", "id": 301} +{"title": "[BUG] 调用知识库进行问答,显存会一直叠加", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/625", "detail": "14G的显存,调用的chatglm-6b-int8模型,进行知识库问答时,最多问答四次就会爆显存了,观察了一下显存使用情况,每一次使用就会增加一次显存,请问这样是正常的吗?是否有什么配置需要开启可以解决这个问题?例如进行一次知识库问答清空上次问题的显存?", "id": 302} +{"title": "[BUG] web页面 重新构建数据库 失败，导致 原来的上传的数据库都没了", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/626", "detail": "web页面 重新构建数据库 失败，导致 原来的上传的数据库都没了", "id": 303} +{"title": "在CPU上运行webui.py报错Tensor on device cpu is not on the expected device meta!", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/627", "detail": "在CPU上运行python webui.py能启动，但最后有：RuntimeError: Tensor on device cpu is not on the expected device meta!", "id": 304} +{"title": "OSError: [WinError 1114] 动态链接库(DLL)初始化例程失败。 Error loading \"E:\\xxx\\envs\\langchain\\lib\\site-packages\\torch\\lib\\caffe2_nvrtc.dll\" or one of its dependencies.哪位大佬知道如何解决吗？", "file": "2023-06-14.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/629", "detail": "**问题描述 / Problem Description**", "id": 305} +{"title": "[BUG] WEBUI删除知识库文档，会导致知识库问答失败", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/632", "detail": "如题，从知识库已有文件中选择要删除的文件，点击删除后，在问答框输入内容回车报错", "id": 306} +{"title": "更新后的版本中，删除知识库中的文件，再提问出现error错误", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/634", "detail": "针对更新版本，识别到一个问题，过程如下：", "id": 307} +{"title": "我配置好了环境，想要实现本地知识库的问答？可是它返回给我的", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/637", "detail": "没有总结，只有相关度的回复，但是我看演示里面表现的，回复是可以实现总结的，我去查询代码", "id": 308} +{"title": "[BUG] NPM run dev can not successfully start the VUE frontend", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/638", "detail": "**问题描述 / Problem Description**", "id": 309} +{"title": "[BUG] 简洁阐述问题 / Concise description of the issue", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/639", "detail": "**问题描述 / Problem Description**", "id": 310} +{"title": "提一个模型加载的bug，我在截图中修复了，你们有空可以看一下。", "file": "2023-06-15.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/642", "detail": "![model_load_bug](https://github.com/imClumsyPanda/langchain-ChatGLM/assets/59411575/4432adc4-ccdd-45d9-aafc-5f2d1963403b)", "id": 311} +{"title": "[求助]关于设置embedding model路径的问题", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/643", "detail": "如题，我之前成功跑起来过一次，但因环境丢失重新配置 再运行webui就总是报错", "id": 312} +{"title": "Lora微调后的模型可以直接使用吗", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/646", "detail": "看model_config.py里是有USE_LORA这个参数的，但是在cli_demo.py和webui.py这两个里面都没有用到，实际测试下来模型没有微调的效果，想问问现在这个功能实现了吗", "id": 313} +{"title": "write_check_file在tmp_files目录下生成的load_file.txt是否需要一直保留，占用空间很大，在建完索引后能否删除", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/647", "detail": "**功能描述 / Feature Description**", "id": 314} +{"title": "[BUG] /local_doc_qa/list_files?knowledge_base_id=test删除知识库bug", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/649", "detail": "1.新建test知识库并上传文件（在vue前端完成并检查后端发现确实生成了test文件夹以及下面的content和vec_store", "id": 315} +{"title": "[BUG] vue webui无法加载知识库", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/650", "detail": "拉取了最新的代码，分别运行了后端api和前端web，点击知识库，始终只能显示simple，无法加载知识库", "id": 316} +{"title": "不能本地加载moss模型吗？", "file": "2023-06-16.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/652", "detail": "手动下载模型设置local_model_path路径依旧提示缺少文件，该如何正确配置？", "id": 317} +{"title": "macos m2 pro docker 安装失败", "file": "2023-06-17.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/654", "detail": "macos m2 pro docker 安装失败", "id": 318} +{"title": " [BUG] mac m1 pro 运行提示 zsh: segmentation fault", "file": "2023-06-17.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/655", "detail": "运行： python webui.py", "id": 319} +{"title": "安装 requirements 报错", "file": "2023-06-17.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/656", "detail": "(langchainchatglm) D:\\github\\langchain-ChatGLM>pip install -r requirements.txt -i https://pypi.tuna.tsinghua.edu.cn/simple/", "id": 320} +{"title": "[BUG] AssertionError", "file": "2023-06-17.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/658", "detail": "**问题描述 / Problem Description**", "id": 321} +{"title": "[FEATURE] 支持AMD win10 本地部署吗？", "file": "2023-06-18.06", "url": "https://github.com/imClumsyPanda/langchain-ChatGLM/issues/660", "detail": "**功能描述 / Feature Description**", "id": 322} diff --git a/samples/isssues_merge/langchain-ChatGLM_open.xlsx b/samples/isssues_merge/langchain-ChatGLM_open.xlsx new file mode 100644 index 0000000000000000000000000000000000000000..3503e2d2b0f35967323f3dc93ce290046e7d68dc Binary files /dev/null and b/samples/isssues_merge/langchain-ChatGLM_open.xlsx differ diff --git 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100644 index 0000000000000000000000000000000000000000..903b4d594cdd0d1965583b92cf49046cd2a60449 --- /dev/null +++ b/ttAyT4oBgHgl3EQfmviR/content/tmp_files/2301.00477v1.pdf.txt @@ -0,0 +1,3414 @@ +A Sequential Quadratic Programming Method with High +Probability Complexity Bounds for Nonlinear Equality +Constrained Stochastic Optimization +Albert S. Berahas† +Miaolan Xie‡ +Baoyu Zhou∗§ +January 3, 2023 +Abstract +A step-search sequential quadratic programming method is proposed for solving +nonlinear equality constrained stochastic optimization problems. It is assumed that +constraint function values and derivatives are available, but only stochastic approxi- +mations of the objective function and its associated derivatives can be computed via +inexact probabilistic zeroth- and first-order oracles. Under reasonable assumptions, a +high-probability bound on the iteration complexity of the algorithm to approximate +first-order stationarity is derived. Numerical results on standard nonlinear optimiza- +tion test problems illustrate the advantages and limitations of our proposed method. +1 +Introduction +In this paper, we propose a step-search1 sequential quadratic programming (SQP) algo- +rithm for solving nonlinear equality-constrained stochastic optimization problems of the +form +min +x∈Rn f(x) +s.t. c(x) = 0, +(1.1) +where f : Rn → R and c : Rn → Rm are both continuously differentiable. We consider +the setting in which exact function and derivative information of the objective function is +unavailable, instead, only random estimates of the objective function ¯f(x; Ξ0(x)) ≈ f(x) +†Dept. of Industrial and Operations Engineering, University of Michigan. (albertberahas@gmail.com) +‡School of Operations Research and Information Engineering, Cornell University. (mx229@cornell.edu) +∗Booth School of Business, The University of Chicago. (baoyu.zhou@chicagobooth.edu) +§Corresponding author. +1We use the term step search methods, coined in [22] to differentiate with line search methods. Step +search methods are similar to line search methods, but the search (step) direction can change during the +back-tracking procedure. +1 +arXiv:2301.00477v1 [math.OC] 1 Jan 2023 + +and its first-order derivative ¯g(x; Ξ1(x)) ≈ ∇f(x) are available via inexact probabilistic +oracles, where Ξ0(x) (with probability space (Ω0, FΩ0, P 0)) and Ξ1(x) (with probability +space (Ω1, FΩ1, P 1)) denote the underlying randomness in the objective function and gra- +dient estimates, respectively. On the other hand, the constraint function value c(x) and its +Jacobian ∇c(x)T are assumed to be available. Such deterministically constrained stochas- +tic optimization problems arise in multiple science and engineering applications, including +but not limited to computer vision [37], multi-stage optimization [39], natural language +processing [30], network optimization [9], and PDE-constrained optimization [35]. +The majority of the methods proposed in the literature for solving deterministically +equality-constrained stochastic optimization problems follow either projection or penalty +approaches. The former type of methods (e.g., stochastic projection methods [21, 23–25]) +require that the feasible region satisfies strict conditions, to ensure well-definedness, that +are not satisfied by general nonlinear functions and thus are not readily applicable. In +contrast, the latter, stochastic penalty methods [14, 34], do not impose such conditions +on the feasible region. These methods transform constrained problems into unconstrained +problems via a constraint penalization term in the objective function and apply stochas- +tic algorithms to solve the transformed unconstrained optimization problems. Stochastic +penalty methods are easy to implement and well-studied, however, the empirical perfor- +mance of such methods is sensitive to parameter choices and ill-conditioning, and is usually +inferior to paradigms that treat constraints as constraints. +Recently, a class of stochastic SQP methods has been developed for solving (1.1). These +methods outperform stochastic penalty methods empirically and have convergence guaran- +tees in expectation [7, 28]. In [7], the authors propose an objective-function-free stochastic +SQP method with adaptive step sizes for the fully stochastic regime. In contrast, in [28], +the authors propose a stochastic step search (referred to as line search in the paper [28]) +SQP method for the setting in which the errors in the function and derivative approxima- +tions can be diminished. We note that several algorithm choices in the two papers [7, 28], +e.g., merit functions and merit parameters, are different. Several other extensions have +been proposed [3, 6, 8, 17, 27, 32], and very few of these works (or others in the literature) +derive worst-case iteration complexity (or sample complexity) due to the difficulties that +arise because of the constrained setting and the stochasticity. Notable exceptions are, [16] +where the authors provide convergence rates (and complexity guarantees) for the algorithm +proposed in [7], and [3, 29] that provide complexity bounds for variants of the stochastic +SQP methods under additional assumptions and in the setting in which the errors can +be diminished. We note that, with the exception of [32], all methods mentioned above +assume access to unbiased estimates of the gradients (and function values where neces- +sary), whereas in this paper, we propose an algorithm that can handle biased function and +gradient estimates. +For all aforementioned methods, the most vital ingredient is the quality and reliability +of the random estimates of the objective function and its derivatives. +In our setting, +neither the objective function nor its derivatives are assumed to be directly accessible, only +2 + +stochastic approximations of them are accessible to the algorithm in the form of inexact +probabilistic zeroth-order and first-order oracles (precise definitions will be introduced in +Section 2.3). Such oracles have been proposed and utilized in several works; e.g., [1, 12, 20, +22]. Moreover, these probabilistic oracles and their variants have been proposed for direct- +search methods [20, 36], trust-region methods [1, 10, 15, 19], and step-search methods +[2, 13, 28, 33]. We note that only [28] considers the setting with (equality) constraints, but +iteration complexity (or sample complexity) results are not provided. +1.1 +Contributions +In this paper, we design, analyze, and implement a step-search SQP (SS-SQP) method for +solving nonlinear equality-constrained stochastic optimization problems where exact con- +straint function values and derivatives are available, but only stochastic approximations +of the objective function and its associated derivatives can be computed. These stochas- +tic approximations are computed via inexact probabilistic zeroth- and first-order oracles, +which are similar to those in [22], with parameters controlling the accuracy and reliability +of the approximations, and allowing for biased approximations. Our proposed algorithm +is inspired by state-of-the-art line search SQP methods [11] in conjunction with the recent +stochastic adaptive step-search framework developed in [22] for the unconstrained stochas- +tic setting. At every iteration, the algorithm constructs a model of the reduction in the +merit function that serves the dual purpose of a measure of sufficient progress (part of the +step size computation) and a proxy for convergence. To mitigate the challenges that arise +due to the noise in the objective function evaluations, our step-search method employs +a relaxed sufficient decrease condition similar to that proposed in [4]. Under reasonable +assumptions, we provide a high probability worst-case iteration complexity bound for the +proposed algorithm. Specifically, we prove that with overwhelmingly high probability, our +proposed algorithm generates a first-order ε-stationary iterate in O(ε−2) iterations, where +ε is bounded away from zero and its lower bound is dictated by the noise and bias in the +zeroth- and first-order oracles. The complexity bound derived matches that of the deter- +ministic algorithm provided in [16]. There are two key differences between our paper and +[16]: (i) our algorithm requires access to the objective function whereas the method in [16] +is objective-function-free; and (ii) our first-order oracle provides estimates with sufficient +accuracy only with some probability and can provide arbitrarily bad estimates otherwise. +Finally, numerical results on standard nonlinear equality-constrained test problems [18] +illustrate the efficiency and efficacy of our proposed algorithm. +1.2 +Notation +Let R denote the set of real numbers, Rn denote the set of n-dimensional real vectors, +Rm×n denote the set of m-by-n-dimensional real matrices, N denote the set of natural +numbers, and Sn denote the set of n-by-n-dimensional real symmetric matrices. For any +3 + +a ∈ R, let R>a (R≥a) denote the set of real numbers strictly larger than (larger than or +equal to) a. We use ∥ · ∥ to denote the ℓ2-norm. We use k ∈ N as the iteration counter +of the algorithm, and for brevity, we use a subscript k for denoting information at the kth +iterate, e.g., fk := f(xk). All quantities with over-bars are stochastic, e.g., ¯f(x; Ξ0(x)) +and ¯g(x; Ξ1(x)) (see Section 2.3), and ¯f(x; ξ0(x)) (resp. ¯g(x; ξ1(x))) denote realizations of +¯f(x; Ξ0(x)) (resp. ¯g(x; Ξ1(x))). +1.3 +Organization +The rest of this paper is organized as follows. The algorithmic framework is introduced +in Section 2. The analysis of the algorithm is established in Section 3. We report numer- +ical results in Section 4. Concluding remarks and future research directions are given in +Section 5. +2 +Algorithm +To solve (1.1), we design an iterative algorithm based on the SQP paradigm that generates: +(i) a primal iterate sequence {xk}, (ii) a primal trial iterate sequence {x+ +k }, (iii) a primal +search direction sequence { ¯dk}, (iv) a dual iterate sequence {¯yk}, (v) a step size sequence +{αk}, (vi) a merit parameter sequence {¯τk}, and, (vii) a trial merit parameter sequence +{¯τ trial +k +}. We discuss each of these sequences in below. We make the following assumption +throughout the remainder of this paper. +Assumption 2.1. Let X ⊆ Rn be an open convex set including iterates {xk} and trial it- +erates {x+ +k }. The objective function f : Rn → R is continuously differentiable and bounded +below over X. The objective gradient function ∇f : Rn → Rn is L-Lipschitz continuous +and bounded over X. The constraint function c : Rn → Rm (where m ≤ n) is continuously +differentiable and bounded over X, and each gradient ∇ci : Rn → Rn is γi-Lipschitz con- +tinuous and bounded over X for all i ∈ {1, . . . , m}. The singular values of J := ∇cT are +bounded away from zero over X. +Assumption 2.1 is a standard assumption in the deterministic constrained optimization +literature [31]. Under Assumption 2.1, there exist constants {κg, κc, κJ, κσ} ⊂ R>0 and +finf ∈ R such that for all k ∈ N, +finf ≤ fk, ∥∇fk∥ ≤ κg, ∥ck∥1 ≤ κc, ∥Jk∥ ≤ κJ, and ∥(JkJT +k )−1∥ ≤ κσ. +We should note that by Assumption 2.1, linear independence constraint qualifications +(LICQ) hold. Moreover, under Assumption 2.1, for all x ∈ Rn, d ∈ Rn and α ∈ R≥0 +it follows that +f(x + αd) ≤ f(x) + α∇f(x)T d + L +2 α2∥d∥2 +and ∥c(x + αd)∥1 ≤ ∥c(x) + α∇c(x)T d∥1 + Γ +2 α2∥d∥2, +where Γ = +m +� +i=1 +γi. +(2.1) +4 + +In this paper, we are particularly interested in finding some primal-dual iterate (x, y) ∈ +Rn × Rm that satisfies the first-order stationarity conditions of (1.1). +To this end, let +L : Rn × Rm → R be the Lagrangian of (1.1), defined as +L(x, y) = f(x) + yT c(x), +(2.2) +where y ∈ Rm are the dual variables. The first-order stationarity conditions for (1.1), +which are necessary by Assumption 2.1 (due to the inclusion of the LICQ), are +0 = +� +∇xL(x, y) +∇yL(x, y) +� += +� +∇f(x) + ∇c(x)y +c(x) +� +. +(2.3) +In the remainder of this section we introduce the key algorithmic components: the merit +function and its associated models, the search direction computation and merit parameter +updating mechanism, and the inexact probabilistic zeroth- and first-order oracles. The +main algorithm is Algorithm 1. +2.1 +Merit function +The merit function φ : Rn × R>0 → R is defined as +φ(x, τ) := τf(x) + ∥c(x)∥1, +(2.4) +where τ ∈ R>0, the merit parameter, acts as a balancing parameter between the objective +function and the constraint violation. Given the gradient (approximation) g ∈ Rn and a +search direction d ∈ Rn, the model of merit function l : Rn ×R>0 ×Rn ×Rn → R is defined +as +l(x, τ, g, d) := τ(f(x) + gT d) + ∥c(x) + ∇c(x)T d∥1. +Given a search direction d ∈ Rn that satisfies linearized feasibility, i.e., c(x)+∇c(x)T d = 0, +the reduction in the model of the merit function ∆l : Rn × R>0 × Rn × Rn → R is defined +as +∆l(x, τ, g, d) :=l(x, τ, g, 0) − l(x, τ, g, d) += − τgT d + ∥c(x)∥1 − ∥c(x) + ∇c(x)T d∥1 += − τgT d + ∥c(x)∥1. +(2.5) +We use the reduction in the model of the merit function (2.5) to monitor the progress made +by our proposed algorithm. We discuss this in more detail in Section 2.2. +2.2 +Algorithmic components +We now establish how to: (i) compute the primal search direction sequence { ¯dk}, (ii) +update the merit parameter sequence {¯τk}, and (iii) update the primal iterate sequence +5 + +{xk}. +These sequences depend on the approximation of the gradient of the objective +function sequence {¯g(xk; Ξ1(xk))}. Let ¯g(xk; ξ1(xk)) denote the realization of ¯g(xk; Ξ1(xk)). +To simplify the notation, in this subsection we drop the dependence on the randomness, +e.g., ¯gk = ¯g(xk; ξ1(xk)). +At each iteration k ∈ N, the primal search direction ¯dk ∈ Rn and the dual variable +¯yk ∈ Rm are computed by solving the linear system of equations +� +Hk +JT +k +Jk +0 +� � ¯dk +¯yk +� += − +� +¯gk +ck +� +, +(2.6) +where {Hk} satisfies the following assumption. +Assumption 2.2. For all k ∈ N, Hk ∈ Sn is chosen independently from ¯gk. Moreover, +there exist constants {κH, ζ} ⊂ R>0 such that for all k ∈ N, ∥Hk∥ ≤ κH and uT Hku ≥ +ζ∥u∥2 for any u ∈ Null(Jk). +It is well known that under Assumptions 2.1 and 2.2, there is a unique solution ( ¯dk, ¯yk) +to (2.6), and, thus, the vectors ¯dk ∈ Rn and ¯yk ∈ Rm are well-defined [31]. +Next, we present the merit parameter updating mechanism. Given constants {ϵτ, σ} ⊂ +(0, 1), for all k ∈ N, we compute ¯τk via +¯τk ← +� +¯τk−1 +if ¯τk−1 ≤ ¯τ trial +k +; +min +� +(1 − ϵτ)¯τk−1, ¯τ trial +k +� +otherwise, +(2.7) +where +¯τ trial +k +← +� +� +� +∞ +if ¯gT +k ¯dk + max +� ¯dT +k Hk ¯dk, 0 +� +≤ 0; +(1−σ)∥ck∥1 +¯gT +k ¯dk+max{ ¯dT +k Hk ¯dk,0} +otherwise. +(2.8) +The merit parameter updating mechanism ensures that the sequence of merit parameter +values is non-increasing. Moreover, the updating mechanism is designed to ensure that the +reduction in the model of the merit function is sufficiently positive. By (2.7) and (2.8), it +follows that (see Lemma 3.7) +∆l(xk, ¯τk, ¯gk, ¯dk) ≥ ¯τk max +� ¯dT +k Hk ¯dk, 0 +� ++ σ∥ck∥1. +(2.9) +In the deterministic setting, the reduction in the model of the merit function is zero only +at iterates that satisfy (2.3). +After updating the merit parameter ¯τk, we evaluate ∆l(xk, ¯τk, ¯gk, ¯dk), the stochastic +model reduction of the merit function, and use it to check for sufficient progress. Specif- +ically, given a step size αk, we compute a candidate iterate x+ +k := xk + αk ¯dk and check +whether sufficient progress can be made via the following modified sufficient decrease con- +dition +¯φ(x+ +k , ¯τk; ξ0(x+ +k )) ≤ ¯φ(xk, ¯τk; ξ0(xk)) − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf, +(2.10) +6 + +where ¯φ(x+ +k , ¯τk; ξ0(x+ +k )) and ¯φ(xk, ¯τk; ξ0(xk)) are merit function estimates, θ ∈ (0, 1) is a +user-defined parameter and ϵf is an upper bound on the expected noise in the objective +function approximations. +We note that ¯φ(x+ +k , ¯τk; ξ0(x+ +k )) and ¯φ(xk, ¯τk; ξ0(xk)) are real- +izations of the zeroth-order oracle described in detail in Section 2.3. The positive term +on the right-hand-side allows for a relaxation in the sufficient decrease condition, i.e., the +merit function may increase after a step, and serves to correct for the noise in the merit +function approximations. If (2.10) is satisfied, we accept the candidate point x+ +k by setting +xk+1 ← x+ +k , and potentially increase the step size for the next iteration, i.e., αk+1 ≥ αk. +If (2.10) is not satisfied, the algorithm does not accept the candidate iterate, instead, it +sets xk+1 ← xk and shrinks the step size for the next iteration, i.e., αk+1 < αk. This step +update rule is the centerpiece of our step-search method, and is fundamentally different +from traditional line-search strategies; see [5, 13, 22] and the references therein. Contrary +to line search methods, which compute a search direction and then look for a step size +along that direction, in our approach the search direction changes in every iteration. +We conclude this section by drawing a few parallels to the unconstrained setting. First, +in the unconstrained setting (with Hk = I), the quantity ∆l(xk, ¯τk, ¯gk, ¯dk) reduces to ∥¯gk∥2, +which provides a sufficient descent measure and is an approximate first-order stationarity +measure. In the constrained setting, the reduction in the model of the merit function will +play a similar role. Second, in the unconstrained optimization setting, (2.10) recovers the +sufficient decrease condition used by some noisy unconstrained optimization algorithm; see +[4, Eq. (3.11)]. +2.3 +Probabilistic oracles +In many real-world applications exact objective function and derivative information cannot +be readily computed. Instead, in lieu of these quantities, approximations are available via +inexact probabilistic zeroth- and first-order oracles. These oracles produce approximations +of different accuracy and reliability, and are formally introduced below. +Oracle 0 (Probabilistic zeroth-order oracle). Given x ∈ Rn, the oracle computes +¯f(x; ξ0(x)), a realization of ¯f(x; Ξ0(x)), which is a (random) estimate of the objective +function value f(x), where Ξ0(x) denotes the underlying randomness (may depend on x) +with associated probability space +� +Ω0, FΩ0, P 0� +. Let e(x; Ξ0(x)) := | ¯f(x; Ξ0(x))−f(x)|. For +any x ∈ Rn, e(x; Ξ0(x)) is a “one-sided” sub-exponential random variable with parameters +{ν, b} ⊂ R≥0, whose mean is bounded by some constant ϵf ∈ R≥0. Specifically, for all +x ∈ Rn and λ ∈ [0, 1/b], +EΞ0(x) +� +e(x; Ξ0(x)) +� +≤ ϵf +and EΞ0(x) +� +exp(λ(e(x; Ξ0(x)) − E +� +e(x; Ξ0(x)) +� +)) +� +≤ exp +� +λ2ν2 +2 +� +. +(2.11) +The stochastic approximation of the merit function value is defined as ¯φ(x, τ; ξ0(x)) = +τ ¯f(x; ξ0(x)) + ∥c(x)∥1. +7 + +Oracle 1 (Probabilistic first-order oracle). Given x ∈ Rn and α ∈ R>0, the oracle +computes ¯g(x; ξ1(x)), a realization of ¯g(x; Ξ1(x)), which is a (random) estimate of the +gradient of the objective function ∇f(x), such that +PΞ1(x) +� +∥¯g(x; Ξ1(x)) − ∇f(x)∥ ≤ +max +� +ϵg, κFOα +� +∆l(x, ¯τ(x; Ξ1(x)), ¯g(x; Ξ1(x)), ¯d(x; Ξ1(x))) +� � +≥ 1 − δ, +where Ξ1(x) denotes the underlying randomness (may depend on x) with associated prob- +ability space (Ω1, FΩ1, P 1), (1 − δ) ∈ ( 1 +2, 1] is the probability that the oracle produces a +gradient estimate that is “sufficiently accurate” (related to the reliability of the oracle) and +{ϵg, κFO} ⊂ R≥0 are constants intrinsic to the oracle (related to the precision of the oracle). +In the rest of the paper, to simplify the notation we drop the dependence on x in ξ0(x) +and ξ1(x). Moreover, we use ξ+ +k to represent ξ0(x+ +k ), the randomness in the zeroth-order +oracle evaluated at the trial point x+ +k . +Remark 2.3. We make a few remarks about Oracles 0 and 1: +• Oracles 0 and 1 are similar to those defined in [12, 22]. For a full discussion and +examples of the oracles, we refer interested readers to [22, Section 5]. +• Oracle 1 is a natural generalization of the ones defined in [12, 22] to the constrained +setting. In particular, the right-hand-side of Oracle 1 reduces to max +� +ϵg, κFOα∥¯g(x; Ξ1)∥ +� +in the unconstrained setting, and is precisely what is used in [12, 22]. +• The presence of ϵg ∈ R≥0 in the max term in Oracle 1 allows the gradient approxi- +mations to be biased; the magnitude of the bias is proportional to ϵg. +2.4 +Algorithmic framework +We are ready to introduce our stochastic step-search SQP method (SS-SQP) in Algorithm 1. +Remark 2.4. We make the following remarks about SS-SQP: +• (Step-search) Algorithm 1 is a step-search algorithm, whose main difference from +traditional line-search methods is that only a single trial iterate is tested at every +iteration. That is, if (2.10) is not satisfied, the step size is reduced and a new search +direction and candidate iterate are computed in the next iteration. This strategy has +been employed in other papers; e.g., see [5, 13, 22, 28]. We should note that at every +iteration, even if the iterate does not change, our algorithm requires new objective +function and gradient estimates in the next iteration. +8 + +Algorithm 1 Adaptive Step-Search SQP (SS-SQP) +Require: initial iterate x0 ∈ Rn; initial merit parameter ¯τ−1 ∈ R>0; maximum step size +αmax ∈ (0, 1]; initial step size α0 ∈ (0, αmax]; parameter ϵf ∈ R≥0 of the zeroth-order +oracle (Oracle 0); and other constant parameters {γ, θ, σ, ϵτ} ⊂ (0, 1) +1: for all k ∈ N do +2: +Generate ¯gk = ¯g(xk; ξ1 +k) via Oracle 1 with α = αk, ¯dk = ¯d(xk; ξ1 +k) as in (2.6), and +¯τk = ¯τ(xk; ξ1 +k) as in (2.7)–(2.8) +3: +Let x+ +k = xk + αk ¯dk, and generate ¯φ(xk, ¯τk; ξ0 +k) and ¯φ(x+ +k , ¯τk; ξ+ +k ) via Oracle 0 +4: +if (2.10) holds then +5: +Set xk+1 ← x+ +k and αk+1 ← min{αmax, γ−1αk} +6: +else +7: +Set xk+1 ← xk and αk+1 ← γαk +8: +end if +9: end for +• (Modified sufficient decrease condition (2.10)) The 2¯τkϵf term on the right-hand- +side of (2.10) is a correction term added to compensate for the inexactness of the +probabilistic zeroth-order oracle (Oracle 0). +This correction provides a relaxation +to the sufficient decrease requirement. In contrast to traditional sufficient decrease +conditions, the modified condition (2.10) allows for a relaxation that is proportional +to the noise level of Oracle 0. +• (Objective function evaluations; Line 3) The randomness associated with the evalu- +ation of the objective function value at the candidate iterate x+ +k (Line 3) is not the +same as that of the evaluation at the current point xk. Moreover, we note that even +for unsuccessful iterations (where the iterates do not change) the objective function +values are re-evaluated. +• (Objective gradient evaluations; Line 2) In order to generate an estimate of the gradi- +ent of the objective function that satisfies the conditions of Oracle 1, one can employ +a procedure (a loop) similar to [38, Algorithm 2]. The idea is to refine the estimate +progressively in order to generate one that satisfies the condition. Indeed, in many +real-world problems, including empirical risk minimization in machine learning, one +can improve the gradient approximation by progressively using a larger number of +samples. +• (Maximum step size αmax) We pick αmax ∈ (0, 1] mainly to simplify our analysis. +That being said, the unit upper bound on αmax is motivated by the deterministic +constraint setting. In the deterministic setting (without any noise), the merit function +decrease is upper bounded by a nonsmooth function, whose only point of nonsmothness +is at α = 1, which complicates the analysis; see [7, Lemma 2.13]. +9 + +Before we proceed, we define the stochastic process related to the algorithm. Let Mk +denote {Ξ0 +k, Ξ+ +k , Ξ1 +k} with realizations {ξ0 +k, ξ+ +k , ξ1 +k}. The algorithm generates a stochastic +process: {(Gk, Dk, Tk, ¯φ(Xk, Tk; Ξ0 +k), ¯φ(X+ +k , Tk; Ξ+ +k ), Xk, Ak)} with realizations +{(¯gk, ¯dk, ¯τk, ¯φ(xk, ¯τk; ξ0 +k), ¯φ(x+ +k , ¯τk; ξ+ +k ), xk, αk)}, adapted to the filtration {Fk : k ≥ 0}, +where Fk = σ(M0, M1, . . . , Mk) and σ denotes the σ-algebra. At iteration k, Gk is the +random gradient, Dk is the random primal search direction, Tk is the random merit param- +eter, ¯φ(Xk, Tk; Ξ0 +k) and ¯φ(X+ +k , Tk; Ξ+ +k ) are the random noisy merit function evaluations at +the current point and the candidate point, respectively, Xk is the random iterate at itera- +tion k and Ak is the random step size. Note that Gk, Dk, Tk are dictated by Ξ1 +k (Oracle 1) +and the noisy merit function evaluations are dictated by Ξ0 +k, Ξ+ +k (Oracle 0). +3 +Theoretical analysis +In this section, we analyze the behavior of Algorithm 1. For brevity, throughout this sec- +tion, we assume Assumptions 2.1 and 2.2 hold and do not restate this fact in every lemma +and theorem. We begin by presenting some preliminary results, definitions, and assump- +tions and then proceed to present a worst-case iteration complexity bound for Algorithm 1. +3.1 +Preliminaries, definitions & assumptions +We first define some deterministic quantities that are used in the analysis of Algorithm 1, +and which are never explicitly computed in the implementation of the algorithm. +Let +(dk, yk) ∈ Rn × Rm be the solution of the deterministic counterpart of (2.6), i.e., +� +Hk +JT +k +Jk +0 +� � +dk +yk +� += − +� +∇fk +ck +� +. +(3.1) +The norm of the gradient of the Lagrangian (defined in (2.2)) of (1.1), which is used as a +first-order stationarity measure, can be upper bounded at every primal-dual iterate (xk, yk) +as +����� +� +∇fk + JT +k yk +ck +������ = +����� +� +−Hkdk +−Jkdk +������ ≤ (κH + κJ)∥dk∥, +(3.2) +where the equality is by (3.1) and the inequality follows by Assumptions 2.1 and 2.2. Thus, +(3.2) implies that dk, the primal search direction, can be used as a proxy of the first-order +stationary measure. The following lemma shows that the tuple (dk, yk) is bounded for all +k ∈ N. +Lemma 3.1. There exist constants {κd, κy} ⊂ R>0 such that ∥dk∥ ≤ κd and ∥yk∥ ≤ κy +for all k ∈ N. +10 + +Proof. By the Cauchy–Schwarz inequality and (3.1), we have +����� +� +dk +yk +������ = +������ +� +Hk +JT +k +Jk +0 +�−1 � +∇fk +ck +������� +≤ +������ +� +Hk +JT +k +Jk +0 +�−1������ +����� +� +∇fk +ck +������ , +where both terms on the right-hand side of the inequality are bounded by Assumptions 2.1 +and 2.2, which concludes the proof. +Moreover, we define τk ∈ R>0 and τ trial +k +∈ R>0, the deterministic counterparts of (2.7) +and (2.8), +τk ← +� +¯τk +if ¯τk ≤ τ trial +k +; +min +� +(1 − ϵτ)¯τk, τ trial +k +� +otherwise, +(3.3) +where +τ trial +k +← +� +� +� +∞ +if ∇fT +k dk + max +� +dT +k Hkdk, 0 +� +≤ 0; +(1−σ)∥ck∥1 +∇fT +k dk+max{dT +k Hkdk,0} +otherwise. +(3.4) +We emphasize again that {(τk, τ trial +k +)}k∈N are introduced only for the purposes of the anal- +ysis, and in Algorithm 1 they are never computed (not even in the setting in which the true +gradient is used, i.e., ¯gk = ∇f(xk)). We also note that this definition is not the same as +that in [7, 16]. The difference is in the fact that in the computation of τk, the comparison +is made to ¯τk instead of ¯τk−1. This is important for the analysis, since this guarantees +τk ≤ ¯τk. +We assume that the merit parameter sequence {¯τk} generated in the stochastic setting +is bounded away from zero (Assumption 3.2). Such an assumption has been adopted in +previous literature [6–8, 16, 17]; we refer readers to [7, Section 3.2] and [16, Section 4.2] +for detailed discussions. Finally, we note that we only assume that {¯τk} is bounded away +from zero, and never require the knowledge of ¯τmin in the algorithm. +Assumption 3.2. Let {¯τk} be the merit parameter sequence generated by Algorithm 1. +There exists a constant ¯τmin ∈ R>0 such that for every realization of Algorithm 1, ¯τk ≥ ¯τmin +for all k ∈ N. +Next, we state and prove a provide a useful property with regards to the deterministic +merit parameter sequence {τk} defined in (3.3). +Lemma 3.3. Suppose Assumption 3.2 holds, then there exists a positive constant τmin ∈ +R>0 such that for every realization of Algorithm 1, τk ≥ τmin for all k ∈ N. +Proof. By [7, Lemma 2.16], {τ trial +k +} ⊂ R>0 ∪ {+∞} is always bounded away from zero. We +define τ trial +min ∈ R>0 such that τ trial +min ≤ τ trial +k +for all k ∈ N. By (3.3)–(3.4) and Assumption 3.2, +one may pick τmin = min{(1 − ϵτ)τ trial +min , ¯τmin} to conclude the proof. +11 + +Our final assumption relates to the zeroth-order oracle (Oracle 0). +Assumption 3.4. Let Ek and E+ +k be the errors in the objective function evaluations from +Oracle 0, i.e., Ek := +�� ¯f(Xk; Ξ0 +k) − f(Xk) +��, and E+ +k +:= +�� ¯f(X+ +k ; Ξ+ +k ) − f(X+ +k ) +��. +We as- +sume that either {Ek} and {E+ +k } are deterministically bounded by ϵf ∈ R≥0, or that the +summation of the errors {Ek + E+ +k } are independent over different iterations. +Next, we introduce several definitions necessary for the analysis of Algorithm 1. Specif- +ically, we define true/false iterations (Definition 1), successful/unsuccessful iterations +(Definition 2) and large/small steps (Definition 3), and introduce three indicator vari- +ables respectively. +Definition 1. An iteration k ∈ N is true if +∥¯gk − ∇fk∥ ≤ max +� +ϵg, κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk) +� +and ek + e+ +k ≤ 2ϵf, +(3.5) +where ∆l(xk, ¯τk, ¯gk, ¯dk) is defined in (2.5) and the constants ϵf, ϵg and κFO are the same +ones as in Oracles 0 and 1. If (3.5) does not hold, we call the iteration a false iteration. +We use the random indicator variable Ik to denote if an iteration is true. +Definition 2. Given a constant θ ∈ (0, 1), let ¯φ(xk, ¯τk; ξk) and ¯φ(x+ +k , ¯τk; ξ+ +k ) be obtained +by Oracle 0. If inequality (2.10) holds, then iteration k is successful, otherwise, it is an +unsuccessful iteration. We use the random indicator variable Θk to denote whether an +iteration is successful. +Definition 3. For any k ∈ N, if min{αk, αk+1} ≥ ˜α where ˜α is some problem-dependent +positive real number (defined explicitly in Lemma 3.15), then we call the step a large step +and set the indicator variable Uk = 1. Otherwise, we call the step k a small step and set +Uk = 0. +We show that under appropriate conditions, if the step is a small step and the iteration +is true, then, the iteration is guaranteed to be successful (see Lemma 3.15). The last +definition is for the stopping time (Tε∆l) and a measure of progress ({Zk}). +Definition +4. For +any +realization +of +Algorithm +1, +define +Tε∆l += +min{k +: +� +∆l(xk, τk, ∇fk, dk) ≤ ε∆l}, the number of iterations required to reach a first-order ε- +stationary iterate, where ε = Ω(ε∆l). We discuss the explicit relationship between ε and +ε∆l in Remark 3.5. Moreover, for all k ∈ N, let Zk := φ(xk, ¯τk) − φmin − (¯τkfinf − ¯τminfinf), +where φmin is a lower bound of φ(·, ¯τmin) over X and ¯τmin is defined in Assumption 3.2. +Remark 3.5. A key ingredient of our algorithm is the stopping time Tε∆l that is related +to ∆l(xk, τk, ∇fk, dk). In fact, by (3.2), Assumption 3.2 and Lemma 3.9 (see below), the +12 + +stopping time Tε∆l defined in Definition 4 is the number of iterations needed to achieve a +first-order ε-stationary iterate, i.e., +max{∥∇fk + JT +k yk∥, +� +∥ck∥} ≤ ε, +where +ε = max{κH,1} +√κlτmin +· ε∆l. +(3.6) +We note that (3.6) is the same stationarity measure as that used in [16, Eq. (5)], and is a +non-standard first-order stationary measure compared to +����� +� +∇fk + JT +k yk +ck +������. That said, one +can show that +����� +� +∇fk + JT +k yk +ck +������ ≤ 2 max{∥∇fk + JT +k yk∥, ∥ck∥} ≤ 2 max{κH,κJ} +√κlτmin +· ε∆l = Ω(ε). +Throughout this paper we focus on (and provide complexity bounds for) (3.6) as it provides +a stronger result for feasibility (∥ck∥) when ε < 1. +3.2 +Main Technical Results +We build toward the main result of the paper (Theorem 3.18) through a sequence of +technical lemmas. Our first lemma shows that Zk (defined in Definition 4) is always non- +negative. +Lemma 3.6. For all k ∈ N, Zk ≥ 0. +Proof. It follows from (2.4) and Definition 4 that +Zk = φ(xk, ¯τk) − φmin − (¯τkfinf − ¯τminfinf) += (¯τk(fk − finf) + ∥ck∥1) − φmin + ¯τminfinf +≥ (¯τmin(fk − finf) + ∥ck∥1) − φmin + ¯τminfinf += (¯τminfk + ∥ck∥1) − φmin += φ(xk, ¯τmin) − φmin ≥ 0, +which concludes the proof. +The next lemma reveals the critical role of the merit parameter update. +Lemma 3.7. For all k ∈ N, (2.9) is satisfied. Furthermore, if ¯τk ̸= ¯τk−1, then 0 < ¯τk ≤ +(1 − ϵτ)¯τk−1. +Proof. By Algorithm 1, we have ¯τk ≤ ¯τ trial +k +. +Moreover, by (2.5), (2.7) and (2.8), it +follows that (2.9) is satisfied for all k ∈ N. +By (2.7), if ¯τk ̸= ¯τk−1, then ¯τk = +min +� +(1 − ϵτ)¯τk−1, ¯τ trial +k +� +≤ (1 − ϵτ)¯τk−1. Moreover, when ck = 0, it follows from Assump- +tion 2.2, (2.6) and (2.8) that ¯dk ∈ Null(Jk) and ¯gT +k ¯dk+max{ ¯dT +k Hk ¯dk, 0} = ¯gT +k ¯dk+ ¯dT +k Hk ¯dk = +cT +k ¯yk = 0, which implies ¯τ trial +k += ∞. Therefore, we have ¯τ trial +k +> 0 for all k ∈ N. Finally, by +¯τ−1 ∈ R>0 and (2.7), we have ¯τk > 0 for all k ∈ N. +13 + +The next lemma provides a useful lower bound for the reduction in the model of the +merit function, ∆l(xk, ¯τk, ¯gk, ¯dk), that is related to the primal search direction (∥ ¯dk∥2) and +a measure of infeasibility (∥ck∥). +Lemma +3.8. There exists some constant κl +∈ +R>0 such that for all k +∈ +N, +∆l(xk, ¯τk, ¯gk, ¯dk) ≥ κl¯τk(∥ ¯dk∥2 + ∥ck∥). +Proof. For any iteration k ∈ N, by [7, Lemma 3.4], there exists some constant κl ∈ R>0 +such that +−¯τk(¯gT +k ¯dk + 1 +2 max{ ¯dT +k Hk ¯dk, 0}) + ∥ck∥1 ≥ κl¯τk(∥ ¯dk∥2 + ∥ck∥1). +By ¯τk ∈ R>0 (from Lemma 3.7), this implies that +∆l(xk, ¯τk, ¯gk, ¯dk) = −¯τk¯gT +k ¯dk + ∥ck∥1 ≥ −¯τk(¯gT +k ¯dk + 1 +2 max{ ¯dT +k Hk ¯dk, 0}) + ∥ck∥1, +which concludes the proof. +Lemma +3.9. There exists some constant κl +∈ +R>0 such that for all k +∈ +N, +∆l(xk, τk, gk, dk) ≥ κlτk(∥dk∥2 + ∥ck∥). +Proof. The proof follows the same logic as that of Lemma 3.8 with the stochastic quantities +replaced by their deterministic counterparts. By [7, Lemma 3.4], the desired inequality is +satisfied for the same constant κl defined in Lemma 3.8. +The next lemma bounds the errors in the stochastic search directions and dual variables, +respectively, with respect to the errors in the gradient approximations. +Lemma 3.10. For all k ∈ N, there exist constants {ζ, ω1} ⊂ R>0 such that ∥ ¯dk − dk∥ ≤ +ζ−1∥¯gk − ∇fk∥ and ∥¯yk − yk∥ ≤ ω1∥¯gk − ∇fk∥, where ζ is defined in Assumption 2.2. +Proof. By the Cauchy–Schwarz inequality, Assumption 2.2, (3.1), and the fact that ( ¯dk − +dk) ∈ Null(Jk), it follows that +∥ ¯dk − dk∥∥¯gk − ∇fk∥ ≥ ( ¯dk − dk)T (∇fk − ¯gk) += ( ¯dk − dk)T (Hk( ¯dk − dk) + JT +k (¯yk − yk)) += ( ¯dk − dk)T Hk( ¯dk − dk) ≥ ζ∥ ¯dk − dk∥2, +which proves that ∥ ¯dk −dk∥ ≤ ζ−1∥¯gk −∇fk∥. Next, by (3.1) and Assumption 2.1 it follows +that +¯yk − yk = −(JkJT +k )−1Jk +� +(¯gk − ∇fk) + Hk( ¯dk − dk) +� +. +14 + +By the triangle inequality, the Cauchy–Schwarz inequality, Assumptions 2.1 and 2.2 and +the fact that ∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇fk∥, it follows that +∥¯yk − yk∥ = ∥(JkJT +k )−1Jk +� +(¯gk − ∇fk) + Hk( ¯dk − dk) +� +∥ +≤ ∥(JkJT +k )−1∥∥Jk∥(∥¯gk − ∇fk∥ + ∥Hk∥∥ ¯dk − dk∥) +≤ κσκJ(1 + κHζ−1)∥¯gk − ∇fk∥. +Setting ω1 = κσκJ(1 + κHζ−1) concludes the proof. +The next lemma relates the inner product of the stochastic gradient and stochastic +search direction to the stochastic reduction in the model of the merit function. We consider +two cases that are related to the two cases in the max term of Oracle 1. +Lemma 3.11. For all k ∈ N: +• If ∥¯gk − ∇fk∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk), then +¯τk|¯gT +k ¯dk| ≤ +� +max{κH,κy} +κl ++ +√¯τk(1+κHζ−1)κFOαk +√κl +� +∆l(xk, ¯τk, ¯gk, ¯dk). +• If ∥¯gk − ∇fk∥ ≤ ϵg, +¯τk|¯gT +k ¯dk| ≤ max{κH,κy}+1 +κl +· ∆l(xk, ¯τk, ¯gk, ¯dk) + +¯τk(1+κHζ−1) +2 +4 +ϵ2 +g. +Proof. If ∥¯gk−∇fk∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk), by the triangle inequality, (2.6), Assump- +tion 2.2, and Lemmas 3.1, 3.8 and 3.10, it follows that +¯τk|¯gT +k ¯dk| = ¯τk|(Hk ¯dk + JT +k yk + JT +k (¯yk − yk))T ¯dk| +≤ ¯τk(| ¯dT +k Hk ¯dk| + |yT +k Jk ¯dk| + |(¯yk − yk)T Jk ¯dk|) +≤ ¯τk(κH∥ ¯dk∥2 + ∥yk∥∥ck∥ + ∥(¯gk − ∇fk) + Hk( ¯dk − dk)∥∥ ¯dk∥) +≤ max{κH, κy} · ¯τk(∥ ¯dk∥2 + ∥ck∥) + ¯τk(∥¯gk − ∇fk∥ + κH∥ ¯dk − dk∥)∥ ¯dk∥ +≤ max{κH,κy} +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk +� +1 + κHζ−1� +∥¯gk − ∇fk∥∥ ¯dk∥ +≤ max{κH,κy} +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + +√¯τk(1+κHζ−1)κFOαk +√κl +∆l(xk, ¯τk, ¯gk, ¯dk), +which completes the first part of the proof. +Using similar logic, if ∥¯gk−∇fk∥ ≤ ϵg, by the triangle inequality, (2.6), Assumption 2.2, +Lemmas 3.1, 3.3, 3.8, 3.10, and the fact that ab ≤ a2+ b2 +4 holds for any {a, b} ⊂ R, it follows +15 + +that +¯τk|¯gT +k ¯dk| ≤ max{κH,κy} +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk +� +1 + κHζ−1� +∥¯gk − ∇fk∥∥ ¯dk∥ +≤ max{κH,κy} +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk +� +1 + κHζ−1� +ϵg∥ ¯dk∥ +≤ max{κH,κy} +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + +√¯τk(1+κHζ−1) +√κl +ϵg +� +∆l(xk, ¯τk, ¯gk, ¯dk) +≤ max{κH,κy}+1 +κl +∆l(xk, ¯τk, ¯gk, ¯dk) + +¯τk(1+κHζ−1) +2 +4 +ϵ2 +g, +which completes the proof. +The next lemma provides a useful upper bounds for the errors related to the stochastic +search directions (and gradients) for the same two cases as in Lemma 3.11. +Lemma 3.12. For all k ∈ N: +• If ∥¯gk − ∇fk∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk), then +|∇fT +k dk − ¯gT +k ¯dk| ≤ +� +(1+κHζ−1)κFOαk +√κl¯τk ++ κ2 +FOα2 +k +ζ +� +∆l(xk, ¯τk, ¯gk, ¯dk) +and |dT +k Hkdk − ¯dT +k Hk ¯dk| ≤ +� +2κHζ−1κFOαk +√κl¯τk ++ κHκ2 +FOα2 +k +ζ2 +� +∆l(xk, ¯τk, ¯gk, ¯dk). +• If ∥¯gk − ∇fk∥ ≤ ϵg, then +|∇fT +k dk − ¯gT +k ¯dk| ≤ (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) + ζ−1ϵ2 +g +and |dT +k Hkdk − ¯dT +k Hk ¯dk| ≤ 2κHζ−1ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) + κHζ−2ϵ2 +g. +Proof. We begin with ∥¯gk − ∇fk∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk). +By the triangle and +Cauchy–Schwarz inequalities, Assumption 2.1, and Lemmas 3.1, 3.8 and 3.10, +|∇fT +k dk − ¯gT +k ¯dk| += |(¯gk − ∇fk)T ¯dk + (∇fk − ¯gk)T ( ¯dk − dk) + ¯gT +k ( ¯dk − dk)| += |(¯gk − ∇fk)T ¯dk + (∇fk − ¯gk)T ( ¯dk − dk) − (Hk ¯dk + JT +k ¯yk)T ( ¯dk − dk)| +≤ |(¯gk − ∇fk)T ¯dk| + |(∇fk − ¯gk)T ( ¯dk − dk)| + | ¯dT +k Hk( ¯dk − dk)| + |¯yT +k Jk( ¯dk − dk)| +≤ ∥¯gk − ∇fk∥∥ ¯dk∥ + ∥∇fk − ¯gk∥∥ ¯dk − dk∥ + κH∥ ¯dk∥∥ ¯dk − dk∥ +≤ (1 + κHζ−1) +� +∆l(xk,¯τk,¯gk, ¯dk) +κl¯τk +∥¯gk − ∇fk∥ + ζ−1∥¯gk − ∇fk∥2 +≤ +� +(1+κHζ−1)κFOαk +√κl¯τk ++ ζ−1κ2 +FOα2 +k +� +∆l(xk, ¯τk, ¯gk, ¯dk). +16 + +Additionally, under Assumption 2.2 it follows that +|dT +k Hkdk − ¯dT +k Hk ¯dk| = |2 ¯dT +k Hk( ¯dk − dk) − ( ¯dk − dk)T Hk( ¯dk − dk)| +≤ 2| ¯dT +k Hk( ¯dk − dk)| + |( ¯dk − dk)T Hk( ¯dk − dk)| +≤ 2κH∥ ¯dk∥∥ ¯dk − dk∥ + κH∥ ¯dk − dk∥2 +≤ 2κHζ−1 +� +∆l(xk,¯τk,¯gk, ¯dk) +κl¯τk +∥¯gk − ∇fk∥ + κHζ−2∥¯gk − ∇fk∥2 +≤ +� +2κHζ−1κFOαk +√κl¯τk ++ κHζ−2κ2 +FOα2 +k +� +∆l(xk, ¯τk, ¯gk, ¯dk), +which completes the first part of the proof. +If ∥¯gk − ∇fk∥ ≤ ϵg, following similar logic as the first part of the proof, by the triangle +and Cauchy–Schwarz inequalities, (3.1), and Lemmas 3.1, 3.9 and 3.10, +|∇fT +k dk − ¯gT +k ¯dk| += |(¯gk − ∇fk)T ( ¯dk − dk) + (¯gk − ∇fk)T dk + ∇fT +k ( ¯dk − dk)| += |(¯gk − ∇fk)T ( ¯dk − dk) + (¯gk − ∇fk)T dk − (Hkdk + JT +k yk)T ( ¯dk − dk)| +≤ |(¯gk − ∇fk)T ( ¯dk − dk)| + |(¯gk − ∇fk)T dk| + |dT +k Hk( ¯dk − dk)| + |yT +k Jk( ¯dk − dk)| +≤ ζ−1∥¯gk − ∇fk∥2 + (1 + κHζ−1)∥dk∥∥¯gk − ∇fk∥ +≤ ζ−1∥¯gk − ∇fk∥2 + 1+κHζ−1 +√κlτk +� +∆l(xk, τk, ∇fk, dk)∥¯gk − ∇fk∥ +≤ ζ−1ϵ2 +g + (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk). +Additionally, under Assumption 2.2 it follows that +|dT +k Hkdk − ¯dT +k Hk ¯dk| = |(dk − ¯dk)T Hk(dk − ¯dk) + 2dT +k Hk( ¯dk − dk)| +≤ κH∥dk − ¯dk∥2 + 2κH∥dk∥∥dk − ¯dk∥ +≤ κHζ−2∥¯gk − ∇fk∥2 + 2κH +√ +∆l(xk,τk,∇fk,dk) +√κlτk +ζ−1∥¯gk − ∇fk∥ +≤ κHζ−2ϵ2 +g + 2κHζ−1ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk), +which completes the proof. +The next lemma provides a bound on the merit function across an iteration. +Lemma 3.13. For all k ∈ N +φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2. +17 + +Proof. By Algorithm 1, for any k ∈ N, 0 < αk ≤ αmax ≤ 1. Moreover, by the triangle +inequality, (2.1), (2.4) and (2.6), it follows that +φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) += ¯τk(f(xk + αk ¯dk) − fk) + (∥c(xk + αk ¯dk)∥1 − ∥ck∥1) +≤ ¯τk(αk∇fT +k ¯dk + L +2 α2 +k∥ ¯dk∥2) + (∥ck + αkJk ¯dk∥1 − ∥ck∥1 + Γ +2 α2 +k∥ ¯dk∥2) +≤ αk¯τk∇fT +k ¯dk + |1 − αk|∥ck∥1 + αk∥ck + Jk ¯dk∥1 − ∥ck∥1 + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 += αk¯τk∇fT +k ¯dk − αk∥ck∥1 + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 += αk¯τk¯gT +k ¯dk − αk∥ck∥1 + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 += − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2, +which completes the proof. +Due to the quality and reliability of the zeroth- and first-order oracles (Oracles 0 and +1), one can only guarantee convergence to a neighborhood of the solution. Assumption 3.14 +provides a lower bound on the size of the convergence neighbourhood in terms of ε (and +ε∆l). +Assumption 3.14. Let +ε > max +� +ϵg +η , √ϵfω7ω8 +� +· max{κH,1} +√κlτmin , +which is equivalent to ε∆l > max +� +ϵg +η , √ϵfω7ω8 +� +by Remark 3.5, where 0 < η < 2(1 − +θ) min +� +1 +η1+η2 , +1 +η3+η4 +� +and {η1, η2, η3, η4} ⊂ R>0 are defined as +η1 = +(1−θ)(1+ϵτ)¯τ−1 +� +1+ κH +ζ +� +√κlτmin +η2 = +� +(1 − θ)2¯τ−1 +� +1 + κH +ζ +�2 � +(1+ϵτ)2¯τ−1 +κlτmin ++ ϵτ +� ++ 4¯τ−1 +� +1+ϵτω2 +κl ++ (1−θ)2(1+ϵτ) +ζ +� +η3 = +(1−θ)¯τ−1 +� +¯τ−1 +� +1+ 3κH +ζ +� ++(1−σ)τmin +� +1+ κH +ζ +�� +(1−σ)τmin√κlτmin +and η4 = +� +� +� +� (1−θ)2¯τ 2 +−1 +� +¯τ−1 +� +1+ 3κH +ζ +� ++(1−σ)τmin +� +1+ κH +ζ +��2 +(1−σ)2τ 3 +minκl ++ 4¯τ−1 +κl ++ 4(1 − θ)2 +� +¯τ 2 +−1 +� +1+ κH +ζ +� +(1−σ)τminζ + ¯τ−1 +ζ +� +18 + +with p ∈ +� 1 +2, 1 +� +, and {ω2, ω3, ω4, ω5, ω6, ω7, ω8} ⊂ R>0 defined as +ω2 = max{κH,κy}+1 +κl +, +ω3 = (1+κHζ−1)κFO +√¯τ−1αmax +√κl ++ ¯τ−1κ2 +FOα2 +max +ζ +, +ω4 = max +� +ϵτ +� +max{κH,κy} +κl ++ +√¯τ−1(1+κHζ−1)κFOαmax +√κl ++ ω3 +� +, +¯τ−1 +(1−σ)τmin +� +(1+3κHζ−1)κFO +√¯τ−1αmax +√κl ++ (1 + κHζ−1) ¯τ−1κ2 +FOα2 +max +ζ +�� +, +ω5 = (1 + ϵτ)¯τ−1 +� +η +ζ + 1+κHζ−1 +√κlτmin +� ++ ϵτ ¯τ−1(1+κHζ−1)2η +4 +, +ω6 = +¯τ 2 +−1· +� +(1+κHζ−1) η +ζ + 1+3κHζ−1 +√κlτmin +� +(1−σ)τmin ++ ¯τ−1 +� +η +ζ + 1+κHζ−1 +√κlτmin +� +, +ω7 = +� +4¯τ−1 +(p− 1 +2 )θ max +� +1+ϵτω2 +1−ηω5 , +1 +1−ηω6 , 1 + ω3 + ω4 +� +, +and +ω8 = +� +� +� +� +� +�max +� +� +� +� +� +� +¯τ−1 +κl κFO+ L +2κl + +Γ +2¯τminκl +1−θ +, +¯τminL+Γ +2¯τminκl +� +1−θ−η +� +¯τ−1 +κl max +�� +1+ϵτω2 +1−ηω5 , +1 +√1−ηω6 +�� +� +� +� +� +� +. +Assumption 3.14 involves many constants and is indeed hard to parse. We make all +constants explicit in order to show the exact dependence on the convergence neighborhood. +That being said, what is important is that the lower bound of ε is proportional to the bias +in the gradient approximations and proportional to the square root of the noise level in +the function approximations. +We are now ready to present the key lemma of this section. +In Lemma 3.15, we +first define (p, ˜α, h(·)), where p ∈ +� 1 +2, 1 +� +is a lower bound on the probability of a true +iteration conditioned on the past (before the stopping time), ˜α ∈ R>0 is the large step +threshold, and h : R>0 → R>0 is a monotonically increasing function (in α) that bounds +the potential progress made at any given iteration. Moreover, we prove five results that can +be summarized as follows: (i) lower bound (proportional to ϵf) on the potential progress +with step size ˜α; (ii) conditioned on the past, the next iteration is true with probability +at least p; (iii) bound the potential progress made in any true and successful iterations; +(iv) true iterations with small step sizes are successful; and, (v) bound (proportional +to ϵf) the damage incurred at any iteration. +Lemma 3.15. Suppose Assumptions 3.2, 3.4 and 3.14 hold. For all k < Tε∆l, let +• p = 1 − δ when the noise is bounded by ϵf, and p = 1 − δ − exp +� +− min{ u2 +2ν2 , u +2b} +� +otherwise (with u = infx∈X {ϵf − E[E(x)]}, where E(x) = | ¯f(x; Ξ0(x)) − f(x)|), +19 + +• ˜α = min +� +� +� +� +� +1−θ +� +¯τ−1 +κl κFO+ L +2κl + +Γ +2¯τminκl +, +2¯τminκl +� +1−θ−η +� +¯τ−1 +κl max +�� +1+ϵτω2 +1−ηω5 , +1 +√1−ηω6 +�� +¯τminL+Γ +� +� +� +� +� +, +• h(α) = αθε2 +∆l min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6, +1 +1+ω3+ω4 +� +. +Then, the following results hold: +(i) h(˜α) > 4¯τ−1 +p− 1 +2 +ϵf. +(ii) P [Ik = 1|Fk−1] ≥ p with some p ∈ +� +1 +2 + 4¯τ−1ϵf +h(˜α) , 1 +� +. +(iii) If iteration k is true and successful, then Zk+1 ≤ Zk − h(αk) + 4¯τ−1ϵf. +(iv) If αk ≤ ˜α and iteration k is true, then iteration k is also successful. +(v) Zk+1 ≤ Zk + 2¯τ−1ϵf + ¯τ−1(ek + e+ +k ). +Proof. First, we note that: (1) due to the constants and the form, p is a valid probability, +i.e., p ∈ ( 1 +2, 1], (2) ˜α > 0 is guaranteed by the restriction on η in Assumption 3.14, and +(3) h : R>0 → R>0 is a positive function that measures the potential progress made +if iterations are true and successful. We proceed with this proof by showing all five +statements separately. +(i) This result follows directly from the definition of h(˜α) and the lower bound on ε∆l; +see Assumption 3.14. +(ii) This proof is essentially the same as that from [22, Proposition 1(ii)]. Let +Jk := 1 +� +∥Gk − ∇f(Xk)∥ ≤ max +� +ϵg, κFOAk +� +∆l(Xk, Tk, Gk, Dk) +�� +. +Clearly, by Definition 1, +P [Ik = 0 | Fk−1] = P +� +Jk = 0 or Ek + E+ +k > 2ϵf | Fk−1 +� +≤ P [Jk = 0 | Fk−1] + P +� +Ek + E+ +k > 2ϵf | Fk−1 +� +. +The first term on the right-hand-side of the inequality is bounded above by δ, by the +first-order probabilistic oracle (Oracle 1). The second term is zero in the case where ϵf +is a deterministic bound on the noise. Otherwise, since Ek and E+ +k individually satisfy +the one-sided sub-exponential bound in (2.11) with parameters ϵf and (ν, b), one can +show that Ek + E+ +k satisfies (2.11) with parameters 2ϵf and (2ν, 2b). Hence by the +20 + +one-sided Bernstein inequality, the second term is bounded above by e +− min +� +u2 +2ν2 , u +2b +� +, +with u = infx∈X {ϵf − E[E(x)]}. As a result, +P [Ik = 1 | Fk−1] ≥ p +for all k, for p as defined in the statement. The range of p ∈ +� +1 +2 + 4¯τ−1ϵf +h(˜α) , 1 +� +follows +from the definitions of h(·) and ˜α in the statement, together with the inequality on +ε∆l in Assumption 3.14. +(iii) Suppose iteration k is true and successful. Since iteration k is true, by Definition 1 +we have +∥¯gk − ∇fk∥ ≤ max +� +ϵg, κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk) +� +, +and we consider the two cases separately. We further subdivide the analysis into the +case where ∇fT +k dk ≤ 0 and ∇fT +k dk > 0. +Case A When ∥¯gk − ∇f(xk)∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk), by Lemma 3.10, +∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇f(xk)∥ ≤ ζ−1κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk). +Case A.1 If ∇fT +k dk ≤ 0, by the fact that ¯τk ≥ τk, the triangle inequality, (2.5) and +Lemma 3.12, it follows that +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) += ¯τk¯gT +k ¯dk − τk∇fT +k dk +≤ ¯τk(¯gT +k ¯dk − ∇fT +k dk) +≤ ¯τk|¯gT +k ¯dk − ∇fT +k dk| +≤ ¯τk +� +(1+κHζ−1)κFOαk +√κl¯τk ++ κ2 +FOα2 +k +ζ +� +∆l(xk, ¯τk, ¯gk, ¯dk). +(3.7) +Case A.2 If ∇fT +k dk > 0, by the triangle inequality, (2.5) and Lemma 3.12, +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) += ¯τk¯gT +k ¯dk − τk∇fT +k dk +≤ |¯τk¯gT +k ¯dk − τk∇fT +k dk| +≤ |(¯τk − τk)∇fT +k dk| + ¯τk|¯gT +k ¯dk − ∇fT +k dk| +≤ |(¯τk − τk)∇fT +k dk| ++ ¯τk +� +(1+κHζ−1)κFOαk +√κl¯τk ++ κ2 +FOα2 +k +ζ +� +∆l(xk, ¯τk, ¯gk, ¯dk). +(3.8) +We proceed to bound the term |(¯τk − τk)∇fT +k dk|; we consider three cases due to +the merit parameter updating formulae ((2.7)–(2.8) and (3.3)–(3.4)). +21 + +Case A.2.1 If τk = ¯τk, then |(¯τk − τk)∇fT +k dk| = 0. +Case A.2.2 If τk = (1 − ϵτ)¯τk, by the triangle inequality and Lemmas 3.11 and 3.12, +|(¯τk − τk)∇fT +k dk| += ϵτ ¯τk|∇fT +k dk| +≤ ϵτ ¯τk(|¯gT +k ¯dk| + |∇fT +k dk − ¯gT +k ¯dk|) +≤ ϵτ +� +max{κH,κy} +κl ++ +√¯τk(1+κHζ−1)κFOαk +√κl +� +∆l(xk, ¯τk, ¯gk, ¯dk) ++ ϵτ ¯τk +� +(1+κHζ−1)κFOαk +√κl¯τk ++ κ2 +FOα2 +k +ζ +� +∆l(xk, ¯τk, ¯gk, ¯dk). +Case A.2.3 If ¯τk > τk = +(1−σ)∥ck∥1 +∇fT +k dk+max{dT +k Hkdk,0}, by (2.7)–(2.8), +∇fT +k dk + max +� +dT +k Hkdk, 0 +� +> (1−σ)∥ck∥1 +¯τk +≥ ¯gT +k ¯dk + max +� ¯dT +k Hk ¯dk, 0 +� +. +(3.9) +By Lemma 3.3, we have τk ≥ τmin for all k ∈ N. Moreover, it follows from (2.5) +and Lemma 3.9 that 0 ≤ ∆l(xk, τk, ∇fk, dk), which implies τk∇fT +k dk ≤ ∥ck∥1. +Using the fact that τk ∈ R>0 and ∇fT +k dk > 0, +|∇fT +k dk| +∥ck∥1 += ∇fT +k dk +∥ck∥1 ≤ 1 +τk . +(3.10) +By Lemma 3.12, (3.9) and (3.10), it follows that +|(¯τk − τk)∇fT +k dk| += +� +¯τk − +(1−σ)∥ck∥1 +∇fT +k dk+max{dT +k Hkdk,0} +� +· |∇fT +k dk| +≤ +(∇fT +k dk+max{dT +k Hkdk,0})−(¯gT +k ¯dk+max{ ¯dT +k Hk ¯dk,0}) +∇fT +k dk+max{dT +k Hkdk,0} +· ¯τk|∇fT +k dk| +≤ +|(∇fT +k dk+max{dT +k Hkdk,0})−(¯gT +k ¯dk+max{ ¯dT +k Hk ¯dk,0})| +(1−σ)∥ck∥1 +· ¯τ 2 +k|∇fT +k dk| +≤ +|∇fT +k dk−¯gT +k ¯dk|+| max{dT +k Hkdk,0}−max{ ¯dT +k Hk ¯dk,0}| +(1−σ)∥ck∥1 +· ¯τ 2 +k|∇fT +k dk| +≤ +¯τ 2 +k +(1−σ)τk · +� +|∇fT +k dk − ¯gT +k ¯dk| + | max +� +dT +k Hkdk, 0 +� +− max{ ¯dT +k Hk ¯dk, 0}| +� +≤ +¯τ 2 +k +(1−σ)τk · +� +|∇fT +k dk − ¯gT +k ¯dk| + |dT +k Hkdk − ¯dT +k Hk ¯dk| +� +≤ +¯τ 2 +k +(1−σ)τmin +� +(1+3κHζ−1)κFOαk +√κl¯τk ++ (1 + κHζ−1)ζ−1κ2 +FOα2 +k +� +∆l(xk, ¯τk, ¯gk, ¯dk). +22 + +Combining (3.7), (3.8) and Cases A.2.1–A.2.3, it follows that +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) +≤ +� +¯τk +� +(1+κHζ−1)κFOαk +√κl¯τk ++ ζ−1κ2 +FOα2 +k +� ++ max +� +ϵτ +� +max{κH,κy} +κl ++ +√¯τk(1+κHζ−1)κFOαk +√κl +� ++ϵτ ¯τk +� +(1+κHζ−1)κFOαk +√κl¯τk ++ ζ−1κ2 +FOα2 +k +� +, +¯τ 2 +k +(1−σ)τmin +� +(1+3κHζ−1)κFOαk +√κl¯τk ++ (1 + κHζ−1)ζ−1κ2 +FOα2 +k +��� +∆l(xk, ¯τk, ¯gk, ¯dk) +≤ (ω3 + ω4) · ∆l(xk, ¯τk, ¯gk, ¯dk), +where {ω3, ω4} ⊂ R>0 are as defined in Assumption 3.14. By {ω3, ω4} ⊂ R>0, +∆l(xk,τk,∇fk,dk) +1+ω3+ω4 +≤ ∆l(xk, ¯τk, ¯gk, ¯dk). +By the fact that iteration k is successful and Definition 2, it follows that +¯φ(x+ +k , ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) ≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf +≤ −αkθ ∆l(xk,τk,∇fk,dk) +1+ω3+ω4 ++ 2¯τ−1ϵf. +Hence, it follows that +Zk+1 − Zk += φ(xk+1, ¯τk+1) − φ(xk, ¯τk) − ¯τk+1finf + ¯τkfinf +≤ φ(xk+1, ¯τk+1) − ¯φ(xk, ¯τk; ξk) − ¯τk+1finf + ¯τkfinf + ¯τkek += φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk; ξ+ +k ) + ¯φ(xk+1, ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) +− ¯τk+1finf + ¯τkfinf + ¯τkek +≤ − αkθ ∆l(xk,τk,∇fk,dk) +1+ω3+ω4 ++ 2¯τ−1ϵf + (¯τk+1 − ¯τk)(f(xk+1) − finf) ++ ¯τk(ek + e+ +k ) +≤ − αkθ ∆l(xk,τk,∇fk,dk) +1+ω3+ω4 ++ 2¯τ−1ϵf + ¯τk(ek + e+ +k ). +(3.11) +Case B When ∥¯gk − ∇f(xk)∥ ≤ ϵg, by the condition that k < Tε∆l and Definition 4, it +follows that +� +∆l(xk, τk, ∇fk, dk) > ε∆l > ϵg +η . By Lemma 3.10, +∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇fk∥ ≤ ζ−1ϵg < ζ−1η +� +∆l(xk, τk, ∇fk, dk). +Case B.1 If ∇fT +k dk ≤ 0, by the fact that ¯τk ≥ τk, the triangle inequality, (2.5) and +23 + +Lemma 3.12, it follows that +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) += ¯τk¯gT +k ¯dk − τk∇fT +k dk +≤ ¯τk(¯gT +k ¯dk − ∇fT +k dk) +≤ ¯τk|¯gT +k ¯dk − ∇fT +k dk| +≤ ¯τk +� +ζ−1ϵ2 +g + (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) +� +≤ ¯τk +� +ζ−1η + 1+κHζ−1 +√κlτk +� +η∆l(xk, τk, ∇fk, dk). +(3.12) +Case B.2 If ∇fT +k dk > 0, by the triangle inequality, (2.5) and Lemma 3.12, +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) += ¯τk¯gT +k ¯dk − τk∇fT +k dk +≤ |¯τk¯gT +k ¯dk − τk∇fT +k dk| +≤ |(¯τk − τk)∇fT +k dk| + ¯τk|¯gT +k ¯dk − ∇fT +k dk| +≤ |(¯τk − τk)∇fT +k dk| ++ ¯τk +� +ζ−1ϵ2 +g + (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) +� +≤ |(¯τk − τk)∇fT +k dk| ++ ¯τk +� +ζ−1η + 1+κHζ−1 +√κlτk +� +η∆l(xk, τk, ∇fk, dk). +(3.13) +We proceed to bound the term |(¯τk − τk)∇fT +k dk|. +Case B.2.1 If τk = ¯τk, then |(¯τk − τk)∇fT +k dk| = 0. +Case B.2.2 If τk = (1 − ϵτ)¯τk, then by Lemmas 3.11 and 3.12 and Assumption 3.14, +|(¯τk − τk)∇fT +k dk| += ϵτ ¯τk|∇fT +k dk| +≤ ϵτ ¯τk +� +|¯gT +k ¯dk| + |∇fT +k dk − ¯gT +k ¯dk| +� +≤ ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τk(1+κHζ−1)2 +4 +ϵ2 +g ++ ϵτ ¯τk +� +ζ−1ϵ2 +g + (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) +� +≤ ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) ++ ϵτ ¯τkη +� +(1+κHζ−1)2η +4 ++ η +ζ + 1+κHζ−1 +√κlτk +� +∆l(xk, τk, ∇fk, dk). +24 + +Case B.2.3 If ¯τk > τk = +(1−σ)∥ck∥1 +∇fT +k dk+max{dT +k Hkdk,0}, following the same logic as in Case A.2.3, +by Lemma 3.12, (3.9) and (3.10), +|(¯τk − τk)∇fT +k dk| +≤ +¯τ 2 +k +(1−σ)τk · +� +|∇fT +k dk − ¯gT +k ¯dk| + |dT +k Hkdk − ¯dT +k Hk ¯dk| +� +≤ +¯τ 2 +k +(1−σ)τmin · +� +ζ−1ϵ2 +g + (1+κHζ−1)ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) ++κHζ−2ϵ2 +g + 2κHζ−1ϵg +√κlτk +� +∆l(xk, τk, ∇fk, dk) +� +≤ +¯τ 2 +k +(1−σ)τmin +� +(1 + κHζ−1)ζ−1η + 1+3κHζ−1 +√κlτk +� +η∆l(xk, τk, ∇fk, dk). +Combining (3.12), (3.13) and Cases B.2.1–B.2.3, it follows that +∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) +≤ max +� +ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τkη +� +(1+κHζ−1)2η +4 ++ η +ζ + 1+κHζ−1 +√κlτk +� +∆l(xk, τk, ∇fk, dk), +η¯τ 2 +k· +� +(1+κHζ−1)ζ−1η+ 1+3κHζ−1 +√κlτk +� +(1−σ)τmin +∆l(xk, τk, ∇fk, dk) +� ++ ¯τk +� +ζ−1η + 1+κHζ−1 +√κlτk +� +η∆l(xk, τk, ∇fk, dk) +≤ max +� +ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ηω5∆l(xk, τk, ∇fk, dk), ηω6∆l(xk, τk, ∇fk, dk) +� +, +(3.14) +where {ω2, ω5, ω6} ⊂ R>0 are defined in Assumption 3.14. Thus, it follows, +∆l(xk, ¯τk, ¯gk, ¯dk) ≥ min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6 +� +· ∆l(xk, τk, ∇fk, dk). +(3.15) +By selecting η following Assumption 3.14, using the fact that iteration k is successful +and Definition 2, +¯φ(x+ +k , ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) +≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf +≤ − αkθ min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6 +� +· ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf. +Hence, following similar logic as in (3.11), it follows that +25 + +Zk+1 − Zk +≤ φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk; ξ+ +k ) + ¯φ(xk+1, ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) +− ¯τk+1finf + ¯τkfinf + ¯τkek +≤ − αkθ min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6 +� +· ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf ++ (¯τk+1 − ¯τk)(f(xk+1) − finf) + ¯τk(ek + e+ +k ) +≤ − αkθ min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6 +� +· ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf + ¯τk(ek + e+ +k ). +Combining the results for Case A and Case B, together with the assumption that +the iteration is true, it follows that +Zk+1 − Zk ≤ − min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6, +1 +1+ω3+ω4 +� +αkθ∆l(xk, τk, ∇fk, dk) ++ 2¯τ−1ϵf + ¯τ−1(ek + e+ +k ) +≤ − h(αk) + 4¯τ−1ϵf, +where the last inequality is from the conditions that ∆l(xk, τk, ∇fk, dk) > ε2 +∆l and +ek + e+ +k ≤ 2ϵf. +(iv) We first show that for any k ∈ N, if αk ≤ ˜α and iteration k is true, then +φ(xk + α ¯dk, ¯τk) ≤ φ(xk, ¯τk) − αkθ∆l(xk, ¯τk, ¯gk, ¯dk). +Since iteration k is true, by Definition 1, it follows that +∥¯gk − ∇fk∥ ≤ max +� +ϵg, κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk) +� +, +and we consider the two cases separately. +Case A When ∥¯gk − ∇fk∥ ≤ κFOαk +� +∆l(xk, ¯τk, ¯gk, ¯dk), by +αk ≤ ˜α ≤ +1−θ +� +¯τ−1 +κl κFO+ L +2κl + +Γ +2¯τminκl +, +26 + +the Cauchy–Schwarz inequality, Assumption 3.2 and Lemmas 3.8 and 3.13, +φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk∥∇fk − ¯gk∥∥ ¯dk∥ + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + +� +¯τk +κl κFOα2 +k∆l(xk, ¯τk, ¯gk, ¯dk) ++ ¯τkL+Γ +2¯τkκl α2 +k∆l(xk, ¯τk, ¯gk, ¯dk) +≤ − +� +1 − +�� +¯τ−1 +κl κFO + +L +2κl + +Γ +2¯τminκl +� +˜α +� +αk∆l(xk, ¯τk, ¯gk, ¯dk) +≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk). +Case B When ∥¯gk − ∇fk∥ ≤ ϵg and iteration k is true, (3.15) holds. Moreover, by the +condition that k < Tε∆l and Definition 4, it follows that +∥¯gk − ∇fk∥ ≤ ϵg < ηε∆l < η +� +∆l(xk, τk, ∇fk, dk). +Therefore, by +αk ≤ ˜α ≤ +2¯τminκl +� +1−θ−η +� +¯τ−1 +κl ·max +�� +1+ϵτω2 +1−ηω5 , +1 +√1−ηω6 +�� +¯τminL+Γ +, +the Cauchy–Schwarz inequality, Assumption 3.2, (3.15) and Lemmas 3.8 +and 3.13, +φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk∥∇fk − ¯gk∥∥ ¯dk∥ + ¯τkL+Γ +2 +α2 +k∥ ¯dk∥2 +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τkL+Γ +2¯τkκl α2 +k∆l(xk, ¯τk, ¯gk, ¯dk) ++ αk¯τk +� +η +� +∆l(xk, τk, ∇fk, dk) +� �� +∆l(xk,¯τk,¯gk, ¯dk) +κl¯τk +� +≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + α2 +k +¯τkL+Γ +2¯τkκl ∆l(xk, ¯τk, ¯gk, ¯dk) ++ αkη +� +¯τk +κl max +�� +1+ϵτω2 +1−ηω5 , +1 +√1−ηω6 +� +∆l(xk, ¯τk, ¯gk, ¯dk) +≤ − αk +�� +1 − η +� +¯τ−1 +κl max +�� +1+ϵτω2 +1−ηω5 , +1 +√1−ηω6 +�� +− ¯τminL+Γ +2¯τminκl ˜α +� +∆l(xk, ¯τk, ¯gk, ¯dk) +≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk). +27 + +Combining Cases A and B, together with the fact the iteration is true, we conclude +the proof of (iv) by +¯φ(xk + αk ¯dk, ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) ≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τkek + ¯τke+ +k +≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf. +(v) If iteration k is unsuccessful, then by definition Zk+1 = Zk, so the inequality holds +trivially. +On the other hand, if iteration k is successful, then starting with the +second equation from (3.11) +Zk+1 − Zk +≤ φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk; ξ+ +k ) + ¯φ(xk+1, ¯τk; ξ+ +k ) − ¯φ(xk, ¯τk; ξk) +− ¯τk+1finf + ¯τkfinf + ¯τkek +≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + (¯τk+1 − ¯τk)(f(xk+1) − finf) ++ 2¯τkϵf + ¯τk(ek + e+ +k ) +≤ 2¯τ−1ϵf + ¯τ−1(ek + e+ +k ). +Therefore, we conclude the proof of (v). +The next two lemmas will be used in the iteration complexity analysis that follows. +Lemma 3.16. For all t ≥ 1, and any ˆp ∈ [0, p), we have +P +�t−1 +� +k=0 +Ik < ˆpt +� +≤ e +− (p−ˆp)2 +2p2 +t. +Proof. The proof is the same as [22, Lemma 3.1]. +Lemma 3.17. For any positive integer t and any ˆp ∈ +� 1 +2, 1 +� +, we have +P +� +Tε∆l > t, +t−1 +� +k=0 +Ik ≥ ˆpt, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2 +� += 0 +where l = max +� +− ln α0−ln ˜α +ln γ +, 0 +� +. +Proof. The proof is the same as [22, Lemma 3.5]. +We are now ready to present the main theorem of the manuscript; the iteration com- +plexity of Algorithm 1. +28 + +Theorem 3.18. Suppose Assumptions 2.1, 2.2, 3.2, 3.4 and 3.14 hold and that the con- +ditions of Oracles 0 and 1 are satisfied. Then, for any s ≥ 0, ˆp ∈ +� +1 +2 + 4¯τ−1ϵf+s +h(˜α) +, p +� +, and +t ≥ +R +ˆp− 1 +2 − 4¯τ−1ϵf+s +h(˜α) +, +P [Tε∆l ≤ t] ≥ 1 − e +− (p−ˆp)2 +2p2 +t − e +− min +� +s2t +2(2¯τ−1ν)2 , +st +2(2¯τ−1b) +� +, +where R = +Z0 +h(˜α) + max +� +ln ˜α−ln α0 +2 ln γ +, 0 +� +, and (p, ˜α, h(·)) are as defined in Lemma 3.15. +Proof. By the law of total probability, +P [Tε∆l > t] =P +� +Tε∆l > t, 1 +t +t−1 +� +k=0 +(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) > 4¯τ−1ϵf + s +� +� +�� +� +A ++ P +� +Tε∆l > t, 1 +t +t−1 +� +k=0 +(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) ≤ 4¯τ−1ϵf + s +� +� +�� +� +B +. +First we bound P[A]. For each iteration k, since Ek and E+ +k satisfy the one-sided sub- +exponential bound (2.11) with parameters (ν, b), one can show that ¯τ−1(Ek + E+ +k ) satisfies +(2.11) with parameters (2¯τ−1ν, 2¯τ−1b). Moreover, since ¯τ−1(Ek + E+ +k ) has mean bounded +by 2¯τ−1ϵf, applying (one-sided) Bernstein’s inequality, for any s ≥ 0 +P[A] ≤ P +� +1 +t +t−1 +� +k=0 +¯τ−1(Ek + E+ +k ) > 2¯τ−1ϵf + s +� +≤ e +− min +� +s2t +2(2¯τ−1ν)2 , +st +2(2¯τ−1b) +� +. +Let l = max +� +− ln α0−ln ˜α +ln γ +, 0 +� +. To bound P[B] we apply the law of total probability, +P[B] = P +� +Tε∆l > t, 1 +t +t−1 +� +k=0 +(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) ≤ 4¯τ−1ϵf + s, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2 +� +� +�� +� +B1 ++ P +� +Tε∆l > t, 1 +t +t−1 +� +k=0 +(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) ≤ 4¯τ−1ϵf + s, +t−1 +� +k=0 +ΘkIkUk ≥ +� +ˆp − 1 +2 +� +t − l +2 +� +� +�� +� +B2 +. +We first show that P[B2] = 0. By Lemma 3.15, for any iteration k < Tε∆l, it follows that +Zk+1 ≤ Zk−h(˜α)+2¯τ−1ϵf +¯τ−1(Ek+E+ +k ) ≤ Zk−h(˜α)+4¯τ−1ϵf if UkIkΘk = 1, and Zk+1 ≤ +29 + +Zk + 2¯τ−1ϵf + ¯τ−1(Ek + E+ +k ) if UkIkΘk = 0. By the definition of the zeroth-order oracle +(Oracle 0), E[Ek] and E[E+ +k ] are bounded above by ϵf for all k. The event Tε∆l > t implies +that Zt > 0 (since Zt = 0 can only happen when Tε∆l ≤ t by the proof of Lemma 3.6). +This together with 1 +t +�t−1 +k=0(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) ≤ 4¯τ−1ϵf + s in turn implies the event +�t−1 +k=0 ΘkIkUk < +� +ˆp − 1 +2 +� +t− l +2. To see this, assume that �t−1 +k=0 ΘkIkUk ≥ +� +ˆp − 1 +2 +� +t− l +2, then +Zt ≤ Z0 − +� +�� +ˆp − 1 +2 +� +t − l +2 +� +h(˜α) − +t−1 +� +k=0 +(2¯τ−1ϵf + ¯τ−1(Ek + E+ +k )) +� +≤ Z0 − +�� +ˆp − 1 +2 +� +t − l +2 +� +h(˜α) + t(4¯τ−1ϵf + s) += Z0 − +�� +ˆp − 1 +2 +� +h(˜α) − (4¯τ−1ϵf + s) +� +t + l +2h(˜α) +≤ 0. +The last inequality above is due to the assumption that ˆp > +1 +2 + 4¯τ−1ϵf+s +h(˜α) +and t ≥ +R +ˆp− 1 +2 − 4¯τ−1ϵf+s +h(˜α) +. Hence, P[B2] = 0. +We now bound P[B1]; by Lemmas 3.16 and 3.17, +P[B1] ≤ P +� +Tε∆l > t, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2 +� += P +� +Tε∆l > t, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2, +t−1 +� +k=0 +Ik < ˆpt +� ++ P +� +Tε∆l > t, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2, +t−1 +� +k=0 +Ik ≥ ˆpt +� +≤ P +�t−1 +� +k=0 +Ik < ˆpt +� ++ P +� +Tε∆l > t, +t−1 +� +k=0 +ΘkIkUk < +� +ˆp − 1 +2 +� +t − l +2, +t−1 +� +k=0 +Ik ≥ ˆpt +� +≤ e +− (p−ˆp)2 +2p2 +t + 0 = e +− (p−ˆp)2 +2p2 +t. +Combining P[A] and P[B], completes the proof. +Corollary +3.19. Under the conditions of Theorem 3.18, +for any s +≥ +0, +ˆp +∈ +� +1 +2 + 4¯τ−1ϵf+s +˜αθωpε2 +∆l , p +� +and t ≥ +ˆR +ˆp− 1 +2 − 4¯τ−1ϵf+s +˜αθωpε2 +∆l +, +P [Tε∆l ≤ t] ≥ 1 − e +− (p−ˆp)2 +2p2 +t − e +− min +� +s2t +2(2¯τ−1ν)2 , +st +2(2¯τ−1b) +� +, +(3.16) +where ˆR = φ(x0,¯τ−1)−φmin−(¯τ−1−¯τmin)finf +˜αθωpε2 +∆l ++ max +� +ln ˜α−ln α0 +2 ln γ +, 0 +� +, equivalently, by Remark 3.5, +ˆR = max{κ2 +H,1} +κlτmin +· φ(x0,¯τ−1)−φmin−(¯τ−1−¯τmin)finf +˜αθωpε2 ++ max +� +ln ˜α−ln α0 +2 ln γ +, 0 +� +, +30 + +ωp = min +� +1−ηω5 +1+ϵτω2 , 1 − ηω6, +1 +1+ω3+ω4 +� +, and the rest of the constants are defined in Assump- +tion 3.14. +Remark 3.20. We make a few remarks about the main theoretical results of the paper +(Theorem 3.18 and Corollary 3.19). +• (Iteration Complexity) By Definition 4 (and Remark 3.5) and Corollary 3.19, we +conclude that, with overwhelmingly high probability, the iteration complexity of Al- +gorithm 1 to generate a primal-dual iterate (xk, yk) ∈ Rn × Rm that satisfies +max{∥∇fk + JT +k yk∥, +� +∥ck∥} ≤ ε is O(ε−2). This iteration complexity is of the same +order in terms of the dependence on ε as the iteration complexity that can be derived +for the deterministic counterpart [16], with the additional restriction that ε is bounded +away from zero (Assumption 3.14) due to the noise and bias in the oracles. +• (Almost Sure Convergence) We note that Algorithm 1 finds an ε-stationary iterate +in a finite number of iterations with probability 1, i.e., P[∩∞ +k=1 ∪∞ +t=k (Tε∆l > t)] = +0. This is a direct consequence of the Borel–Cantelli lemma, since it follows from +(3.16) that the probability of failure events is summable, i.e., �∞ +t=1 P[Tε∆l > t] = +�∞ +t=1 (1 − P[Tε∆l ≤ t]) < ∞. +• (Unconstrained Setting) The high probability complexity bound in this paper is a gen- +eralization of the unconstrained version. In the unconstrained setting, the parameters +reduce to σ = 0, ω1 = 0, ω2 = 1, Γ = 0, ζ = 1, κH = 1, κl = 1, ϵτ = 0, and ¯τk = 1 +for all k ∈ N. Using these values in the results of Corollary 3.19 does not exactly +recover the result from the unconstrained setting [22]. That being said, the order of +the results is the same in terms of the dependence on ε. The existence of the gap is +due to complications that arise in the constrained setting related to the adaptivity of +the merit parameter. We conclude by emphasizing again that though there is a con- +stant difference in function h and value ˜α comparing to [22], our algorithm recovers +the complexity bound of the deterministic variant algorithm [16]. +4 +Numerical Results +In this section, we present numerical results for our proposed algorithm on standard equal- +ity constrained nonlinear optimization problems. The goal of the numerical experiments +is to investigate the efficiency and robustness of the SS-SQP algorithm across a diverse set +of test problems with different levels of noise in the objective function and gradient eval- +uations. All experiments were conducted in MATLAB. Before we present the numerical +results, we describe the test problems, implementation details, and evaluation metrics. +31 + +4.1 +Test Problems +We ran the numerical experiments on a subset of the equality-constrained optimization +problems from the CUTEst collection [18]. We selected the problems that satisfy the fol- +lowing criteria: (i) the objective function is not a constant function, (ii) the total number +of variables and constraints are not larger than 103, and (iii) the singular values of Jaco- +bians of the constraints at all iterates in all runs were greater than 10−8. This resulted in +35 test problems of various dimensions. +We considered noisy (noisy objective function and gradient evaluations) versions +of the 35 CUTEst problems. +Specifically, whenever an objective function or objec- +tive gradient evaluation was required, approximations, +¯f(x; ξ) = N +� +f(x), ϵ2 +f,N +� +and +¯g(x; ξ′) = N +� +∇f(x), +ϵ2 +g,N +n I +� +, respectively, were utilized. +We considered 4 different +noise levels in the objective function and gradient evaluations, dictated by the con- +stants ϵf,N ∈ +� +0, 10−4, 10−2, 10−1� +and ϵg,N ∈ +� +0, 10−4, 10−2, 10−1� +, respectively. Each +CUTEst problem has a unique initial starting point, which was used as the starting +point of all runs of all algorithms. +Moreover, for each selected tuple of noise levels +(ϵf,N, ϵg,N) ∈ +� +0, 10−4, 10−2, 10−1� +× +� +10−4, 10−2, 10−1� +∪ {0} × {0}, where appropriate, +we ran each problem with five different random seeds. +4.2 +Implementation Details +We compared SS-SQP (Algorithm 1) to the adaptive stochastic SQP algorithm proposed +in [7] (which we call AS-SQP) on the previously described noisy CUTEst problems. We set +user-defined parameters for SS-SQP as follows: ϵf = ϵf,N, ϵg = ϵg,N, ϵτ = 10−2, ¯τ−1 = σ = +0.1, γ = 0.5, θ = 10−4, α0 = αmax = 1, and Hk = I for all k ∈ N. For AS-SQP [7] we set +the parameters as follows (this parameter selection was guided by the choice of parameters +in [7]): ¯τ−1 = σ = 0.1, ¯ξ−1 = 1, ϵ = 10−2, θ = 104, Hk = I and βk = 1 for all k ∈ N. The +AS-SQP step size rule requires knowledge (or estimates) of the Lipschitz constants L and Γ. +To this end, we estimated these constants using gradient differences near the initial point, +and set Lk = L and Γk = Γ for all k ∈ N. We note that while the analysis of the SS-SQP +algorithm requires that the condition of Oracles 1 hold, such conditions are not enforced +or checked, and rather in each experiment, the algorithms were given random gradient +estimates with the same, fixed, pre-specified accuracy (as described above). That being +said, a clear distinction between SS-SQP and AS-SQP is the fact that the former requires +function evaluations of the objective function (for the step search) whereas AS-SQP does +not (AS-SQP is an objective-function-free method). We discuss this further when presenting +the numerical results. +32 + +4.3 +Termination Conditions and Evaluation Metrics +In all of our experiments, results are given in terms of infeasibility (∥c(xk)∥∞) and station- +arity (KKT) (max{∥c(xk)∥∞, miny∈Rm ∥∇f(xk) + ∇c(xk)y∥∞}) with respect to different +evaluation metrics (iterations and work). We ran all algorithms with a budget of iterations +(103), and only terminated a run early if an approximate stationary point was found, which +we define as x∗ ∈ Rn such that ∥c(x∗)∥∞ ≤ 10−6 and miny∈Rm ∥∇f(x∗) + ∇c(x∗)y∥∞ ≤ +10−4. +We present results in the form of performance profiles with respect to iterations and +work (defined as the number of function and gradient evaluations), and use the convergence +metric as described in [26], i.e., m(x0) − m(x) ≥ (1 − ϵpp)(m(x0) − mb), where m(x) is +either ∥c(x)∥∞ (infeasibility) or max{∥c(x)∥∞, miny∈Rm ∥∇f(x)+∇c(x)y∥∞} (stationarity +(KKT)), x0 is the initial iterate, and mb is the best value of the metric found by any +algorithm for a given problem instance within the budget, and ϵpp ∈ (0, 1) is the tolerance. +For all experiments presented, we chose ϵpp = 10−3. +4.4 +Noisy Gradients, Exact Functions (ϵf = 0) +In our first set of experiments, we consider problems with exact objective function eval- +uations and noisy objective gradient evaluations and compare SS-SQP and AS-SQP. The +goal of this experiment is to show the effect of noise in the gradient and the advantages of +using (exact) function values. Each row in Figure 1 shows performance profiles for a dif- +ferent noise level in the gradient (bottom row, highest noise level) and each column shows +a different evaluation metric. Starting from the noise-less benchmark case (ϵf = 0 and +ϵg = 0, the first row of Figure 1), it is clear that the performance of the methods in terms +of both infeasibility error and KKT error is similar with a slight advantage in effectiveness +(total problems that can be solved) for SS-SQP in terms of KKT error. As the noise in +the gradient is increased, the gap between the performance of the two methods (in terms +of all metrics) increases favoring SS-SQP. This, of course, is not surprising as SS-SQP uses +additional information (exact function values). These results highlight the effect reliable +function information can have on the performance of the methods. +4.5 +Noisy Functions and Gradients +Here we present results with noise in both the objective function and gradient evaluations. +As in Figure 1, in Figure 2 different rows show results for different noise levels in the gradi- +ent (the bottom row has the highest noise) and different columns show results for different +evaluation metrics. Each performance profile has 4 lines: the AS-SQP (that is objective- +function-free and is not affected by the noise in the function evaluations) and three variants +of the SS-SQP method with different levels of noise in the objective function evaluations. +One can make the following observations. First, not surprisingly, the performance of the +33 + +2 +4 +6 +8 +10 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( f = 0, +g = 0) +AS-SQP +SS-SQP +5 +10 +15 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( f = 0, +g = 0) +AS-SQP +SS-SQP +10 +20 +30 +40 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( f = 0, +g = 0) +AS-SQP +SS-SQP +5 +10 +15 +20 +25 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( f = 0, +g = 0) +AS-SQP +SS-SQP +2 +4 +6 +8 +10 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( f = 0, +g = 10-4) +AS-SQP +SS-SQP +5 +10 +15 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( f = 0, +g = 10-4) +AS-SQP +SS-SQP +10 +20 +30 +40 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( f = 0, +g = 10-4) +AS-SQP +SS-SQP +5 +10 +15 +20 +25 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( f = 0, +g = 10-4) +AS-SQP +SS-SQP +10 +20 +30 +40 +50 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( f = 0, +g = 10-2) +AS-SQP +SS-SQP +5 +10 +15 +20 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( f = 0, +g = 10-2) +AS-SQP +SS-SQP +10 +20 +30 +40 +50 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( f = 0, +g = 10-2) +AS-SQP +SS-SQP +5 +10 +15 +20 +25 +30 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( f = 0, +g = 10-2) +AS-SQP +SS-SQP +20 +40 +60 +80 +100 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( f = 0, +g = 10-1) +AS-SQP +SS-SQP +20 +40 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( f = 0, +g = 10-1) +AS-SQP +SS-SQP +10 +20 +30 +40 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( f = 0, +g = 10-1) +AS-SQP +SS-SQP +5 +10 +15 +20 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( f = 0, +g = 10-1) +AS-SQP +SS-SQP +Figure 1: Performance profiles for AS-SQP and SS-SQP on CUTEst collection [18] with +deterministic objective function evaluations (ϵf = 0) and noisy objective gradient evalu- +ations. Each column corresponds to a different evaluation metric (infeasibility and KKT +errors vs. iterations and work). The noise in the objective gradient evaluations ϵg increases +from top to bottom (First row: ϵg = 0; Second row: ϵg = 10−4; Third row: ϵg = 10−2; +Fourth row: ϵg = 10−1). +SS-SQP method degrades as the noise in the objective function evaluations increases. Sec- +ond, AS-SQP and SS-SQP are competitive and achieve similar robustness levels with respect +to infeasibility errors. Third, and most interestingly, the performance of the methods de- +pends on the relative errors of the function and gradient evaluations. In particular, when +the objective function noise level is sufficiently small compared to the objective gradient +bias, SS-SQP performs better. On the other hand, when the function estimations are too +noisy compared to the noise level in the gradient evaluations, AS-SQP performs slightly +better. These results highlight the power of objective-function-free optimization methods +34 + +in the presence of noise (especially high noise in the objective function evaluations) and the +value of quality (or at least relative quality) function evaluations in methods that require +zeroth-order information. +2 +4 +6 +8 +10 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( g = 10-4) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +10 +20 +30 +40 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( g = 10-4) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +10 +20 +30 +40 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( g = 10-4) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +5 +10 +15 +20 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( g = 10-4) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +10 +20 +30 +40 +50 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( g = 10-2) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +5 +10 +15 +20 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( g = 10-2) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +10 +20 +30 +40 +50 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( g = 10-2) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +5 +10 +15 +20 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( g = 10-2) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +20 +40 +60 +80 +100 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Iterations +( g = 10-1) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +20 +40 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +Infeas. Error/Work +( g = 10-1) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +20 +40 +60 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Iterations +( g = 10-1) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +5 +10 +15 +20 +25 +30 +Performance Ratio +0 +0.2 +0.4 +0.6 +0.8 +1 +KKT Error/Work +( g = 10-1) +AS-SQP +SS-SQP ( f = 10-1) +SS-SQP ( f = 10-2) +SS-SQP ( f = 10-4) +Figure 2: Performance profiles for AS-SQP and SS-SQP on CUTEst collection [18] with +noise in both the objective function and gradient evaluations. Each column corresponds +to a different evaluation metric (infeasibility and KKT vs. +iterations and work). +The +noise in the objective gradient evaluations ϵg increases from top to bottom (First row: +ϵg = 10−4; Second row: ϵg = 10−2; Third row: ϵg = 10−1). The different variants of +SS-SQP correspond to different levels of noise in the objective function evaluations. +5 +Conclusion +We have proposed a step-search SQP algorithm (SS-SQP) for solving stochastic optimiza- +tion problems with deterministic equality constraints, i.e., the setting in which constraint +function values and derivatives are available, but only stochastic estimates of the objec- +tive function and its associated derivatives can be computed. +We showed that under +reasonable assumptions on the inexact probabilistic zeroth- and first-order oracles, with +overwhelmingly high probability, in O(ε−2) iterations our algorithm can produce an iterate +that satisfies the first-order ε-stationarity, which matches the iteration complexity of the +35 + +deterministic counterparts of the SQP algorithm [16]. Numerical results provide strong +evidence for the efficiency and efficacy of the proposed method. Some future directions +include but are not limited to, (1) incorporating stochastic constraint evaluations into the +algorithm design and analysis, and (2) extending the framework to the setting with inequal- +ity constraints. Both avenues above are subjects of future work as they require significant +adaptations in the design, analysis, and implementation of the algorithm. +Acknowledgments +This material is based upon work supported by the Office of Naval Research under award +number N00014-21-1-2532. We would like to thank Professors Frank E. Curtis and Katya +Scheinberg for their invaluable support and feedback. +References +[1] Afonso S Bandeira, Katya Scheinberg, and Luis Nunes Vicente. Convergence of trust- +region methods based on probabilistic models. +SIAM J. 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SIAM, 2021. +39 + diff --git a/ttAyT4oBgHgl3EQfmviR/content/tmp_files/load_file.txt b/ttAyT4oBgHgl3EQfmviR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9673fc62022cf78d2cda64f65db468887d500f3f --- /dev/null +++ b/ttAyT4oBgHgl3EQfmviR/content/tmp_files/load_file.txt @@ -0,0 +1,1294 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf,len=1293 +page_content='A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization Albert S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Berahas† Miaolan Xie‡ Baoyu Zhou∗§ January 3, 2023 Abstract A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' It is assumed that constraint function values and derivatives are available, but only stochastic approxi- mations of the objective function and its associated derivatives can be computed via inexact probabilistic zeroth- and first-order oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Under reasonable assumptions, a high-probability bound on the iteration complexity of the algorithm to approximate first-order stationarity is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Numerical results on standard nonlinear optimiza- tion test problems illustrate the advantages and limitations of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 Introduction In this paper, we propose a step-search1 sequential quadratic programming (SQP) algo- rithm for solving nonlinear equality-constrained stochastic optimization problems of the form min x∈Rn f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' c(x) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1) where f : Rn → R and c : Rn → Rm are both continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We consider the setting in which exact function and derivative information of the objective function is unavailable, instead, only random estimates of the objective function ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) ≈ f(x) †Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' of Industrial and Operations Engineering, University of Michigan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (albertberahas@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='com) ‡School of Operations Research and Information Engineering, Cornell University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (mx229@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='edu) ∗Booth School of Business, The University of Chicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (baoyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='zhou@chicagobooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='edu) §Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1We use the term step search methods, coined in [22] to differentiate with line search methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Step search methods are similar to line search methods, but the search (step) direction can change during the back-tracking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='00477v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='OC] 1 Jan 2023 and its first-order derivative ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)) ≈ ∇f(x) are available via inexact probabilistic oracles, where Ξ0(x) (with probability space (Ω0, FΩ0, P 0)) and Ξ1(x) (with probability space (Ω1, FΩ1, P 1)) denote the underlying randomness in the objective function and gra- dient estimates, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' On the other hand, the constraint function value c(x) and its Jacobian ∇c(x)T are assumed to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Such deterministically constrained stochas- tic optimization problems arise in multiple science and engineering applications, including but not limited to computer vision [37], multi-stage optimization [39], natural language processing [30], network optimization [9], and PDE-constrained optimization [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The majority of the methods proposed in the literature for solving deterministically equality-constrained stochastic optimization problems follow either projection or penalty approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The former type of methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', stochastic projection methods [21, 23–25]) require that the feasible region satisfies strict conditions, to ensure well-definedness, that are not satisfied by general nonlinear functions and thus are not readily applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In contrast, the latter, stochastic penalty methods [14, 34], do not impose such conditions on the feasible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These methods transform constrained problems into unconstrained problems via a constraint penalization term in the objective function and apply stochas- tic algorithms to solve the transformed unconstrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Stochastic penalty methods are easy to implement and well-studied, however, the empirical perfor- mance of such methods is sensitive to parameter choices and ill-conditioning, and is usually inferior to paradigms that treat constraints as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Recently, a class of stochastic SQP methods has been developed for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These methods outperform stochastic penalty methods empirically and have convergence guaran- tees in expectation [7, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In [7], the authors propose an objective-function-free stochastic SQP method with adaptive step sizes for the fully stochastic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In contrast, in [28], the authors propose a stochastic step search (referred to as line search in the paper [28]) SQP method for the setting in which the errors in the function and derivative approxima- tions can be diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We note that several algorithm choices in the two papers [7, 28], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', merit functions and merit parameters, are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Several other extensions have been proposed [3, 6, 8, 17, 27, 32], and very few of these works (or others in the literature) derive worst-case iteration complexity (or sample complexity) due to the difficulties that arise because of the constrained setting and the stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Notable exceptions are, [16] where the authors provide convergence rates (and complexity guarantees) for the algorithm proposed in [7], and [3, 29] that provide complexity bounds for variants of the stochastic SQP methods under additional assumptions and in the setting in which the errors can be diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We note that, with the exception of [32], all methods mentioned above assume access to unbiased estimates of the gradients (and function values where neces- sary), whereas in this paper, we propose an algorithm that can handle biased function and gradient estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all aforementioned methods, the most vital ingredient is the quality and reliability of the random estimates of the objective function and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In our setting, neither the objective function nor its derivatives are assumed to be directly accessible, only 2 stochastic approximations of them are accessible to the algorithm in the form of inexact probabilistic zeroth-order and first-order oracles (precise definitions will be introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Such oracles have been proposed and utilized in several works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', [1, 12, 20, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, these probabilistic oracles and their variants have been proposed for direct- search methods [20, 36], trust-region methods [1, 10, 15, 19], and step-search methods [2, 13, 28, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We note that only [28] considers the setting with (equality) constraints, but iteration complexity (or sample complexity) results are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 Contributions In this paper, we design, analyze, and implement a step-search SQP (SS-SQP) method for solving nonlinear equality-constrained stochastic optimization problems where exact con- straint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These stochas- tic approximations are computed via inexact probabilistic zeroth- and first-order oracles, which are similar to those in [22], with parameters controlling the accuracy and reliability of the approximations, and allowing for biased approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Our proposed algorithm is inspired by state-of-the-art line search SQP methods [11] in conjunction with the recent stochastic adaptive step-search framework developed in [22] for the unconstrained stochas- tic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' At every iteration, the algorithm constructs a model of the reduction in the merit function that serves the dual purpose of a measure of sufficient progress (part of the step size computation) and a proxy for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To mitigate the challenges that arise due to the noise in the objective function evaluations, our step-search method employs a relaxed sufficient decrease condition similar to that proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Under reasonable assumptions, we provide a high probability worst-case iteration complexity bound for the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Specifically, we prove that with overwhelmingly high probability, our proposed algorithm generates a first-order ε-stationary iterate in O(ε−2) iterations, where ε is bounded away from zero and its lower bound is dictated by the noise and bias in the zeroth- and first-order oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The complexity bound derived matches that of the deter- ministic algorithm provided in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' There are two key differences between our paper and [16]: (i) our algorithm requires access to the objective function whereas the method in [16] is objective-function-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' and (ii) our first-order oracle provides estimates with sufficient accuracy only with some probability and can provide arbitrarily bad estimates otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Finally, numerical results on standard nonlinear equality-constrained test problems [18] illustrate the efficiency and efficacy of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 Notation Let R denote the set of real numbers, Rn denote the set of n-dimensional real vectors, Rm×n denote the set of m-by-n-dimensional real matrices, N denote the set of natural numbers, and Sn denote the set of n-by-n-dimensional real symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any 3 a ∈ R, let R>a (R≥a) denote the set of real numbers strictly larger than (larger than or equal to) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We use ∥ · ∥ to denote the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We use k ∈ N as the iteration counter of the algorithm, and for brevity, we use a subscript k for denoting information at the kth iterate, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', fk := f(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' All quantities with over-bars are stochastic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) and ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)) (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3), and ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1(x))) denote realizations of ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3 Organization The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The algorithmic framework is introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The analysis of the algorithm is established in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We report numer- ical results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Concluding remarks and future research directions are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2 Algorithm To solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), we design an iterative algorithm based on the SQP paradigm that generates: (i) a primal iterate sequence {xk}, (ii) a primal trial iterate sequence {x+ k }, (iii) a primal search direction sequence { ¯dk}, (iv) a dual iterate sequence {¯yk}, (v) a step size sequence {αk}, (vi) a merit parameter sequence {¯τk}, and, (vii) a trial merit parameter sequence {¯τ trial k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We discuss each of these sequences in below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We make the following assumption throughout the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let X ⊆ Rn be an open convex set including iterates {xk} and trial it- erates {x+ k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The objective function f : Rn → R is continuously differentiable and bounded below over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The objective gradient function ∇f : Rn → Rn is L-Lipschitz continuous and bounded over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The constraint function c : Rn → Rm (where m ≤ n) is continuously differentiable and bounded over X, and each gradient ∇ci : Rn → Rn is γi-Lipschitz con- tinuous and bounded over X for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The singular values of J := ∇cT are bounded away from zero over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 is a standard assumption in the deterministic constrained optimization literature [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, there exist constants {κg, κc, κJ, κσ} ⊂ R>0 and finf ∈ R such that for all k ∈ N, finf ≤ fk, ∥∇fk∥ ≤ κg, ∥ck∥1 ≤ κc, ∥Jk∥ ≤ κJ, and ∥(JkJT k )−1∥ ≤ κσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We should note that by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, linear independence constraint qualifications (LICQ) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, for all x ∈ Rn, d ∈ Rn and α ∈ R≥0 it follows that f(x + αd) ≤ f(x) + α∇f(x)T d + L 2 α2∥d∥2 and ∥c(x + αd)∥1 ≤ ∥c(x) + α∇c(x)T d∥1 + Γ 2 α2∥d∥2, where Γ = m � i=1 γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1) 4 In this paper, we are particularly interested in finding some primal-dual iterate (x, y) ∈ Rn × Rm that satisfies the first-order stationarity conditions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To this end, let L : Rn × Rm → R be the Lagrangian of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), defined as L(x, y) = f(x) + yT c(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2) where y ∈ Rm are the dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The first-order stationarity conditions for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), which are necessary by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 (due to the inclusion of the LICQ), are 0 = � ∇xL(x, y) ∇yL(x, y) � = � ∇f(x) + ∇c(x)y c(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3) In the remainder of this section we introduce the key algorithmic components: the merit function and its associated models, the search direction computation and merit parameter updating mechanism, and the inexact probabilistic zeroth- and first-order oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The main algorithm is Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 Merit function The merit function φ : Rn × R>0 → R is defined as φ(x, τ) := τf(x) + ∥c(x)∥1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4) where τ ∈ R>0, the merit parameter, acts as a balancing parameter between the objective function and the constraint violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given the gradient (approximation) g ∈ Rn and a search direction d ∈ Rn, the model of merit function l : Rn ×R>0 ×Rn ×Rn → R is defined as l(x, τ, g, d) := τ(f(x) + gT d) + ∥c(x) + ∇c(x)T d∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given a search direction d ∈ Rn that satisfies linearized feasibility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', c(x)+∇c(x)T d = 0, the reduction in the model of the merit function ∆l : Rn × R>0 × Rn × Rn → R is defined as ∆l(x, τ, g, d) :=l(x, τ, g, 0) − l(x, τ, g, d) = − τgT d + ∥c(x)∥1 − ∥c(x) + ∇c(x)T d∥1 = − τgT d + ∥c(x)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) We use the reduction in the model of the merit function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) to monitor the progress made by our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We discuss this in more detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 Algorithmic components We now establish how to: (i) compute the primal search direction sequence { ¯dk}, (ii) update the merit parameter sequence {¯τk}, and (iii) update the primal iterate sequence 5 {xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These sequences depend on the approximation of the gradient of the objective function sequence {¯g(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(xk))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let ¯g(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1(xk)) denote the realization of ¯g(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(xk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To simplify the notation, in this subsection we drop the dependence on the randomness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', ¯gk = ¯g(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1(xk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' At each iteration k ∈ N, the primal search direction ¯dk ∈ Rn and the dual variable ¯yk ∈ Rm are computed by solving the linear system of equations � Hk JT k Jk 0 � � ¯dk ¯yk � = − � ¯gk ck � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6) where {Hk} satisfies the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N, Hk ∈ Sn is chosen independently from ¯gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, there exist constants {κH, ζ} ⊂ R>0 such that for all k ∈ N, ∥Hk∥ ≤ κH and uT Hku ≥ ζ∥u∥2 for any u ∈ Null(Jk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' It is well known that under Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, there is a unique solution ( ¯dk, ¯yk) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6), and, thus, the vectors ¯dk ∈ Rn and ¯yk ∈ Rm are well-defined [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Next, we present the merit parameter updating mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given constants {ϵτ, σ} ⊂ (0, 1), for all k ∈ N, we compute ¯τk via ¯τk ← � ¯τk−1 if ¯τk−1 ≤ ¯τ trial k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' min � (1 − ϵτ)¯τk−1, ¯τ trial k � otherwise, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) where ¯τ trial k ← � � � ∞ if ¯gT k ¯dk + max � ¯dT k Hk ¯dk, 0 � ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (1−σ)∥ck∥1 ¯gT k ¯dk+max{ ¯dT k Hk ¯dk,0} otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) The merit parameter updating mechanism ensures that the sequence of merit parameter values is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, the updating mechanism is designed to ensure that the reduction in the model of the merit function is sufficiently positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8), it follows that (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) ∆l(xk, ¯τk, ¯gk, ¯dk) ≥ ¯τk max � ¯dT k Hk ¯dk, 0 � + σ∥ck∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) In the deterministic setting, the reduction in the model of the merit function is zero only at iterates that satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' After updating the merit parameter ¯τk, we evaluate ∆l(xk, ¯τk, ¯gk, ¯dk), the stochastic model reduction of the merit function, and use it to check for sufficient progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Specif- ically, given a step size αk, we compute a candidate iterate x+ k := xk + αk ¯dk and check whether sufficient progress can be made via the following modified sufficient decrease con- dition ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x+ k )) ≤ ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(xk)) − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) 6 where ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x+ k )) and ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(xk)) are merit function estimates, θ ∈ (0, 1) is a user-defined parameter and ϵf is an upper bound on the expected noise in the objective function approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We note that ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x+ k )) and ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(xk)) are real- izations of the zeroth-order oracle described in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The positive term on the right-hand-side allows for a relaxation in the sufficient decrease condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', the merit function may increase after a step, and serves to correct for the noise in the merit function approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) is satisfied, we accept the candidate point x+ k by setting xk+1 ← x+ k , and potentially increase the step size for the next iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', αk+1 ≥ αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) is not satisfied, the algorithm does not accept the candidate iterate, instead, it sets xk+1 ← xk and shrinks the step size for the next iteration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', αk+1 < αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This step update rule is the centerpiece of our step-search method, and is fundamentally different from traditional line-search strategies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' see [5, 13, 22] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Contrary to line search methods, which compute a search direction and then look for a step size along that direction, in our approach the search direction changes in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We conclude this section by drawing a few parallels to the unconstrained setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' First, in the unconstrained setting (with Hk = I), the quantity ∆l(xk, ¯τk, ¯gk, ¯dk) reduces to ∥¯gk∥2, which provides a sufficient descent measure and is an approximate first-order stationarity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In the constrained setting, the reduction in the model of the merit function will play a similar role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Second, in the unconstrained optimization setting, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) recovers the sufficient decrease condition used by some noisy unconstrained optimization algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' see [4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3 Probabilistic oracles In many real-world applications exact objective function and derivative information cannot be readily computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Instead, in lieu of these quantities, approximations are available via inexact probabilistic zeroth- and first-order oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These oracles produce approximations of different accuracy and reliability, and are formally introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Oracle 0 (Probabilistic zeroth-order oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given x ∈ Rn, the oracle computes ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x)), a realization of ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)), which is a (random) estimate of the objective function value f(x), where Ξ0(x) denotes the underlying randomness (may depend on x) with associated probability space � Ω0, FΩ0, P 0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) := | ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x))−f(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any x ∈ Rn, e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) is a “one-sided” sub-exponential random variable with parameters {ν, b} ⊂ R≥0, whose mean is bounded by some constant ϵf ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Specifically, for all x ∈ Rn and λ ∈ [0, 1/b], EΞ0(x) � e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) � ≤ ϵf and EΞ0(x) � exp(λ(e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) − E � e(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) � )) � ≤ exp � λ2ν2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) The stochastic approximation of the merit function value is defined as ¯φ(x, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x)) = τ ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0(x)) + ∥c(x)∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 7 Oracle 1 (Probabilistic first-order oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given x ∈ Rn and α ∈ R>0, the oracle computes ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1(x)), a realization of ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)), which is a (random) estimate of the gradient of the objective function ∇f(x), such that PΞ1(x) � ∥¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)) − ∇f(x)∥ ≤ max � ϵg, κFOα � ∆l(x, ¯τ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)), ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x)), ¯d(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1(x))) � � ≥ 1 − δ, where Ξ1(x) denotes the underlying randomness (may depend on x) with associated prob- ability space (Ω1, FΩ1, P 1), (1 − δ) ∈ ( 1 2, 1] is the probability that the oracle produces a gradient estimate that is “sufficiently accurate” (related to the reliability of the oracle) and {ϵg, κFO} ⊂ R≥0 are constants intrinsic to the oracle (related to the precision of the oracle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In the rest of the paper, to simplify the notation we drop the dependence on x in ξ0(x) and ξ1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, we use ξ+ k to represent ξ0(x+ k ), the randomness in the zeroth-order oracle evaluated at the trial point x+ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We make a few remarks about Oracles 0 and 1: Oracles 0 and 1 are similar to those defined in [12, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For a full discussion and examples of the oracles, we refer interested readers to [22, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Oracle 1 is a natural generalization of the ones defined in [12, 22] to the constrained setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In particular, the right-hand-side of Oracle 1 reduces to max � ϵg, κFOα∥¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ1)∥ � in the unconstrained setting, and is precisely what is used in [12, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The presence of ϵg ∈ R≥0 in the max term in Oracle 1 allows the gradient approxi- mations to be biased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' the magnitude of the bias is proportional to ϵg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 Algorithmic framework We are ready to introduce our stochastic step-search SQP method (SS-SQP) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We make the following remarks about SS-SQP: (Step-search) Algorithm 1 is a step-search algorithm, whose main difference from traditional line-search methods is that only a single trial iterate is tested at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That is, if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) is not satisfied, the step size is reduced and a new search direction and candidate iterate are computed in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This strategy has been employed in other papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', see [5, 13, 22, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We should note that at every iteration, even if the iterate does not change, our algorithm requires new objective function and gradient estimates in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 8 Algorithm 1 Adaptive Step-Search SQP (SS-SQP) Require: initial iterate x0 ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' initial merit parameter ¯τ−1 ∈ R>0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' maximum step size αmax ∈ (0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' initial step size α0 ∈ (0, αmax];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' parameter ϵf ∈ R≥0 of the zeroth-order oracle (Oracle 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' and other constant parameters {γ, θ, σ, ϵτ} ⊂ (0, 1) 1: for all k ∈ N do 2: Generate ¯gk = ¯g(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1 k) via Oracle 1 with α = αk, ¯dk = ¯d(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1 k) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6), and ¯τk = ¯τ(xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ1 k) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) 3: Let x+ k = xk + αk ¯dk, and generate ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0 k) and ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) via Oracle 0 4: if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) holds then 5: Set xk+1 ← x+ k and αk+1 ← min{αmax, γ−1αk} 6: else 7: Set xk+1 ← xk and αk+1 ← γαk 8: end if 9: end for (Modified sufficient decrease condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10)) The 2¯τkϵf term on the right-hand- side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) is a correction term added to compensate for the inexactness of the probabilistic zeroth-order oracle (Oracle 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This correction provides a relaxation to the sufficient decrease requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In contrast to traditional sufficient decrease conditions, the modified condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) allows for a relaxation that is proportional to the noise level of Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Objective function evaluations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Line 3) The randomness associated with the evalu- ation of the objective function value at the candidate iterate x+ k (Line 3) is not the same as that of the evaluation at the current point xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, we note that even for unsuccessful iterations (where the iterates do not change) the objective function values are re-evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Objective gradient evaluations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Line 2) In order to generate an estimate of the gradi- ent of the objective function that satisfies the conditions of Oracle 1, one can employ a procedure (a loop) similar to [38, Algorithm 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The idea is to refine the estimate progressively in order to generate one that satisfies the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Indeed, in many real-world problems, including empirical risk minimization in machine learning, one can improve the gradient approximation by progressively using a larger number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Maximum step size αmax) We pick αmax ∈ (0, 1] mainly to simplify our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That being said, the unit upper bound on αmax is motivated by the deterministic constraint setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In the deterministic setting (without any noise), the merit function decrease is upper bounded by a nonsmooth function, whose only point of nonsmothness is at α = 1, which complicates the analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' see [7, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 9 Before we proceed, we define the stochastic process related to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let Mk denote {Ξ0 k, Ξ+ k , Ξ1 k} with realizations {ξ0 k, ξ+ k , ξ1 k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The algorithm generates a stochastic process: {(Gk, Dk, Tk, ¯φ(Xk, Tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0 k), ¯φ(X+ k , Tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ+ k ), Xk, Ak)} with realizations {(¯gk, ¯dk, ¯τk, ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ0 k), ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ), xk, αk)}, adapted to the filtration {Fk : k ≥ 0}, where Fk = σ(M0, M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' , Mk) and σ denotes the σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' At iteration k, Gk is the random gradient, Dk is the random primal search direction, Tk is the random merit param- eter, ¯φ(Xk, Tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0 k) and ¯φ(X+ k , Tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ+ k ) are the random noisy merit function evaluations at the current point and the candidate point, respectively, Xk is the random iterate at itera- tion k and Ak is the random step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Note that Gk, Dk, Tk are dictated by Ξ1 k (Oracle 1) and the noisy merit function evaluations are dictated by Ξ0 k, Ξ+ k (Oracle 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 3 Theoretical analysis In this section, we analyze the behavior of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For brevity, throughout this sec- tion, we assume Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 hold and do not restate this fact in every lemma and theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We begin by presenting some preliminary results, definitions, and assump- tions and then proceed to present a worst-case iteration complexity bound for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 Preliminaries, definitions & assumptions We first define some deterministic quantities that are used in the analysis of Algorithm 1, and which are never explicitly computed in the implementation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let (dk, yk) ∈ Rn × Rm be the solution of the deterministic counterpart of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', � Hk JT k Jk 0 � � dk yk � = − � ∇fk ck � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1) The norm of the gradient of the Lagrangian (defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2)) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), which is used as a first-order stationarity measure, can be upper bounded at every primal-dual iterate (xk, yk) as ����� � ∇fk + JT k yk ck ������ = ����� � −Hkdk −Jkdk ������ ≤ (κH + κJ)∥dk∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2) where the equality is by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1) and the inequality follows by Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2) implies that dk, the primal search direction, can be used as a proxy of the first-order stationary measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The following lemma shows that the tuple (dk, yk) is bounded for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' There exist constants {κd, κy} ⊂ R>0 such that ∥dk∥ ≤ κd and ∥yk∥ ≤ κy for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the Cauchy–Schwarz inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), we have ����� � dk yk ������ = ������ � Hk JT k Jk 0 �−1 � ∇fk ck ������� ≤ ������ � Hk JT k Jk 0 �−1������ ����� � ∇fk ck ������ , where both terms on the right-hand side of the inequality are bounded by Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, we define τk ∈ R>0 and τ trial k ∈ R>0, the deterministic counterparts of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8), τk ← � ¯τk if ¯τk ≤ τ trial k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' min � (1 − ϵτ)¯τk, τ trial k � otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3) where τ trial k ← � � � ∞ if ∇fT k dk + max � dT k Hkdk, 0 � ≤ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (1−σ)∥ck∥1 ∇fT k dk+max{dT k Hkdk,0} otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4) We emphasize again that {(τk, τ trial k )}k∈N are introduced only for the purposes of the anal- ysis, and in Algorithm 1 they are never computed (not even in the setting in which the true gradient is used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', ¯gk = ∇f(xk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We also note that this definition is not the same as that in [7, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The difference is in the fact that in the computation of τk, the comparison is made to ¯τk instead of ¯τk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This is important for the analysis, since this guarantees τk ≤ ¯τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We assume that the merit parameter sequence {¯τk} generated in the stochastic setting is bounded away from zero (Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Such an assumption has been adopted in previous literature [6–8, 16, 17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' we refer readers to [7, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2] and [16, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2] for detailed discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Finally, we note that we only assume that {¯τk} is bounded away from zero, and never require the knowledge of ¯τmin in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let {¯τk} be the merit parameter sequence generated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' There exists a constant ¯τmin ∈ R>0 such that for every realization of Algorithm 1, ¯τk ≥ ¯τmin for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Next, we state and prove a provide a useful property with regards to the deterministic merit parameter sequence {τk} defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Suppose Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 holds, then there exists a positive constant τmin ∈ R>0 such that for every realization of Algorithm 1, τk ≥ τmin for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By [7, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='16], {τ trial k } ⊂ R>0 ∪ {+∞} is always bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We define τ trial min ∈ R>0 such that τ trial min ≤ τ trial k for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4) and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, one may pick τmin = min{(1 − ϵτ)τ trial min , ¯τmin} to conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 11 Our final assumption relates to the zeroth-order oracle (Oracle 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let Ek and E+ k be the errors in the objective function evaluations from Oracle 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', Ek := �� ¯f(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0 k) − f(Xk) ��, and E+ k := �� ¯f(X+ k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ+ k ) − f(X+ k ) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We as- sume that either {Ek} and {E+ k } are deterministically bounded by ϵf ∈ R≥0, or that the summation of the errors {Ek + E+ k } are independent over different iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Next, we introduce several definitions necessary for the analysis of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Specif- ically, we define true/false iterations (Definition 1), successful/unsuccessful iterations (Definition 2) and large/small steps (Definition 3), and introduce three indicator vari- ables respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' An iteration k ∈ N is true if ∥¯gk − ∇fk∥ ≤ max � ϵg, κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk) � and ek + e+ k ≤ 2ϵf, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) where ∆l(xk, ¯τk, ¯gk, ¯dk) is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and the constants ϵf, ϵg and κFO are the same ones as in Oracles 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) does not hold, we call the iteration a false iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We use the random indicator variable Ik to denote if an iteration is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Given a constant θ ∈ (0, 1), let ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) and ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) be obtained by Oracle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) holds, then iteration k is successful, otherwise, it is an unsuccessful iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We use the random indicator variable Θk to denote whether an iteration is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any k ∈ N, if min{αk, αk+1} ≥ ˜α where ˜α is some problem-dependent positive real number (defined explicitly in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15), then we call the step a large step and set the indicator variable Uk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Otherwise, we call the step k a small step and set Uk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We show that under appropriate conditions, if the step is a small step and the iteration is true, then, the iteration is guaranteed to be successful (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The last definition is for the stopping time (Tε∆l) and a measure of progress ({Zk}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any realization of Algorithm 1, define Tε∆l = min{k : � ∆l(xk, τk, ∇fk, dk) ≤ ε∆l}, the number of iterations required to reach a first-order ε- stationary iterate, where ε = Ω(ε∆l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We discuss the explicit relationship between ε and ε∆l in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, for all k ∈ N, let Zk := φ(xk, ¯τk) − φmin − (¯τkfinf − ¯τminfinf), where φmin is a lower bound of φ(·, ¯τmin) over X and ¯τmin is defined in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' A key ingredient of our algorithm is the stopping time Tε∆l that is related to ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In fact, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2), Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9 (see below), the 12 stopping time Tε∆l defined in Definition 4 is the number of iterations needed to achieve a first-order ε-stationary iterate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', max{∥∇fk + JT k yk∥, � ∥ck∥} ≤ ε, where ε = max{κH,1} √κlτmin ε∆l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6) We note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6) is the same stationarity measure as that used in [16, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (5)], and is a non-standard first-order stationary measure compared to ����� � ∇fk + JT k yk ck ������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That said, one can show that ����� � ∇fk + JT k yk ck ������ ≤ 2 max{∥∇fk + JT k yk∥, ∥ck∥} ≤ 2 max{κH,κJ} √κlτmin ε∆l = Ω(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Throughout this paper we focus on (and provide complexity bounds for) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6) as it provides a stronger result for feasibility (∥ck∥) when ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 Main Technical Results We build toward the main result of the paper (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='18) through a sequence of technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Our first lemma shows that Zk (defined in Definition 4) is always non- negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N, Zk ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4) and Definition 4 that Zk = φ(xk, ¯τk) − φmin − (¯τkfinf − ¯τminfinf) = (¯τk(fk − finf) + ∥ck∥1) − φmin + ¯τminfinf ≥ (¯τmin(fk − finf) + ∥ck∥1) − φmin + ¯τminfinf = (¯τminfk + ∥ck∥1) − φmin = φ(xk, ¯τmin) − φmin ≥ 0, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next lemma reveals the critical role of the merit parameter update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Furthermore, if ¯τk ̸= ¯τk−1, then 0 < ¯τk ≤ (1 − ϵτ)¯τk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By Algorithm 1, we have ¯τk ≤ ¯τ trial k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8), it follows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) is satisfied for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7), if ¯τk ̸= ¯τk−1, then ¯τk = min � (1 − ϵτ)¯τk−1, ¯τ trial k � ≤ (1 − ϵτ)¯τk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, when ck = 0, it follows from Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) that ¯dk ∈ Null(Jk) and ¯gT k ¯dk+max{ ¯dT k Hk ¯dk, 0} = ¯gT k ¯dk+ ¯dT k Hk ¯dk = cT k ¯yk = 0, which implies ¯τ trial k = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Therefore, we have ¯τ trial k > 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Finally, by ¯τ−1 ∈ R>0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7), we have ¯τk > 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 13 The next lemma provides a useful lower bound for the reduction in the model of the merit function, ∆l(xk, ¯τk, ¯gk, ¯dk), that is related to the primal search direction (∥ ¯dk∥2) and a measure of infeasibility (∥ck∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' There exists some constant κl ∈ R>0 such that for all k ∈ N, ∆l(xk, ¯τk, ¯gk, ¯dk) ≥ κl¯τk(∥ ¯dk∥2 + ∥ck∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any iteration k ∈ N, by [7, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4], there exists some constant κl ∈ R>0 such that −¯τk(¯gT k ¯dk + 1 2 max{ ¯dT k Hk ¯dk, 0}) + ∥ck∥1 ≥ κl¯τk(∥ ¯dk∥2 + ∥ck∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By ¯τk ∈ R>0 (from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7), this implies that ∆l(xk, ¯τk, ¯gk, ¯dk) = −¯τk¯gT k ¯dk + ∥ck∥1 ≥ −¯τk(¯gT k ¯dk + 1 2 max{ ¯dT k Hk ¯dk, 0}) + ∥ck∥1, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' There exists some constant κl ∈ R>0 such that for all k ∈ N, ∆l(xk, τk, gk, dk) ≥ κlτk(∥dk∥2 + ∥ck∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The proof follows the same logic as that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 with the stochastic quantities replaced by their deterministic counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By [7, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4], the desired inequality is satisfied for the same constant κl defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next lemma bounds the errors in the stochastic search directions and dual variables, respectively, with respect to the errors in the gradient approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N, there exist constants {ζ, ω1} ⊂ R>0 such that ∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇fk∥ and ∥¯yk − yk∥ ≤ ω1∥¯gk − ∇fk∥, where ζ is defined in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the Cauchy–Schwarz inequality, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), and the fact that ( ¯dk − dk) ∈ Null(Jk), it follows that ∥ ¯dk − dk∥∥¯gk − ∇fk∥ ≥ ( ¯dk − dk)T (∇fk − ¯gk) = ( ¯dk − dk)T (Hk( ¯dk − dk) + JT k (¯yk − yk)) = ( ¯dk − dk)T Hk( ¯dk − dk) ≥ ζ∥ ¯dk − dk∥2, which proves that ∥ ¯dk −dk∥ ≤ ζ−1∥¯gk −∇fk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Next, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1) and Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 it follows that ¯yk − yk = −(JkJT k )−1Jk � (¯gk − ∇fk) + Hk( ¯dk − dk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 14 By the triangle inequality, the Cauchy–Schwarz inequality, Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 and the fact that ∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇fk∥, it follows that ∥¯yk − yk∥ = ∥(JkJT k )−1Jk � (¯gk − ∇fk) + Hk( ¯dk − dk) � ∥ ≤ ∥(JkJT k )−1∥∥Jk∥(∥¯gk − ∇fk∥ + ∥Hk∥∥ ¯dk − dk∥) ≤ κσκJ(1 + κHζ−1)∥¯gk − ∇fk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Setting ω1 = κσκJ(1 + κHζ−1) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next lemma relates the inner product of the stochastic gradient and stochastic search direction to the stochastic reduction in the model of the merit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We consider two cases that are related to the two cases in the max term of Oracle 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N: If ∥¯gk − ∇fk∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk), then ¯τk|¯gT k ¯dk| ≤ � max{κH,κy} κl + √¯τk(1+κHζ−1)κFOαk √κl � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If ∥¯gk − ∇fk∥ ≤ ϵg, ¯τk|¯gT k ¯dk| ≤ max{κH,κy}+1 κl ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk(1+κHζ−1) 2 4 ϵ2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If ∥¯gk−∇fk∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk), by the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6), Assump- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' it follows that ¯τk|¯gT k ¯dk| = ¯τk|(Hk ¯dk + JT k yk + JT k (¯yk − yk))T ¯dk| ≤ ¯τk(| ¯dT k Hk ¯dk| + |yT k Jk ¯dk| + |(¯yk − yk)T Jk ¯dk|) ≤ ¯τk(κH∥ ¯dk∥2 + ∥yk∥∥ck∥ + ∥(¯gk − ∇fk) + Hk( ¯dk − dk)∥∥ ¯dk∥) ≤ max{κH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' κy} · ¯τk(∥ ¯dk∥2 + ∥ck∥) + ¯τk(∥¯gk − ∇fk∥ + κH∥ ¯dk − dk∥)∥ ¯dk∥ ≤ max{κH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='κy} κl ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + ¯τk � 1 + κHζ−1� ∥¯gk − ∇fk∥∥ ¯dk∥ ≤ max{κH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='κy} κl ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + √¯τk(1+κHζ−1)κFOαk √κl ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' which completes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Using similar logic, if ∥¯gk−∇fk∥ ≤ ϵg, by the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6), Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10, and the fact that ab ≤ a2+ b2 4 holds for any {a, b} ⊂ R, it follows 15 that ¯τk|¯gT k ¯dk| ≤ max{κH,κy} κl ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk � 1 + κHζ−1� ∥¯gk − ∇fk∥∥ ¯dk∥ ≤ max{κH,κy} κl ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk � 1 + κHζ−1� ϵg∥ ¯dk∥ ≤ max{κH,κy} κl ∆l(xk, ¯τk, ¯gk, ¯dk) + √¯τk(1+κHζ−1) √κl ϵg � ∆l(xk, ¯τk, ¯gk, ¯dk) ≤ max{κH,κy}+1 κl ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τk(1+κHζ−1) 2 4 ϵ2 g, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next lemma provides a useful upper bounds for the errors related to the stochastic search directions (and gradients) for the same two cases as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N: If ∥¯gk − ∇fk∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk), then |∇fT k dk − ¯gT k ¯dk| ≤ � (1+κHζ−1)κFOαk √κl¯τk + κ2 FOα2 k ζ � ∆l(xk, ¯τk, ¯gk, ¯dk) and |dT k Hkdk − ¯dT k Hk ¯dk| ≤ � 2κHζ−1κFOαk √κl¯τk + κHκ2 FOα2 k ζ2 � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If ∥¯gk − ∇fk∥ ≤ ϵg, then |∇fT k dk − ¯gT k ¯dk| ≤ (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) + ζ−1ϵ2 g and |dT k Hkdk − ¯dT k Hk ¯dk| ≤ 2κHζ−1ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) + κHζ−2ϵ2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We begin with ∥¯gk − ∇fk∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the triangle and Cauchy–Schwarz inequalities, Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' |∇fT k dk − ¯gT k ¯dk| = |(¯gk − ∇fk)T ¯dk + (∇fk − ¯gk)T ( ¯dk − dk) + ¯gT k ( ¯dk − dk)| = |(¯gk − ∇fk)T ¯dk + (∇fk − ¯gk)T ( ¯dk − dk) − (Hk ¯dk + JT k ¯yk)T ( ¯dk − dk)| ≤ |(¯gk − ∇fk)T ¯dk| + |(∇fk − ¯gk)T ( ¯dk − dk)| + | ¯dT k Hk( ¯dk − dk)| + |¯yT k Jk( ¯dk − dk)| ≤ ∥¯gk − ∇fk∥∥ ¯dk∥ + ∥∇fk − ¯gk∥∥ ¯dk − dk∥ + κH∥ ¯dk∥∥ ¯dk − dk∥ ≤ (1 + κHζ−1) � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) κl¯τk ∥¯gk − ∇fk∥ + ζ−1∥¯gk − ∇fk∥2 ≤ � (1+κHζ−1)κFOαk √κl¯τk + ζ−1κ2 FOα2 k � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 16 Additionally, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 it follows that |dT k Hkdk − ¯dT k Hk ¯dk| = |2 ¯dT k Hk( ¯dk − dk) − ( ¯dk − dk)T Hk( ¯dk − dk)| ≤ 2| ¯dT k Hk( ¯dk − dk)| + |( ¯dk − dk)T Hk( ¯dk − dk)| ≤ 2κH∥ ¯dk∥∥ ¯dk − dk∥ + κH∥ ¯dk − dk∥2 ≤ 2κHζ−1 � ∆l(xk,¯τk,¯gk, ¯dk) κl¯τk ∥¯gk − ∇fk∥ + κHζ−2∥¯gk − ∇fk∥2 ≤ � 2κHζ−1κFOαk √κl¯τk + κHζ−2κ2 FOα2 k � ∆l(xk, ¯τk, ¯gk, ¯dk), which completes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' If ∥¯gk − ∇fk∥ ≤ ϵg, following similar logic as the first part of the proof, by the triangle and Cauchy–Schwarz inequalities, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10, |∇fT k dk − ¯gT k ¯dk| = |(¯gk − ∇fk)T ( ¯dk − dk) + (¯gk − ∇fk)T dk + ∇fT k ( ¯dk − dk)| = |(¯gk − ∇fk)T ( ¯dk − dk) + (¯gk − ∇fk)T dk − (Hkdk + JT k yk)T ( ¯dk − dk)| ≤ |(¯gk − ∇fk)T ( ¯dk − dk)| + |(¯gk − ∇fk)T dk| + |dT k Hk( ¯dk − dk)| + |yT k Jk( ¯dk − dk)| ≤ ζ−1∥¯gk − ∇fk∥2 + (1 + κHζ−1)∥dk∥∥¯gk − ∇fk∥ ≤ ζ−1∥¯gk − ∇fk∥2 + 1+κHζ−1 √κlτk � ∆l(xk, τk, ∇fk, dk)∥¯gk − ∇fk∥ ≤ ζ−1ϵ2 g + (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Additionally, under Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 it follows that |dT k Hkdk − ¯dT k Hk ¯dk| = |(dk − ¯dk)T Hk(dk − ¯dk) + 2dT k Hk( ¯dk − dk)| ≤ κH∥dk − ¯dk∥2 + 2κH∥dk∥∥dk − ¯dk∥ ≤ κHζ−2∥¯gk − ∇fk∥2 + 2κH √ ∆l(xk,τk,∇fk,dk) √κlτk ζ−1∥¯gk − ∇fk∥ ≤ κHζ−2ϵ2 g + 2κHζ−1ϵg √κlτk � ∆l(xk, τk, ∇fk, dk), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next lemma provides a bound on the merit function across an iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k ∈ N φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) ≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ 2 α2 k∥ ¯dk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By Algorithm 1, for any k ∈ N, 0 < αk ≤ αmax ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, by the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' it follows that φ(xk + αk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk) − φ(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk) = ¯τk(f(xk + αk ¯dk) − fk) + (∥c(xk + αk ¯dk)∥1 − ∥ck∥1) ≤ ¯τk(αk∇fT k ¯dk + L 2 α2 k∥ ¯dk∥2) + (∥ck + αkJk ¯dk∥1 − ∥ck∥1 + Γ 2 α2 k∥ ¯dk∥2) ≤ αk¯τk∇fT k ¯dk + |1 − αk|∥ck∥1 + αk∥ck + Jk ¯dk∥1 − ∥ck∥1 + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 = αk¯τk∇fT k ¯dk − αk∥ck∥1 + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 = αk¯τk¯gT k ¯dk − αk∥ck∥1 + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 = − αk∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ 2 α2 k∥ ¯dk∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Due to the quality and reliability of the zeroth- and first-order oracles (Oracles 0 and 1), one can only guarantee convergence to a neighborhood of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14 provides a lower bound on the size of the convergence neighbourhood in terms of ε (and ε∆l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let ε > max � ϵg η , √ϵfω7ω8 � max{κH,1} √κlτmin , which is equivalent to ε∆l > max � ϵg η , √ϵfω7ω8 � by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' where 0 < η < 2(1 − θ) min � 1 η1+η2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 η3+η4 � and {η1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' η2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' η3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' η4} ⊂ R>0 are defined as η1 = (1−θ)(1+ϵτ)¯τ−1 � 1+ κH ζ � √κlτmin η2 = � (1 − θ)2¯τ−1 � 1 + κH ζ �2 � (1+ϵτ)2¯τ−1 κlτmin + ϵτ � + 4¯τ−1 � 1+ϵτω2 κl + (1−θ)2(1+ϵτ) ζ � η3 = (1−θ)¯τ−1 � ¯τ−1 � 1+ 3κH ζ � +(1−σ)τmin � 1+ κH ζ �� (1−σ)τmin√κlτmin and η4 = � � � � (1−θ)2¯τ 2 −1 � ¯τ−1 � 1+ 3κH ζ � +(1−σ)τmin � 1+ κH ζ ��2 (1−σ)2τ 3 minκl + 4¯τ−1 κl + 4(1 − θ)2 � ¯τ 2 −1 � 1+ κH ζ � (1−σ)τminζ + ¯τ−1 ζ � 18 with p ∈ � 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' and {ω2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω8} ⊂ R>0 defined as ω2 = max{κH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='κy}+1 κl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω3 = (1+κHζ−1)κFO √¯τ−1αmax √κl + ¯τ−1κ2 FOα2 max ζ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω4 = max � ϵτ � max{κH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='κy} κl + √¯τ−1(1+κHζ−1)κFOαmax √κl + ω3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τ−1 (1−σ)τmin � (1+3κHζ−1)κFO √¯τ−1αmax √κl + (1 + κHζ−1) ¯τ−1κ2 FOα2 max ζ �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω5 = (1 + ϵτ)¯τ−1 � η ζ + 1+κHζ−1 √κlτmin � + ϵτ ¯τ−1(1+κHζ−1)2η 4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω6 = ¯τ 2 −1· � (1+κHζ−1) η ζ + 1+3κHζ−1 √κlτmin � (1−σ)τmin + ¯τ−1 � η ζ + 1+κHζ−1 √κlτmin � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ω7 = � 4¯τ−1 (p− 1 2 )θ max � 1+ϵτω2 1−ηω5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 1−ηω6 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 + ω3 + ω4 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' and ω8 = � � � � � �max � � � � � � ¯τ−1 κl κFO+ L 2κl + Γ 2¯τminκl 1−θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τminL+Γ 2¯τminκl � 1−θ−η � ¯τ−1 κl max �� 1+ϵτω2 1−ηω5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 √1−ηω6 �� � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14 involves many constants and is indeed hard to parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We make all constants explicit in order to show the exact dependence on the convergence neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That being said, what is important is that the lower bound of ε is proportional to the bias in the gradient approximations and proportional to the square root of the noise level in the function approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We are now ready to present the key lemma of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15, we first define (p, ˜α, h(·)), where p ∈ � 1 2, 1 � is a lower bound on the probability of a true iteration conditioned on the past (before the stopping time), ˜α ∈ R>0 is the large step threshold, and h : R>0 → R>0 is a monotonically increasing function (in α) that bounds the potential progress made at any given iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, we prove five results that can be summarized as follows: (i) lower bound (proportional to ϵf) on the potential progress with step size ˜α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (ii) conditioned on the past, the next iteration is true with probability at least p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iii) bound the potential progress made in any true and successful iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iv) true iterations with small step sizes are successful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' and, (v) bound (proportional to ϵf) the damage incurred at any iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Suppose Assumptions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all k < Tε∆l, let p = 1 − δ when the noise is bounded by ϵf, and p = 1 − δ − exp � − min{ u2 2ν2 , u 2b} � otherwise (with u = infx∈X {ϵf − E[E(x)]}, where E(x) = | ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Ξ0(x)) − f(x)|), 19 ˜α = min � � � � � 1−θ � ¯τ−1 κl κFO+ L 2κl + Γ 2¯τminκl , 2¯τminκl � 1−θ−η � ¯τ−1 κl max �� 1+ϵτω2 1−ηω5 , 1 √1−ηω6 �� ¯τminL+Γ � � � � � , h(α) = αθε2 ∆l min � 1−ηω5 1+ϵτω2 , 1 − ηω6, 1 1+ω3+ω4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Then, the following results hold: (i) h(˜α) > 4¯τ−1 p− 1 2 ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (ii) P [Ik = 1|Fk−1] ≥ p with some p ∈ � 1 2 + 4¯τ−1ϵf h(˜α) , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iii) If iteration k is true and successful, then Zk+1 ≤ Zk − h(αk) + 4¯τ−1ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iv) If αk ≤ ˜α and iteration k is true, then iteration k is also successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (v) Zk+1 ≤ Zk + 2¯τ−1ϵf + ¯τ−1(ek + e+ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' First, we note that: (1) due to the constants and the form, p is a valid probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', p ∈ ( 1 2, 1], (2) ˜α > 0 is guaranteed by the restriction on η in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14, and (3) h : R>0 → R>0 is a positive function that measures the potential progress made if iterations are true and successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We proceed with this proof by showing all five statements separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (i) This result follows directly from the definition of h(˜α) and the lower bound on ε∆l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' see Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (ii) This proof is essentially the same as that from [22, Proposition 1(ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let Jk := 1 � ∥Gk − ∇f(Xk)∥ ≤ max � ϵg, κFOAk � ∆l(Xk, Tk, Gk, Dk) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Clearly, by Definition 1, P [Ik = 0 | Fk−1] = P � Jk = 0 or Ek + E+ k > 2ϵf | Fk−1 � ≤ P [Jk = 0 | Fk−1] + P � Ek + E+ k > 2ϵf | Fk−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The first term on the right-hand-side of the inequality is bounded above by δ, by the first-order probabilistic oracle (Oracle 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The second term is zero in the case where ϵf is a deterministic bound on the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Otherwise, since Ek and E+ k individually satisfy the one-sided sub-exponential bound in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) with parameters ϵf and (ν, b), one can show that Ek + E+ k satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) with parameters 2ϵf and (2ν, 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Hence by the 20 one-sided Bernstein inequality, the second term is bounded above by e − min � u2 2ν2 , u 2b � , with u = infx∈X {ϵf − E[E(x)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' As a result, P [Ik = 1 | Fk−1] ≥ p for all k, for p as defined in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The range of p ∈ � 1 2 + 4¯τ−1ϵf h(˜α) , 1 � follows from the definitions of h(·) and ˜α in the statement, together with the inequality on ε∆l in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iii) Suppose iteration k is true and successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Since iteration k is true, by Definition 1 we have ∥¯gk − ∇fk∥ ≤ max � ϵg, κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk) � , and we consider the two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We further subdivide the analysis into the case where ∇fT k dk ≤ 0 and ∇fT k dk > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case A When ∥¯gk − ∇f(xk)∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10, ∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇f(xk)∥ ≤ ζ−1κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 If ∇fT k dk ≤ 0, by the fact that ¯τk ≥ τk, the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, it follows that ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) = ¯τk¯gT k ¯dk − τk∇fT k dk ≤ ¯τk(¯gT k ¯dk − ∇fT k dk) ≤ ¯τk|¯gT k ¯dk − ∇fT k dk| ≤ ¯τk � (1+κHζ−1)κFOαk √κl¯τk + κ2 FOα2 k ζ � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7) Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 If ∇fT k dk > 0, by the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) = ¯τk¯gT k ¯dk − τk∇fT k dk ≤ |¯τk¯gT k ¯dk − τk∇fT k dk| ≤ |(¯τk − τk)∇fT k dk| + ¯τk|¯gT k ¯dk − ∇fT k dk| ≤ |(¯τk − τk)∇fT k dk| + ¯τk � (1+κHζ−1)κFOαk √κl¯τk + κ2 FOα2 k ζ � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) We proceed to bound the term |(¯τk − τk)∇fT k dk|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' we consider three cases due to the merit parameter updating formulae ((2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 21 Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 If τk = ¯τk, then |(¯τk − τk)∇fT k dk| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 If τk = (1 − ϵτ)¯τk, by the triangle inequality and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, |(¯τk − τk)∇fT k dk| = ϵτ ¯τk|∇fT k dk| ≤ ϵτ ¯τk(|¯gT k ¯dk| + |∇fT k dk − ¯gT k ¯dk|) ≤ ϵτ � max{κH,κy} κl + √¯τk(1+κHζ−1)κFOαk √κl � ∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τk � (1+κHζ−1)κFOαk √κl¯τk + κ2 FOα2 k ζ � ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3 If ¯τk > τk = (1−σ)∥ck∥1 ∇fT k dk+max{dT k Hkdk,0}, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8), ∇fT k dk + max � dT k Hkdk, 0 � > (1−σ)∥ck∥1 ¯τk ≥ ¯gT k ¯dk + max � ¯dT k Hk ¯dk, 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3, we have τk ≥ τmin for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9 that 0 ≤ ∆l(xk, τk, ∇fk, dk), which implies τk∇fT k dk ≤ ∥ck∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Using the fact that τk ∈ R>0 and ∇fT k dk > 0, |∇fT k dk| ∥ck∥1 = ∇fT k dk ∥ck∥1 ≤ 1 τk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' it follows that |(¯τk − τk)∇fT k dk| = � ¯τk − (1−σ)∥ck∥1 ∇fT k dk+max{dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0} � |∇fT k dk| ≤ (∇fT k dk+max{dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0})−(¯gT k ¯dk+max{ ¯dT k Hk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0}) ∇fT k dk+max{dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0} ¯τk|∇fT k dk| ≤ |(∇fT k dk+max{dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0})−(¯gT k ¯dk+max{ ¯dT k Hk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0})| (1−σ)∥ck∥1 ¯τ 2 k|∇fT k dk| ≤ |∇fT k dk−¯gT k ¯dk|+| max{dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0}−max{ ¯dT k Hk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='0}| (1−σ)∥ck∥1 ¯τ 2 k|∇fT k dk| ≤ ¯τ 2 k (1−σ)τk · � |∇fT k dk − ¯gT k ¯dk| + | max � dT k Hkdk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 0 � − max{ ¯dT k Hk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 0}| � ≤ ¯τ 2 k (1−σ)τk · � |∇fT k dk − ¯gT k ¯dk| + |dT k Hkdk − ¯dT k Hk ¯dk| � ≤ ¯τ 2 k (1−σ)τmin � (1+3κHζ−1)κFOαk √κl¯τk + (1 + κHζ−1)ζ−1κ2 FOα2 k � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 22 Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8) and Cases A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3, it follows that ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) ≤ � ¯τk � (1+κHζ−1)κFOαk √κl¯τk + ζ−1κ2 FOα2 k � + max � ϵτ � max{κH,κy} κl + √¯τk(1+κHζ−1)κFOαk √κl � +ϵτ ¯τk � (1+κHζ−1)κFOαk √κl¯τk + ζ−1κ2 FOα2 k � , ¯τ 2 k (1−σ)τmin � (1+3κHζ−1)κFOαk √κl¯τk + (1 + κHζ−1)ζ−1κ2 FOα2 k ��� ∆l(xk, ¯τk, ¯gk, ¯dk) ≤ (ω3 + ω4) · ∆l(xk, ¯τk, ¯gk, ¯dk), where {ω3, ω4} ⊂ R>0 are as defined in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By {ω3, ω4} ⊂ R>0, ∆l(xk,τk,∇fk,dk) 1+ω3+ω4 ≤ ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the fact that iteration k is successful and Definition 2, it follows that ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) ≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf ≤ −αkθ ∆l(xk,τk,∇fk,dk) 1+ω3+ω4 + 2¯τ−1ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Hence, it follows that Zk+1 − Zk = φ(xk+1, ¯τk+1) − φ(xk, ¯τk) − ¯τk+1finf + ¯τkfinf ≤ φ(xk+1, ¯τk+1) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) − ¯τk+1finf + ¯τkfinf + ¯τkek = φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) + ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) − ¯τk+1finf + ¯τkfinf + ¯τkek ≤ − αkθ ∆l(xk,τk,∇fk,dk) 1+ω3+ω4 + 2¯τ−1ϵf + (¯τk+1 − ¯τk)(f(xk+1) − finf) + ¯τk(ek + e+ k ) ≤ − αkθ ∆l(xk,τk,∇fk,dk) 1+ω3+ω4 + 2¯τ−1ϵf + ¯τk(ek + e+ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) Case B When ∥¯gk − ∇f(xk)∥ ≤ ϵg, by the condition that k < Tε∆l and Definition 4, it follows that � ∆l(xk, τk, ∇fk, dk) > ε∆l > ϵg η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10, ∥ ¯dk − dk∥ ≤ ζ−1∥¯gk − ∇fk∥ ≤ ζ−1ϵg < ζ−1η � ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 If ∇fT k dk ≤ 0, by the fact that ¯τk ≥ τk, the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and 23 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, it follows that ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) = ¯τk¯gT k ¯dk − τk∇fT k dk ≤ ¯τk(¯gT k ¯dk − ∇fT k dk) ≤ ¯τk|¯gT k ¯dk − ∇fT k dk| ≤ ¯τk � ζ−1ϵ2 g + (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) � ≤ ¯τk � ζ−1η + 1+κHζ−1 √κlτk � η∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12) Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 If ∇fT k dk > 0, by the triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) = ¯τk¯gT k ¯dk − τk∇fT k dk ≤ |¯τk¯gT k ¯dk − τk∇fT k dk| ≤ |(¯τk − τk)∇fT k dk| + ¯τk|¯gT k ¯dk − ∇fT k dk| ≤ |(¯τk − τk)∇fT k dk| + ¯τk � ζ−1ϵ2 g + (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) � ≤ |(¯τk − τk)∇fT k dk| + ¯τk � ζ−1η + 1+κHζ−1 √κlτk � η∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13) We proceed to bound the term |(¯τk − τk)∇fT k dk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 If τk = ¯τk, then |(¯τk − τk)∇fT k dk| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 If τk = (1 − ϵτ)¯τk, then by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12 and Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14, |(¯τk − τk)∇fT k dk| = ϵτ ¯τk|∇fT k dk| ≤ ϵτ ¯τk � |¯gT k ¯dk| + |∇fT k dk − ¯gT k ¯dk| � ≤ ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τk(1+κHζ−1)2 4 ϵ2 g + ϵτ ¯τk � ζ−1ϵ2 g + (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) � ≤ ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τkη � (1+κHζ−1)2η 4 + η ζ + 1+κHζ−1 √κlτk � ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 24 Case B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3 If ¯τk > τk = (1−σ)∥ck∥1 ∇fT k dk+max{dT k Hkdk,0}, following the same logic as in Case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='10), |(¯τk − τk)∇fT k dk| ≤ ¯τ 2 k (1−σ)τk · � |∇fT k dk − ¯gT k ¯dk| + |dT k Hkdk − ¯dT k Hk ¯dk| � ≤ ¯τ 2 k (1−σ)τmin · � ζ−1ϵ2 g + (1+κHζ−1)ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) +κHζ−2ϵ2 g + 2κHζ−1ϵg √κlτk � ∆l(xk, τk, ∇fk, dk) � ≤ ¯τ 2 k (1−σ)τmin � (1 + κHζ−1)ζ−1η + 1+3κHζ−1 √κlτk � η∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13) and Cases B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3, it follows that ∆l(xk, τk, ∇fk, dk) − ∆l(xk, ¯τk, ¯gk, ¯dk) ≤ max � ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ϵτ ¯τkη � (1+κHζ−1)2η 4 + η ζ + 1+κHζ−1 √κlτk � ∆l(xk, τk, ∇fk, dk), η¯τ 2 k· � (1+κHζ−1)ζ−1η+ 1+3κHζ−1 √κlτk � (1−σ)τmin ∆l(xk, τk, ∇fk, dk) � + ¯τk � ζ−1η + 1+κHζ−1 √κlτk � η∆l(xk, τk, ∇fk, dk) ≤ max � ϵτω2∆l(xk, ¯τk, ¯gk, ¯dk) + ηω5∆l(xk, τk, ∇fk, dk), ηω6∆l(xk, τk, ∇fk, dk) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14) where {ω2, ω5, ω6} ⊂ R>0 are defined in Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Thus, it follows, ∆l(xk, ¯τk, ¯gk, ¯dk) ≥ min � 1−ηω5 1+ϵτω2 , 1 − ηω6 � ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15) By selecting η following Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14, using the fact that iteration k is successful and Definition 2, ¯φ(x+ k , ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) ≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf ≤ − αkθ min � 1−ηω5 1+ϵτω2 , 1 − ηω6 � ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Hence, following similar logic as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11), it follows that 25 Zk+1 − Zk ≤ φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) + ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) − ¯τk+1finf + ¯τkfinf + ¯τkek ≤ − αkθ min � 1−ηω5 1+ϵτω2 , 1 − ηω6 � ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf + (¯τk+1 − ¯τk)(f(xk+1) − finf) + ¯τk(ek + e+ k ) ≤ − αkθ min � 1−ηω5 1+ϵτω2 , 1 − ηω6 � ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf + ¯τk(ek + e+ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Combining the results for Case A and Case B, together with the assumption that the iteration is true, it follows that Zk+1 − Zk ≤ − min � 1−ηω5 1+ϵτω2 , 1 − ηω6, 1 1+ω3+ω4 � αkθ∆l(xk, τk, ∇fk, dk) + 2¯τ−1ϵf + ¯τ−1(ek + e+ k ) ≤ − h(αk) + 4¯τ−1ϵf, where the last inequality is from the conditions that ∆l(xk, τk, ∇fk, dk) > ε2 ∆l and ek + e+ k ≤ 2ϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (iv) We first show that for any k ∈ N, if αk ≤ ˜α and iteration k is true, then φ(xk + α ¯dk, ¯τk) ≤ φ(xk, ¯τk) − αkθ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Since iteration k is true, by Definition 1, it follows that ∥¯gk − ∇fk∥ ≤ max � ϵg, κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk) � , and we consider the two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case A When ∥¯gk − ∇fk∥ ≤ κFOαk � ∆l(xk, ¯τk, ¯gk, ¯dk), by αk ≤ ˜α ≤ 1−θ � ¯τ−1 κl κFO+ L 2κl + Γ 2¯τminκl , 26 the Cauchy–Schwarz inequality, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13, φ(xk + αk ¯dk, ¯τk) − φ(xk, ¯τk) ≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 ≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + αk¯τk∥∇fk − ¯gk∥∥ ¯dk∥ + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 ≤ − αk∆l(xk, ¯τk, ¯gk, ¯dk) + � ¯τk κl κFOα2 k∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τkL+Γ 2¯τkκl α2 k∆l(xk, ¯τk, ¯gk, ¯dk) ≤ − � 1 − �� ¯τ−1 κl κFO + L 2κl + Γ 2¯τminκl � ˜α � αk∆l(xk, ¯τk, ¯gk, ¯dk) ≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Case B When ∥¯gk − ∇fk∥ ≤ ϵg and iteration k is true, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, by the condition that k < Tε∆l and Definition 4, it follows that ∥¯gk − ∇fk∥ ≤ ϵg < ηε∆l < η � ∆l(xk, τk, ∇fk, dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Therefore, by αk ≤ ˜α ≤ 2¯τminκl � 1−θ−η � ¯τ−1 κl ·max �� 1+ϵτω2 1−ηω5 , 1 √1−ηω6 �� ¯τminL+Γ , the Cauchy–Schwarz inequality, Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15) and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' φ(xk + αk ¯dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk) − φ(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk) ≤ − αk∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + αk¯τk(∇fk − ¯gk)T ¯dk + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 ≤ − αk∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + αk¯τk∥∇fk − ¯gk∥∥ ¯dk∥ + ¯τkL+Γ 2 α2 k∥ ¯dk∥2 ≤ − αk∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + ¯τkL+Γ 2¯τkκl α2 k∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + αk¯τk � η � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ∇fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' dk) � �� ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) κl¯τk � ≤ − αk∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + α2 k ¯τkL+Γ 2¯τkκl ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) + αkη � ¯τk κl max �� 1+ϵτω2 1−ηω5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 √1−ηω6 � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) ≤ − αk �� 1 − η � ¯τ−1 κl max �� 1+ϵτω2 1−ηω5 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 1 √1−ηω6 �� − ¯τminL+Γ 2¯τminκl ˜α � ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk) ≤ − αkθ∆l(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯τk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ¯dk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 27 Combining Cases A and B, together with the fact the iteration is true, we conclude the proof of (iv) by ¯φ(xk + αk ¯dk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) ≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + ¯τkek + ¯τke+ k ≤ −αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + 2¯τkϵf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (v) If iteration k is unsuccessful, then by definition Zk+1 = Zk, so the inequality holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' On the other hand, if iteration k is successful, then starting with the second equation from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) Zk+1 − Zk ≤ φ(xk+1, ¯τk+1) − ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) + ¯φ(xk+1, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ+ k ) − ¯φ(xk, ¯τk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξk) − ¯τk+1finf + ¯τkfinf + ¯τkek ≤ − αkθ∆l(xk, ¯τk, ¯gk, ¯dk) + (¯τk+1 − ¯τk)(f(xk+1) − finf) + 2¯τkϵf + ¯τk(ek + e+ k ) ≤ 2¯τ−1ϵf + ¯τ−1(ek + e+ k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Therefore, we conclude the proof of (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The next two lemmas will be used in the iteration complexity analysis that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all t ≥ 1, and any ˆp ∈ [0, p), we have P �t−1 � k=0 Ik < ˆpt � ≤ e − (p−ˆp)2 2p2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The proof is the same as [22, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For any positive integer t and any ˆp ∈ � 1 2, 1 � , we have P � Tε∆l > t, t−1 � k=0 Ik ≥ ˆpt, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2 � = 0 where l = max � − ln α0−ln ˜α ln γ , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The proof is the same as [22, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We are now ready to present the main theorem of the manuscript;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' the iteration com- plexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 28 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Suppose Assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14 hold and that the con- ditions of Oracles 0 and 1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Then, for any s ≥ 0, ˆp ∈ � 1 2 + 4¯τ−1ϵf+s h(˜α) , p � , and t ≥ R ˆp− 1 2 − 4¯τ−1ϵf+s h(˜α) , P [Tε∆l ≤ t] ≥ 1 − e − (p−ˆp)2 2p2 t − e − min � s2t 2(2¯τ−1ν)2 , st 2(2¯τ−1b) � , where R = Z0 h(˜α) + max � ln ˜α−ln α0 2 ln γ , 0 � , and (p, ˜α, h(·)) are as defined in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the law of total probability, P [Tε∆l > t] =P � Tε∆l > t, 1 t t−1 � k=0 (2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) > 4¯τ−1ϵf + s � � �� � A + P � Tε∆l > t, 1 t t−1 � k=0 (2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) ≤ 4¯τ−1ϵf + s � � �� � B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' First we bound P[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For each iteration k, since Ek and E+ k satisfy the one-sided sub- exponential bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) with parameters (ν, b), one can show that ¯τ−1(Ek + E+ k ) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='11) with parameters (2¯τ−1ν, 2¯τ−1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, since ¯τ−1(Ek + E+ k ) has mean bounded by 2¯τ−1ϵf, applying (one-sided) Bernstein’s inequality, for any s ≥ 0 P[A] ≤ P � 1 t t−1 � k=0 ¯τ−1(Ek + E+ k ) > 2¯τ−1ϵf + s � ≤ e − min � s2t 2(2¯τ−1ν)2 , st 2(2¯τ−1b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Let l = max � − ln α0−ln ˜α ln γ , 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To bound P[B] we apply the law of total probability, P[B] = P � Tε∆l > t, 1 t t−1 � k=0 (2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) ≤ 4¯τ−1ϵf + s, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2 � � �� � B1 + P � Tε∆l > t, 1 t t−1 � k=0 (2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) ≤ 4¯τ−1ϵf + s, t−1 � k=0 ΘkIkUk ≥ � ˆp − 1 2 � t − l 2 � � �� � B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We first show that P[B2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='15, for any iteration k < Tε∆l, it follows that Zk+1 ≤ Zk−h(˜α)+2¯τ−1ϵf +¯τ−1(Ek+E+ k ) ≤ Zk−h(˜α)+4¯τ−1ϵf if UkIkΘk = 1, and Zk+1 ≤ 29 Zk + 2¯τ−1ϵf + ¯τ−1(Ek + E+ k ) if UkIkΘk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' By the definition of the zeroth-order oracle (Oracle 0), E[Ek] and E[E+ k ] are bounded above by ϵf for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The event Tε∆l > t implies that Zt > 0 (since Zt = 0 can only happen when Tε∆l ≤ t by the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This together with 1 t �t−1 k=0(2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) ≤ 4¯τ−1ϵf + s in turn implies the event �t−1 k=0 ΘkIkUk < � ˆp − 1 2 � t− l 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To see this, assume that �t−1 k=0 ΘkIkUk ≥ � ˆp − 1 2 � t− l 2, then Zt ≤ Z0 − � �� ˆp − 1 2 � t − l 2 � h(˜α) − t−1 � k=0 (2¯τ−1ϵf + ¯τ−1(Ek + E+ k )) � ≤ Z0 − �� ˆp − 1 2 � t − l 2 � h(˜α) + t(4¯τ−1ϵf + s) = Z0 − �� ˆp − 1 2 � h(˜α) − (4¯τ−1ϵf + s) � t + l 2h(˜α) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The last inequality above is due to the assumption that ˆp > 1 2 + 4¯τ−1ϵf+s h(˜α) and t ≥ R ˆp− 1 2 − 4¯τ−1ϵf+s h(˜α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Hence, P[B2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We now bound P[B1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='17, P[B1] ≤ P � Tε∆l > t, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2 � = P � Tε∆l > t, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2, t−1 � k=0 Ik < ˆpt � + P � Tε∆l > t, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2, t−1 � k=0 Ik ≥ ˆpt � ≤ P �t−1 � k=0 Ik < ˆpt � + P � Tε∆l > t, t−1 � k=0 ΘkIkUk < � ˆp − 1 2 � t − l 2, t−1 � k=0 Ik ≥ ˆpt � ≤ e − (p−ˆp)2 2p2 t + 0 = e − (p−ˆp)2 2p2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Combining P[A] and P[B], completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='18, for any s ≥ 0, ˆp ∈ � 1 2 + 4¯τ−1ϵf+s ˜αθωpε2 ∆l , p � and t ≥ ˆR ˆp− 1 2 − 4¯τ−1ϵf+s ˜αθωpε2 ∆l , P [Tε∆l ≤ t] ≥ 1 − e − (p−ˆp)2 2p2 t − e − min � s2t 2(2¯τ−1ν)2 , st 2(2¯τ−1b) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='16) where ˆR = φ(x0,¯τ−1)−φmin−(¯τ−1−¯τmin)finf ˜αθωpε2 ∆l + max � ln ˜α−ln α0 2 ln γ , 0 � , equivalently, by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5, ˆR = max{κ2 H,1} κlτmin φ(x0,¯τ−1)−φmin−(¯τ−1−¯τmin)finf ˜αθωpε2 + max � ln ˜α−ln α0 2 ln γ , 0 � , 30 ωp = min � 1−ηω5 1+ϵτω2 , 1 − ηω6, 1 1+ω3+ω4 � , and the rest of the constants are defined in Assump- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We make a few remarks about the main theoretical results of the paper (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='18 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Iteration Complexity) By Definition 4 (and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5) and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='19, we conclude that, with overwhelmingly high probability, the iteration complexity of Al- gorithm 1 to generate a primal-dual iterate (xk, yk) ∈ Rn × Rm that satisfies max{∥∇fk + JT k yk∥, � ∥ck∥} ≤ ε is O(ε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This iteration complexity is of the same order in terms of the dependence on ε as the iteration complexity that can be derived for the deterministic counterpart [16], with the additional restriction that ε is bounded away from zero (Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='14) due to the noise and bias in the oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Almost Sure Convergence) We note that Algorithm 1 finds an ε-stationary iterate in a finite number of iterations with probability 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', P[∩∞ k=1 ∪∞ t=k (Tε∆l > t)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This is a direct consequence of the Borel–Cantelli lemma, since it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='16) that the probability of failure events is summable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', �∞ t=1 P[Tε∆l > t] = �∞ t=1 (1 − P[Tε∆l ≤ t]) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' (Unconstrained Setting) The high probability complexity bound in this paper is a gen- eralization of the unconstrained version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In the unconstrained setting, the parameters reduce to σ = 0, ω1 = 0, ω2 = 1, Γ = 0, ζ = 1, κH = 1, κl = 1, ϵτ = 0, and ¯τk = 1 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Using these values in the results of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='19 does not exactly recover the result from the unconstrained setting [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That being said, the order of the results is the same in terms of the dependence on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The existence of the gap is due to complications that arise in the constrained setting related to the adaptivity of the merit parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We conclude by emphasizing again that though there is a con- stant difference in function h and value ˜α comparing to [22], our algorithm recovers the complexity bound of the deterministic variant algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 4 Numerical Results In this section, we present numerical results for our proposed algorithm on standard equal- ity constrained nonlinear optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The goal of the numerical experiments is to investigate the efficiency and robustness of the SS-SQP algorithm across a diverse set of test problems with different levels of noise in the objective function and gradient eval- uations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' All experiments were conducted in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Before we present the numerical results, we describe the test problems, implementation details, and evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1 Test Problems We ran the numerical experiments on a subset of the equality-constrained optimization problems from the CUTEst collection [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We selected the problems that satisfy the fol- lowing criteria: (i) the objective function is not a constant function, (ii) the total number of variables and constraints are not larger than 103, and (iii) the singular values of Jaco- bians of the constraints at all iterates in all runs were greater than 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This resulted in 35 test problems of various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We considered noisy (noisy objective function and gradient evaluations) versions of the 35 CUTEst problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Specifically, whenever an objective function or objec- tive gradient evaluation was required, approximations, ¯f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ) = N � f(x), ϵ2 f,N � and ¯g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' ξ′) = N � ∇f(x), ϵ2 g,N n I � , respectively, were utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We considered 4 different noise levels in the objective function and gradient evaluations, dictated by the con- stants ϵf,N ∈ � 0, 10−4, 10−2, 10−1� and ϵg,N ∈ � 0, 10−4, 10−2, 10−1� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Each CUTEst problem has a unique initial starting point, which was used as the starting point of all runs of all algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Moreover, for each selected tuple of noise levels (ϵf,N, ϵg,N) ∈ � 0, 10−4, 10−2, 10−1� × � 10−4, 10−2, 10−1� ∪ {0} × {0}, where appropriate, we ran each problem with five different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 Implementation Details We compared SS-SQP (Algorithm 1) to the adaptive stochastic SQP algorithm proposed in [7] (which we call AS-SQP) on the previously described noisy CUTEst problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We set user-defined parameters for SS-SQP as follows: ϵf = ϵf,N, ϵg = ϵg,N, ϵτ = 10−2, ¯τ−1 = σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5, θ = 10−4, α0 = αmax = 1, and Hk = I for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For AS-SQP [7] we set the parameters as follows (this parameter selection was guided by the choice of parameters in [7]): ¯τ−1 = σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='1, ¯ξ−1 = 1, ϵ = 10−2, θ = 104, Hk = I and βk = 1 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The AS-SQP step size rule requires knowledge (or estimates) of the Lipschitz constants L and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' To this end, we estimated these constants using gradient differences near the initial point, and set Lk = L and Γk = Γ for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We note that while the analysis of the SS-SQP algorithm requires that the condition of Oracles 1 hold, such conditions are not enforced or checked, and rather in each experiment, the algorithms were given random gradient estimates with the same, fixed, pre-specified accuracy (as described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' That being said, a clear distinction between SS-SQP and AS-SQP is the fact that the former requires function evaluations of the objective function (for the step search) whereas AS-SQP does not (AS-SQP is an objective-function-free method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We discuss this further when presenting the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='3 Termination Conditions and Evaluation Metrics In all of our experiments, results are given in terms of infeasibility (∥c(xk)∥∞) and station- arity (KKT) (max{∥c(xk)∥∞, miny∈Rm ∥∇f(xk) + ∇c(xk)y∥∞}) with respect to different evaluation metrics (iterations and work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We ran all algorithms with a budget of iterations (103), and only terminated a run early if an approximate stationary point was found, which we define as x∗ ∈ Rn such that ∥c(x∗)∥∞ ≤ 10−6 and miny∈Rm ∥∇f(x∗) + ∇c(x∗)y∥∞ ≤ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We present results in the form of performance profiles with respect to iterations and work (defined as the number of function and gradient evaluations), and use the convergence metric as described in [26], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', m(x0) − m(x) ≥ (1 − ϵpp)(m(x0) − mb), where m(x) is either ∥c(x)∥∞ (infeasibility) or max{∥c(x)∥∞, miny∈Rm ∥∇f(x)+∇c(x)y∥∞} (stationarity (KKT)), x0 is the initial iterate, and mb is the best value of the metric found by any algorithm for a given problem instance within the budget, and ϵpp ∈ (0, 1) is the tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' For all experiments presented, we chose ϵpp = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 Noisy Gradients, Exact Functions (ϵf = 0) In our first set of experiments, we consider problems with exact objective function eval- uations and noisy objective gradient evaluations and compare SS-SQP and AS-SQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The goal of this experiment is to show the effect of noise in the gradient and the advantages of using (exact) function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Each row in Figure 1 shows performance profiles for a dif- ferent noise level in the gradient (bottom row, highest noise level) and each column shows a different evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Starting from the noise-less benchmark case (ϵf = 0 and ϵg = 0, the first row of Figure 1), it is clear that the performance of the methods in terms of both infeasibility error and KKT error is similar with a slight advantage in effectiveness (total problems that can be solved) for SS-SQP in terms of KKT error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' As the noise in the gradient is increased, the gap between the performance of the two methods (in terms of all metrics) increases favoring SS-SQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' This, of course, is not surprising as SS-SQP uses additional information (exact function values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These results highlight the effect reliable function information can have on the performance of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='5 Noisy Functions and Gradients Here we present results with noise in both the objective function and gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' As in Figure 1, in Figure 2 different rows show results for different noise levels in the gradi- ent (the bottom row has the highest noise) and different columns show results for different evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Each performance profile has 4 lines: the AS-SQP (that is objective- function-free and is not affected by the noise in the function evaluations) and three variants of the SS-SQP method with different levels of noise in the objective function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' One can make the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' First, not surprisingly, the performance of the 33 2 4 6 8 10 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( f = 0, g = 0) AS-SQP SS-SQP 5 10 15 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( f = 0, g = 0) AS-SQP SS-SQP 10 20 30 40 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( f = 0, g = 0) AS-SQP SS-SQP 5 10 15 20 25 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( f = 0, g = 0) AS-SQP SS-SQP 2 4 6 8 10 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( f = 0, g = 10-4) AS-SQP SS-SQP 5 10 15 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( f = 0, g = 10-4) AS-SQP SS-SQP 10 20 30 40 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( f = 0, g = 10-4) AS-SQP SS-SQP 5 10 15 20 25 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( f = 0, g = 10-4) AS-SQP SS-SQP 10 20 30 40 50 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( f = 0, g = 10-2) AS-SQP SS-SQP 5 10 15 20 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( f = 0, g = 10-2) AS-SQP SS-SQP 10 20 30 40 50 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( f = 0, g = 10-2) AS-SQP SS-SQP 5 10 15 20 25 30 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( f = 0, g = 10-2) AS-SQP SS-SQP 20 40 60 80 100 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( f = 0, g = 10-1) AS-SQP SS-SQP 20 40 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( f = 0, g = 10-1) AS-SQP SS-SQP 10 20 30 40 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( f = 0, g = 10-1) AS-SQP SS-SQP 5 10 15 20 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( f = 0, g = 10-1) AS-SQP SS-SQP Figure 1: Performance profiles for AS-SQP and SS-SQP on CUTEst collection [18] with deterministic objective function evaluations (ϵf = 0) and noisy objective gradient evalu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Each column corresponds to a different evaluation metric (infeasibility and KKT errors vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' iterations and work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The noise in the objective gradient evaluations ϵg increases from top to bottom (First row: ϵg = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Second row: ϵg = 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Third row: ϵg = 10−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Fourth row: ϵg = 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' SS-SQP method degrades as the noise in the objective function evaluations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Sec- ond, AS-SQP and SS-SQP are competitive and achieve similar robustness levels with respect to infeasibility errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Third, and most interestingly, the performance of the methods de- pends on the relative errors of the function and gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' In particular, when the objective function noise level is sufficiently small compared to the objective gradient bias, SS-SQP performs better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' On the other hand, when the function estimations are too noisy compared to the noise level in the gradient evaluations, AS-SQP performs slightly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' These results highlight the power of objective-function-free optimization methods 34 in the presence of noise (especially high noise in the objective function evaluations) and the value of quality (or at least relative quality) function evaluations in methods that require zeroth-order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 2 4 6 8 10 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( g = 10-4) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 10 20 30 40 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( g = 10-4) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 10 20 30 40 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( g = 10-4) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 5 10 15 20 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( g = 10-4) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 10 20 30 40 50 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( g = 10-2) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 5 10 15 20 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( g = 10-2) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 10 20 30 40 50 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( g = 10-2) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 5 10 15 20 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( g = 10-2) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 20 40 60 80 100 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Iterations ( g = 10-1) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 20 40 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 Infeas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Error/Work ( g = 10-1) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 20 40 60 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Iterations ( g = 10-1) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) 5 10 15 20 25 30 Performance Ratio 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='8 1 KKT Error/Work ( g = 10-1) AS-SQP SS-SQP ( f = 10-1) SS-SQP ( f = 10-2) SS-SQP ( f = 10-4) Figure 2: Performance profiles for AS-SQP and SS-SQP on CUTEst collection [18] with noise in both the objective function and gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Each column corresponds to a different evaluation metric (infeasibility and KKT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' iterations and work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The noise in the objective gradient evaluations ϵg increases from top to bottom (First row: ϵg = 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Second row: ϵg = 10−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Third row: ϵg = 10−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' The different variants of SS-SQP correspond to different levels of noise in the objective function evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' 5 Conclusion We have proposed a step-search SQP algorithm (SS-SQP) for solving stochastic optimiza- tion problems with deterministic equality constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=', the setting in which constraint function values and derivatives are available, but only stochastic estimates of the objec- tive function and its associated derivatives can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We showed that under reasonable assumptions on the inexact probabilistic zeroth- and first-order oracles, with overwhelmingly high probability, in O(ε−2) iterations our algorithm can produce an iterate that satisfies the first-order ε-stationarity, which matches the iteration complexity of the 35 deterministic counterparts of the SQP algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Numerical results provide strong evidence for the efficiency and efficacy of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Some future directions include but are not limited to, (1) incorporating stochastic constraint evaluations into the algorithm design and analysis, and (2) extending the framework to the setting with inequal- ity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Both avenues above are subjects of future work as they require significant adaptations in the design, analysis, and implementation of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Acknowledgments This material is based upon work supported by the Office of Naval Research under award number N00014-21-1-2532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' We would like to thank Professors Frank E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' Curtis and Katya Scheinberg for their invaluable support and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttAyT4oBgHgl3EQfmviR/content/2301.00477v1.pdf'} +page_content=' References [1] Afonso S Bandeira, Katya Scheinberg, and Luis Nunes Vicente.' 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diff --git a/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/2301.03266v1.pdf.txt b/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/2301.03266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b1e0ef85233a15d0c6ba5f4af2c3c13f2dba1fc --- /dev/null +++ b/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/2301.03266v1.pdf.txt @@ -0,0 +1,594 @@ +Doc2Query--: When Less is More +Mitko Gospodinov1, Sean MacAvaney2, and Craig Macdonald2 +University of Glasgow +12024810G@student.gla.ac.uk +2{first}.{last}@glasgow.ac.uk +Abstract. Doc2Query — the process of expanding the content of a +document before indexing using a sequence-to-sequence model — has +emerged as a prominent technique for improving the first-stage retrieval +effectiveness of search engines. However, sequence-to-sequence models are +known to be prone to “hallucinating” content that is not present in the +source text. We argue that Doc2Query is indeed prone to hallucination, +which ultimately harms retrieval effectiveness and inflates the index size. +In this work, we explore techniques for filtering out these harmful queries +prior to indexing. We find that using a relevance model to remove poor- +quality queries can improve the retrieval effectiveness of Doc2Query by +up to 16%, while simultaneously reducing mean query execution time by +30% and cutting the index size by 48%. We release the code, data, and +a live demonstration to facilitate reproduction and further exploration.1 +1 +Introduction +Neural network models, particularly those based on contextualised language +models, have been shown to improve search effectiveness [3]. While some ap- +proaches focus on re-ranking document sets from a first-stage retrieval function +to improve precision [27], others aim to improve the first stage itself [4]. In this +work, we focus on one of these first-stage approaches: Doc2Query [29]. This ap- +proach trains a sequence-to-sequence model (e.g., T5 [33]) to predict queries that +may be relevant to a particular text. Then, when indexing, this model is used +to expand the document by generating a collection of queries and appending +them to the document. Though computationally expensive at index time [34], +this approach has been shown to be remarkably effective even when retrieving +using simple lexical models like BM25 [28]. Numerous works have shown that +the approach can produce a high-quality pool of results that are effective for +subsequent stages in the ranking pipeline [19,20,23,40]. +However, sequence-to-sequence models are well-known to be prone to gener- +ate content that does not reflect the input text – a defect known in literature +as “hallucination” [25]. We find that existing Doc2Query models are no excep- +tion. Figure 1 provides example generated queries from the state-of-the-art T5 +Doc2Query model [28]. In this example, we see that many of the generated +queries cannot actually be answered by the source passage (score ≤ 1). +1 https://github.com/terrierteam/pyterrier_doc2query +arXiv:2301.03266v1 [cs.IR] 9 Jan 2023 + +2 +Gospodinov et al. +Original Passage: Barley (Hordeum vulgare L.), a +member of the grass family, is a major cereal grain. It +was one of the first cultivated grains and is now grown +widely. Barley grain is a staple in Tibetan cuisine and +was eaten widely by peasants in Medieval Europe. Bar- +ley has also been used as animal fodder, as a source +of fermentable material for beer and certain distilled +beverages, and as a component of various health foods. +Generated Queries: (1) where does barley originate +from · (2) what is the name of the cereal grain used +in tibetan cooking? · (3) what is barley used for · (1) +what is barley in food · (0) what is bare wheat · (3) +what family of organisms is barley in · (1) why is bar- +ley important in tibetan diet · (3) what is barley · +(2) where is barley grown · (1) where was barley first +grown and eaten · (1) where was barley first used ... +Fig. 1. Example passage from MS MARCO and generated queries using the T5 +Doc2Query model. The relevance of each query to the passage is scored by the au- +thors on a scale of 0–3 using the TREC Deep Learning passage relevance criteria. +Based on this observation, we hypothesise that retrieval performance of +Doc2Query would improve if hallucinated queries were removed. In this paper, we +conduct experiments where we apply a new filtering phase that aims to remove +poor queries prior to indexing. Given that this approach removes queries, we +call the approach Doc2Query-- (Doc2Query-minus-minus). Rather than training +a new model for this task, we identify that relevance models are already fit for +this purpose: they estimate how relevant a passage is to a query. We therefore +explore filtering strategies that make use of existing neural relevance models. +Through experimentation on the MS MARCO dataset, we find that our fil- +tering approach can improve the retrieval effectiveness of indexes built using +Doc2Query-- by up to 16%; less can indeed be more. Meanwhile, filtering nat- +urally reduces the index size, lowering storage and query-time computational +costs. Finally, we conduct an exploration of the index-time overheads introduced +by the filtering process and conclude that the gains from filtering more than make +up for the additional time spent generating more queries. The approach also has +a positive impact on the environmental costs of applying Doc2Query; the same +retrieval effectiveness can be achieved with only about a third of the compu- +tational cost when indexing. To facilitate last-metre, last-mile, and complete +reproduction efforts [36], we release the code, indices, and filtering scores.1 In +summary, we contribute a technique to improve the effectiveness and efficiency +of Doc2Query by filtering out queries that do not reflect the original passage. +2 +Related Work +The classical lexical mismatch problem is a key one in information retrieval - +documents that do not contain the query terms may not be retrieved. In the +literature, various approaches have addressed this: query reformulation – includ- +ing stemming, query expansion models (e.g. Rocchio, Bo1 [1], RM3 [12]) – and +document expansion [9,30,35]. Classically, query expansion models have been +popular, as they avoid the costs associated with making additional processing +for each document needed for document expansion. However, query expansion +may result in reduced performance [11], as queries are typically short and the +necessary evidence to understand the context of the user is limited. + +Doc2Query--: When Less is More +3 +The application of latent representations of queries and documents, such as +using latent semantic indexing [8] allow retrieval using to not be driven directly +by lexical signals. More recently, transformer-based language models (such as +BERT [6]) have resulted in representations of text where the contextualised +meaning of words are accounted for. In particular, in dense retrieval, queries +and documents are represented in embeddings spaces [14,37], often facilitated +by Approximate Nearest Neighbour (ANN) data structures [13]. However, even +when using ANN, retrieval can still be inefficient or insufficiently effective [15]. +Others have explored approaches for augmenting lexical representations with +additional terms that may be relevant. In this work, we explore Doc2Query [29], +which uses a sequence-to-sequence model that maps a document to queries that +it might be able to answer. By appending these generated queries to a docu- +ment’s content before indexing, the document is more likely to be retrieved for +user queries when using a model like BM25. An alternative style of document +expansion, proposed by MacAvaney et al. [19] and since used by several other +models (e.g., [10,39,40]), uses the built-in Masked Language Modelling (MLM) +mechanism. MLM expansion generates individual tokens to append to the docu- +ment as a bag of words (rather than as a sequence). Although MLM expansion is +also prone to hallucination,2 the bag-of-words nature of MLM expansion means +that individual expansion tokens may not have sufficient context to apply fil- +tering effectively. We therefore focus only on sequence-style expansion and leave +the exploration of MLM expansion for future work. +3 +Doc2Query-- +Doc2Query-- consists of two phases: a generation phrase and a filtering phase. +In the generation phase, a Doc2Query model generates a set of n queries that +each document might be able to answer. However, as shown in Figure 1, not +all of the queries are necessarily relevant to the document. To mitigate this +problem, Doc2Query-- then proceeds to a filtering phase, which is responsible +for eliminating the generated queries that are least relevant to the source doc- +ument. Because hallucinated queries contain details not present in the original +text (by definition), we argue that hallucinated queries are less useful for re- +trieval than non-hallucinated ones. Filtering is accomplished by retaining only +the most relevant p proportion of generated queries over the entire corpus. The +retained queries are then concatenated to their corresponding documents prior +to indexing, as per the existing Doc2Query approach. +More formally, consider an expansion function e that maps a document to n +queries: e : D �→ Qn. In Doc2Query, each document in corpus D are concate- +nated with their expansion queries, forming a new corpus D′ = {Concat(d, e(d)) | +d ∈ D}, which is then indexed by a retrieval system. Doc2Query-- adds a filtering +mechanism that uses a relevance model that maps a query and document to a +real-valued relevance score s : Q × D �→ R (with larger values indicating higher +2 For instance, we find that SPLADE [10] generates the following seemingly-unrelated +terms for the passage in Figure 1 in the top 20 expansion terms: reed, herb, and troy. + +4 +Gospodinov et al. +relevance). The relevance scoring function is used to filter down the queries to +those that meet a certain score threshold t as follows: +D′ = +� +Concat +� +d, +� +q | q ∈ e(d) ∧ s(q, d) ≥ t +�� +| d ∈ D +� +(1) +The relevance threshold t is naturally dependent upon the relevance scoring +function. It can be set empirically, chosen based on operational criteria (e.g., +target index size), or (for a well-calibrated relevance scoring function) determined +a priori. In this work, we combine the first two strategies: we pick t based on +the distribution of relevance scores across all expansion queries. For instance, +at p = 0.3 we only keep queries with relevance scores in the top 30%, which is +t = 3.215 for the ELECTRA [31] scoring model on the MS MARCO dataset [26]. +4 +Experimental Setup +We conduct experiments to answer the following research questions: +RQ1 Does Doc2Query-- improve the effectiveness of document expansion? +RQ2 What are the trade-offs in terms of effectiveness, efficiency, and storage when +using Doc2Query--? +Datasets and Measures. We conduct tests using the MS MARCO [26] v1 +passage corpus. We use five test collections:3 (1) the MS MARCO Dev (small) +collection, consisting of 6,980 queries (1.1 qrels/query); (2) the Dev2 collection, +consisting of 4,281 (1.1 qrels/query); (3) the MS MARCO Eval set, consisting of +6,837 queries (held-out leaderboard set); (4/5) the TREC DL’19/’20 collections, +consisting of 43/54 queries (215/211 qrels/query). We evaluate using the official +task evaluation measures: Reciprocal Rank at 10 (RR@10) for Dev/Dev2/Eval, +nDCG@10 for DL’19/’20. We tune systems4 on Dev, leaving the remaining col- +lections as held-out test sets. +Models. We use the T5 Doc2Query model from Nogueira and Lin [28], mak- +ing use of the inferred queries released by the authors (80 per passage). To the +best of our knowledge, this is the highest-performing Doc2Query model avail- +able. We consider three neural relevance models for filtering: ELECTRA5 [31], +MonoT56 [32], and TCT-ColBERT7 [16], covering two strong cross-encoder mod- +els and one strong bi-encoder model. We also explored filters that use the prob- +abilities from the generation process itself but found them to be ineffective and +therefore omit these results due to space constraints. +Tools and Environment. We use the PyTerrier toolkit [22] with a PISA [24,17] +index to conduct our experiments. We deploy PISA’s Block-Max WAND [7] im- +plementation for BM25 retrieval. Inference was conducted on an NVIDIA 3090 +GPU. Evaluation was conducted using the ir-measures package [18]. +3 ir-datasets +[21] +IDs: +msmarco-passage/dev/small, +msmarco-passage/dev/2, +msmarco-passage/eval/small, +msmarco-passage/trec-dl-2019/judged, +msmarco-passage/trec-dl-2020/judged +4 BM25’s +k1, +b, +and +whether +to +remove +stopwords +were +tuned +for +all +systems; +the +filtering +percentage +(p) +was +also +tuned +for +filtered +systems. +5 crystina-z/monoELECTRA_LCE_nneg31 +6 castorini/monot5-base-msmarco +7 castorini/tct_colbert-v2-hnp-msmarco + +Doc2Query--: When Less is More +5 +Table 1. Effectiveness and efficiency measurements for Doc2Query-- and baselines. +Significant differences between Doc2Query and their corresponding filtered versions +for Dev, Dev2, DL’19 and DL’20 are indicated with * (paired t-test, p < 0.05). Values +marked with † are taken from the corresponding submissions to the public leaderboard. +RR@10 +nDCG@10 +ms/q +GB +System +Dev +Dev2 +Eval +DL’19 +DL’20 +MRT +Index +BM25 +0.185 +0.182 +†0.186 +0.499 +0.479 +5 +0.71 +Doc2Query (n = 40) +0.277 +0.265 +†0.272 +0.626 +0.607 +30 +1.17 +w/ ELECTRA Filter (30%) +*0.316 +*0.310 +- +0.667 +0.611 +23 +0.89 +w/ MonoT5 Filter (40%) +*0.308 +*0.298 +0.306 +0.650 +0.611 +29 +0.93 +w/ TCT Filter (50%) +*0.287 +*0.280 +- +0.640 +0.599 +30 +0.94 +Doc2Query (n = 80) +0.279 +0.267 +- +0.627 +0.605 +30 +1.41 +w/ ELECTRA Filter (30%) +*0.323 +*0.316 +0.325 +0.670 +0.614 +23 +0.95 +w/ MonoT5 Filter (40%) +*0.311 +*0.298 +- +0.665 +0.609 +28 +1.04 +w/ TCT Filter (50%) +*0.293 +*0.283 +- +0.642 +0.588 +28 +1.05 +5 +Results +We first explore RQ1: whether relevance filtering can improve the retrieval of +Doc2Query models. Table 1 compares the effectiveness of Doc2Query with var- +ious filters. We observe that all the filters significantly improve the retrieval +effectiveness on the Dev and Dev2 datasets at both n = 40 and n = 80. We also +observe a large boost in performance on the Eval dataset.8 Though the differ- +ences in DL’19 and DL’20 appear to be considerable (e.g., 0.627 to 0.670), these +differences are not statistically significant. +Digging a little deeper, Figure 2 shows the retrieval effectiveness of Doc2Query +with various numbers of generated queries (in dotted black) and the correspond- +ing performance when filtering using the top-performing ELECTRA scorer (in +solid blue). We observe that performing relevance filtering at each value of n +improves the retrieval effectiveness. For instance, keeping only 30% of expan- +sion queries at n = 80, performance is increased from 0.279 to 0.323 – a 16% +improvement. +In aggregate, results from Table 1 and Figure 2 answer RQ1: Doc2Query-- +filtering can significantly improve the retrieval effectiveness of Doc2Query across +various scoring models, numbers of generated queries (n) and thresholds (p). +Next, we explore the trade-offs in terms of effectiveness, efficiency, and storage +when using Doc2Query--. Table 1 includes the mean response time and index +sizes for each of the settings. As expected, filtering reduces the index size since +fewer terms are stored. For the best-performing setting (n = 80 with ELECTRA +8 Significance cannot be determined due to the held-out nature of the dataset. Further, +due to restrictions on the number of submissions to the leaderboard, we only are able +to submit two runs. The first aims to be a fair comparison with the existing Doc2Query +Eval result, using the same number of generated queries and same base T5 model for +scoring. The second is our overall best-performing setting, using the ELECTRA filter +at n = 80 generated queries. + +6 +Gospodinov et al. +0 +1 +2 +3 +4 +5 +Total Tokens +1e9 +0.225 +0.250 +0.275 +0.300 +0.325 +RR@10 +90% +80% +70% +60% +50% +40% +30% +n=5 +n=10 +n=20 +n=40 +n=80 +Generation Phase +Filtering Phase +Fig. 2. Effectiveness (RR@10) on the Dev set, compared with the total number of +indexed tokens. The generation phase is shown in dotted black (at various values of +n), and the ELECTRA filtering phase is shown in solid blue (at various values of p). +filter), this amounts to a 48% reduction in index size (1.41 GB down to 0.95 GB). +Naturally, such a reduction has an impact on query processing time as well; it +yields a 30% reduction in mean response time (30ms down to 23ms). +Doc2Query-- filtering adds substantial cost an indexing time, mostly due to +scoring each of the generated queries. Table 2 reports the cost (in hours of GPU +time) of the generation and filtering phases. We observe that ELECTRA filter- +ing can yield up to a 78% increase in GPU time (n = 10). However, we find that +the improved effectiveness makes up for this cost. To demonstrate this, we al- +locate the time spent filtering to generating additional queries for each passage. +For instance, the 15 hours spent scoring n = 5 queries could instead be spent +generating 6 more queries per passage (for a total of n = 11). We find that when +comparing against an unfiltered n that closely approximates the total time when +Table 2. Retrieval effectiveness comparison for comparable indexing computational +budgets (in hours of GPU time). Values of n without a filter are chosen to best approx- +imate the total compute hours or the Dev effectiveness of the corresponding filtered +version. Significant differences between in RR@10 performance are indicated with * +(paired t-test, p < 0.05). +GPU Hours +RR@10 +n +Filter +Gen+Filt=Tot +Dev +Dev2 +Comment +5 +ELECTRA +20 + 15 = +34 +0.273 +0.270 +11 +None +34 + +0 = +34 +*0.261 +*0.256 +−4% Dev RR for sim. GPU hrs +31 +None +99 + +0 = +99 +0.273 +0.265 +×2.9 GPU hrs to match Dev RR +10 +ELECTRA +32 + 25 = +57 +0.292 +0.292 +18 +None +59 + +0 = +59 +*0.270 +*0.260 +−8% Dev RR for sim. GPU hrs +20 +ELECTRA +66 + 47 = 113 +0.307 +0.303 +36 +None +113 + +0 = 113 +*0.275 +*0.265 +−10% Dev RR for sim. GPU hrs +40 +ELECTRA +128 + 86 = 214 +0.316 +0.310 +68 +None +216 + +0 = 216 +*0.279 +*0.267 +−12% Dev RR for sim. GPU hrs + +Doc2Query--: When Less is More +7 +filtering, the filtered results consistently yield significantly higher retrieval effec- +tiveness. As the computational budget increases, so does the margin between +Doc2Query and Doc2Query--, from 4% at 34 hours up to 12% at 216 hours. +From the opposite perspective, Doc2Query consumes 2.9× or more GPU +time than Doc2Query-- to achieve similar effectiveness (n = 13 with no filter +vs. n = 5 with ELECTRA filter). Since the effectiveness of Doc2Query flattens +out between n = 40 and n = 80 (as seen in Figure 2), it likely requires a +massive amount of additional compute to reach the effectiveness of Doc2Query-- +at n ≥ 10, if that effectiveness is achievable at all. These comparisons show that +if a deployment is targeting a certain level of effectiveness (rather than a target +compute budget), Doc2Query-- is also preferable to Doc2Query. +These results collectively answer RQ2: Doc2Query-- provides higher effective- +ness at lower query-time costs, even when controlling for the additional compute +required at index time. +6 +Conclusions +This work demonstrated that there are untapped advantages in generating natural- +language for document expansion. Specifically, we presented Doc2Query--, which +is a new approach for improving the effectiveness and efficiency of the Doc2Query +model by filtering out the least relevant queries. We observed that a 16% im- +provement in retrieval effectiveness can be achieved, while reducing the index +size by 48% and mean query execution time by 30%. +The technique of filtering text generated from language models using rel- +evance scoring is ripe for future work. For instance, relevance filtering could +potentially apply to approaches that generate alternative forms of queries [38], +training data [2], or natural language responses to queries [5] — all of which +are potentially affected by hallucinated content. Furthermore, future work could +explore approaches for relevance filtering over masked language modelling ex- +pansion [19], rather than sequence-to-sequence expansion. +References +1. Amati, G., Van Rijsbergen, C.J.: Probabilistic models of information retrieval +based on measuring the divergence from randomness. ACM Trans. Inf. Syst. 20(4) +(2002) +2. 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In: Proceedings of SIGIR (2021) + diff --git a/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/load_file.txt b/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9a1fb9f281431c3a8663e4493e2b08feba416bb --- /dev/null +++ b/ttE1T4oBgHgl3EQfjwR5/content/tmp_files/load_file.txt @@ -0,0 +1,524 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf,len=523 +page_content='Doc2Query--: When Less is More Mitko Gospodinov1, Sean MacAvaney2, and Craig Macdonald2 University of Glasgow 12024810G@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='gla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='uk 2{first}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' {last}@glasgow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='uk Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Doc2Query — the process of expanding the content of a document before indexing using a sequence-to-sequence model — has emerged as a prominent technique for improving the first-stage retrieval effectiveness of search engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, sequence-to-sequence models are known to be prone to “hallucinating” content that is not present in the source text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We argue that Doc2Query is indeed prone to hallucination, which ultimately harms retrieval effectiveness and inflates the index size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this work, we explore techniques for filtering out these harmful queries prior to indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We find that using a relevance model to remove poor- quality queries can improve the retrieval effectiveness of Doc2Query by up to 16%, while simultaneously reducing mean query execution time by 30% and cutting the index size by 48%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We release the code, data, and a live demonstration to facilitate reproduction and further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='1 1 Introduction Neural network models, particularly those based on contextualised language models, have been shown to improve search effectiveness [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' While some ap- proaches focus on re-ranking document sets from a first-stage retrieval function to improve precision [27], others aim to improve the first stage itself [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this work, we focus on one of these first-stage approaches: Doc2Query [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' This ap- proach trains a sequence-to-sequence model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=', T5 [33]) to predict queries that may be relevant to a particular text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Then, when indexing, this model is used to expand the document by generating a collection of queries and appending them to the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Though computationally expensive at index time [34], this approach has been shown to be remarkably effective even when retrieving using simple lexical models like BM25 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Numerous works have shown that the approach can produce a high-quality pool of results that are effective for subsequent stages in the ranking pipeline [19,20,23,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, sequence-to-sequence models are well-known to be prone to gener- ate content that does not reflect the input text – a defect known in literature as “hallucination” [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We find that existing Doc2Query models are no excep- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Figure 1 provides example generated queries from the state-of-the-art T5 Doc2Query model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this example, we see that many of the generated queries cannot actually be answered by the source passage (score ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='com/terrierteam/pyterrier_doc2query arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='03266v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='IR] 9 Jan 2023 2 Gospodinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Original Passage: Barley (Hordeum vulgare L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='), a member of the grass family, is a major cereal grain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' It was one of the first cultivated grains and is now grown widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Barley grain is a staple in Tibetan cuisine and was eaten widely by peasants in Medieval Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Bar- ley has also been used as animal fodder, as a source of fermentable material for beer and certain distilled beverages, and as a component of various health foods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Generated Queries: (1) where does barley originate from · (2) what is the name of the cereal grain used in tibetan cooking?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' · (3) what is barley used for · (1) what is barley in food · (0) what is bare wheat · (3) what family of organisms is barley in · (1) why is bar- ley important in tibetan diet · (3) what is barley · (2) where is barley grown · (1) where was barley first grown and eaten · (1) where was barley first used .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Example passage from MS MARCO and generated queries using the T5 Doc2Query model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The relevance of each query to the passage is scored by the au- thors on a scale of 0–3 using the TREC Deep Learning passage relevance criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Based on this observation, we hypothesise that retrieval performance of Doc2Query would improve if hallucinated queries were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this paper, we conduct experiments where we apply a new filtering phase that aims to remove poor queries prior to indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Given that this approach removes queries, we call the approach Doc2Query-- (Doc2Query-minus-minus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Rather than training a new model for this task, we identify that relevance models are already fit for this purpose: they estimate how relevant a passage is to a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We therefore explore filtering strategies that make use of existing neural relevance models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Through experimentation on the MS MARCO dataset, we find that our fil- tering approach can improve the retrieval effectiveness of indexes built using Doc2Query-- by up to 16%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' less can indeed be more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Meanwhile, filtering nat- urally reduces the index size, lowering storage and query-time computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Finally, we conduct an exploration of the index-time overheads introduced by the filtering process and conclude that the gains from filtering more than make up for the additional time spent generating more queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The approach also has a positive impact on the environmental costs of applying Doc2Query;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' the same retrieval effectiveness can be achieved with only about a third of the compu- tational cost when indexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' To facilitate last-metre, last-mile, and complete reproduction efforts [36], we release the code, indices, and filtering scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='1 In summary, we contribute a technique to improve the effectiveness and efficiency of Doc2Query by filtering out queries that do not reflect the original passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 2 Related Work The classical lexical mismatch problem is a key one in information retrieval - documents that do not contain the query terms may not be retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In the literature, various approaches have addressed this: query reformulation – includ- ing stemming, query expansion models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Rocchio, Bo1 [1], RM3 [12]) – and document expansion [9,30,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Classically, query expansion models have been popular, as they avoid the costs associated with making additional processing for each document needed for document expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, query expansion may result in reduced performance [11], as queries are typically short and the necessary evidence to understand the context of the user is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Doc2Query--: When Less is More 3 The application of latent representations of queries and documents, such as using latent semantic indexing [8] allow retrieval using to not be driven directly by lexical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' More recently, transformer-based language models (such as BERT [6]) have resulted in representations of text where the contextualised meaning of words are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In particular, in dense retrieval, queries and documents are represented in embeddings spaces [14,37], often facilitated by Approximate Nearest Neighbour (ANN) data structures [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, even when using ANN, retrieval can still be inefficient or insufficiently effective [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Others have explored approaches for augmenting lexical representations with additional terms that may be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this work, we explore Doc2Query [29], which uses a sequence-to-sequence model that maps a document to queries that it might be able to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' By appending these generated queries to a docu- ment’s content before indexing, the document is more likely to be retrieved for user queries when using a model like BM25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' An alternative style of document expansion, proposed by MacAvaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' [19] and since used by several other models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=', [10,39,40]), uses the built-in Masked Language Modelling (MLM) mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' MLM expansion generates individual tokens to append to the docu- ment as a bag of words (rather than as a sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Although MLM expansion is also prone to hallucination,2 the bag-of-words nature of MLM expansion means that individual expansion tokens may not have sufficient context to apply fil- tering effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We therefore focus only on sequence-style expansion and leave the exploration of MLM expansion for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 3 Doc2Query-- Doc2Query-- consists of two phases: a generation phrase and a filtering phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In the generation phase, a Doc2Query model generates a set of n queries that each document might be able to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, as shown in Figure 1, not all of the queries are necessarily relevant to the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' To mitigate this problem, Doc2Query-- then proceeds to a filtering phase, which is responsible for eliminating the generated queries that are least relevant to the source doc- ument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Because hallucinated queries contain details not present in the original text (by definition), we argue that hallucinated queries are less useful for re- trieval than non-hallucinated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Filtering is accomplished by retaining only the most relevant p proportion of generated queries over the entire corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The retained queries are then concatenated to their corresponding documents prior to indexing, as per the existing Doc2Query approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' More formally, consider an expansion function e that maps a document to n queries: e : D �→ Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In Doc2Query, each document in corpus D are concate- nated with their expansion queries, forming a new corpus D′ = {Concat(d, e(d)) | d ∈ D}, which is then indexed by a retrieval system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Doc2Query-- adds a filtering mechanism that uses a relevance model that maps a query and document to a real-valued relevance score s : Q × D �→ R (with larger values indicating higher 2 For instance, we find that SPLADE [10] generates the following seemingly-unrelated terms for the passage in Figure 1 in the top 20 expansion terms: reed, herb, and troy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 4 Gospodinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' relevance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The relevance scoring function is used to filter down the queries to those that meet a certain score threshold t as follows: D′ = � Concat � d, � q | q ∈ e(d) ∧ s(q, d) ≥ t �� | d ∈ D � (1) The relevance threshold t is naturally dependent upon the relevance scoring function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' It can be set empirically, chosen based on operational criteria (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=', target index size), or (for a well-calibrated relevance scoring function) determined a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In this work, we combine the first two strategies: we pick t based on the distribution of relevance scores across all expansion queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' For instance, at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='3 we only keep queries with relevance scores in the top 30%, which is t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='215 for the ELECTRA [31] scoring model on the MS MARCO dataset [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 4 Experimental Setup We conduct experiments to answer the following research questions: RQ1 Does Doc2Query-- improve the effectiveness of document expansion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' RQ2 What are the trade-offs in terms of effectiveness, efficiency, and storage when using Doc2Query--?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Datasets and Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We conduct tests using the MS MARCO [26] v1 passage corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We use five test collections:3 (1) the MS MARCO Dev (small) collection, consisting of 6,980 queries (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='1 qrels/query);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' (2) the Dev2 collection, consisting of 4,281 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='1 qrels/query);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' (3) the MS MARCO Eval set, consisting of 6,837 queries (held-out leaderboard set);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' (4/5) the TREC DL’19/’20 collections, consisting of 43/54 queries (215/211 qrels/query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We evaluate using the official task evaluation measures: Reciprocal Rank at 10 (RR@10) for Dev/Dev2/Eval, nDCG@10 for DL’19/’20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We tune systems4 on Dev, leaving the remaining col- lections as held-out test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We use the T5 Doc2Query model from Nogueira and Lin [28], mak- ing use of the inferred queries released by the authors (80 per passage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' To the best of our knowledge, this is the highest-performing Doc2Query model avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We consider three neural relevance models for filtering: ELECTRA5 [31], MonoT56 [32], and TCT-ColBERT7 [16], covering two strong cross-encoder mod- els and one strong bi-encoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We also explored filters that use the prob- abilities from the generation process itself but found them to be ineffective and therefore omit these results due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Tools and Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We use the PyTerrier toolkit [22] with a PISA [24,17] index to conduct our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We deploy PISA’s Block-Max WAND [7] im- plementation for BM25 retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Inference was conducted on an NVIDIA 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Evaluation was conducted using the ir-measures package [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 3 ir-datasets [21] IDs: msmarco-passage/dev/small, msmarco-passage/dev/2, msmarco-passage/eval/small, msmarco-passage/trec-dl-2019/judged, msmarco-passage/trec-dl-2020/judged 4 BM25’s k1, b, and whether to remove stopwords were tuned for all systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' the filtering percentage (p) was also tuned for filtered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 5 crystina-z/monoELECTRA_LCE_nneg31 6 castorini/monot5-base-msmarco 7 castorini/tct_colbert-v2-hnp-msmarco Doc2Query--: When Less is More 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Effectiveness and efficiency measurements for Doc2Query-- and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Significant differences between Doc2Query and their corresponding filtered versions for Dev, Dev2, DL’19 and DL’20 are indicated with * (paired t-test, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Values marked with † are taken from the corresponding submissions to the public leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' RR@10 nDCG@10 ms/q GB System Dev Dev2 Eval DL’19 DL’20 MRT Index BM25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='182 †0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='479 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='71 Doc2Query (n = 40) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='265 †0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='607 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='17 w/ ELECTRA Filter (30%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='611 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='89 w/ MonoT5 Filter (40%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='611 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='93 w/ TCT Filter (50%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='599 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='94 Doc2Query (n = 80) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='605 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='41 w/ ELECTRA Filter (30%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='614 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='95 w/ MonoT5 Filter (40%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='609 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='04 w/ TCT Filter (50%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='588 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='05 5 Results We first explore RQ1: whether relevance filtering can improve the retrieval of Doc2Query models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Table 1 compares the effectiveness of Doc2Query with var- ious filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We observe that all the filters significantly improve the retrieval effectiveness on the Dev and Dev2 datasets at both n = 40 and n = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We also observe a large boost in performance on the Eval dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='8 Though the differ- ences in DL’19 and DL’20 appear to be considerable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='627 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='670), these differences are not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Digging a little deeper, Figure 2 shows the retrieval effectiveness of Doc2Query with various numbers of generated queries (in dotted black) and the correspond- ing performance when filtering using the top-performing ELECTRA scorer (in solid blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We observe that performing relevance filtering at each value of n improves the retrieval effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' For instance, keeping only 30% of expan- sion queries at n = 80, performance is increased from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='279 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='323 – a 16% improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' In aggregate, results from Table 1 and Figure 2 answer RQ1: Doc2Query-- filtering can significantly improve the retrieval effectiveness of Doc2Query across various scoring models, numbers of generated queries (n) and thresholds (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Next, we explore the trade-offs in terms of effectiveness, efficiency, and storage when using Doc2Query--.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Table 1 includes the mean response time and index sizes for each of the settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' As expected, filtering reduces the index size since fewer terms are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' For the best-performing setting (n = 80 with ELECTRA 8 Significance cannot be determined due to the held-out nature of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Further, due to restrictions on the number of submissions to the leaderboard, we only are able to submit two runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The first aims to be a fair comparison with the existing Doc2Query Eval result, using the same number of generated queries and same base T5 model for scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The second is our overall best-performing setting, using the ELECTRA filter at n = 80 generated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 6 Gospodinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 0 1 2 3 4 5 Total Tokens 1e9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='325 RR@10 90% 80% 70% 60% 50% 40% 30% n=5 n=10 n=20 n=40 n=80 Generation Phase Filtering Phase Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Effectiveness (RR@10) on the Dev set, compared with the total number of indexed tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The generation phase is shown in dotted black (at various values of n), and the ELECTRA filtering phase is shown in solid blue (at various values of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' filter), this amounts to a 48% reduction in index size (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='41 GB down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='95 GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Naturally, such a reduction has an impact on query processing time as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' it yields a 30% reduction in mean response time (30ms down to 23ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Doc2Query-- filtering adds substantial cost an indexing time, mostly due to scoring each of the generated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Table 2 reports the cost (in hours of GPU time) of the generation and filtering phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We observe that ELECTRA filter- ing can yield up to a 78% increase in GPU time (n = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' However, we find that the improved effectiveness makes up for this cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' To demonstrate this, we al- locate the time spent filtering to generating additional queries for each passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' For instance, the 15 hours spent scoring n = 5 queries could instead be spent generating 6 more queries per passage (for a total of n = 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We find that when comparing against an unfiltered n that closely approximates the total time when Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Retrieval effectiveness comparison for comparable indexing computational budgets (in hours of GPU time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Values of n without a filter are chosen to best approx- imate the total compute hours or the Dev effectiveness of the corresponding filtered version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Significant differences between in RR@10 performance are indicated with * (paired t-test, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' GPU Hours RR@10 n Filter Gen+Filt=Tot Dev Dev2 Comment 5 ELECTRA 20 + 15 = 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='270 11 None 34 + 0 = 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='256 −4% Dev RR for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' GPU hrs 31 None 99 + 0 = 99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='265 ×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='9 GPU hrs to match Dev RR 10 ELECTRA 32 + 25 = 57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='292 18 None 59 + 0 = 59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='260 −8% Dev RR for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' GPU hrs 20 ELECTRA 66 + 47 = 113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='303 36 None 113 + 0 = 113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='265 −10% Dev RR for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' GPU hrs 40 ELECTRA 128 + 86 = 214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='310 68 None 216 + 0 = 216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='267 −12% Dev RR for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' GPU hrs Doc2Query--: When Less is More 7 filtering, the filtered results consistently yield significantly higher retrieval effec- tiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' As the computational budget increases, so does the margin between Doc2Query and Doc2Query--, from 4% at 34 hours up to 12% at 216 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' From the opposite perspective, Doc2Query consumes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content='9× or more GPU time than Doc2Query-- to achieve similar effectiveness (n = 13 with no filter vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' n = 5 with ELECTRA filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Since the effectiveness of Doc2Query flattens out between n = 40 and n = 80 (as seen in Figure 2), it likely requires a massive amount of additional compute to reach the effectiveness of Doc2Query-- at n ≥ 10, if that effectiveness is achievable at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' These comparisons show that if a deployment is targeting a certain level of effectiveness (rather than a target compute budget), Doc2Query-- is also preferable to Doc2Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' These results collectively answer RQ2: Doc2Query-- provides higher effective- ness at lower query-time costs, even when controlling for the additional compute required at index time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' 6 Conclusions This work demonstrated that there are untapped advantages in generating natural- language for document expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' Specifically, we presented Doc2Query--, which is a new approach for improving the effectiveness and efficiency of the Doc2Query model by filtering out the least relevant queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' We observed that a 16% im- provement in retrieval effectiveness can be achieved, while reducing the index size by 48% and mean query execution time by 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE1T4oBgHgl3EQfjwR5/content/2301.03266v1.pdf'} +page_content=' The technique of 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in the community. Even +as the community races to provide and verify complete Open +RAN systems, the importance of verification of systems based on +Open RAN under real-world conditions has become clear, and +testbed facilities for general use have been envisioned, in addition +to private testing facilities. Aerial robots, including autonomous +ones, are among the increasingly important and interesting clients +of RAN systems, but also present a challenge for testbeds. Based +on our experience in architecting and operating an advanced +wireless testbed with aerial robots as a primary citizen, we +present considerations relevant to the design of Open RAN +testbeds, with particular attention to making such a testbed +capable of controlled experimentation with aerial clients. We also +present representative results from the NSF AERPAW testbed on +Open RAN slicing, programmable vehicles, and programmable +radios. +Index +Terms—Open +RAN, +Interoperability +Testing, +IOT, +Testbed, open-source, eNB, gNB, aerial, UAV, drone. +I. INTRODUCTION +Open Radio Access Network (specifications defined in the +O-RAN Alliance) has emerged as a serious and perhaps +critically necessary alternative to the proprietary radio access +network (RAN) solutions that have characterized cellular net- +works. In particular, Open RAN provides a richer eco-system +based on the virtualization of network functions providing +greater economies of scale and reduced cost. The open archi- +tecture of Open RAN, and the definition of interfaces among +modules that have been thus far treated as essentially mono- +lithic, are expected to ensure inter-operation between products +from different providers, and a competitive market, leading to +improved quality and lower cost of ownership. It also enables +the inclusion of commodity controllers, and the ability of +operators to develop their custom control applications on top +of those controllers, bringing the power of software-defined +networking to RANs on an open-interface basis. +Such disaggregation comes at the cost of increased over- +head, and early Open RAN systems are widely expected to +have higher overheads and lower efficiency compared to extant +single-vendor systems that, after all, have evolved and been in- +tegrated for decades. Optimistic views consist of expectations +of workable, if inefficient, implementations soon, followed +by rapid improvements in performance. Pessimistic views +incline to doubts regarding how long such a process might +This work was supported in part by the NSF Award CNS-1939334. M. +Mushi, Y. Liu, R. Dutta are with the Dept. Computer Science, NC State +University, Raleigh, NC; S. Sreenivasa, O. Ozdemir, I. Guvenc, M. Sichitiu +are with the Dept. Electrical and Computer Engineering, NC State University, +Raleigh, NC; Russ Gyurek is with Cisco Systems, Raleigh, NC. +take, or whether such systems can approach the efficiency of +proprietary monolithic systems, or even be workable at scale. +However, there are significant gains in terms of economies of +scale through virtualization as well as additional functionality +that provides a much richer set of capabilities (e.g., RIC apps, +etc). +To dispassionately and pragmatically assess the worka- +bility of Open RAN, the community must move beyond +early experiments and greenfield deployments to demonstrable +repeatability of predictable system performance. Designing +dependable test facilities for Open RAN components and +systems, therefore, is among the most important outstanding +tasks of the Open RAN community at this time. A key +promise of Open RAN is interoperability (multi-vendor), and +the key to verifying such claims is through interoperability +testing (IOT). Recognizing the importance of IOT, the O- +RAN Alliance has dedicated two entire work groups (WG4 +and WG5) to specifying interfaces, and both groups have +published specifications on interoperability testing and profiles +in addition to unit test specifications (see [1], [2] and other +specifications of WG4 and WG5). Such profiles allow the +interoperability of any set of components to be tested in test +configurations that can be realized in lab-environment test +benches. However, to engender the above-mentioned growing +confidence, Open RAN ecosystem players (contributors, as +well as vendors, operators, and users) need to be able to test +components in a comprehensive end-to-end test facility - one +that is embedded in a realistic setting and span in the real +world, including at least in part an outdoor setting, with a non- +trivial number of UEs interacting with a non-trivial number of +base stations. In the rest of this paper, we reserve the term +“testbed” to indicate facilities capable of such complete RAN +system tests. +Unpeopled Aerial Vehicles (UAVs), have long been gener- +ally acknowledged as important clients of any future wide-area +wireless communications system. However, the full scope of +such devices as denizens of the wireless communication world +is only coming to be appreciated recently. A key observation +is that UAVs are not only wireless communications clients for +command-and-control (the most obvious use case), but play +roles in at least two other ways in the wireless ecosystem. +First, trivially, as such devices increase in intelligence, and are +tasked with increasingly more sophisticated missions, these +missions are likely to pose additional – and likely much +heavier – communication requirements; for example streaming +live on-site video back to the cloud, or engaging in other +data-heavy cloud-assisted distributed computation tasks. More +importantly, and more significantly in the current context, with +increasing on-board compute intelligence, such devices are +capable of engaging not just as clients, but as crucial parts +arXiv:2301.11365v1 [cs.NI] 26 Jan 2023 + +2 +of the wireless communication infrastructure itself. This is es- +pecially important in an open interoperable ecosystem such as +Open RAN aspires to be, as open competition spurs innovative +contributors to explore previously unoccupied ecosystem roles. +The visioning and design exercise for an Open RAN testbed +that aspires to provide interoperability and system testing +capabilities, if such a facility expects to support the full +evolutionary arc of aerial devices, must include reflection +specific to these considerations. In this paper, we leverage our +joint experience in (i) architecting and operating an advanced +wireless testbed with aerial robots as primary citizens, and (ii) +industry Open RAN testing and dependability expectations, to +provide a starting point that we hope will be useful to such +architects and designers. In the next section, we briefly review +some existing test facilities with the capability (or potential +near capability) of acting as Open RAN testing resources and +juxtapose them with industry Open RAN testing norms, as +well as basic support requirements for UAVs. In Section III, +we discuss in further detail the class of use cases that represent +the potential synergistic use of UAVs in Open RAN systems. +Finally, we provide a deep consideration of one extant testbed +– our own NSF AERPAW platform at NC State University – +to showcase the process of reviewing testbed capabilities to +articulate both strengths and shortcomings in light of an ideal +Open RAN testbed with native UAV support. +II. VISIONING AN OPEN RAN/UAV TESTBED +A. Existing Wireless Testbeds and System Testing +There are numerous testbeds that are accessible to re- +searchers to experiment with wireless technologies including +5G, Open RAN, and UAVs. In Table I, we provide a list a +few of these testbed facilities that are accessible to researchers +from academia, government, and industry. Note that we do +not intend to present Table I as either comprehensive or +authoritative. There are likely many facilities that we are +unaware of, or for which no information is publicly available +to us. Even for those we have surveyed, Table I represents our +best knowledge as obtained from publicly available sources +(as cited); we regret any unintended mischaracterization. Our +survey was also heavily biased toward facilities in the USA. +Nevertheless, since our focus is on test facilities publicly or +generally available to researchers and practitioners in the US, +and on facilities sizeable enough for UAVs to be practically +a part of the test ecosystem, we believe that Table I provides +representative, and meaningfully extensive, information for the +Open RAN testbed designer of the near future. We have chosen +to characterize each facility listed by means of a few high-level +considerations. Obviously, explicit currently stated support of +Open RAN testing, and UAV support/integration, are features +we looked for. Related to Open RAN, we also looked at the +RF spectrum the facility is capable of and allowed to operate +in, by noting if it lists an Innovation Zone (IZ) license from the +Federal Communications Commission (see for example [3]), +and also its deployment context (indoor facilities may be able +to use isolation such as Faraday cages and operate without an +FCC Innovation Zone or experimental licenses). +Related to UAV support, we also looked at whether such +UAVs (or any component of the testbed, for those without +UAVs) support controlled mobility. We consider this feature +an important one for future Open RAN testbeds. A signif- +icant proportion of wireless communications system com- +plexity arises from (or is exacerbated by) the mobility of +system components, most usually that of User Equipment +(UE); therefore it it important for the testbed to support +experiments involving mobility, hand-over, and disconnect- +reconnect events. However, the core of the scientific method +is the repeatability of experiments and the reproduction of +experimental results. To provide this for experiments related +to mobility, the relative motion of various system components +must be possible to precisely reproduce on demand, for as +many runs of an experiment as necessary. +Another key feature we looked for was emulation support. +The single most valuable characteristic of actual wireless test +facilities is the availability of a real Radio Frequency (RF) +environment, providing real-world challenges such as fading, +multi-path, and statistical uncertainty, simultaneously with the +experiment repeatability. The simulation of RF environments +by means of mathematical models, no matter how sophisti- +cated, abstracts a measure of realism from test results; further, +the experimenter has no need of an experimental facility (or +even actual radios) for simulation exercises, which are an +appropriate earlier stage in proving research before considering +testbed validation. The exercise of emulation, on the other +hand, provides an important added value to a testbed, in that +it is a digital twin of a real RF system, capable of operating +in real-time, in which physical radio equipment can actually +be immersed. Emulation systems are driven by calibration (to +some real RF environment) rather than modeling and may be +realized by digital twinning, or more often by analog RF cir- +cuitry. In extreme cases, a test facility may be entirely based on +emulation, as in the case of the Colosseum system (originally +created by DARPA and currently operated at Northeastern +University under NSF aegis; see Table I). More typically, +emulation support is an adjunct part of a physical test facility +that can serve as an early and less costly stage of full testbed +validation. +Even before moving on to discussions of testing require- +ments specific to Open RAN or UAVs, we can note a few +points from Table I. Naturally, those we were able to survey +were largely public-use testbeds, since those are the ones that +are most likely to provide information publicly about them- +selves. This dovetails with our focus since the interoperability +focus of Open RAN implies that for engendering maximum +confidence, the testbed facility should be open to anybody that +is interested in repeating experiments and verifying results. +Unsurprisingly, there is no testbed on the list that provides +full Open RAN as well as UAV support today, even without +considering controlled mobility. Less obviously, we find that +the combination of UAV support and controlled mobility is +rather rare; only a handful of testbeds on our list provide even +partial mobility control in conjunction with UAV support. +Interestingly, we note that a number of testbeds provide +emulation support, in keeping with our expectation that this +is a key required feature of wireless testbeds. However, when +emulation is considered jointly with mobility control, a non- +obvious consideration may be worth mentioning. For a testbed + +3 +TABLE I: Existing testbeds with advanced wireless and UAV experimentation capabilities. Public testbeds indicated with an +asterisk (*) may be open only to partners or require contacting testbed operators rather than being generally available through +an experimentation portal. Features for which public information could not be found are marked as Not Known (NK). +Testbeds (alphabeti- +cal) +Location +Emulation +Support +Open +RAN +Support +UAV +Support +Controlled +Mobility +FCC- +IZ +Main +Focus +Area +Deployment +Environment +Access +AERPAW [4] +Raleigh, NC +✓ +Partial +✓ +✓ +✓ +UAVs, SDRs +Rural, Urban +Public +ARA [5] +Central Iowa, IA +✓ +× +× +× +× +Rural wireless +Rural +Public +Arena [6] +Boston, MA +✓ +✓ +× +× +✓ +SDRs +Indoor grid +Public* +ARLIS [7] +College +Park, +MD +× +× +× +× +× +5G security +Virtual +Public* +ARM / Tech Mahin- +dra 5G Lab [8] +NK +NK +✓ +× +× +× +5G testing +NK +Private +Booz +Allen +5G +Lab [9] +Annapolis Junc- +tion, MD +NK +NK +× +× +× +Mission critical +5G +NK +Private +CCI xG Testbed [10] +Arlington, VA +NK +✓ +× +× +× +SDRs, AI +Indoor +Public* +Colosseum [11] +Burlington, MA +✓ +✓ +× +× +✓ +Emulation, +SDRs +Cloud +Public +CORNET [12] +Blacksburg, VA +× +✓ +× +× +× +SDRs +Indoor, +Rooftop +Public +COSMOS [13] +Manhattan, NY +✓ +✓ +× +× +✓ +mmWave, back- +haul +Urban +Public +Drexel Grid [14] +Philadelphia, PA +✓ +× +× +× +× +Emulation, +SDRs +Indoor grid +Public* +Ericsson +Open +Lab [15] +NK +✓ +✓ +× +× +× +CloudRAN, vir- +tualized 5G +Indoor +Private +INL +Wireless +Testbed [16] +Idaho Falls, ID +× +× +✓ +Partial +× +Wireless +security +Rural +Private +IRIS [17] +Los +Angeles, +CA +× +× +× +✓ +× +Robotic wireless +networks +Indoor +Public* +LinQuest Labs [18] +Chantilly, VA +✓ +NK +✓ +NK +× +5G +security, +UAV, NTN +Cloud, indoor +Public* +NASA MTBs [19] +× +× +× +✓ +× +× +Multirotor UAV +testing +Indoor +Public* +New York UAS Test +Site [20] +Rome, NY +× +× +✓ +Partial +× +BVLOS +UAV +testing +Rural, Urban +Public* +NIST 5G Coexistence +Testbed [21] +Boulder, CO +✓ +NK +× +× +× +5G +coexistence +testing +Indoor +Public* +NIST +NBIT +Testbed [22] +NK +× +× +× +× +Spectrum +shar- +ing +Indoor +Public* +NITOS [23] +Volos, Greece +× +✓ +× +× +× +Cloud-based +Wireless +services +Rooftop +Public +Northeastern +UAS +Chamber [24] +Burlington, MA +× +× +✓ +NK +× +Drone flights +Drone +cage, +anechoic +chamber +Public* +ORBIT [25] +N. +Brunswick, +NJ +✓ +× +× +× +× +SDRs +Indoor grid +Public +PNNL 5G Innovation +Studio [26] +Richland, WA +× +× +× +× +× +Commercial 5G +Indoor +Private +POWDER-RENEW +[27] +Salt Lake City, +UT +✓ +✓ +× +× +✓ +SDRs, +massive +MIMO +Urban +Public +RELLIS 5G testbed +[28] +Bryan, TX +× +NK +NK +NK +5G (AT&T) +Outdoor +Public* +Cyber Living Innova- +tion Lab [29] +Fairfax, VA +NK +✓ +NK +NK +× +5G +security, +robotics +Indoor +Public* +SOAR [30] +Buffalo, NY +× +× +✓ +Partial +× +Drone flights +Drone cage +Public* +TIP +Community +Lab [31] +Overland +Park, +Kansas +NK +✓ +× +× +× +O-RAN 5G NR +(Sprint) +NK +Private +UNH Interoperability +Lab [32] +Durham, NH +× +✓ +× +× +× +Interoperability +testing +Indoor +Public* +Virginia Tech Drone +Park [33] +Blacksburg, VA +× +× +✓ +Partial +× +Drone flights +Drone cage +Public* +that provides mobile airborne components, any emulation +system must not only emulate the physical RF environment, +but also the physics of airflow and aerial navigation, including +wind gusts and other disturbing factors (analogous to noise and +interference in the RF environment), as well as the dynamics, +features, and constraints of a specific UAV. The ability to au- +tonomously navigate one or more UAVs in the 3D space based +on RF observations in the environment is also an important +capability with various use cases. Furthemore, subtle moves +of the UAVs (e.g., a multicopter pitching to move forward) +can change the orientation of highly directional RF antennas +(especially relevant for mmWave transmissions). With this + +4 +in mind, it is perhaps unsurprising that the combination of +emulation support and mobility control is quite rare in the +extant testbeds. +B. Extant Industry Open RAN Testing Practices +The facilities listed in Table I are largely those focused +on system testing, some of which currently already support +deploying some particular Open RAN system in part or in full. +Researchers or ecosystem developers may find this sufficient +since it is possible for them to test or study their products +or innovations in contiguous areas supported by “some” Open +RAN implementation. However, vendors, carriers, and other +ecosystem players who are involved in the business of actually +building or operating a data network as a service need to +focus far more deeply on component testing, and (critically for +Open RAN) cross-vendor interoperability testing - especially +the large swathes of new interoperability modes enabled by +Open RAN’s disaggregation modes. +Such testing proceeds by identifying Key Performance +Indicators (KPIs) of interest, and then measuring them for +Devices Under Test (DUT) or System Under Test (SUT) for +comparison purposes, as well as possible absolute acceptance +criteria. It would seem a reasonable expectation that an Open +RAN system testbed should enable such KPIs to be measured, +not just end-to-end, but at interoperation points or interfaces +(and for specific O-RAN alliance defined interfaces, including +F1/W1/E1/X2/Xn). +However, once one enters the domain of detailed KPIs, there +is little standardization of what to measure. To an extent, +the detailed definition of KPIs is part of the specialized +knowledge of vendors, operators, and testing service providers +that are perceived to provide a competitive advantage, and +hence considered confidential. Because many of the KPIs +may be specific to specific vendors, there are also a very +large number of them. Commercial 5G networks test and +validate literally thousands of KPIs; the testing regime of well- +known mobile operators actually includes over ten thousand +KPIs. Many KPIs have sub-KPIs and the RF optimization +KPIs are substantial. This will only increase further with the +greater use of disaggregation in Open RAN networks. There +are numerous Open RAN interoperability and validation labs +today. There are private and public testbeds supported by +vendors, consortia, universities, and the government. Not all +labs concentrate on all parts of the toolchain and ecosystem, +most focus on specific aspects; validation testing will be +greatly dependent on the use case and focus of the lab. In the +Open RAN ecosystem, the RAN Intelligent Controllers (RICs) +allow for x-Apps and r-Apps to use the RIC framework as an +engine, but with custom functionality. This implies that every +such app can be expected to have a fairly large number of KPIs +associated with it depending on its particular functionality. +There is the potential for cross-KPIs between the different +apps as well. +In light of this, we are forced to go back to fundamentals +in recommending KPI capabilities for Open RAN testbeds. At +the highest level of abstraction, there are certain priority KPIs +that are foundational for a validation environment, and detailed +TABLE II: Example components for an Open RAN validation +environment testbed. +Open RAN Components +Test/Evaluation Components +• 5G core access and/or edge +• A Faraday cage / environment +• O-RAN Radios: gNB/eNB +(some at controllable UAVs) +• 5G signal analyzer – test and validate +measurements +• vRAN SW +• RTSA: Real-time spectrum analyzer +• GPS system(s)/Antenna- for +synchronization +• Network analyzer- antenna system +and cable measurements +• Forward Error Correction +(FEC) +• Antenna testing: anechoic chamber- +measure patterns +• Edge /Server, part of the core +network in a box +• Smaller Shielded enclosures, Faraday +cages for individual UAV testing +• (Open) RIC platform +• Traffic generator +• rApps, xApps +• Interferers – for testing purposes +• UEs (some at controllable +UAV for certain use cases) +• Various Adapters: need for every type +of connector +• ToR switch +• Jumper cables +• Cell site routers (CSR) +• Attenuators +• Acceleration for Open RAN +• Power splitters / power dividers +consideration of many custom KPIs for various operators and +vendors (although we are not in a position to list them here) +can be seen to trace back to one or the other of these few +foundational KPIs: +• Ability for UE to attach to the network; +• UE link quality – uplink and downlink; +• UE throughput – uplink and downlink; +• Latency; +• Retainability; +• Accessibility; and +• Optimization +Each of these KPIs drives multiple other test parameters and +features such as performance, load testing, and RF design +and optimization. At this time, practical Open RAN testing in +the real world is largely confined to component testing and +using KPIs related to the top few items in the above list; +in the future, more testing related to the Accessibility and +Optimization KPIs is likely to proceed. +Finally, an Open RAN testbed must include at least one +complete reference Open RAN implementation, both to serve +as a benchmark for other components to be tested against, +and also to enable system tests to proceed for experimenters +who wish to innovate in some, but not all, parts of the Open +RAN ecosystem. While Open RAN provides for a multi- +vendor environment in building a network from radios, vRAN +software, hardware servers, and related software and services, +it is important to note that “open” does not automatically +or necessarily equate to “interoperable”. The same need for +system integration of multi-vendor Open RAN networks that +has driven the need for open test environments must inform the +testbed designer in choosing such reference implementations +that are actually workable, and hopefully as compliant with +O-RAN interface definitions as possible, so as to be broadly +compatible with components and devices that testbed clients +may bring in the future. In Table II we have summarized what +we perceive to be key high-level components for an Open RAN +validation environment testbed. + +5 +C. Supporting UAVs in a Testbed +In its simplest form, any aerial robot (i.e. an airborne +device that stays aloft for significant periods of time and is +capable of directed motion) can be considered a UAV, but +the term is usually reserved for devices that are capable of +full (or at least a high degree of) autonomous operation. A +UAV can therefore exhibit not only primitive autonomous +behavior (pre-programmed/way-point trajectory, heat-seeking, +collision avoidance, auto-return-to-launch on predetermined +conditions such as GPS-lock-loss), but also more complex +operations such as computed conditional sensor-driven on-the- +fly trajectory control (such as search-and-rescue), participation +in coordinated trajectory control locally (platoon or swarm +behavior) or globally (such as UTM – the US Federal Aviation +Authority’s Unmanned Aircraft System Traffic Management +– or similar), or dynamic self-aware re-tasking (such as +degrading mission parameters for safety if battery reserves fall +to risky levels). +In distinguishing between testbed support of UAVs, it is +important to realize that a UAV implies close integration of +the onboard computing and communication equipment with +the vehicle’s command and control. It is helpful to think +of two extreme cases as representative of the two classes. +On the one hand, we can mount a computing/communication +device (such as an ordinary smartphone) on a UAV. The +UAV’s autonomy, trajectory computation, or command and +control, remain completely as before. The coupling between +the UAV and the cellphone it carries as a payload is simply +mechanical (but may include antenna mounts or high-gain +antennas custom-positioned for the UAV, and common power +supply). At the other extreme, the UAV contains only a single +computing/communication device, which is capable of being +tasked with complex missions (such as air quality analysis, +image analysis based search-and-rescue), and also subsumes +the trajectory computation (whether autonomous, command- +and-control-based, or based on some coordination) for the +UAV; in this case, the vehicle becomes in effect a peripheral +of the onboard computer. +First, we consider the task of integrating support in a +wireless testbed for UAVs only, used as vehicles for an +airborne UE. This includes the case where the air vehicle has +no autonomy and is controlled by a ground-based operator +using a handheld or other radio remote control equipment; +and even the case where the air vehicle does not have any +controlled mobility (such as free-floating balloons) or any +mobility at all (such as tethered aerostats or helikites). The +basic challenge for a wireless testbed to support UAVs is posed +not only by the fact that they are mobile (which, after all, +ground UEs also exhibit, when users walk or drive), but the +fact that they have a widely varied altitude as well as azimuth +compared to traditional UEs on the ground. Both spectrum +and latency are KPIs of interest for a UE. The front-haul and +mid-haul latencies must provide very low latency to maintain +system synchronization and function under a varying altitude +of the UE, and the spectrum used for communication can +significantly affect the achievable coverage and throughput. +A further challenge is that of antenna occlusion, which some +Proposed Stages for Open RAN Drone Validation +Stage 1: Basic UAV +functionality and +performance testing +Single UAV-single Cell (Attach, +Connect, Active/Inactive, +Throughput, Latency +Stage 2: Single Cell +Loaded +Performance Testing +Multiple UAV (UEs)-single Cell +(Attach, Connect, Active/Inactive, +Cell Throughput, Latency, AFR, DCR +Stage 3: Multiple +Cell Functional and +Performance Testing +Single UAV-Multi-Cell (Attach, +Active/Inactive, UAV Capacity, +Latency, AFR, DCR, Cell Capacity +Stage 4: Multiple +Cell Loaded +Performance +Validation +Multiple UAV-Cell (Attach, Connect, +Active, Inactive, UAV Throughput, +Latency, AFR, DCR, Cell Capacity +KPIs for the Stages: +• UAV throughput +• UE latency +• Cell Throughput +• Accessibility (AFR) +• Retainability (DCR) +• Control +Stage 5: Component +Capacity Validation +Multiple RUs per DU, Multiple DUs, +CPRI, etc +including UE/UAV Loading +Stage 6: Advanced +System testing +RIC platform applications for UAV +Control and System Optimization +on a Loaded System +Fig. 1: Proposed stages for Open RAN UAV validation. +UAVs attempt to mitigate by multiple antenna locations around +their bodies. Some UAVs mount antennas on gimbals in an +effort to maintain constant directional properties, others allow +for servos to allow controlled pointing of antennas. These +challenges are exacerbated by the fact that most base stations, +whether commercial or built out of commodity open technol- +ogy, exhibit their own antenna coverage patterns, which are +optimized for ground coverage. Studies have shown that the +consequence of this optimization is the formation of multiple +lobes at increasing altitudes, in complex patterns, that cannot +be predicted easily as a function of the altitude of the UAV. +The UAV will have to be tested in a controlled environment +to ensure the network functions and meet O-RAN specifica- +tions. Creating a Faraday environment to do the controlled +validation testing will pose challenges compared to traditional +Open RAN lab Faraday environments. Then the testing will +need to be expanded to an open environment and optimized +based on interferers, physical obstacles, and spectrum bands +used – as the propagation and throughput are connected to +the spectrum band used for communications. In Fig. 1, we +summarize six proposed stages for Open RAN UAV validation. +While UAVs allow intelligent control of position and tra- +jectory jointly with RAN intelligence (Apps executing at the +RICs), the softwarized character of Open RAN also opens up +exciting possibilities of allowing the onboard computer to take +part in the Open RAN ecosystem in ways other than just as a +UE. We devote the next section to these considerations. +III. USE CASES FOR UAVS IN OPEN RAN +Considering the aerial controlled mobility and communi- +cation among fixed and portable nodes, UAVs will facilitate +enhancements to Open RAN with flexible deployments and + +6 +on-demand, on-time network access. Several use case exam- +ples on Open RAN-based air mobility scenarios are provided +as follows (see Fig. 2). +Scenario 1. UAVs serve as UEs: This use case focuses on +exploring the functionalities of O-RAN RICs for managing +and orchestrating network components aimed at 3D critical +mission operations (e.g., secure, search and rescue) assisted by +UAVs, as they are able to exhibit agile, fast, and autonomous +behavior by organizing themselves to exchange information. +Considering a scenario involving UAVs connected to an Open +RAN ground BS, UAVs as UEs can carry high-resolution cam- +eras and/or sensors, collecting real-time video and transmitting +it back to the ground BS, e.g., to be used to identify possible +targets of interest through deep neural network object detection +model, and in addition report information about application +performance to rApps. In the meantime, the E2 nodes of O- +RAN are responsible for updating UAV control with insights +produced by their applications (xApps and rApps) to support +the RAN optimization process. In this context, Open RAN +is able to support the demands of highly dynamic scenarios +of critical-mission operations integrated with UAVs due to its +flexibility and characteristics of component dissociation. +Scenario 2. UAVs act as O-RUs: As described in O- +RAN specifications [34], [35], UAVs can play a role as O- +RUs and process several simple tasks. As the extension, this +scenario focuses on the use of UAVs as O-RUs to handle +more complicated network tasks, e.g., to quickly deploy an +aerial network to assist or extend the terrestrial network +where communication and computing resources can move +closer to users to meet diverse and stringent 5G application +requirements, such as ultra-low latency and ultra-high reliable +connectivity. Considering a scenario in which each UAV-BS +is equipped with an O-RU to serve ground mobile users, the +objective is to optimize the performance of serving offloading +tasks via both controlling UAV-BSs to guarantee the quality of +communication channels to ground users and efficiently dis- +tributing offloading tasks to appropriate Open RAN elements +according to the current association. Because of the 3D air +mobility capability of UAVs and disaggregation of Open RAN +architecture, they may potentially deliver better data offloading +capabilities and better resource utilization. +Scenario 3. UAVs act as O-DUs and O-CUs: 1) Using UAVs +as O-DUs allows for flexibly hosting RLC/MAC/High-PHY +layers based on a lower layer functional split, where UAVs can +dynamically connect to multiple O-RUs allowing on-demand +resource pooling for virtual baseband functions of high PHY +layer, MAC, RLC, and synchronization; 2) using UAVs as +O-CUs helps to easily control the operation of multiple O- +DUs within/beyond the coverage area, e.g., the radio resource +control for flexibly managing the life cycle of the connection, +routing or duplication for split bearers, and the service data +adaptation for managing the QoS of the traffic flows through +autonomous 3D air mobility capability of UAVs. +Scenario 4. Drone swarm based Open RAN: This use case +envisions multi-role drones without ground facilities that forms +an ad-hoc/swarm based Open RAN. Based on Scenarios 2-3, +we can consider a set of containers to virtualize different O- +RAN elements such as O-RUs, O-DUs, and O-CUs deployed +in drones and distributed computing nodes of the network. +Given these containers with different functions, the objective +is to create a robust Open RAN testbed in a swarm of drones +towards full decentralization and controlled air mobility. +Scenario 5. Flying wireless backhaul in Open RAN: Wire- +less backhaul as an economically sustainable solution has +been included by 3GPP as part of the integrated access and +backhaul study item [36], [37] for the 5G NR standard. As an +extension in Open RAN architecture, this scenario focuses on +building a large-scale, self-organizing network of drones that +are connected using a wireless mesh backhaul, which caters to +dynamic bandwidth-hungry and latency-sensitive applications. +Based on Scenario 4 with role-specific operations, drones can +hover above or close to the O-RU and serve as an airborne last- +hop link connecting RAN to the core network. Additionally, +they can act as relays between two O-RUs separated by +a longer distance to extend coverage forming a multi-hop +mesh network for communications and control. Multi-drone +backhaul in Open RAN is capable of flexibly adapting itself +to cater to highly dynamic applications and events, and easily +being scaled up to cover urban scenarios using long-range +radios. +Scenario 6. D2D communications underlaying drone- +assisted Open RAN: Implementation of device-to-device +(D2D) communication such as sidelink can be an extension +of the network into areas that traditional propagation of the +fixed O-RU cannot reach. Particularly, drones can serve as +UEs or relays deployed much more swiftly and improve the +network throughput performance by dynamically adjusting +their locations to provide direct or relayed D2D links to any +out-of-coverage users. Additional sidelink capabilities such as +multi-hop [38] and multi-link (in 3GPP Rel. 19) can provide +higher resiliency in this mode, especially offering a valuable +set of capabilities for mission-critical services such as disaster +response rescue and operation. +Testbed Considerations: The above poses a rich and varie- +gated set of potential operational scenarios, and it is impracti- +cal to attempt to enumerate specific design issues. Instead, we +again propose foundational considerations and hark back to +our discussion in Section II-A. The general capabilities of the +testbed that we can identify in order to support such innovative +scenarios are: +• The capability of mobility control of custom air vehicles, +• The ability to emulate not only the RF environment, but +of airflow and UAV flight, and +• The inclusion of onboard computers, suitable for inte- +gration into UAVs, that can support user programming to +create software components of the Open RAN ecosystem. +IV. AERPAW TESTBED REVIEW FOR OPEN RAN +Thus far, we have reflected on general requirements of +an Open RAN testbed that is able to integrate UAVs with +controlled mobility. In the remainder of this paper, we take a +deep dive into the AERPAW testbed, reviewing it in light of the +considerations we have derived above. We choose AERPAW +because we are intimately familiar with it; the authors of +this paper include the PIs of the AERPAW project, and key + +7 +UE Layer +Drone swarm based +O-RAN +Non-real-time +RIC +Near-real- +time RIC +A1 +O-CU +O-DU +O-RU +E2 +O1 +Sync +O-RU +O-RU +O-RU +O-DU +O-DU +O-DU +O-CU + O-DU +O-CU +O-CU +core +Offloading +tasks +O-RU +O-RU +O-RU +core +O-CU +Region Cloud +O-DU +Edge Cloud +O-CU +Region Cloud +O-DU +Region Cloud +O-RU +O-RU +O-RU +core +O-CU +Region Cloud +O-DU +Edge Cloud +O-CU +Region Cloud +O-DU +Region Cloud +Routing Path +O-RU +O-RU +UAV Relay +core +UAV BS +Application +UAV +UAV Relay +UGV +O-DU +Edge Cloud +(a) Scenario 1. +(b) Scenario 2. +(c) Scenario 3. +(d) Scenario 4. +(e) Scenario 5. +O-RAN Layer +O-DU +Region Cloud +(f) Scenario 6. +O-RU +Direct D2D Link +Relayed D2D Link from Drones +D2D UE +D2D UE +Cellular UE +Drone UE +D2D UE +D2D UE +Drone relay +Fig. 2: Use case examples for Open RAN-based air mobility: (a) UAVs as UEs; (b) UAVs as O-RUs; (c) UAVs as O-DUs +and O-CUs; (d) UAV swarms in O-RAN; (e) Flying wireless backhaul in O-RAN; (f) D2D communications underlaying +UAV-assisted O-RAN. +architects and DevOps personnel working on the AERPAW +facility. However, it is also true that AERPAW was conceived +and built to support controlled air mobility in a testbed for +use by a national community of researchers. Thus, it is a +reasonable facility in which to conduct such a thought exercise +of how a fully-featured Open RAN testbed may be built +up along the same lines. AERPAW has the foundation for +becoming a highly valuable Open RAN UAS test-bed. +AERPAW is the third testbed funded under the PAWR initia- +tive to support advanced and emerging wireless research. It is a +multi-year, multi-phase project that started in September 2019 +and it is expected to be finalized by 2025. AERPAW experi- +mentation capabilities became generally available with initial +set of resources and features in November 2021. Additional +platform resources, sample experiments, and experimentation +capabilities are expected to be released at the end of Phase- +2 (by May 2023) and Phase-3 (by May 2024). AERPAW +is primarily and essentially a testbed of physical resources, +not computing resources. The crucial part of these physical +resources are: (i) the RF environment and the airspace that the +AERPAW operating areas represent; (ii) the physical equip- +ment (SDRs, commercial RF equipment, UAVs, and UGVs) +that AERPAW provides to leverage those environments for +experimental studies; and (iii) the expertise (and consequent +exemptions) in conducting such studies in compliance with +FCC and FAA regulations that AERPAW represents. +Physically, the testbed is hosted at sites in and around +the NC State campus in Raleigh, NC. Central to AERPAW’s +unique characteristic is the availability of UAVs and UGVs in +the testbed that can be placed under the direct programmatic +control (of trajectories) of the researcher. In conjunction with +the programmable USRPs that are also available for direct +programming by the researchers, as well as other real-world, +commercial radio equipment, this provides the NextG wireless +researcher a facility for research experiments not practicable +in any other facility at this time. +Fixed Nodes, Portable Nodes, and Vehicles: +At a very +high level, the facility includes a number of tower locations +(fixed nodes), at each of which some combination of AERPAW +programmable SDRs and commercial radio equipment are +permanently installed. The SDRs are controlled by servers, +or companion computer (CCs), installed in each location that +also represent edge-computing capabilities. These fixed node +locations are distributed over the extensive Lake Wheeler +Agricultural Fields of NC State (see Fig. 3a), and some nodes +are also installed in the Centennial Campus (see Fig. 3b). +The complement of these fixed nodes are AERPAW’s portable +nodes, also consisting of a computer and SDR(s), but smaller +ones so that an AERPAW portable node can be mounted on +a UAV/UGV. The CC on a portable node, an Intel NUC, also +controls the UAV/UGV itself. A smaller version of the portable +node that can get carried at the smaller UAV is also available, +to do experiments with mobile phones and LoRa sensors that +are connected to a LattePanda as the CC. +More information on AERPAW is available at the AERPAW +Facility website and User Manual linked therefrom, and previ- +ous publications (also listed on the website). In what follows, +we attempt not a comprehensive overview of AERPAW, but +rather a review in light of the desirable characteristics we +identified above. +A. Span, Scale, Access +Fig. 3a and Fig. 3b show the outdoor deployment footprint +of AERPAW’s fixed nodes in NC State Lake Wheeler and NC +State Centennial Campus, respectively. The equipment that are +expected to be available publicly for experimentation by the +end of (AERPAW’s Phase-2 (expected May 2023) are also +illustrated. Currently, it is possible to experiment with UAVs + +8 +8 +LW-2 +LW-1 +LW-3 +LW-5 +LW-4 +• +4 NI USRPs +• +1 Ericsson 4G/5G BS (NSA) +• +1 Keysight RF Sensor +• +1 LoRa Gateway +• +4 NI USRPs +• +1 Keysight RF Sensor +• +4 NI USRPs +• +1 LoRa Gateway +• +4 NI USRPs +• +1 Keysight RF Sensor +• +4 NI USRPs +• +1 Keysight RF Sensor +(a) Since Nov. 2021, LW-1 is publicly available for experimentation, +and LW-2, LW-3, LW-4, LW5 are expected to be publicly available by +May, 2023. +70 +• +Facebook TG +Radios (60 GHz) +• +Facebook TG +Radios (60 GHz) +• +Facebook TG +Radios (60 GHz) +• +6 Facebook TG +Radios (60 GHz) +CC3 +CC2 +CC1 +• +4 NI USRPs +• +1 Keysight RF Sensor +• +1 LoRa Gateway +• +4 NI USRPs +• +4 NI USRPs +• +1 LoRa Gateway +CC4 +CC5 +CC6 +• +Fixed wireless SDR +experiments +• +Portable node +experiments at carts +(b) Since Nov. 2022, CC1 and CC2 are publicly available for experi- +mentation, and CC-3 is expected to be publicly available by May, 2023. +CC3, CC4, CC5, and CC6 each also has Terragraph radios from Meta +operating at 60 GHz. +Fig. 3: AERPAW fixed node deployments at (a) NC State +University Lake Wheeler Field Labs, Raleigh, NC; and (b) +NC State University Centennial Campus, Raleigh, NC. +at Lake Wheeler Field Labs; AERPAW does not currently sup- +port UAV operation by experimenters in Centennial Campus +but supports UGV operation, and UAV operation will likely +become available in the future for experimenters. +This geographical span is reasonable for an Open RAN +testbed, even with experiments including UAVs. However, +scale is a different matter. With nine fixed nodes, six +portable nodes, eight programmable UAVs, and some non- +programmable commercial radio systems such as an Ericsson +base station and five Keysight RF sensors, AERPAW can sup- +port a large variety of meaningful advanced wireless research +– including proof-of-concept Open RAN experiments at small +scales. But to support the full gamut of Open RAN testing +and Open RAN related research experiments, AERPAW would +need to add a large number and variety of commercial or stock +UEs, and a larger number of programmable UAVs; a few more +programmable fixed and portable nodes would also likely be +useful. +In Open RAN, the potential softwarization or virtualization +of various system components is a particularly attractive +feature for innovators. This requires allowing experimenters +Platform Resources +Platform Control +Experimenter +Portal +Website +AERPAW Ops +Human +Platform Control +Software tools +Virtual Resources +(Development mode) +Physical Resources +(Testbed mode) +Servers, SDN, orchestration, +custom emulation +Towers, SDRs, UAVs, UGVs +(New in Phase 2: +Ericsson, Keysight, FB TG, LoRa) +Safety Pilots +Configure, Orchestrate, Monitor +AVNs +ARNs +(a) Interaction of an AERPAW experimenter with platform control and +platform resources (development mode and testbed mode). +Platform Resources +Platform Control +1 Register, supply credentials +2 Create experiment, request develop +3 Trigger virtual experiment request +4 Instantiate virtual experiment +7 Login to virtual nodes, code, test +8 Save experiment, submit to testbed +9 Trigger testbed experiment request +10 Retrieve experiment from virtual +11 Install experiment on testbed +12 Handover to pilots/operators +13 Retrieve experiment, set complete +14 Notify experimenter of status +15 Request develop returned expmt. +20 Login to virtual nodes, view +1 2 +3 +4 +7 +8 +10 +11 +12 +13 +14 +15 +16 +17 +20 +5 Notify virtual experiment ready +5 +6 +6 Provide virtual experiment access +Experimenter +Portal +AERPAW +Ops +Control +AVNs +ARNs +18 Change status +19 Notify virtual experiment ready +18 +19 +17 Re-instantiate virtual experiment +16 Trigger virtual experiment request +9 +13 +13 +10 +11 +13 +11 +10 +(b) Steps for carrying out an experiment in AERPAW.. +Fig. 4: Experiment workflow for users of AERPAW. +direct programming access to all parts of the facility, and at the +highest levels of access. Managing such access while ensuring +the safety and regulatory compliance of the facility is a distinct +challenge for any testbed that aspires to achieve this. +On this front, AERPAW is already well positioned, hav- +ing been designed from the outset as a batch-mode facility. +Experimenters develop experiments in a virtual environment +and submit experiments for execution on the physical testbed +once development is complete. AERPAW Operations person- +nel (Ops) then execute these submitted experiments in the +physical testbed environment and collect the output of the +experiments as designed by the Experimenters, which are +available for Experimenters to view and analyze back in the +virtual environment. +This is not an arbitrarily decided constraint, but a considered +architectural choice. In operating a facility with programmable +radios and programmable air vehicles, we are obligated to +make, and uphold, certain guarantees to the FCC and FAA. +However, we also want to allow Experimenters the ability to +program those radios and air vehicles, ideally without needing +to become fully conversant with FCC and FAA regulation +details, obtain exemptions, or expertise in techniques for ensur- +ing compliance. Batch mode operation allows us to interpose +critical filters and monitors into the Experiment code execution +flow that allow us to guarantee safe and compliant operation. It +is one of the most valuable features of the AERPAW platform +that we assume this guarantee ourselves, rather than passing + +()(Q)(Q)(0)NCSTATEUNIVERSITYAnimal Health +Building. +NCSUG +z) +The Oval(Q)AtnCamupunOr +PartnersWay +ParthersWayNC STATE UNIVERSITYWilson College +aa +James BHunt J.Library +artnersWay +Fitts-Woolard +Hall +Wolf Ridge Apartments口 +口 +口 +口 +口 +口 +口 +口口 +口 +口 +口 +口 +口 +口 +口0 +00E22.0...9 +on the responsibility for compliant operations (and liability for +non-compliance) to the Experimenter. +Figure 4a and 4b show the entity relationships in AER- +PAW, and the experimenter’s experiment design workflow. +Experimenters request “Development Sessions” in which they +program a virtual environment that is programmatically indis- +tinguishable from the computing environment in the physical +testbed. Once completed, they submit such experiments for +“Testbed Execution Sessions”. The containers housing the +experimenter’s code is bodily moved to the corresponding +nodes in the physical testbed, where they are executed as +before, but with additional supervisory containers monitoring +for any RF violation or unsafe air-vehicle operating conditions, +overriding as necessary. As an additional line of defense, +human operators in the field are able to issue aborts if the +automated system should fail to override. +B. Spectrum and Licenses +AERPAW supports multiple frequencies for experimentation +with its fixed and portable nodes and vehicles. In particular, +AERPAW is one of the few FCC Innovation Zones (FCC-IZs) +in the United States [39, §1.6] with frequency bands that are +highlighted in Table III. The maximum effective isotropically +radiated power (EIRP) limits for fixed stations (FSs) and +mobile stations (MSs) are also specified in the table. The FCC- +IZ for Lake Wheeler Field Labs site for AERPAW covers an +area of approximately 10.5 square miles, while the Centennial +Campus FCC-IZ covers an area of approximately 3 square +miles. Experimenters can also port their FCC experimental +licenses at AERPAW’s FCC Innovation Zone. As noted in +Table III, due to the sensitivities of certain bands and the wide +interference footprint of transmissions from an aerial vehicle, +FCC does not allow airborne use in certain bands [40]. +AERPAW currently supports a subset of the frequency +bands through additional FCC experimental licenses (FCC +Call Sign: WK2XQH [41]), which are offered to AERPAW’s +users to carry out over-the-air experiments on the platform. In +particular, for SDR experiments, AERPAW has experimental +licenses at 3.3-3.55 GHz and 902-928 MHz, with plans +to incorporate this band into the AERPAW FCC-IZ in the +future. The experimental licenses for the Ericsson network +include 1.7/2.1 GHz for the LTE system and 3.4 GHz for +the 5G system. AERPAW also has plans to support generally +available experiments using its mmWave SDR framework by +the end of Phase-3 using Sivers phased arrays operating at +28 GHz. Spectrum monitoring and passive I/Q data collection +experiments can be supported using USRPs and Keysight RF +sensors between 100 MHz to 6 GHz. +A particular spectrum band that is of recent interest to +safety and navigation related command-and-control commu- +nications for UAVs, and that AERPAW will explore experi- +mental licenses in the future, is 5030-5091 MHz for which +FCC recently released a Notice of Proposed Rule Making +(NPRM) [42]. Another band that may potentially be used for +ensuring vehicle-to-vehicle (V2V) separation with cooperative +surveillance in the future for urban air mobility (UAM) scenar- +ios is 1104 MHz (also known as UAT2) [43]–[45]. Additional +TABLE III: AERPAW’s FCC Innovation Zone frequencies. +Footnotes: 1) Commission rules do not permit airborne use on +all or portions of these bands. 2) Any experimental use must be +coordinated with authorized users and registered receive-only +fixed satellite earth stations. 3) Operations must be coordinated +with a spectrum access system administrator. +Frequency +Band +Type +of +Operation +Allocation +FS +Max +EIRP +MS Max +EIRP +617-634.5 +MHz (DL) +Fixed +Non-federal +65 +- +663-698 +MHz (UL) +Mobile +Non-federal +- +20 (dBm) +907.5-912.5 +MHz +Fixed +and +Mobile +Shared +65 (dBm) +20 (dBm) +1755-1760 +MHz (UL) +Mobile +Shared +- +20 (dBm) +2155-2160 +MHz (DL) +Fixed +Non-federal +65 (dBm) +- +2390-2483.5 +MHz +Fixed +and +Mobile +Shared +65 (dBm) +20 (dBm) +2500-2690 +MHz1,2 +Fixed +and +Mobile +Non-federal +65 (dBm) +20 (dBm) +3550-3700 +MHz1,2,3 +Fixed +and +Mobile +Shared +65 (dBm) +20 (dBm) +3700-3980 +MHz1,2 +Mobile +Non-federal +- +20 (dBm) +5850-5925 +MHz +Fixed +and +Mobile +Shared +65 (dBm) +20 (dBm) +5925-7125 +MHz2 +Fixed +and +Mobile +Non-federal +65 (dBm) +20 (dBm) +27.5-28.35 +GHz +Fixed +and +Mobile +Non-federal +65 (dBm) +20 (dBm) +38.6-40.0 +GHz +Fixed +and +Mobile +Non-federal +65 (dBm) +20 (dBm) +spectrum bands that are specifically of interest for UAV/UAM +scenarios can be found in [40]. +C. Mobility Control +AERPAW is also, by its original design, already adequate +in providing controlled mobility, both for repeatability of +experiments and for experimentation with programmatic tra- +jectory control by experimenters; and both for aerial vehicles +as well as ground vehicles. Figure 5 shows the AERPAW +vehicle control stack. In AERPAW the main autopilot we +support at this time is ArduPilot [46] as it is open source +and well-trusted. ArduPilot is supporting MAVLink [47] as a +communication protocol, and, therefore, all AERPAW vehicle +software sends and receives MAVLink commands. For the +safety of the testbed and of the AERPAW operators, only a +reduced subset of MAVLink commands is allowed to pass +through the MAVLink Filter and reach the autopilot. +Keeping in mind the caveat on the reduced subset of +MAVLink commands allowed passing to the autopilot, at one +extreme, an experienced AERPAW user can, however, discard +the entire stack shown at the top of Fig. 5 and write their +own MAVLink application using any other framework they +wish (e.g., they could use MAVSDK [48] if they prefer a C++ +based library). +However, to smooth the learning curve, we implemented a +vehicle library named aerpawlib [49], which features a finite +state machine model, with hooks for vehicle (and/or radio) +actions at each state. Several examples are available either to + +10 +Fig. 5: AERPAW vehicle control stack. +Fig. 6: Sample vehicle experiment with two coordinated +drones: the tracer (red) goes through a list of waypoints, +while the orbiter (yellow) orbits around the tracer while at +the waypoint. +be used as-is or to be modified by experimenters to fit their +needs. The most popular example at the moment is the pre- +determined trajectory sample application, where users specify +a series of 3D waypoints to be traversed in order, including +choices of the speed and wait times at each waypoint. +The AERPAW framework also allows the experimenter’s +programs to take decisions on the fly, thus enabling au- +tonomous applications, such as a radio-based search and rescue +(SAR), where the next direction of movement can be chosen +based on the current radio measurements. +Autonomous Coordinated Multi-UAV Experiments: An ad- +ditional feature supported by the application programming +library provided by AERPAW is the ability of applications to +synchronize the control of multiple vehicles. This is achieved +either by using centralized control (where a coordinator pro- +gram sends synchronized commands to multiple vehicles), or +decentralized applications, (where programs on the compan- +ion computer of each of the vehicles coordinate without a +centralized conductor). This ability can be leveraged to allow +for swarm control. Fig. 6 shows the traces followed by two +drones in a coordinated drone experiment, where one drone +(the tracer) follows a list of waypoints, while the second drone +(the orbiter) shadows the tracer by moving at the same time +in the same direction, and upon reaching the target waypoint, +it orbits around the tracer once before they both move to the +next waypoint. +This experiment is initially designed and tested in the em- +ulation environment and subsequently executed in the testbed +environment. More complicated swarm experiments with a +larger number of drones and including communication links +with SDRs can be easily carried out using the same workflow. +Autonomous decisions can be integrated into the experiment, +where the drones can make next waypoint decisions based on +the observations of wireless signals. +Other testbeds can, of course, use alternate methodologies +for providing programmatic online trajectory control to experi- +menters, and repeatability of mobility profiles for experiments. +We have described AERPAW’s approach above not to advocate +it as the only way, but rather to articulate the level of +programmability and repeatability that experimenters should +be able to expect from a testbed facility. +D. Emulation Support +AERPAW has well-articulated emulation support for both +RF and air/mobility aspects of experiments. In the “Devel- +opment session” mentioned earlier, users can prepare their +experiments with perfectly repeatable trajectories and wire- +less propagation. The main goal of providing the emulation +environment is to allow users to develop their experiments in +a safe and fully repeatable environment. +Fig. 7a depicts an example experiment comprising a +portable node on the left and a fixed node on the right +while deployed in the emulation environment. In emulation +mode, the experimenters’ code (encapsulated in the two E- +VM, and shown in green in the picture), is running with no +modifications in comparison with an experiment in testbed +mode. In contrast, in emulation mode, the vehicle and the +wireless channel are emulated, thus allowing for a full software +emulation, amenable to cloud deployment. +For vehicle emulation, we use an open-source available +emulator that has been developed by the ArduPilot community, +which features as its main characteristic the use of the same +firmware as the autopilot we use on all our vehicles (at this +time, drones, rovers, helikite, and a push-cart). Careful com- +parisons between the performance of the emulated vehicles +and the testbed vehicles show that the vehicle emulator is +performing very realistically. +In contrast, for the wireless channel emulator (CHEM), to +the best of our knowledge, there is no open-source solution +that satisfies all our requirements; therefore, we developed our +own solution. Fig. 7b shows the main components involved in +the CHEM. In general, each radio-enabled node in the testbed +is capable of both transmitting and receiving radio signals, +which we capture at baseband, IQ level. The IQ samples are +sent to the channel emulator, which then “propagates” them +to the corresponding receivers. The propagation in CHEM is + +11 +(a) AERPAW emulation environment overview for one mobile node +and one fixed node. +(b) AERPAW wireless channel emulator overview. +Fig. 7: AERPAW emulation environment overview. +controlled by the channel control module, which dynamically +computes a channel matrix based on both dynamic information +(e.g., the current mobile node positions and orientations), as +well as static information (e.g., position of the fixed nodes, +antenna patterns, transmitter gains, etc.). +The CHEM supports several features, including free space +and two-ray ground propagation models, two noise models, +MIMO channels, up to 100 MHz of instantaneous bandwidth, +multi-rate processing, different antenna patterns, multiple fre- +quencies, and, importantly for efficiency, suppressing silences +for bursty traffic. +Once again, we have described AERPAW’s approach above +not to advocate it as the only way, but to articulate the level of +emulation support we find required for an Open RAN testbed. +Regarding AERPAW itself, while it has a good base from +which to provide emulation support for Open RAN experi- +ments, it would remain a non-trivial task to develop/procure +and incorporate the large volume of software modules that +would be required to be integrated into this framework in order +to provide emulation support for a comprehensive complement +of Open RAN experiments. In the next section, we return to +this topic briefly. +E. Programmability, Radios, Software Stack +AERPAW does not currently incorporate a full reference O- +RAN implementation, although some component parts exist. +TABLE IV: AERPAW example experiments with SDRs. +Software +Sample Experi- +ment +Comments +srsRAN +SE1: Multi-node +LTE SISO +Complete end-to-end LTE network +with +multiple +srsUE, +and +one +srsENB and srsEPC +SE2: +LTE +Cell +Scan +Search for LTE cells and capture key +parameters of interest +SE3: +Two-Node +LTE MIMO +Complete end-to-end 2x2 MIMO +LTE +network, +using +srsUE +with +srsENB and srsEPC +SE4: Multi-Node +IoT +Basic NB-IoT signalling between the +eNB and UE nodes +SE5: LTE Han- +dover +Complete end-to-end LTE network +with S1 handover, using srsUE with +srsENB and open5GS +SE6: +Single- +Node 5G SA +Complete end-to-end 5G SA net- +work, using srsUE with srsENB and +open5GS +OAI +OE1: Two-Node +LTE SISO +Complete end-to-end LTE network, +using OAIENB and srsUE +OE2: +Single- +Node 5G SA +complete end-to-end LTE network, +using OAIGNB and srsUE +GNU +Radio +GE1: +OFDM +TX-RX +Send and receive data using an +OFDM waveform +GE2: +Channel +Sounder +Pseudo-random +sequence +of +bits +are transmitted/received for channel +sounding +GE3: LoRa PHY +TX0RX +LoRa transceiver with all the neces- +sary receiver components +UHD +Python- +API +UHD1: Spectrum +Monitoring +Sweep based spectrum monitoring +between 87 MHz and 6 GHz +UHD2: IQ Col- +lection +IQ samples are collected at desired +center frequencies with some sam- +pling rate for a specified amount of +duration +The edge-cloud model of companion computers at every AER- +PAW Radio Node (including both fixed and portable nodes) +allows for an easy transition into Open RAN softwarized radio +modules, as such modules become available and integrated into +the testbed. +The Software Defined Radios of AERPAW represent a po- +tential strength in a possible transition path to full Open RAN +support since experimenting with evolving or innovative radio +protocols is reduced to an exercise of software development +and integration. +AERPAW team provides a variety of SDR sample exper- +iments for experimenters to work with using open-source +software and USRP SDRs from NI. Any AERPAW user can +start with one of these experiments and develop their code +further to research e.g. different protocols and waveforms. +AERPAW presently supports four different sets of open- +source software for SDR experiments: srsRAN [39, §4.1.1], +OpenAirInterface [39, §4.1.2], GNURadio [39, §4.1.3], and +Python scripts [39, §4.1.4]. A variety of sample experiments +are provided in AERPAW’s user manual for each case under +Section 4.1 [39, §4.1]. +In Table IV, we provide a list of SDR sample experiments +that are currently available or to be available by the end of +AERPAW’s Phase-2 (May 2023). An additional set of SDR +experiments is expected to be added for general availability +by the end of Phase-3 (expected May 2024). All these experi- +ments are tested both in the development environment and the +testbed environment of AERPAW. While experimenters can + +TNATIONAL +INSTRUMENTS +NIUSRP-2930 +N0MME.22Gl +REFIN +PSN +GSETHERET +POWER: +I1:: +=: +":12 +TABLE V: AERPAW example experiments with commercial +RF hardware. +Software +Sample +Experi- +ment +Comments +Ericsson +EE1: 5G Modem +RF Logging and +Throughput +Quectel modem logs various KPIs +from 4G/5G Ericsson network +Keysight +RF +Sensors +KRSE1: Spectrum +monitoring +Monitor and record spectrum up to 6 +GHz +KRSE2: +Signal +classification +Classify and detect a variety of sig- +nals based on RF signature +KRSE3: +Signal +source tracking +TDOA based localization of a signal +source by passive monitoring of its +RF signature +also bring their own software to the platform, AERPAW can +not guarantee that they will work smoothly with the existing +AERPAW hardware and software, and the development envi- +ronment. For further details, readers are referred to AERPAW’s +user manual [39, §4.1]. +AERPAW also includes similar prepared experiment profiles +for commercial radio equipment available in the testbed (see +Table V), but they are relevant in the Open RAN context +mainly as potential support equipment, so we do not discuss +them further here. +F. Summary - Open RAN Related Components of AERPAW +While AERPAW has not been designed initially as an Open +RAN testbed, its open, modular, and flexible design allows +possible expanded support for Open RAN use cases as a living +lab for UAVs with comparative ease. The AERPAW team +filled out, upon request, a survey in November 2022 developed +by the recently established Open RAN working group of the +National Spectrum Consortium (NSC) [51]. This survey was +shared by NSC members with existing testbed platforms that +may potentially support Open RAN experiments in the future. +In Table VI, we present a revised version of NSC’s Open +RAN survey and included comments on AERPAW’s features +and capabilities that can support Open RAN experiments with +controlled aerial mobility. In particular, we highlight open +and programmable end-to-end network capabilities as well +as commercial 5G equipment deployments in AERPAW, on- +site access to wireless spectrum, different experimentation +capabilities supported, compute nodes, unique use case testing +scenarios, testing types, among other related platform features. +The information provided in Table VI relates specifically +to the match and extensibility of AERPAW as a meaningful +Open RAN testbed for use cases with controlled air mobility. +However, the exercise of preparing this table affords us prac- +tical insights into designing and building such an Open RAN +testbed, to complement our observations in Section II, and we +pass these on to the community here. +V. REPRESENTATIVE RESULTS RELATED TO OPEN RAN +AND CONTROLLED AIR MOBILITY +In this section, we present two early representative exper- +iments from AERPAW that are of relevance for Open RAN +experiments. We also elaborate on other possible experiments +of relevance to Open RAN that may be supported in AERPAW +in the future. +0 +20 +40 +60 +80 +100 +120 +140 +160 +Time Interval (seconds) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Throughput Achieved per Slice (MBps) +100 PRBs +15 PRBs +80:20 +Configuration +20:80 +Configuration +50:50 +Configuration +Fig. 8: Representative results on O-RAN slicing xApp using +srsRAN with two UEs. +A. RAN Slicing xApp Experiments +In this section, we provide representative results using the +RAN slicing xApp and srsRAN, using the framework by +the NSF POWDER Wireless platform [52], executed at the +AERPAW testbed. (Note that these features have not yet been +integrated into the AERPAW’s development and transition-to- +testbed environments; we are exploring integration options at +this time). The goal is to dynamically create network slices and +observe the effects of slice reconfiguration with a TCP stream +on the performance of a UE. A near Real-time RIC is deployed +as part of two separate Kubernetes clusters. Detailed steps +are provided in [53], we will provide a high-level overview +of the architecture. The RIC cluster is used for deploying +the platform and applications which are part of the RIC, +whereas the Aux cluster is used to deploy other auxiliary +functions. The RIC Kubernetes cluster installation is done +through configuration scripts and pre-generated helm charts +for each of the RIC components. Once the process is done, +we created a persistent volume through a storage class for +the influxDB on the RIC platform namespace. Once the RIC +platform is deployed, a modified E2 termination is created +which has few services enabled to communicate and exchange +messages between RIC and E2 Agent [53]. +Once the Kubernetes clusters are deployed, we can deploy +the Near Real-time RIC using a RECIPE file which provides +customized parameters for the configuration of a particular +deployment group. This Recipe file can be tinkered with if +we want to change any configuration to suit our requirements. +Next is the installation of srsRAN components such as srsUE, +srsEnB, and srsEPC which use ZeroMQ networking libraries. +Since we use ZeroMQ mode, the 4G/5G network can be set +up using a single machine that hosts both the RIC and srsRAN +components. Finally, the xAPP is onboarded and deployed on +top of the Near real-time RIC and full integration is completed. +Using this setup, we create two network slices in a work- +conserving mode and bind two srsUEs to these network slices. +Some representative results are presented in Fig. 8 for two + +13 +TABLE VI: AERPAW features and capabilities related to Open RAN. +Capability +O-RAN Related Components +AERPAW Availability +Open and +Programmable +End-to-End +Network +Multiple SDRs connected to power and network backhaul +USRPs, Keysight RF sensors +Indoor wireless operations in a lab +N/A +Outdoor wireless operations +Rural farm and urban campus +Open 5G mobile cores +Open5GS +Open fronthaul interface for testing open RUs +Not currently available +Open source software stacks ready to use with or without +additional software development +srsRAN, OAI, GNURadio, I/Q collection with sample experi- +ments [39] +Open source RIC implementation +Not currently available +BYOD operation +Yes (on a case-by-case basis) +BYOS operations +Yes (on a case-by-case basis) +Bare metal for software installations +Not currently available +Containers for software installations +Yes – both in emulation and testbed modes +Remote access to network resources +Yes during development (emulation) mode, not normally during +testbed mode +End-to-End +Network with +Commercial +Equipment and +Swappable +Components +Commercial equipment +Ericsson 4G/5G network +Indoor wireless operations +N/A +Outdoor wireless operations +Rural farm area +Commercial 5G mobile cores +Ericsson NSA core network (Release-15) +Includes one or more of a commercial RIC, CU, DU, and RU +Not currently available +Open fronthaul interface enabling testing of open RUs to +support different physical layers +Not currently supported +On-site Access +to Spectrum +Unlicensed or ISM band +900 MHz for aerial communications with SDR front ends +CBRS spectrum and CBRS SAS features +N/A +Licensed spectrum from a spectrum owner +N/A +Experimental or Innovation Zone licensed spectrum +Yes – FCC Innovation Zone with 13 bands in 0.6-40 GHz [39] +Techniques +Channel emulation systems +Software emulation available now [39], Keysight Propsim (32 ports) +channel emulator in the process of integration +Multiple modes of massive MIMO +Not presently available – mmWave UAV capabilities with 4x4 Sivers +phased arrays in development +Emulation capabilities for the RIC, CU, DU, RU, and UE +Presently not available +Compute +Capacity +One optical hop +Yes +Edge compute +Yes – Dell 5820 Server at fixed nodes, Intel NUC (i9) at portable +nodes carried by AERPAW vehicles +Public cloud computing +Not presently supported +Unique Use +Case Testing +Drone support +Multiple different custom drones for different use cases +Rural and urban environment +Yes (autonomous drone experiments available only in rural) +Military base +N/A +Smart agriculture +Deployment in Lake Wheeler agricultural farm of NC State [39] +Testing Types +Research and development +Free access by NSF-funded academic researchers, charge-based +access for other researchers +Compliance (3GPP, ETSI, O-RAN, etc.) +3GPP +compliant +open-source +and +commercial +4G/5G +hard- +ware/software +Interoperability +Partial +Security +Partial +Performance/stress testing +Partial +Others +Research staff availability +Yes (multiple research associates/students for research support) +Operational staff availability +Yes (multiple research associates/students to support experiments) +Wireless certification program +Not presently supported +Established connections to standards/specifications organiza- +tions +NextG Alliance, Open Generation Alliance, GUTMA, Linux Foun- +dation InterUSS Platform [50] +different bandwidths, which show the throughput of one of +the UEs. We configure the slice scheduler in steps to alter the +proportionate scheduling in different ways and observe the +effects on the TCP stream for the UE [54], [55]. An Iperf +server is created on the UE namespace to observe the effects +of dynamic RAN slicing and a corresponding Iperf client [56]. +We create two slices, referred to as fast and slow, where each +slice can be dynamically configured to share the bandwidth. +For the baseline scenario, the full bandwidth of 15 PRBs (100 +PRBs) is initially allocated to the unsliced UE which gives a +throughput of around 35-40 MBps (170 MBps) as illustrated +in Fig. 8. +After this, the resources are distributed with the 80:20 +configuration among the two UEs. The results in Fig. 8 show +that the UE’s throughput falls to 27 MBps (140 MBps) for this +configuration, and when the priorities are inverted between the +fast and slow slices to 20:80, the throughput further reduces +to 6-7 MBps (40 MBps). Finally, when the priorities are +equalized to 50:50 configuration, the throughput increases to +16-17 MBps (70 MBps) for the first UE. The results can +be easily extended to a larger number of UEs and more +complicated resource configurations. +Our future work includes implementing this same scenario +in AERPAW’s development and testbed environments with +multiple controllable vehicles. The throughput needs and the +link qualities of UEs will change dynamically over time as +the vehicles move around, and there is a need to have a +dynamic slicing mechanism that satisfies the requirements of +individual network slices. AERPAW can support development +and testing in such dynamic RAN slicing scenarios, first in +the emulation environment, and then in the testbed mode +with realistic propagation conditions. Programmable mobility + +14 +Fig. 9: I/Q sample experiments representative results: LTE +reference signal received power (RSRP) at five different UAV +altitudes. +with multiple vehicles in both environments and will make it +possible to have a testing environment that provides repeatable +measurements involving precise mobility control for the UEs, +and in some cases, mobile relays and mobile base stations with +wireless backhaul. +B. I/Q Sample Collection Experiments +In Fig. 9, we provide representative results for the UHD2: +IQ collection sample experiment shown in Table IV. The +UAV is programmed to fly at five different altitudes and the +USRP B205mini at the UAV collects IQ samples centered at +3.51 GHz with a sampling rate of 2 MHz. The only signal that +can be observed in the spectrogram in the same band is an LTE +signal of 1.4 MHz bandwidth, transmitted from a USRP B205 +mini that runs srsRAN at our LW1 fixed node. We post-process +the collected I/Q samples using Matlab’s 4G toolbox, obtain +RSRP for each I/Q sample location, and plot the RSRP over +the trajectory. Additional details of the measurement setup +and representative results are available in [57] using further +post-processing with Matlab’s 4G toolbox, such as coherence +time and coherence bandwidth with respect to the distance +between the UAV and the fixed node, kriging interpolation +of the received signal across the whole 3D volume, channel +estimation, synchronization procedures, among others. +A similar experiment can be carried out to capture I/Q +samples and evaluate the KPIs for any Open RAN based 5G +system with varying locations of UAVs and UGVs. One or +more of the SDR, commercial wireless, or vehicle control +sample experiments from AERPAW’s sample vehicle experi- +ment repository, such as the one illustrated in Figure 6 above, +can be used simultaneously with the I/Q sample collection +experiment, to collect the raw I/Q data at the finest granularity +and post-process them in Matlab’s 4G and 5G toolboxes +to generate desired KPIs. Such data collected in realistic +propagation conditions can be made publicly available to the +research community for furthering the research in controlled +aerial mobility technologies. +VI. CONCLUSION +Open RAN expands the capabilities of 5G to support fea- +tures and functions tied directly to use cases. Disaggregation +and virtualization are well suited to UAVs/drones which will +continue to grow and become a much greater part of the 5G +network from a UE or acting as an O-RU, O-DU, or O-CU +component of the network architecture. However, testing and +validation are critical to successful integration into 5G and the +expansion of Open RAN network capabilities. +Creating a testbed that supports UAVs poses challenges to +meeting all the demands from the physical network to Open +RAN interoperability needs. For the UAV market to grow +and flourish testing and validation are necessary. As rules +and regulations remain volatile in the immediate future, a +UAV Open RAN lab can provide extremely valuable technical +results to inform such actions. +In this paper, we have provided conclusions drawn from our +experience and expertise gained from designing AERPAW, a +one-of-a-kind public advanced wireless testbed that provides +programmable radio and vehicle control in a realistic outdoor +area of considerable span, and also reflected on its fit as a +possible Open RAN / UAV testbed in future. 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Mushi, “AERIQ: SDR-Based LTE I/Q Measurement and Analysis +Framework for Air-to-Ground Propagation Modeling,” to appear in +IEEE Aerospace Conf., arXiv preprint arXiv:2210.07433, 2022. + diff --git a/w9FIT4oBgHgl3EQfzit1/content/tmp_files/load_file.txt b/w9FIT4oBgHgl3EQfzit1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38ad05e14e2ecf039cdbcd3d4d3f2f4235efdfeb --- /dev/null +++ b/w9FIT4oBgHgl3EQfzit1/content/tmp_files/load_file.txt @@ -0,0 +1,1228 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf,len=1227 +page_content='1 Open RAN Testbeds with Controlled Air Mobility Magreth Mushi, Yuchen Liu, Member, IEEE, Shreyas Sreenivasa, Ozgur Ozdemir, Ismail Guvenc, Fellow, IEEE, Mihail Sichitiu, Member, IEEE, Rudra Dutta, Senior Member, IEEE, and Russ Gyurek Abstract—With its promise of increasing softwarization, im- proving disaggregability, and creating an open-source based ecosystem in the area of Radio Access Networks, the idea of Open RAN has generated rising interest in the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Even as the community races to provide and verify complete Open RAN systems, the importance of verification of systems based on Open RAN under real-world conditions has become clear, and testbed facilities for general use have been envisioned, in addition to private testing facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Aerial robots, including autonomous ones, are among the increasingly important and interesting clients of RAN systems, but also present a challenge for testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Based on our experience in architecting and operating an advanced wireless testbed with aerial robots as a primary citizen, we present considerations relevant to the design of Open RAN testbeds, with particular attention to making such a testbed capable of controlled experimentation with aerial clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We also present representative results from the NSF AERPAW testbed on Open RAN slicing, programmable vehicles, and programmable radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Index Terms—Open RAN, Interoperability Testing, IOT, Testbed, open-source, eNB, gNB, aerial, UAV, drone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' INTRODUCTION Open Radio Access Network (specifications defined in the O-RAN Alliance) has emerged as a serious and perhaps critically necessary alternative to the proprietary radio access network (RAN) solutions that have characterized cellular net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In particular, Open RAN provides a richer eco-system based on the virtualization of network functions providing greater economies of scale and reduced cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The open archi- tecture of Open RAN, and the definition of interfaces among modules that have been thus far treated as essentially mono- lithic, are expected to ensure inter-operation between products from different providers, and a competitive market, leading to improved quality and lower cost of ownership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' It also enables the inclusion of commodity controllers, and the ability of operators to develop their custom control applications on top of those controllers, bringing the power of software-defined networking to RANs on an open-interface basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Such disaggregation comes at the cost of increased over- head, and early Open RAN systems are widely expected to have higher overheads and lower efficiency compared to extant single-vendor systems that, after all, have evolved and been in- tegrated for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Optimistic views consist of expectations of workable, if inefficient, implementations soon, followed by rapid improvements in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Pessimistic views incline to doubts regarding how long such a process might This work was supported in part by the NSF Award CNS-1939334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Mushi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Dutta are with the Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Computer Science, NC State University, Raleigh, NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Sreenivasa, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Ozdemir, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Guvenc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Sichitiu are with the Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Electrical and Computer Engineering, NC State University, Raleigh, NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Russ Gyurek is with Cisco Systems, Raleigh, NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' take, or whether such systems can approach the efficiency of proprietary monolithic systems, or even be workable at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, there are significant gains in terms of economies of scale through virtualization as well as additional functionality that provides a much richer set of capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', RIC apps, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' To dispassionately and pragmatically assess the worka- bility of Open RAN, the community must move beyond early experiments and greenfield deployments to demonstrable repeatability of predictable system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Designing dependable test facilities for Open RAN components and systems, therefore, is among the most important outstanding tasks of the Open RAN community at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A key promise of Open RAN is interoperability (multi-vendor), and the key to verifying such claims is through interoperability testing (IOT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Recognizing the importance of IOT, the O- RAN Alliance has dedicated two entire work groups (WG4 and WG5) to specifying interfaces, and both groups have published specifications on interoperability testing and profiles in addition to unit test specifications (see [1], [2] and other specifications of WG4 and WG5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Such profiles allow the interoperability of any set of components to be tested in test configurations that can be realized in lab-environment test benches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, to engender the above-mentioned growing confidence, Open RAN ecosystem players (contributors, as well as vendors, operators, and users) need to be able to test components in a comprehensive end-to-end test facility - one that is embedded in a realistic setting and span in the real world, including at least in part an outdoor setting, with a non- trivial number of UEs interacting with a non-trivial number of base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the rest of this paper, we reserve the term “testbed” to indicate facilities capable of such complete RAN system tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Unpeopled Aerial Vehicles (UAVs), have long been gener- ally acknowledged as important clients of any future wide-area wireless communications system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, the full scope of such devices as denizens of the wireless communication world is only coming to be appreciated recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A key observation is that UAVs are not only wireless communications clients for command-and-control (the most obvious use case), but play roles in at least two other ways in the wireless ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' First, trivially, as such devices increase in intelligence, and are tasked with increasingly more sophisticated missions, these missions are likely to pose additional – and likely much heavier – communication requirements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' for example streaming live on-site video back to the cloud, or engaging in other data-heavy cloud-assisted distributed computation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' More importantly, and more significantly in the current context, with increasing on-board compute intelligence, such devices are capable of engaging not just as clients, but as crucial parts arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='11365v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='NI] 26 Jan 2023 2 of the wireless communication infrastructure itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This is es- pecially important in an open interoperable ecosystem such as Open RAN aspires to be, as open competition spurs innovative contributors to explore previously unoccupied ecosystem roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The visioning and design exercise for an Open RAN testbed that aspires to provide interoperability and system testing capabilities, if such a facility expects to support the full evolutionary arc of aerial devices, must include reflection specific to these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In this paper, we leverage our joint experience in (i) architecting and operating an advanced wireless testbed with aerial robots as primary citizens, and (ii) industry Open RAN testing and dependability expectations, to provide a starting point that we hope will be useful to such architects and designers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the next section, we briefly review some existing test facilities with the capability (or potential near capability) of acting as Open RAN testing resources and juxtapose them with industry Open RAN testing norms, as well as basic support requirements for UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Section III, we discuss in further detail the class of use cases that represent the potential synergistic use of UAVs in Open RAN systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Finally, we provide a deep consideration of one extant testbed – our own NSF AERPAW platform at NC State University – to showcase the process of reviewing testbed capabilities to articulate both strengths and shortcomings in light of an ideal Open RAN testbed with native UAV support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VISIONING AN OPEN RAN/UAV TESTBED A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Existing Wireless Testbeds and System Testing There are numerous testbeds that are accessible to re- searchers to experiment with wireless technologies including 5G, Open RAN, and UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Table I, we provide a list a few of these testbed facilities that are accessible to researchers from academia, government, and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Note that we do not intend to present Table I as either comprehensive or authoritative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' There are likely many facilities that we are unaware of, or for which no information is publicly available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Even for those we have surveyed, Table I represents our best knowledge as obtained from publicly available sources (as cited);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' we regret any unintended mischaracterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Our survey was also heavily biased toward facilities in the USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Nevertheless, since our focus is on test facilities publicly or generally available to researchers and practitioners in the US, and on facilities sizeable enough for UAVs to be practically a part of the test ecosystem, we believe that Table I provides representative, and meaningfully extensive, information for the Open RAN testbed designer of the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We have chosen to characterize each facility listed by means of a few high-level considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Obviously, explicit currently stated support of Open RAN testing, and UAV support/integration, are features we looked for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Related to Open RAN, we also looked at the RF spectrum the facility is capable of and allowed to operate in, by noting if it lists an Innovation Zone (IZ) license from the Federal Communications Commission (see for example [3]), and also its deployment context (indoor facilities may be able to use isolation such as Faraday cages and operate without an FCC Innovation Zone or experimental licenses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Related to UAV support, we also looked at whether such UAVs (or any component of the testbed, for those without UAVs) support controlled mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We consider this feature an important one for future Open RAN testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A signif- icant proportion of wireless communications system com- plexity arises from (or is exacerbated by) the mobility of system components, most usually that of User Equipment (UE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' therefore it it important for the testbed to support experiments involving mobility, hand-over, and disconnect- reconnect events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, the core of the scientific method is the repeatability of experiments and the reproduction of experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' To provide this for experiments related to mobility, the relative motion of various system components must be possible to precisely reproduce on demand, for as many runs of an experiment as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Another key feature we looked for was emulation support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The single most valuable characteristic of actual wireless test facilities is the availability of a real Radio Frequency (RF) environment, providing real-world challenges such as fading, multi-path, and statistical uncertainty, simultaneously with the experiment repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The simulation of RF environments by means of mathematical models, no matter how sophisti- cated, abstracts a measure of realism from test results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' further, the experimenter has no need of an experimental facility (or even actual radios) for simulation exercises, which are an appropriate earlier stage in proving research before considering testbed validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The exercise of emulation, on the other hand, provides an important added value to a testbed, in that it is a digital twin of a real RF system, capable of operating in real-time, in which physical radio equipment can actually be immersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Emulation systems are driven by calibration (to some real RF environment) rather than modeling and may be realized by digital twinning, or more often by analog RF cir- cuitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In extreme cases, a test facility may be entirely based on emulation, as in the case of the Colosseum system (originally created by DARPA and currently operated at Northeastern University under NSF aegis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' More typically, emulation support is an adjunct part of a physical test facility that can serve as an early and less costly stage of full testbed validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Even before moving on to discussions of testing require- ments specific to Open RAN or UAVs, we can note a few points from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Naturally, those we were able to survey were largely public-use testbeds, since those are the ones that are most likely to provide information publicly about them- selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This dovetails with our focus since the interoperability focus of Open RAN implies that for engendering maximum confidence, the testbed facility should be open to anybody that is interested in repeating experiments and verifying results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Unsurprisingly, there is no testbed on the list that provides full Open RAN as well as UAV support today, even without considering controlled mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Less obviously, we find that the combination of UAV support and controlled mobility is rather rare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' only a handful of testbeds on our list provide even partial mobility control in conjunction with UAV support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Interestingly, we note that a number of testbeds provide emulation support, in keeping with our expectation that this is a key required feature of wireless testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, when emulation is considered jointly with mobility control, a non- obvious consideration may be worth mentioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For a testbed 3 TABLE I: Existing testbeds with advanced wireless and UAV experimentation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Public testbeds indicated with an asterisk (*) may be open only to partners or require contacting testbed operators rather than being generally available through an experimentation portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Features for which public information could not be found are marked as Not Known (NK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Testbeds (alphabeti- cal) Location Emulation Support Open RAN Support UAV Support Controlled Mobility FCC- IZ Main Focus Area Deployment Environment Access AERPAW [4] Raleigh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NC ✓ Partial ✓ ✓ ✓ UAVs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' SDRs Rural,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Urban Public ARA [5] Central Iowa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' IA ✓ × × × × Rural wireless Rural Public Arena [6] Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' MA ✓ ✓ × × ✓ SDRs Indoor grid Public* ARLIS [7] College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' MD × × × × × 5G security Virtual Public* ARM / Tech Mahin- dra 5G Lab [8] NK NK ✓ × × × 5G testing NK Private Booz Allen 5G Lab [9] Annapolis Junc- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' MD NK NK × × × Mission critical 5G NK Private CCI xG Testbed [10] Arlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VA NK ✓ × × × SDRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AI Indoor Public* Colosseum [11] Burlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' MA ✓ ✓ × × ✓ Emulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' SDRs Cloud Public CORNET [12] Blacksburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VA × ✓ × × × SDRs Indoor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Rooftop Public COSMOS [13] Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NY ✓ ✓ × × ✓ mmWave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' back- haul Urban Public Drexel Grid [14] Philadelphia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' PA ✓ × × × × Emulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' SDRs Indoor grid Public* Ericsson Open Lab [15] NK ✓ ✓ × × × CloudRAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' vir- tualized 5G Indoor Private INL Wireless Testbed [16] Idaho Falls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ID × × ✓ Partial × Wireless security Rural Private IRIS [17] Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CA × × × ✓ × Robotic wireless networks Indoor Public* LinQuest Labs [18] Chantilly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VA ✓ NK ✓ NK × 5G security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NTN Cloud,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' indoor Public* NASA MTBs [19] × × × ✓ × × Multirotor UAV testing Indoor Public* New York UAS Test Site [20] Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NY × × ✓ Partial × BVLOS UAV testing Rural,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Urban Public* NIST 5G Coexistence Testbed [21] Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CO ✓ NK × × × 5G coexistence testing Indoor Public* NIST NBIT Testbed [22] NK × × × × Spectrum shar- ing Indoor Public* NITOS [23] Volos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Greece × ✓ × × × Cloud-based Wireless services Rooftop Public Northeastern UAS Chamber [24] Burlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' MA × × ✓ NK × Drone flights Drone cage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' anechoic chamber Public* ORBIT [25] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Brunswick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NJ ✓ × × × × SDRs Indoor grid Public PNNL 5G Innovation Studio [26] Richland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' WA × × × × × Commercial 5G Indoor Private POWDER-RENEW [27] Salt Lake City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UT ✓ ✓ × × ✓ SDRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' massive MIMO Urban Public RELLIS 5G testbed [28] Bryan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' TX × NK NK NK 5G (AT&T) Outdoor Public* Cyber Living Innova- tion Lab [29] Fairfax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VA NK ✓ NK NK × 5G security,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' robotics Indoor Public* SOAR [30] Buffalo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NY × × ✓ Partial × Drone flights Drone cage Public* TIP Community Lab [31] Overland Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Kansas NK ✓ × × × O-RAN 5G NR (Sprint) NK Private UNH Interoperability Lab [32] Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NH × ✓ × × × Interoperability testing Indoor Public* Virginia Tech Drone Park [33] Blacksburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VA × × ✓ Partial × Drone flights Drone cage Public* that provides mobile airborne components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' any emulation system must not only emulate the physical RF environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' but also the physics of airflow and aerial navigation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' including wind gusts and other disturbing factors (analogous to noise and interference in the RF environment),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' as well as the dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and constraints of a specific UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The ability to au- tonomously navigate one or more UAVs in the 3D space based on RF observations in the environment is also an important capability with various use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Furthemore, subtle moves of the UAVs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', a multicopter pitching to move forward) can change the orientation of highly directional RF antennas (especially relevant for mmWave transmissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' With this 4 in mind, it is perhaps unsurprising that the combination of emulation support and mobility control is quite rare in the extant testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Extant Industry Open RAN Testing Practices The facilities listed in Table I are largely those focused on system testing, some of which currently already support deploying some particular Open RAN system in part or in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Researchers or ecosystem developers may find this sufficient since it is possible for them to test or study their products or innovations in contiguous areas supported by “some” Open RAN implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, vendors, carriers, and other ecosystem players who are involved in the business of actually building or operating a data network as a service need to focus far more deeply on component testing, and (critically for Open RAN) cross-vendor interoperability testing - especially the large swathes of new interoperability modes enabled by Open RAN’s disaggregation modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Such testing proceeds by identifying Key Performance Indicators (KPIs) of interest, and then measuring them for Devices Under Test (DUT) or System Under Test (SUT) for comparison purposes, as well as possible absolute acceptance criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' It would seem a reasonable expectation that an Open RAN system testbed should enable such KPIs to be measured, not just end-to-end, but at interoperation points or interfaces (and for specific O-RAN alliance defined interfaces, including F1/W1/E1/X2/Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, once one enters the domain of detailed KPIs, there is little standardization of what to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' To an extent, the detailed definition of KPIs is part of the specialized knowledge of vendors, operators, and testing service providers that are perceived to provide a competitive advantage, and hence considered confidential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Because many of the KPIs may be specific to specific vendors, there are also a very large number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Commercial 5G networks test and validate literally thousands of KPIs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' the testing regime of well- known mobile operators actually includes over ten thousand KPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Many KPIs have sub-KPIs and the RF optimization KPIs are substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This will only increase further with the greater use of disaggregation in Open RAN networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' There are numerous Open RAN interoperability and validation labs today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' There are private and public testbeds supported by vendors, consortia, universities, and the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Not all labs concentrate on all parts of the toolchain and ecosystem, most focus on specific aspects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' validation testing will be greatly dependent on the use case and focus of the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the Open RAN ecosystem, the RAN Intelligent Controllers (RICs) allow for x-Apps and r-Apps to use the RIC framework as an engine, but with custom functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This implies that every such app can be expected to have a fairly large number of KPIs associated with it depending on its particular functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' There is the potential for cross-KPIs between the different apps as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In light of this, we are forced to go back to fundamentals in recommending KPI capabilities for Open RAN testbeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' At the highest level of abstraction, there are certain priority KPIs that are foundational for a validation environment, and detailed TABLE II: Example components for an Open RAN validation environment testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Open RAN Components Test/Evaluation Components 5G core access and/or edge A Faraday cage / environment O-RAN Radios: gNB/eNB (some at controllable UAVs) 5G signal analyzer – test and validate measurements vRAN SW RTSA: Real-time spectrum analyzer GPS system(s)/Antenna- for synchronization Network analyzer- antenna system and cable measurements Forward Error Correction (FEC) Antenna testing: anechoic chamber- measure patterns Edge /Server,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' part of the core network in a box Smaller Shielded enclosures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Faraday cages for individual UAV testing (Open) RIC platform Traffic generator rApps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' xApps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Interferers – for testing purposes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UEs (some at controllable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UAV for certain use cases) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Various Adapters: need for every type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='of connector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='ToR switch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Jumper cables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Cell site routers (CSR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Attenuators ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Acceleration for Open RAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Power splitters / power dividers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='consideration of many custom KPIs for various operators and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='vendors (although we are not in a position to list them here) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='can be seen to trace back to one or the other of these few ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='foundational KPIs: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Ability for UE to attach to the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UE link quality – uplink and downlink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UE throughput – uplink and downlink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Latency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Retainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Accessibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and Optimization Each of these KPIs drives multiple other test parameters and features such as performance, load testing, and RF design and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' At this time, practical Open RAN testing in the real world is largely confined to component testing and using KPIs related to the top few items in the above list;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' in the future, more testing related to the Accessibility and Optimization KPIs is likely to proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Finally, an Open RAN testbed must include at least one complete reference Open RAN implementation, both to serve as a benchmark for other components to be tested against, and also to enable system tests to proceed for experimenters who wish to innovate in some, but not all, parts of the Open RAN ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' While Open RAN provides for a multi- vendor environment in building a network from radios, vRAN software, hardware servers, and related software and services, it is important to note that “open” does not automatically or necessarily equate to “interoperable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The same need for system integration of multi-vendor Open RAN networks that has driven the need for open test environments must inform the testbed designer in choosing such reference implementations that are actually workable, and hopefully as compliant with O-RAN interface definitions as possible, so as to be broadly compatible with components and devices that testbed clients may bring in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Table II we have summarized what we perceive to be key high-level components for an Open RAN validation environment testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Supporting UAVs in a Testbed In its simplest form, any aerial robot (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' an airborne device that stays aloft for significant periods of time and is capable of directed motion) can be considered a UAV, but the term is usually reserved for devices that are capable of full (or at least a high degree of) autonomous operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A UAV can therefore exhibit not only primitive autonomous behavior (pre-programmed/way-point trajectory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' heat-seeking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' collision avoidance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' auto-return-to-launch on predetermined conditions such as GPS-lock-loss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' but also more complex operations such as computed conditional sensor-driven on-the- fly trajectory control (such as search-and-rescue),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' participation in coordinated trajectory control locally (platoon or swarm behavior) or globally (such as UTM – the US Federal Aviation Authority’s Unmanned Aircraft System Traffic Management – or similar),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' or dynamic self-aware re-tasking (such as degrading mission parameters for safety if battery reserves fall to risky levels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In distinguishing between testbed support of UAVs, it is important to realize that a UAV implies close integration of the onboard computing and communication equipment with the vehicle’s command and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' It is helpful to think of two extreme cases as representative of the two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' On the one hand, we can mount a computing/communication device (such as an ordinary smartphone) on a UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The UAV’s autonomy, trajectory computation, or command and control, remain completely as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The coupling between the UAV and the cellphone it carries as a payload is simply mechanical (but may include antenna mounts or high-gain antennas custom-positioned for the UAV, and common power supply).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' At the other extreme, the UAV contains only a single computing/communication device, which is capable of being tasked with complex missions (such as air quality analysis, image analysis based search-and-rescue), and also subsumes the trajectory computation (whether autonomous, command- and-control-based, or based on some coordination) for the UAV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' in this case, the vehicle becomes in effect a peripheral of the onboard computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' First, we consider the task of integrating support in a wireless testbed for UAVs only, used as vehicles for an airborne UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This includes the case where the air vehicle has no autonomy and is controlled by a ground-based operator using a handheld or other radio remote control equipment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and even the case where the air vehicle does not have any controlled mobility (such as free-floating balloons) or any mobility at all (such as tethered aerostats or helikites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The basic challenge for a wireless testbed to support UAVs is posed not only by the fact that they are mobile (which, after all, ground UEs also exhibit, when users walk or drive), but the fact that they have a widely varied altitude as well as azimuth compared to traditional UEs on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Both spectrum and latency are KPIs of interest for a UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The front-haul and mid-haul latencies must provide very low latency to maintain system synchronization and function under a varying altitude of the UE, and the spectrum used for communication can significantly affect the achievable coverage and throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A further challenge is that of antenna occlusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' which some Proposed Stages for Open RAN Drone Validation Stage 1: Basic UAV functionality and performance testing Single UAV-single Cell (Attach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Connect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Active/Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Throughput,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Latency Stage 2: Single Cell Loaded Performance Testing Multiple UAV (UEs)-single Cell (Attach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Connect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Active/Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Cell Throughput,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AFR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' DCR Stage 3: Multiple Cell Functional and Performance Testing Single UAV-Multi-Cell (Attach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Active/Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAV Capacity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AFR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' DCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Cell Capacity Stage 4: Multiple Cell Loaded Performance Validation Multiple UAV-Cell (Attach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Connect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Active,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAV Throughput,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Latency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AFR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' DCR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Cell Capacity KPIs for the Stages: UAV throughput UE latency Cell Throughput Accessibility (AFR) Retainability (DCR) Control Stage 5: Component Capacity Validation Multiple RUs per DU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Multiple DUs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CPRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' etc including UE/UAV Loading Stage 6: Advanced System testing RIC platform applications for UAV Control and System Optimization on a Loaded System Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 1: Proposed stages for Open RAN UAV validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAVs attempt to mitigate by multiple antenna locations around their bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Some UAVs mount antennas on gimbals in an effort to maintain constant directional properties, others allow for servos to allow controlled pointing of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' These challenges are exacerbated by the fact that most base stations, whether commercial or built out of commodity open technol- ogy, exhibit their own antenna coverage patterns, which are optimized for ground coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Studies have shown that the consequence of this optimization is the formation of multiple lobes at increasing altitudes, in complex patterns, that cannot be predicted easily as a function of the altitude of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The UAV will have to be tested in a controlled environment to ensure the network functions and meet O-RAN specifica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Creating a Faraday environment to do the controlled validation testing will pose challenges compared to traditional Open RAN lab Faraday environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Then the testing will need to be expanded to an open environment and optimized based on interferers, physical obstacles, and spectrum bands used – as the propagation and throughput are connected to the spectrum band used for communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 1, we summarize six proposed stages for Open RAN UAV validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' While UAVs allow intelligent control of position and tra- jectory jointly with RAN intelligence (Apps executing at the RICs), the softwarized character of Open RAN also opens up exciting possibilities of allowing the onboard computer to take part in the Open RAN ecosystem in ways other than just as a UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We devote the next section to these considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' USE CASES FOR UAVS IN OPEN RAN Considering the aerial controlled mobility and communi- cation among fixed and portable nodes, UAVs will facilitate enhancements to Open RAN with flexible deployments and 6 on-demand, on-time network access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Several use case exam- ples on Open RAN-based air mobility scenarios are provided as follows (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAVs serve as UEs: This use case focuses on exploring the functionalities of O-RAN RICs for managing and orchestrating network components aimed at 3D critical mission operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', secure, search and rescue) assisted by UAVs, as they are able to exhibit agile, fast, and autonomous behavior by organizing themselves to exchange information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Considering a scenario involving UAVs connected to an Open RAN ground BS, UAVs as UEs can carry high-resolution cam- eras and/or sensors, collecting real-time video and transmitting it back to the ground BS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', to be used to identify possible targets of interest through deep neural network object detection model, and in addition report information about application performance to rApps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the meantime, the E2 nodes of O- RAN are responsible for updating UAV control with insights produced by their applications (xApps and rApps) to support the RAN optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In this context, Open RAN is able to support the demands of highly dynamic scenarios of critical-mission operations integrated with UAVs due to its flexibility and characteristics of component dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAVs act as O-RUs: As described in O- RAN specifications [34], [35], UAVs can play a role as O- RUs and process several simple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' As the extension, this scenario focuses on the use of UAVs as O-RUs to handle more complicated network tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', to quickly deploy an aerial network to assist or extend the terrestrial network where communication and computing resources can move closer to users to meet diverse and stringent 5G application requirements, such as ultra-low latency and ultra-high reliable connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Considering a scenario in which each UAV-BS is equipped with an O-RU to serve ground mobile users, the objective is to optimize the performance of serving offloading tasks via both controlling UAV-BSs to guarantee the quality of communication channels to ground users and efficiently dis- tributing offloading tasks to appropriate Open RAN elements according to the current association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Because of the 3D air mobility capability of UAVs and disaggregation of Open RAN architecture, they may potentially deliver better data offloading capabilities and better resource utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAVs act as O-DUs and O-CUs: 1) Using UAVs as O-DUs allows for flexibly hosting RLC/MAC/High-PHY layers based on a lower layer functional split, where UAVs can dynamically connect to multiple O-RUs allowing on-demand resource pooling for virtual baseband functions of high PHY layer, MAC, RLC, and synchronization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2) using UAVs as O-CUs helps to easily control the operation of multiple O- DUs within/beyond the coverage area, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', the radio resource control for flexibly managing the life cycle of the connection, routing or duplication for split bearers, and the service data adaptation for managing the QoS of the traffic flows through autonomous 3D air mobility capability of UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Drone swarm based Open RAN: This use case envisions multi-role drones without ground facilities that forms an ad-hoc/swarm based Open RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Based on Scenarios 2-3, we can consider a set of containers to virtualize different O- RAN elements such as O-RUs, O-DUs, and O-CUs deployed in drones and distributed computing nodes of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Given these containers with different functions, the objective is to create a robust Open RAN testbed in a swarm of drones towards full decentralization and controlled air mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Flying wireless backhaul in Open RAN: Wire- less backhaul as an economically sustainable solution has been included by 3GPP as part of the integrated access and backhaul study item [36], [37] for the 5G NR standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' As an extension in Open RAN architecture, this scenario focuses on building a large-scale, self-organizing network of drones that are connected using a wireless mesh backhaul, which caters to dynamic bandwidth-hungry and latency-sensitive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Based on Scenario 4 with role-specific operations, drones can hover above or close to the O-RU and serve as an airborne last- hop link connecting RAN to the core network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Additionally, they can act as relays between two O-RUs separated by a longer distance to extend coverage forming a multi-hop mesh network for communications and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Multi-drone backhaul in Open RAN is capable of flexibly adapting itself to cater to highly dynamic applications and events, and easily being scaled up to cover urban scenarios using long-range radios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Scenario 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' D2D communications underlaying drone- assisted Open RAN: Implementation of device-to-device (D2D) communication such as sidelink can be an extension of the network into areas that traditional propagation of the fixed O-RU cannot reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Particularly, drones can serve as UEs or relays deployed much more swiftly and improve the network throughput performance by dynamically adjusting their locations to provide direct or relayed D2D links to any out-of-coverage users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Additional sidelink capabilities such as multi-hop [38] and multi-link (in 3GPP Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 19) can provide higher resiliency in this mode, especially offering a valuable set of capabilities for mission-critical services such as disaster response rescue and operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Testbed Considerations: The above poses a rich and varie- gated set of potential operational scenarios, and it is impracti- cal to attempt to enumerate specific design issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Instead, we again propose foundational considerations and hark back to our discussion in Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The general capabilities of the testbed that we can identify in order to support such innovative scenarios are: The capability of mobility control of custom air vehicles, The ability to emulate not only the RF environment, but of airflow and UAV flight, and The inclusion of onboard computers, suitable for inte- gration into UAVs, that can support user programming to create software components of the Open RAN ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW TESTBED REVIEW FOR OPEN RAN Thus far, we have reflected on general requirements of an Open RAN testbed that is able to integrate UAVs with controlled mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the remainder of this paper, we take a deep dive into the AERPAW testbed, reviewing it in light of the considerations we have derived above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We choose AERPAW because we are intimately familiar with it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' the authors of this paper include the PIs of the AERPAW project,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and key ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UE Layer ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Sync ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU + O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='core ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Offloading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Region Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='core ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Region Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Edge Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-CU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Region Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Region Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Routing Path ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-RU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UAV Relay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='core ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UAV BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Application ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UAV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UAV Relay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UGV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='O-DU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Edge Cloud ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='(a) Scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (b) Scenario 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (c) Scenario 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (d) Scenario 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (e) Scenario 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' O-RAN Layer O-DU Region Cloud (f) Scenario 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' O-RU Direct D2D Link Relayed D2D Link from Drones D2D UE D2D UE Cellular UE Drone UE D2D UE D2D UE Drone relay Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2: Use case examples for Open RAN-based air mobility: (a) UAVs as UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (b) UAVs as O-RUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (c) UAVs as O-DUs and O-CUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (d) UAV swarms in O-RAN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (e) Flying wireless backhaul in O-RAN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (f) D2D communications underlaying UAV-assisted O-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' architects and DevOps personnel working on the AERPAW facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, it is also true that AERPAW was conceived and built to support controlled air mobility in a testbed for use by a national community of researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Thus, it is a reasonable facility in which to conduct such a thought exercise of how a fully-featured Open RAN testbed may be built up along the same lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW has the foundation for becoming a highly valuable Open RAN UAS test-bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW is the third testbed funded under the PAWR initia- tive to support advanced and emerging wireless research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' It is a multi-year, multi-phase project that started in September 2019 and it is expected to be finalized by 2025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW experi- mentation capabilities became generally available with initial set of resources and features in November 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Additional platform resources, sample experiments, and experimentation capabilities are expected to be released at the end of Phase- 2 (by May 2023) and Phase-3 (by May 2024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW is primarily and essentially a testbed of physical resources, not computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The crucial part of these physical resources are: (i) the RF environment and the airspace that the AERPAW operating areas represent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (ii) the physical equip- ment (SDRs, commercial RF equipment, UAVs, and UGVs) that AERPAW provides to leverage those environments for experimental studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and (iii) the expertise (and consequent exemptions) in conducting such studies in compliance with FCC and FAA regulations that AERPAW represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Physically, the testbed is hosted at sites in and around the NC State campus in Raleigh, NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Central to AERPAW’s unique characteristic is the availability of UAVs and UGVs in the testbed that can be placed under the direct programmatic control (of trajectories) of the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In conjunction with the programmable USRPs that are also available for direct programming by the researchers, as well as other real-world, commercial radio equipment, this provides the NextG wireless researcher a facility for research experiments not practicable in any other facility at this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fixed Nodes, Portable Nodes, and Vehicles: At a very high level, the facility includes a number of tower locations (fixed nodes), at each of which some combination of AERPAW programmable SDRs and commercial radio equipment are permanently installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The SDRs are controlled by servers, or companion computer (CCs), installed in each location that also represent edge-computing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' These fixed node locations are distributed over the extensive Lake Wheeler Agricultural Fields of NC State (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3a), and some nodes are also installed in the Centennial Campus (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The complement of these fixed nodes are AERPAW’s portable nodes, also consisting of a computer and SDR(s), but smaller ones so that an AERPAW portable node can be mounted on a UAV/UGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The CC on a portable node, an Intel NUC, also controls the UAV/UGV itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A smaller version of the portable node that can get carried at the smaller UAV is also available, to do experiments with mobile phones and LoRa sensors that are connected to a LattePanda as the CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' More information on AERPAW is available at the AERPAW Facility website and User Manual linked therefrom, and previ- ous publications (also listed on the website).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In what follows, we attempt not a comprehensive overview of AERPAW, but rather a review in light of the desirable characteristics we identified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Span, Scale, Access Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3b show the outdoor deployment footprint of AERPAW’s fixed nodes in NC State Lake Wheeler and NC State Centennial Campus, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The equipment that are expected to be available publicly for experimentation by the end of (AERPAW’s Phase-2 (expected May 2023) are also illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Currently, it is possible to experiment with UAVs 8 8 LW-2 LW-1 LW-3 LW-5 LW-4 4 NI USRPs 1 Ericsson 4G/5G BS (NSA) 1 Keysight RF Sensor 1 LoRa Gateway 4 NI USRPs 1 Keysight RF Sensor 4 NI USRPs 1 LoRa Gateway 4 NI USRPs 1 Keysight RF Sensor 4 NI USRPs 1 Keysight RF Sensor (a) Since Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2021, LW-1 is publicly available for experimentation, and LW-2, LW-3, LW-4, LW5 are expected to be publicly available by May, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 70 Facebook TG Radios (60 GHz) Facebook TG Radios (60 GHz) Facebook TG Radios (60 GHz) 6 Facebook TG Radios (60 GHz) CC3 CC2 CC1 4 NI USRPs 1 Keysight RF Sensor 1 LoRa Gateway 4 NI USRPs 4 NI USRPs 1 LoRa Gateway CC4 CC5 CC6 Fixed wireless SDR experiments Portable node experiments at carts (b) Since Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2022, CC1 and CC2 are publicly available for experi- mentation, and CC-3 is expected to be publicly available by May, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CC3, CC4, CC5, and CC6 each also has Terragraph radios from Meta operating at 60 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3: AERPAW fixed node deployments at (a) NC State University Lake Wheeler Field Labs, Raleigh, NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and (b) NC State University Centennial Campus, Raleigh, NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' at Lake Wheeler Field Labs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW does not currently sup- port UAV operation by experimenters in Centennial Campus but supports UGV operation, and UAV operation will likely become available in the future for experimenters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This geographical span is reasonable for an Open RAN testbed, even with experiments including UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, scale is a different matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' With nine fixed nodes, six portable nodes, eight programmable UAVs, and some non- programmable commercial radio systems such as an Ericsson base station and five Keysight RF sensors, AERPAW can sup- port a large variety of meaningful advanced wireless research – including proof-of-concept Open RAN experiments at small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' But to support the full gamut of Open RAN testing and Open RAN related research experiments, AERPAW would need to add a large number and variety of commercial or stock UEs, and a larger number of programmable UAVs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' a few more programmable fixed and portable nodes would also likely be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Open RAN, the potential softwarization or virtualization of various system components is a particularly attractive feature for innovators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This requires allowing experimenters Platform Resources Platform Control Experimenter Portal Website AERPAW Ops Human Platform Control Software tools Virtual Resources (Development mode) Physical Resources (Testbed mode) Servers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' SDN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' orchestration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' custom emulation Towers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' SDRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UAVs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' UGVs (New in Phase 2: Ericsson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Keysight,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' FB TG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' LoRa) Safety Pilots Configure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Orchestrate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Monitor AVNs ARNs (a) Interaction of an AERPAW experimenter with platform control and platform resources (development mode and testbed mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Platform Resources Platform Control 1 Register,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' supply credentials 2 Create experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' request develop 3 Trigger virtual experiment request 4 Instantiate virtual experiment 7 Login to virtual nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' test 8 Save experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' submit to testbed 9 Trigger testbed experiment request 10 Retrieve experiment from virtual 11 Install experiment on testbed 12 Handover to pilots/operators 13 Retrieve experiment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' set complete 14 Notify experimenter of status 15 Request develop returned expmt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 20 Login to virtual nodes, view 1 2 3 4 7 8 10 11 12 13 14 15 16 17 20 5 Notify virtual experiment ready 5 6 6 Provide virtual experiment access Experimenter Portal AERPAW Ops Control AVNs ARNs 18 Change status 19 Notify virtual experiment ready 18 19 17 Re-instantiate virtual experiment 16 Trigger virtual experiment request 9 13 13 10 11 13 11 10 (b) Steps for carrying out an experiment in AERPAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='. Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 4: Experiment workflow for users of AERPAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' direct programming access to all parts of the facility, and at the highest levels of access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Managing such access while ensuring the safety and regulatory compliance of the facility is a distinct challenge for any testbed that aspires to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' On this front, AERPAW is already well positioned, hav- ing been designed from the outset as a batch-mode facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Experimenters develop experiments in a virtual environment and submit experiments for execution on the physical testbed once development is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW Operations person- nel (Ops) then execute these submitted experiments in the physical testbed environment and collect the output of the experiments as designed by the Experimenters, which are available for Experimenters to view and analyze back in the virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This is not an arbitrarily decided constraint, but a considered architectural choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In operating a facility with programmable radios and programmable air vehicles, we are obligated to make, and uphold, certain guarantees to the FCC and FAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, we also want to allow Experimenters the ability to program those radios and air vehicles, ideally without needing to become fully conversant with FCC and FAA regulation details, obtain exemptions, or expertise in techniques for ensur- ing compliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Batch mode operation allows us to interpose critical filters and monitors into the Experiment code execution flow that allow us to guarantee safe and compliant operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' It is one of the most valuable features of the AERPAW platform that we assume this guarantee ourselves, rather than passing ()(Q)(Q)(0)NCSTATEUNIVERSITYAnimal Health Building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' NCSUG z) The Oval(Q)AtnCamupunOr PartnersWay ParthersWayNC STATE UNIVERSITYWilson College aa James BHunt J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Library artnersWay Fitts-Woolard Hall Wolf Ridge Apartments口 口 口 口 口 口 口 口口 口 口 口 口 口 口 口0 00E22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='9 on the responsibility for compliant operations (and liability for non-compliance) to the Experimenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Figure 4a and 4b show the entity relationships in AER- PAW, and the experimenter’s experiment design workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Experimenters request “Development Sessions” in which they program a virtual environment that is programmatically indis- tinguishable from the computing environment in the physical testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Once completed, they submit such experiments for “Testbed Execution Sessions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The containers housing the experimenter’s code is bodily moved to the corresponding nodes in the physical testbed, where they are executed as before, but with additional supervisory containers monitoring for any RF violation or unsafe air-vehicle operating conditions, overriding as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' As an additional line of defense, human operators in the field are able to issue aborts if the automated system should fail to override.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Spectrum and Licenses AERPAW supports multiple frequencies for experimentation with its fixed and portable nodes and vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In particular, AERPAW is one of the few FCC Innovation Zones (FCC-IZs) in the United States [39, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='6] with frequency bands that are highlighted in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The maximum effective isotropically radiated power (EIRP) limits for fixed stations (FSs) and mobile stations (MSs) are also specified in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The FCC- IZ for Lake Wheeler Field Labs site for AERPAW covers an area of approximately 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5 square miles, while the Centennial Campus FCC-IZ covers an area of approximately 3 square miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Experimenters can also port their FCC experimental licenses at AERPAW’s FCC Innovation Zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' As noted in Table III, due to the sensitivities of certain bands and the wide interference footprint of transmissions from an aerial vehicle, FCC does not allow airborne use in certain bands [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW currently supports a subset of the frequency bands through additional FCC experimental licenses (FCC Call Sign: WK2XQH [41]), which are offered to AERPAW’s users to carry out over-the-air experiments on the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In particular, for SDR experiments, AERPAW has experimental licenses at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='55 GHz and 902-928 MHz, with plans to incorporate this band into the AERPAW FCC-IZ in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The experimental licenses for the Ericsson network include 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='7/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1 GHz for the LTE system and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='4 GHz for the 5G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW also has plans to support generally available experiments using its mmWave SDR framework by the end of Phase-3 using Sivers phased arrays operating at 28 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Spectrum monitoring and passive I/Q data collection experiments can be supported using USRPs and Keysight RF sensors between 100 MHz to 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A particular spectrum band that is of recent interest to safety and navigation related command-and-control commu- nications for UAVs, and that AERPAW will explore experi- mental licenses in the future, is 5030-5091 MHz for which FCC recently released a Notice of Proposed Rule Making (NPRM) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Another band that may potentially be used for ensuring vehicle-to-vehicle (V2V) separation with cooperative surveillance in the future for urban air mobility (UAM) scenar- ios is 1104 MHz (also known as UAT2) [43]–[45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Additional TABLE III: AERPAW’s FCC Innovation Zone frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Footnotes: 1) Commission rules do not permit airborne use on all or portions of these bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 2) Any experimental use must be coordinated with authorized users and registered receive-only fixed satellite earth stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 3) Operations must be coordinated with a spectrum access system administrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Frequency Band Type of Operation Allocation FS Max EIRP MS Max EIRP 617-634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5 MHz (DL) Fixed Non-federal 65 663-698 MHz (UL) Mobile Non-federal 20 (dBm) 907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5-912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5 MHz Fixed and Mobile Shared 65 (dBm) 20 (dBm) 1755-1760 MHz (UL) Mobile Shared 20 (dBm) 2155-2160 MHz (DL) Fixed Non-federal 65 (dBm) 2390-2483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5 MHz Fixed and Mobile Shared 65 (dBm) 20 (dBm) 2500-2690 MHz1,2 Fixed and Mobile Non-federal 65 (dBm) 20 (dBm) 3550-3700 MHz1,2,3 Fixed and Mobile Shared 65 (dBm) 20 (dBm) 3700-3980 MHz1,2 Mobile Non-federal 20 (dBm) 5850-5925 MHz Fixed and Mobile Shared 65 (dBm) 20 (dBm) 5925-7125 MHz2 Fixed and Mobile Non-federal 65 (dBm) 20 (dBm) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='5-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='35 GHz Fixed and Mobile Non-federal 65 (dBm) 20 (dBm) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='6-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='0 GHz Fixed and Mobile Non-federal 65 (dBm) 20 (dBm) spectrum bands that are specifically of interest for UAV/UAM scenarios can be found in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Mobility Control AERPAW is also, by its original design, already adequate in providing controlled mobility, both for repeatability of experiments and for experimentation with programmatic tra- jectory control by experimenters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and both for aerial vehicles as well as ground vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Figure 5 shows the AERPAW vehicle control stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In AERPAW the main autopilot we support at this time is ArduPilot [46] as it is open source and well-trusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ArduPilot is supporting MAVLink [47] as a communication protocol, and, therefore, all AERPAW vehicle software sends and receives MAVLink commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For the safety of the testbed and of the AERPAW operators, only a reduced subset of MAVLink commands is allowed to pass through the MAVLink Filter and reach the autopilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Keeping in mind the caveat on the reduced subset of MAVLink commands allowed passing to the autopilot, at one extreme, an experienced AERPAW user can, however, discard the entire stack shown at the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 5 and write their own MAVLink application using any other framework they wish (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', they could use MAVSDK [48] if they prefer a C++ based library).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, to smooth the learning curve, we implemented a vehicle library named aerpawlib [49], which features a finite state machine model, with hooks for vehicle (and/or radio) actions at each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Several examples are available either to 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 5: AERPAW vehicle control stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 6: Sample vehicle experiment with two coordinated drones: the tracer (red) goes through a list of waypoints, while the orbiter (yellow) orbits around the tracer while at the waypoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' be used as-is or to be modified by experimenters to fit their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The most popular example at the moment is the pre- determined trajectory sample application, where users specify a series of 3D waypoints to be traversed in order, including choices of the speed and wait times at each waypoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The AERPAW framework also allows the experimenter’s programs to take decisions on the fly, thus enabling au- tonomous applications, such as a radio-based search and rescue (SAR), where the next direction of movement can be chosen based on the current radio measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Autonomous Coordinated Multi-UAV Experiments: An ad- ditional feature supported by the application programming library provided by AERPAW is the ability of applications to synchronize the control of multiple vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This is achieved either by using centralized control (where a coordinator pro- gram sends synchronized commands to multiple vehicles), or decentralized applications, (where programs on the compan- ion computer of each of the vehicles coordinate without a centralized conductor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This ability can be leveraged to allow for swarm control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 6 shows the traces followed by two drones in a coordinated drone experiment, where one drone (the tracer) follows a list of waypoints, while the second drone (the orbiter) shadows the tracer by moving at the same time in the same direction, and upon reaching the target waypoint, it orbits around the tracer once before they both move to the next waypoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This experiment is initially designed and tested in the em- ulation environment and subsequently executed in the testbed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' More complicated swarm experiments with a larger number of drones and including communication links with SDRs can be easily carried out using the same workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Autonomous decisions can be integrated into the experiment, where the drones can make next waypoint decisions based on the observations of wireless signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Other testbeds can, of course, use alternate methodologies for providing programmatic online trajectory control to experi- menters, and repeatability of mobility profiles for experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We have described AERPAW’s approach above not to advocate it as the only way, but rather to articulate the level of programmability and repeatability that experimenters should be able to expect from a testbed facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Emulation Support AERPAW has well-articulated emulation support for both RF and air/mobility aspects of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the “Devel- opment session” mentioned earlier, users can prepare their experiments with perfectly repeatable trajectories and wire- less propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The main goal of providing the emulation environment is to allow users to develop their experiments in a safe and fully repeatable environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 7a depicts an example experiment comprising a portable node on the left and a fixed node on the right while deployed in the emulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In emulation mode, the experimenters’ code (encapsulated in the two E- VM, and shown in green in the picture), is running with no modifications in comparison with an experiment in testbed mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In contrast, in emulation mode, the vehicle and the wireless channel are emulated, thus allowing for a full software emulation, amenable to cloud deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For vehicle emulation, we use an open-source available emulator that has been developed by the ArduPilot community, which features as its main characteristic the use of the same firmware as the autopilot we use on all our vehicles (at this time, drones, rovers, helikite, and a push-cart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Careful com- parisons between the performance of the emulated vehicles and the testbed vehicles show that the vehicle emulator is performing very realistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In contrast, for the wireless channel emulator (CHEM), to the best of our knowledge, there is no open-source solution that satisfies all our requirements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' therefore, we developed our own solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 7b shows the main components involved in the CHEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In general, each radio-enabled node in the testbed is capable of both transmitting and receiving radio signals, which we capture at baseband, IQ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The IQ samples are sent to the channel emulator, which then “propagates” them to the corresponding receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The propagation in CHEM is 11 (a) AERPAW emulation environment overview for one mobile node and one fixed node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (b) AERPAW wireless channel emulator overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 7: AERPAW emulation environment overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' controlled by the channel control module, which dynamically computes a channel matrix based on both dynamic information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', the current mobile node positions and orientations), as well as static information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', position of the fixed nodes, antenna patterns, transmitter gains, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The CHEM supports several features, including free space and two-ray ground propagation models, two noise models, MIMO channels, up to 100 MHz of instantaneous bandwidth, multi-rate processing, different antenna patterns, multiple fre- quencies, and, importantly for efficiency, suppressing silences for bursty traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Once again, we have described AERPAW’s approach above not to advocate it as the only way, but to articulate the level of emulation support we find required for an Open RAN testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Regarding AERPAW itself, while it has a good base from which to provide emulation support for Open RAN experi- ments, it would remain a non-trivial task to develop/procure and incorporate the large volume of software modules that would be required to be integrated into this framework in order to provide emulation support for a comprehensive complement of Open RAN experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In the next section, we return to this topic briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Programmability, Radios, Software Stack AERPAW does not currently incorporate a full reference O- RAN implementation, although some component parts exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' TABLE IV: AERPAW example experiments with SDRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Software Sample Experi- ment Comments srsRAN SE1: Multi-node LTE SISO Complete end-to-end LTE network with multiple srsUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and one srsENB and srsEPC SE2: LTE Cell Scan Search for LTE cells and capture key parameters of interest SE3: Two-Node LTE MIMO Complete end-to-end 2x2 MIMO LTE network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' using srsUE with srsENB and srsEPC SE4: Multi-Node IoT Basic NB-IoT signalling between the eNB and UE nodes SE5: LTE Han- dover Complete end-to-end LTE network with S1 handover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' using srsUE with srsENB and open5GS SE6: Single- Node 5G SA Complete end-to-end 5G SA net- work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' using srsUE with srsENB and open5GS OAI OE1: Two-Node LTE SISO Complete end-to-end LTE network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' using OAIENB and srsUE OE2: Single- Node 5G SA complete end-to-end LTE network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='using OAIGNB and srsUE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='GNU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Radio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='GE1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='OFDM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='TX-RX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Send and receive data using an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='OFDM waveform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='GE2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Sounder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Pseudo-random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='are transmitted/received for channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='sounding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='GE3: LoRa PHY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='TX0RX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='LoRa transceiver with all the neces- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='sary receiver components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UHD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Python- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='API ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UHD1: Spectrum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Sweep based spectrum monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='between 87 MHz and 6 GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='UHD2: IQ Col- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='lection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='IQ samples are collected at desired ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='center frequencies with some sam- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='pling rate for a specified amount of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='duration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='The edge-cloud model of companion computers at every AER- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='PAW Radio Node (including both fixed and portable nodes) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='allows for an easy transition into Open RAN softwarized radio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='modules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' as such modules become available and integrated into the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The Software Defined Radios of AERPAW represent a po- tential strength in a possible transition path to full Open RAN support since experimenting with evolving or innovative radio protocols is reduced to an exercise of software development and integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW team provides a variety of SDR sample exper- iments for experimenters to work with using open-source software and USRP SDRs from NI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Any AERPAW user can start with one of these experiments and develop their code further to research e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' different protocols and waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW presently supports four different sets of open- source software for SDR experiments: srsRAN [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1], OpenAirInterface [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='2], GNURadio [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='3], and Python scripts [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A variety of sample experiments are provided in AERPAW’s user manual for each case under Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1 [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Table IV, we provide a list of SDR sample experiments that are currently available or to be available by the end of AERPAW’s Phase-2 (May 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' An additional set of SDR experiments is expected to be added for general availability by the end of Phase-3 (expected May 2024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' All these experi- ments are tested both in the development environment and the testbed environment of AERPAW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' While experimenters can TNATIONAL INSTRUMENTS NIUSRP-2930 N0MME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='22Gl REFIN PSN GSETHERET POWER: I1:: =: ":12 TABLE V: AERPAW example experiments with commercial RF hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Software ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Sample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Experi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='ment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Comments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Ericsson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='EE1: 5G Modem ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='RF Logging and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Throughput ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Quectel modem logs various KPIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='from 4G/5G Ericsson network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Keysight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='RF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Sensors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='KRSE1: Spectrum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='monitoring ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Monitor and record spectrum up to 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='GHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='KRSE2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Classify and detect a variety of sig- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='nals based on RF signature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='KRSE3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='source tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='TDOA based localization of a signal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='source by passive monitoring of its ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='RF signature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='also bring their own software to the platform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW can not guarantee that they will work smoothly with the existing AERPAW hardware and software,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and the development envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For further details, readers are referred to AERPAW’s user manual [39, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW also includes similar prepared experiment profiles for commercial radio equipment available in the testbed (see Table V), but they are relevant in the Open RAN context mainly as potential support equipment, so we do not discuss them further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Summary - Open RAN Related Components of AERPAW While AERPAW has not been designed initially as an Open RAN testbed, its open, modular, and flexible design allows possible expanded support for Open RAN use cases as a living lab for UAVs with comparative ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The AERPAW team filled out, upon request, a survey in November 2022 developed by the recently established Open RAN working group of the National Spectrum Consortium (NSC) [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This survey was shared by NSC members with existing testbed platforms that may potentially support Open RAN experiments in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In Table VI, we present a revised version of NSC’s Open RAN survey and included comments on AERPAW’s features and capabilities that can support Open RAN experiments with controlled aerial mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In particular, we highlight open and programmable end-to-end network capabilities as well as commercial 5G equipment deployments in AERPAW, on- site access to wireless spectrum, different experimentation capabilities supported, compute nodes, unique use case testing scenarios, testing types, among other related platform features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The information provided in Table VI relates specifically to the match and extensibility of AERPAW as a meaningful Open RAN testbed for use cases with controlled air mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, the exercise of preparing this table affords us prac- tical insights into designing and building such an Open RAN testbed, to complement our observations in Section II, and we pass these on to the community here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' REPRESENTATIVE RESULTS RELATED TO OPEN RAN AND CONTROLLED AIR MOBILITY In this section, we present two early representative exper- iments from AERPAW that are of relevance for Open RAN experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We also elaborate on other possible experiments of relevance to Open RAN that may be supported in AERPAW in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 Time Interval (seconds) 0 20 40 60 80 100 120 140 160 180 Throughput Achieved per Slice (MBps) 100 PRBs 15 PRBs 80:20 Configuration 20:80 Configuration 50:50 Configuration Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 8: Representative results on O-RAN slicing xApp using srsRAN with two UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' RAN Slicing xApp Experiments In this section, we provide representative results using the RAN slicing xApp and srsRAN, using the framework by the NSF POWDER Wireless platform [52], executed at the AERPAW testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' (Note that these features have not yet been integrated into the AERPAW’s development and transition-to- testbed environments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' we are exploring integration options at this time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The goal is to dynamically create network slices and observe the effects of slice reconfiguration with a TCP stream on the performance of a UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A near Real-time RIC is deployed as part of two separate Kubernetes clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Detailed steps are provided in [53], we will provide a high-level overview of the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The RIC cluster is used for deploying the platform and applications which are part of the RIC, whereas the Aux cluster is used to deploy other auxiliary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The RIC Kubernetes cluster installation is done through configuration scripts and pre-generated helm charts for each of the RIC components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Once the process is done, we created a persistent volume through a storage class for the influxDB on the RIC platform namespace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Once the RIC platform is deployed, a modified E2 termination is created which has few services enabled to communicate and exchange messages between RIC and E2 Agent [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Once the Kubernetes clusters are deployed, we can deploy the Near Real-time RIC using a RECIPE file which provides customized parameters for the configuration of a particular deployment group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' This Recipe file can be tinkered with if we want to change any configuration to suit our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Next is the installation of srsRAN components such as srsUE, srsEnB, and srsEPC which use ZeroMQ networking libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Since we use ZeroMQ mode, the 4G/5G network can be set up using a single machine that hosts both the RIC and srsRAN components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Finally, the xAPP is onboarded and deployed on top of the Near real-time RIC and full integration is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Using this setup, we create two network slices in a work- conserving mode and bind two srsUEs to these network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Some representative results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 8 for two 13 TABLE VI: AERPAW features and capabilities related to Open RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Capability O-RAN Related Components AERPAW Availability Open and Programmable End-to-End Network Multiple SDRs connected to power and network backhaul USRPs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Keysight RF sensors Indoor wireless operations in a lab N/A Outdoor wireless operations Rural farm and urban campus Open 5G mobile cores Open5GS Open fronthaul interface for testing open RUs Not currently available Open source software stacks ready to use with or without additional software development srsRAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' OAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' GNURadio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' I/Q collection with sample experi- ments [39] Open source RIC implementation Not currently available BYOD operation Yes (on a case-by-case basis) BYOS operations Yes (on a case-by-case basis) Bare metal for software installations Not currently available Containers for software installations Yes – both in emulation and testbed modes Remote access to network resources Yes during development (emulation) mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' not normally during testbed mode End-to-End Network with Commercial Equipment and Swappable Components Commercial equipment Ericsson 4G/5G network Indoor wireless operations N/A Outdoor wireless operations Rural farm area Commercial 5G mobile cores Ericsson NSA core network (Release-15) Includes one or more of a commercial RIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' DU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and RU Not currently available Open fronthaul interface enabling testing of open RUs to support different physical layers Not currently supported On-site Access to Spectrum Unlicensed or ISM band 900 MHz for aerial communications with SDR front ends CBRS spectrum and CBRS SAS features N/A Licensed spectrum from a spectrum owner N/A Experimental or Innovation Zone licensed spectrum Yes – FCC Innovation Zone with 13 bands in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='6-40 GHz [39] Techniques Channel emulation systems Software emulation available now [39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Keysight Propsim (32 ports) channel emulator in the process of integration Multiple modes of massive MIMO Not presently available – mmWave UAV capabilities with 4x4 Sivers phased arrays in development Emulation capabilities for the RIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' DU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' RU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' and UE Presently not available Compute Capacity One optical hop Yes Edge compute Yes – Dell 5820 Server at fixed nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Intel NUC (i9) at portable nodes carried by AERPAW vehicles Public cloud computing Not presently supported Unique Use Case Testing Drone support Multiple different custom drones for different use cases Rural and urban environment Yes (autonomous drone experiments available only in rural) Military base N/A Smart agriculture Deployment in Lake Wheeler agricultural farm of NC State [39] Testing Types Research and development Free access by NSF-funded academic researchers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' charge-based access for other researchers Compliance (3GPP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ETSI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' O-RAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='3GPP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='compliant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='open-source ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='commercial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='4G/5G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='hard- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='ware/software ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Interoperability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Partial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Security ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Partial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Performance/stress testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Partial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Others ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Research staff availability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Yes (multiple research associates/students for research support) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Operational staff availability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Yes (multiple research associates/students to support experiments) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Wireless certification program ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Not presently supported ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='Established connections to standards/specifications organiza- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='tions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='NextG Alliance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Open Generation Alliance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' GUTMA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Linux Foun- dation InterUSS Platform [50] different bandwidths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' which show the throughput of one of the UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We configure the slice scheduler in steps to alter the proportionate scheduling in different ways and observe the effects on the TCP stream for the UE [54], [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' An Iperf server is created on the UE namespace to observe the effects of dynamic RAN slicing and a corresponding Iperf client [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We create two slices, referred to as fast and slow, where each slice can be dynamically configured to share the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For the baseline scenario, the full bandwidth of 15 PRBs (100 PRBs) is initially allocated to the unsliced UE which gives a throughput of around 35-40 MBps (170 MBps) as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' After this, the resources are distributed with the 80:20 configuration among the two UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 8 show that the UE’s throughput falls to 27 MBps (140 MBps) for this configuration, and when the priorities are inverted between the fast and slow slices to 20:80, the throughput further reduces to 6-7 MBps (40 MBps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Finally, when the priorities are equalized to 50:50 configuration, the throughput increases to 16-17 MBps (70 MBps) for the first UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The results can be easily extended to a larger number of UEs and more complicated resource configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Our future work includes implementing this same scenario in AERPAW’s development and testbed environments with multiple controllable vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The throughput needs and the link qualities of UEs will change dynamically over time as the vehicles move around, and there is a need to have a dynamic slicing mechanism that satisfies the requirements of individual network slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' AERPAW can support development and testing in such dynamic RAN slicing scenarios, first in the emulation environment, and then in the testbed mode with realistic propagation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Programmable mobility 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 9: I/Q sample experiments representative results: LTE reference signal received power (RSRP) at five different UAV altitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' with multiple vehicles in both environments and will make it possible to have a testing environment that provides repeatable measurements involving precise mobility control for the UEs, and in some cases, mobile relays and mobile base stations with wireless backhaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' I/Q Sample Collection Experiments In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' 9, we provide representative results for the UHD2: IQ collection sample experiment shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The UAV is programmed to fly at five different altitudes and the USRP B205mini at the UAV collects IQ samples centered at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='51 GHz with a sampling rate of 2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' The only signal that can be observed in the spectrogram in the same band is an LTE signal of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='4 MHz bandwidth, transmitted from a USRP B205 mini that runs srsRAN at our LW1 fixed node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We post-process the collected I/Q samples using Matlab’s 4G toolbox, obtain RSRP for each I/Q sample location, and plot the RSRP over the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Additional details of the measurement setup and representative results are available in [57] using further post-processing with Matlab’s 4G toolbox, such as coherence time and coherence bandwidth with respect to the distance between the UAV and the fixed node, kriging interpolation of the received signal across the whole 3D volume, channel estimation, synchronization procedures, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' A similar experiment can be carried out to capture I/Q samples and evaluate the KPIs for any Open RAN based 5G system with varying locations of UAVs and UGVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' One or more of the SDR, commercial wireless, or vehicle control sample experiments from AERPAW’s sample vehicle experi- ment repository, such as the one illustrated in Figure 6 above, can be used simultaneously with the I/Q sample collection experiment, to collect the raw I/Q data at the finest granularity and post-process them in Matlab’s 4G and 5G toolboxes to generate desired KPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Such data collected in realistic propagation conditions can be made publicly available to the research community for furthering the research in controlled aerial mobility technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' CONCLUSION Open RAN expands the capabilities of 5G to support fea- tures and functions tied directly to use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Disaggregation and virtualization are well suited to UAVs/drones which will continue to grow and become a much greater part of the 5G network from a UE or acting as an O-RU, O-DU, or O-CU component of the network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' However, testing and validation are critical to successful integration into 5G and the expansion of Open RAN network capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Creating a testbed that supports UAVs poses challenges to meeting all the demands from the physical network to Open RAN interoperability needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' For the UAV market to grow and flourish testing and validation are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' As rules and regulations remain volatile in the immediate future, a UAV Open RAN lab can provide extremely valuable technical results to inform such actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' In this paper, we have provided conclusions drawn from our experience and expertise gained from designing AERPAW, a one-of-a-kind public advanced wireless testbed that provides programmable radio and vehicle control in a realistic outdoor area of considerable span, and also reflected on its fit as a possible Open RAN / UAV testbed in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' We hope these observations may be helpful to the community of designers of other such facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors would like to thank the PAWR Project Office (PPO) and AERPAW project partners including project per- sonnel from Mississippi State University, Wireless Research Center, RENCI, University of South Carolina, and Purdue University, for their contributions to developing the AERPAW infrastructure and for their feedback on this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' REFERENCES [1] O-RAN Fronthaul Working Group, “Fronthaul interoperability test specification (iot),” revision 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='0, August 3, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=' Available: 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SDR-Based LTE I/Q Measurement and Analysis Framework for Air-to-Ground Propagation Modeling,” to appear in IEEE Aerospace Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content=', arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} +page_content='07433, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FIT4oBgHgl3EQfzit1/content/2301.11365v1.pdf'} diff --git a/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/2301.13586v1.pdf.txt b/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/2301.13586v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..64441f9e7b641317f1b3b5d2dbd559292a16ebd2 --- /dev/null +++ b/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/2301.13586v1.pdf.txt @@ -0,0 +1,1613 @@ +arXiv:2301.13586v1 [math.PR] 31 Jan 2023 +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF UNIFORM RANDOM VECTORS +IN LARGE INTEGER DOMAINS +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Abstract. For a wide class of sequences of integer domains Dn ⊂ Nd, n ∈ N, we prove distribu- +tional limit theorems for F(X(n) +1 ,...,X(n) +d ), where F is a multivariate multiplicative function and +(X(n) +1 ,...,X(n) +d ) is a random vector with uniform distribution on Dn. As a corollary, we obtain +limit theorems for the greatest common divisor and least common multiple of the random set +{X(n) +1 ,...,X(n) +d }. This generalizes previously known limit results for Dn being either a discrete +cube or a discrete hyperbolic region. +1. Introduction +Let F : Nd → C be an arithmetic function of d ≥ 1 integer arguments, with N = {1,2,3,...}. A +standard problem in analytic number theory is the estimation of the multivariate sum +n1 +� +x1=1 +··· +nd +� +xd=1 +F(x1,...,xd) +for large values of (n1,...,nd) ∈ Nd. A particular instance of this problem consists in establish- +ing existence of the so-called mean value of F, which is defined via +(1) +M(f ) := +lim +n1,...,nd→∞ +1 +n1 ···nd +n1 +� +x1=1 +··· +nd +� +xd=1 +F(x1,...,xd). +In the probabilistic language, (1) may be recast as follows. Let (U(n1) +1 +,...,U(nd) +d +) be a random +vector defined on some probability space (Ω,F ,P) and which has the uniform distribution on +the finite rectangular set +(2) +Rn1,...,nd := + + +d +� +i=1 +[1,ni] + + +� +Nd. +2020 Mathematics Subject Classification. Primary: 11A05, 60F05; secondary: 11N60. +Key words and phrases. Distribution of arithmetic functions; greatest common divisor; least common multiple; +multivariate multiplicative function; regular growth of integer domains; van Hove condition. +1 + +2 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Then, with E denoting the expectation with respect to P, +(3) +M(F) = +lim +n1,...,nd→∞EF(U(n1) +1 +,...,U(nd) +d +). +A general result on existence of M(F) is due to Ushiroya [21]. +A multivariate arithmetic function F : Nd → C is called multiplicative, see [20, 21, 22], if +F(1,...,1) = 1 +and +F(m1n1,...,mdnd) = F(m1,...,md)F(n1,...,nd), +for all (m1,...,md) ∈ Nd and (n1,...,nd) ∈ Nd such that +GCD(m1 ···md,n1 ···nd) = 1. +A specialization of Ushiroya’s results from [21] to a multiplicative function F implies that under +a mild summability assumption on F, the mean value M(F) exists and is equal to +(4) +M(F) := +� +p∈P +� +1 − 1 +p +�d +∞ +� +i1=0 +··· +∞ +� +id=0 +F(pi1,...,pid) +pi1+···+id +, +where P stands for the set of prime numbers. +In the last years, there has been a lot of activity around various generalizations and exten- +sions of the aforementioned results. In a probabilistic direction, one may ask about the asymp- +totic behavior of distributions of the random variable F(U(n1) +1 +,...,U(nd) +d +), as n1,...,nd → ∞ in (2). +This question has been addressed in [4] for a particular choice of F, namely, for F(x1,...,xd) = +G(LCM(x1,...,xd)), with G being a univariate multiplicative arithmetic function. The univari- +ate case d = 1 is the classical Erd˝os-Wintner theorem, see [11], which provides necessary and +sufficient conditions for the distributional convergence of F(U(n) +1 ) as n → ∞. In another, more +analytic direction, the rectangular domains Rn1,...,nd in (2) are replaced by more sophisticated +domains of summation Dn ⊂ Nd, which grow to Nd as n → ∞. In particular, in the recent +work [17], the case of spherical summation over the regions +Sn := {(x1,...,xd) ∈ Nd : x2 +1 + ··· + x2 +d ≤ n}, +has been analyzed, whereas the papers [14, 15, 16] were devoted to the study of summation +over hyperbolic regions +Hn := {(x1,...,xd) ∈ Nd : x1 ···xd ≤ n} +and their generalizations. A surprising phenomenon revealed in the cited works is that the +mean value M(F) given by (4) is universal for rectangular, spherical and hyperbolic domains. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +3 +More specifically, let Dn be either Rn,...,n, Sn or Hn. For every n ∈ N, let (X(n) +1 ,...,X(n) +d ) be a +random vector defined on (Ω,F ,P) and having the uniform distribution on Dn, that is, +P{(X(n) +1 ,...,X(n) +d ) = (i1,...,id)} = +1 +#Dn +, +(i1,...,id) ∈ Dn, +where #Dn denotes the cardinality of Dn. Then, under the same summability assumption on F +as in Ushiroya’s result, we have +(5) +lim +n→∞EF(X(n) +1 ,...,X(n) +d ) = lim +n→∞ +1 +#Dn +� +(x1,...,xd)∈Dn +F(x1,...,xd) += M(F) = +� +p∈P +� +1 − 1 +p +�d ∞ +� +i1=0 +··· +∞ +� +id=0 +F(pi1,...,pid) +pi1+···+id +. +The purpose of the present paper is two-fold. First, we shall provide a probabilistic explana- +tion which lies in the core of (5), by providing sufficient conditions on F for the distributional +convergence of F(X(n) +1 ,...,X(n) +d ) as n → ∞. Second, we shall do this not only for the three types +of regions mentioned before, but for a quite general class of integer domains Dn satisfying mild +assumptions. +The paper is organized as follows. In Section 2, we formulate our standing assumptions on +Dn and present our main results, which are distributional limit theorems for F(X(n) +1 ,...,X(n) +d ). +The proofs are collected in Section 3. In Section 4, we provide various examples of domains +Dn satisfying our standing assumptions. In particular, the aforementioned domains Rn1,...,nd, +Sn and Hn are covered. In Section 5 we discuss how to construct new domains satisfying our +conditions, using standard set-theoretic operations. Some auxiliary results are collected in +Appendix A. +Throughout the paper we use the following standard notation: +w +−→ denotes the convergence +in distribution (weak convergence of probability measures); Int(A), cl(A) and ∂A are the topo- +logical interior, closure and boundary of a set A ⊂ Rd, respectively; a(n) ∼ b(n), n → ∞, means +that limn→∞(a(n)/b(n)) = 1. +2. Main results +2.1. Preliminaries. Throughout the paper, we assume that F is a multivariate multiplicative +arithmetic function of d ≥ 2 variables. Every multivariate multiplicative function is completely +determined by its values on the powers of primes. More precisely, let λp(n) denote the power + +4 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +of prime p ∈ P in the prime decomposition of n ∈ N. Then +xi = +� +p∈P +pλp(xi), +i = 1,...,d, +implies +F(x1,...,xd) = +� +p∈P +F(pλp(x1),...,pλp(xd)). +The crucial observation for everything to follow is the representation for M(F) in (4) via +independent geometric random variables. Let (G1(p),...,Gd(p))p∈P be an array of mutually in- +dependent random variables with geometric distributions +P{Gk(p) ≥ j} = 1 +pj , +j ∈ N0, +p ∈ P , +k = 1,...,d, +where N0 := N ∪ {0}. Then +M(F) = E + + +� +p∈P +F(pG1(p),...,pGd(p)) + +. +The main result of our paper gives sufficient conditions on F which ensure the convergence in +distribution +(6) +F(X(n) +1 ,...,X(n) +d ) = +� +p∈P +F(pλp(X(n) +1 ),...,pλp(X(n) +d )) +w +−→ +n→∞ +� +p∈P +F(pG1(p),...,pGd(p)) =: F∞, +for a general class of integer domains Dn, which we are now going to introduce. +Let (Dn)n∈N be a sequence of finite, non-empty subsets of Nd. Assume that for every fixed +c ∈ Zd, where Z = {0,±1,±2,...}, the following condition is fulfilled: +(7) +lim +n→∞ +#((Dn + c) ∩ Dn) +#Dn += 1. +Note that (7) is equivalent to saying that for all c ∈ Zd, +lim +n→∞ +δn(c) +#Dn += 0, +where, denoting ∆ the symmetric difference of two sets, +(8) +δn(c) := #(Dn∆(Dn + c)). +Condition (7) is known in the literature as the regular growth condition; see Chapter 3 in [5]. +Several equivalent versions of (7) can be found in Appendix A below. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +5 +2.2. Convergence of prime powers to geometric laws. Our first main result states that, solely +under assumption (7), the array of random vectors (λp(X(n) +1 ),...,λp(X(n) +d ))p∈P converges in dis- +tribution to an array of independent geometric variables, thereby providing the first evidence +supporting (6). +Theorem 2.1. Assume that (7) holds. Then +� +λp(X(n) +1 ),...,λp(X(n) +d ) +� +p∈P +w +−→ +n→∞ (G1(p),...,Gd(p))p∈P , +in the space (Rd)∞ endowed with the product topology. +Remark 2.2. In the rectangular case Dn = Rn1,...,nd, Theorem 2.1 is well known in probabilistic +number theory and has a long history, see, for instance, Eqs. (2.5)–(2.7) in [19] and [2]. Note that +in this case, the components X(n) +1 ,...,X(n) +d +are independent and X(n) +j +has the uniform distribution +on {1,...,nj}, for every j = 1,...,d. +2.3. Limit theorems for F. We start with finding conditions ensuring a.s. finiteness of F∞ in +(6). Recall that we assume d ≥ 2. According to Eq. (20) in [4] (or just by an appeal to the +Borel-Cantelli lemma), we have +� +p∈P +1{�d +k=1 Gk(p)≥2} < ∞ +a.s. +Furthermore, because F is multiplicative, F(1,1,...,1) = 1. Thus, a.s. finiteness of F∞ is equiva- +lent to the a.s. convergence of the product +�F∞ := +� +p∈P : �d +k=1 Gk(p)=1 +F(pG1(p),...,pGd(p)). +For i = 1,...,d, put +Fi(x) := logF(1,...,1,x,1,...,1), +where x ∈ N on the right-hand side is on the i-th position and log is the principal branch of the +logarithm (a branch which satisfies log(1) = 0 and has a branch cut along (−∞,0]). We assume +that for all i = 1,...,d, there are only finitely many p ∈ P such that F(1,...,1,p,1,...,1) falls in- +side the branch cut. Otherwise, we stipulate that the series diverges. Thus, the a.s. convergence +of �F∞, hence of F∞, is equivalent to the a.s. convergence of the series +(9) +� +p∈P + + +d +� +i=1 +Fi(p) +1{Gi(p)=1,Gj(p)=0 for j�i} + +, +comprised of independent random variables. An application of Kolmogorov’s three series the- +orem immediately yields the following: + +6 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Proposition 2.3. The infinite product F∞ converges a.s. if and only if the following series converge +for every A > 0: +(10) +� +p∈P +1 +p +d +� +i=1 +1{|Fi(p)|>A}, +� +p∈P +1 +p +d +� +i=1 +Fi(p) +1{|Fi(p)|≤A}, +� +p∈P +1 +p +d +� +i=1 +|Fi(p)|2 +1{|Fi(p)|≤A} . +It is clear that the convergence of the three series (10) is a necessary condition for (6). Prov- +ing (6) under (10) alone seems to be a very difficult task, even for simple regions Dn as Rn,...,n. +In this paper, we restrict our attention to a subclass of multivariate multiplicative functions +satisfying (10). Namely, we shall assume that, for all i = 1,...,d, +(11) +� +p∈P +1 +p +1{|Fi(p)|>A} < ∞ +and +� +p∈P +1 +p|Fi(p)| +1{|Fi(p)|≤A} < ∞. +It is obvious that (11) implies (10). The difference between conditions (10) and (11) is that (11) +is necessary and sufficient for the a.s. absolute convergence of the series (9), whereas under (10) +the a.s. convergence of the series (9) is, in general, only conditional. +In order to prove (6) under (11), we shall impose a mild additional assumption on Dn. For +i = 1,...,d and a ∈ N, put +Zi(a) := {(x1,...,xd) ∈ Zd : xi is divisible by a}. +As we shall see below in Lemma 3.1, solely under assumption (7), one has +(12) +lim +n→∞ +#(Dn ∩ Zi(a) ∩ Zj(b)) +#Dn += 1 +ab, +for every fixed a,b ∈ N and i,j = 1,...,d, i � j. However, we shall need a further assumption +that refines the above limit relation, providing a kind of uniformity in (12). Namely, we assume +that there exists K > 0 such that for all i,j = 1,...,d, i � j, a,b ∈ N and n ∈ N, +(13) +#(Dn ∩ Zi(a) ∩ Zj(b)) +#Dn +≤ K +ab . +Recall that (X(n) +1 ,...,X(n) +d ) is a random vector picked uniformly at random from Dn. Below is +our main result. +Theorem 2.4. Assume that F : Nd → C is a multiplicative arithmetic function such that condi- +tions (11) hold. Let Dn, n ∈ N, be a sequence of subsets of Nd such that (7) and (13) hold. Then +F(X(n) +1 ,...,X(n) +d ) +w +−→ +n→∞ +� +p∈P +F(pG1(p),...,pGd(p)). + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +7 +Examples of integer domains satisfying (7) and (13) will be presented in Section 4. +The following functions F +Nd ∋ (x1,...,xd) �→ GCD(x1,...,xd) +and +Nd ∋ (x1,...,xd) �→ LCM(x1,...,xd) +x1 ···xd +are multiplicative and satisfy Fi(x) ≡ 0 for every i = 1,...,d. Thus, Theorem 2.4 is applicable, +leading to the following corollaries. +Corollary 2.5. Assume that (7) and (13) hold. Then +GCD(X(n) +1 ,...,X(n) +d ) +w +−→ +n→∞ +� +p∈P +pmink=1,...,d Gk(p). +The limiting random variable has the following distribution +(14) +P + +� +p∈P +pmink=1,...,d Gk(p) = j + += +1 +ζ(d) +1 +jd , +j ∈ N, +where ζ is the Riemann zeta function. +Corollary 2.6. Assume that (7) and (13) hold. Then +LCM(X(n) +1 ,...,X(n) +d ) +X(n) +1 ···X(n) +d +w +−→ +n→∞ +� +p∈P +pmaxk=1,...,d Gk(p)−�d +k=1 Gk(p). +Remark 2.7 (Bibliographic comments). Below is a comparison of our results with the existing +ones. +Case Dn = Rn,...,n. In this case Corollaries 2.5 and 2.6 are known, with Corollary 2.5 having a +long history. The fact that two independent random integers picked uniformly at random from +{1,...,n} are asymptotically co-prime with probability 1/ζ(2) = 6/π2, that is +lim +n→∞P{GCD(X(n) +1 ,X(n) +2 ) = 1} = 6 +π2 +goes back to Dirichlet [10], and generalizations of this relation to d > 2 integers are due to +Ces`aro [6, 7]. To the best of our knowledge, Corollary 2.5 is due to Christopher [8], see also [9]. +Formula (14) follows from the following chain of equalities. For s < d − 1, by Euler’s product +formula +E + + +� +p∈P +pmink=1,...,d Gk(p) + + +s += +� +p∈P +Epsmink=1,...,d Gk(p) = +� +p∈P +� +1 − 1 +pd +� +1 +1 − ps−d = ζ(d − s) +ζ(d) += +1 +ζ(d) +d +� +j=1 +js +jd . +Corollary 2.6 can be extracted from Theorem 2.1 in [18] and is given explicitly in Remark 2.4 +in [4]. Further pointers to literature related to Corollaries 2.5 and 2.6 in case Dn = Rn,...,n + +8 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +can be found in the introduction [4] and in the survey [13]. In [4] a version of Theorem 2.4 +was proved assuming that F(x1,...,xd) = G(LCM(x1,...,xd)) for some univariate multiplicative +function G : N → C. Asymptotics of moments accompanying the aforementioned distribu- +tional convergences have been derived in [18, 20, 21]. +Case Dn = Hn (and more general hyperbolic regions, see Example 4.5 below). In this case, +Corollaries 2.5 and 2.6 can be found in Theorems 3.5 and 3.7 in [14]. The corresponding +asymptotics of moments has been derived in [15, 16]. +Case Dn = Sn. The distributional convergence is completely new. The asymptotics of moments +has been analyzed in [17]. +3. Proof of the main results +3.1. Proof of Theorem 2.1. We first need an auxiliary lemma. +Lemma 3.1. Fix m1,...,md ∈ N and jk ∈ {0,...,mk − 1}, k = 1,...,d. Put +D(j1,m1,...,jd,md) +n +:= {(i1,...,id) ∈ Dn : ik ≡ jk (modmk) for all k = 1,...,d}. +If (7) holds, then +(15) +lim +n→∞ +#D(j1,m1,...,jd,md) +n +#Dn += +1 +m1 ···md +. +Proof. Note that +(16) +Dn = +m1−1 +� +j1=0 +··· +md−1 +� +jd=0 +D(j1,m1,...,jd,md) +n +, +and the sets on the right-hand side are pairwise disjoint. Furthermore, +D(j1,m1,...,jd,md) +n += Dn ∩(j1 +m1Z,...,jd +mdZ) = (j1,...,jd)+(Dn −(j1,...,jd))∩(m1Z,...,mdZ). +Thus, +����#D(0,m1,...,0,md) +n +− #D(j1,m1,...,jd,md) +n +���� += |#(Dn ∩ (m1Z,...,mdZ)) − #((Dn − (j1,...,jd)) ∩ (m1Z,...,mdZ))| +≤ #((Dn ∩ (m1Z,...,mdZ))∆((Dn − (j1,...,jd)) ∩ (m1Z,...,mdZ))) +≤ #(Dn∆(Dn − (j1,...,jd))), +and we have proved that (with δn introduced in (8)) +(17) +����#D(0,m1,...,0,md) +n +− #D(j1,m1,...,jd,md) +n +���� ≤ δn(−(j1,...,jd)). + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +9 +Plugging this into (16) yields +����#Dn − m1 ···md#D(0,m1,...,0,md) +n +���� ≤ +m1−1 +� +j1=0 +··· +md−1 +� +jd=0 +δn(−(j1,...,jd)). +Dividing both sides by #Dn and sending n → ∞ implies (15) for j1 = ··· = jd = 0. Using the +estimate (17), we obtain (15) for arbitrary j1,...,jd. +□ +Proof of Theorem 2.1. Fix pairwise distinct prime numbers p1,...,pm ∈ P , nonnegative integers +jk,t, k = 1,...,d, t = 1,...,m, and write +P{λpt(X(n) +k ) ≥ jk,t for all k = 1,...,d and t = 1,...,m} += P{X(n) +k +is divisible by pjk,t +t +for all k = 1,...,d and t = 1,...,m} += P{X(n) +k +is divisible by +m +� +t=1 +p +jk,t +t +=: µk for all k = 1,...,d} += +1 +#Dn +∞ +� +i1=1 +··· +∞ +� +id=1 +1�(i1,...,id) ∈ Dn : ik ≡ 0 (modµk),k = 1,...,d�. +By Lemma 3.1 applied with mk = µk and jk = 0, k = 1,...,d, we see that the right-hand side +converges to (µ1 ···µd)−1 as n → ∞. It remains to note that +1 +µ1 ···µd += +d +� +k=1 +m +� +t=1 +1 +p +jk,t +t += P{Gk(pt) ≥ jk,t for all k = 1,...,d and t = 1,...,m}. +The proof of Theorem 2.1 is complete. +□ +3.2. Proof of Theorem 2.4. Fix a large positive constant M and note that +F(X(n) +1 ,...,X(n) +d ) = +� +p∈P +F(pλp(X(n) +1 ),...,pλp(X(n) +d )) += + + +� +p∈P ,p≤M +F(pλp(X(n) +1 ),...,pλp(X(n) +d )) + + + + +� +p∈P ,p>M +F(pλp(X(n) +1 ),...,pλp(X(n) +d )) + + =: Y1(M,n)Y2(M,n). +By Theorem 2.1, one has +Y1(M,n) +w +−→ +n→∞ +� +p∈P ,p≤M +F(pG1(p),...,pGd(p)). +Furthermore, the right-hand side of the latter converges a.s. to F∞ as M → ∞, which is a.s. fi- +nite. According to Theorem 3.2 in [1], it remains to check that for every fixed ε > 0, +(18) +lim +M→∞limsup +n→∞ +P{|Y2(M,n) − 1| ≥ ε} = 0. + +10 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Note that +(19) +P{|Y2(M,n) − 1| ≥ ε} ≤ P +for all p ∈ P ,p > M, +d +� +i=1 +λp(X(n) +i +) ≤ 1,|Y2(M,n) − 1| ≥ ε + ++ P +for some p ∈ P ,p > M, +d +� +i=1 +λp(X(n) +i +) ≥ 2 +. +The second term in (19) can be estimated as follows: +P{for some p ∈ P ,p > M, +d +� +i=1 +λp(X(n) +i +) ≥ 2} +≤ P{there exist p ∈ P ,p > M and i = 1,...,d such that λp(X(n) +i +) ≥ 2} ++ P{there exist p ∈ P ,p > M and i,j = 1,...,d,i � j such that λp(X(n) +i +) ≥ 1,λp(X(n) +j ) ≥ 1} += P{there exist p ∈ P ,p > M and i = 1,...,d such that p2 divides X(n) +i +} ++ P{there exist p ∈ P ,p > M and i,j = 1,...,d,i � j such that p divides X(n) +i +and X(n) +j +} +≤ +d +� +i=1 +� +p∈P ,p>M +P{p2 divides X(n) +i +} + +d +� +i,j=1,i�j +� +p∈P ,p>M +P{p divides X(n) +i +and X(n) +j } += +d +� +i=1 +� +p∈P ,p>M +#(Dn ∩ Zi(p2)) +#Dn ++ +d +� +i,j=1,i�j +� +p∈P ,p>M +#(Dn ∩ Zi(p) ∩ Zj(p)) +#Dn +. +The double limit (n → ∞, M → ∞) of the first term is equal to zero by an appeal to (13) with +a = p2 and b = 1, since +lim +M→∞ +� +p∈P ,p>M +1 +p2 = 0. +Similarly, the double limit of the second term is equal to zero by an appeal to (13) with a = b = p. +In order to deal with the first summand in (19), we first observe that on the event +for all p ∈ P ,p > M, +d +� +i=1 +λp(X(n) +i +) ≤ 1 +, +we may pass to the logarithm of Y2(M,n). Thus, it suffices to prove that, for every ε > 0, +lim +M→∞limsup +n→∞ +P + +for all p ∈ P ,p > M, +d +� +i=1 +λp(X(n) +i +) ≤ 1, +�������� +� +p∈P ,p>M +logF(pλp(X(n) +1 ),...,pλp(X(n) +d )) +�������� +≥ ε + += 0. +Introduce, for n ∈ N, i = 1,...,d and p ∈ P , the events +Cn,i,p := {λp(X(n) +i +) = 1,λp(X(n) +j +) = 0,j � i}, + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +11 +and note that Cn,i,p ∩ Cn,j,p = ∅ as soon as i � j. On the event Cn,i,p, we have +logF(pλp(X(n) +1 ),...,pλp(X(n) +d )) = Fi(p) +and, therefore, it suffices to show that, for every fixed ε > 0, +(20) +lim +M→∞limsup +n→∞ +P + +�������� +� +p∈P ,p>M +d +� +i=1 +Fi(p) +1Cn,i,p +�������� +≥ ε + += 0. +Fix some A > 0 and note that, for every ε > 0, +P + +�������� +� +p∈P ,p>M +d +� +i=1 +Fi(p) +1{|Fi(p)|>A,Cn,i,p} +�������� +≥ ε + +≤ P{for some p ∈ P and i = 1,...,d, |Fi(p)| > A and Cn,i,p holds} +≤ +� +p∈P ,p>M +d +� +i=1 +1{|Fi(p)|>A} P{Cn,i,p} ≤ +� +p∈P ,p>M +d +� +i=1 +1{|Fi(p)|>A} P{λp(X(n) +i +) ≥ 1} += +� +p∈P ,p>M +d +� +i=1 +1{|Fi(p)|>A} +#(Dn ∩ Zi(p)) +#Dn +≤ K +� +p∈P ,p>M +1 +p +d +� +i=1 +1{|Fi(p)|>A}, +where we used (13) with a = p and b = 1 for the last passage. The right-hand side converges to +zero as M → ∞, in view of the first relation in (10). So, in order to prove (20), we need to check +that +(21) +lim +M→∞limsup +n→∞ +P + +�������� +� +p∈P ,p>M +d +� +i=1 +Fi(p) +1{|Fi(p)|≤A,Cn,i,p} +�������� +≥ ε + += 0. +This is accomplished by an appeal to Markov’s inequality as follows: +P + +�������� +� +p∈P ,p>M +d +� +i=1 +Fi(p) +1{|Fi(p)|≤A,Cn,i,p} +�������� +≥ ε + +≤ 1 +ε +� +p∈P ,p>M +d +� +i=1 +|Fi(p)| +1{|Fi(p)|≤A} P{Cn,i,p} +≤ 1 +ε +� +p∈P ,p>M +d +� +i=1 +|Fi(p)| +1{|Fi(p)|≤A} P{λp(X(n) +i +) ≥ 1} += 1 +ε +� +p∈P ,p>M +d +� +i=1 +Fi(p) +1{|Fi(p)|≤A} +#(Dn ∩ Zi(p)) +#Dn +≤ K +ε +� +p∈P ,p>M +1 +p +d +� +i=1 +Fi(p) +1{|Fi(p)|≤A}, + +12 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +where we have utilized (13) with a = p and b = 1 for the last inequality. The proof of The- +orem 2.4 is complete, since the right-hand side converges to zero, as M → ∞, by the second +relation in (11). +4. Examples of suitable integer domains +In this section we provide a series of examples of domains Dn that satisfy (7) and (13). In +particular, we show that Rn1,n2,...,nd in (2), Sn and Hn mentioned in the introduction, are all +admissible. Thus, under assumption (11) on F, the distributional convergence (6) holds true +for all domains listed below. +4.1. Sublevels of monotone functions. +Proposition 4.1. Assume that f : [1,∞)d → R is a coordinate-wise nondecreasing function such +that, for every j = 1,...,d, +lim +xj→∞f (x1,...,xd) = ∞, +provided xi ≥ 1, i � j, are fixed. Put +Dn := Df +n = {(x1,...,xd) ∈ Nd : f (x1,...,xd) ≤ n} +and +Dn,i := Df +n,i = {(x1,...,xi−1,xi+1,...,xd) ∈ Nd−1 : f (x1,...,xi−1,1,xi+1,...,xd) ≤ n}, +for i = 1,...,d. If, for every i = 1,...,d, +(22) +lim +n→∞ +#Dn,i +#Dn += 0, +then the sequence Dn, n ∈ N, satisfies (7) and (13). +Proof. Let us first verify (7). According to Proposition A.2 in Appendix A, it is sufficient to +check (7) for c = ei, i = 1,...,d, where e1,...,ed denotes the standard basis of Rd. Note that +Dn \ (Dn + ei) = Dn,i. Thus, (22) yields that for i = 1,...,d, +lim +n→∞ +#(Dn \ (Dn + ei)) +#Dn += 0. +It remains to check that for i = 1,...,d, +(23) +lim +n→∞ +#((Dn + ei) \ Dn) +#Dn += 0. +Without loss of generality, we shall do this for i = 1. Note that +(Dn + e1) \ Dn = {(x1,...,xd) ∈ Nd : x1 ≥ 2,f (x1 − 1,x2,...,xd) ≤ n,f (x1,...,xd) > n}. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +13 +For every fixed collection (x2,...,xd) ∈ Nd−1 and n ∈ N, there exists at most one x1 ≥ 2, x1 ∈ N, +such that +f (x1 − 1,x2,...,xd) ≤ n +and +f (x1,...,xd) > n, +since f is monotone in x1. Therefore, +#((Dn + e1) \ Dn) = +∞ +� +x2=1 +··· +∞ +� +xd=1 +1{there exists x1≥2 such that f (x1−1,x2,...,xd)≤n,f (x1,...,xd)>n} +≤ +∞ +� +x2=1 +··· +∞ +� +xd=1 +1{there exists x1≥2 such that f (x1−1,x2,...,xd)≤n} = +∞ +� +x2=1 +··· +∞ +� +xd=1 +1{f (1,x2,...,xd)≤n} += #Dn,1. +This proves (23) for i = 1. +We shall now prove that (13) holds, for all i,j = 1,...,d, with K = 1. For notational simplicity, +we shall do this only for i = 1 and j = 2. The monotonicity of f implies that, for all a,b ∈ N, +#Dn = +a−1 +� +j=0 +b−1 +� +k=0 + + +∞ +� +x1=1 +∞ +� +x2=1 +··· +∞ +� +xd=1 +1{f (ax1−j,bx2−k,x3,...,xd)≤n} + + +≥ ab +∞ +� +x1=1 +∞ +� +x2=1 +··· +∞ +� +xd=1 +1{f (ax1,bx2,x3,...,xd)≤n} += ab#(Dn ∩ Z1(a) ∩ Z2(b)). +The proof of Proposition 4.1 is complete. +□ +Proposition 4.1 yields the following explicit examples. +Example 4.2 (Rectangular domains). Let f1,...,fd : [1,∞) → [1,∞) be strictly increasing continu- +ous functions. Putting f (x1,...,xd) := max(f −1 +1 (x1),...,f −1 +d (xd)), we obtain +Dn = Rf1(n),...,fd(n) = ([1,f1(n)] × ··· × [1,fd(n)]) ∩ Nd. +Condition (22) is fulfilled if limx→∞ fi(x) = ∞, for every i = 1,...,d. +Example 4.3 (Tetrahedral domains). Let a1,...,ad > 0 be fixed positive real numbers. The sequence +of tetrahedral sets +Dn = Tn := {(x1,...,xd) ∈ Nd : a1x1 + ··· + adxd ≤ n} +satisfies (7) and (13). Indeed, +#Tn ∼ +1 +d!a1 ···ad +nd, +n → ∞, + +14 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +whereas, for i = 1,...,d, +#Tn,i ∼ +ai +(d − 1)!a1···ad +nd−1, +n → ∞. +Thus, Proposition 4.1 is applicable. +Example 4.4 (Hyperbolic domains). Let f (x1,...,xd) = x1 ···xd. Then the sequence of sets +Dn = Hn := {(x1,...,xd) ∈ Nd : x1 ···xd ≤ n} +satisfies (7) and (13). Indeed, according to Proposition 4.1 in [14]. +#Dn ∼ nlogd−1 n +(d − 1)! , +n → ∞, +and, for every i = 1,...,d, +#Dn,i ∼ nlogd−2 n +(d − 2)! , +n → ∞. +Thus, Proposition 4.1 is applicable. +Example 4.5 (Further hyperbolic domains). Fix 2 ≤ ℓ ≤ d. Define the ℓ-th standard symmetric +polynomial in d variables by +f (x1,...,xd) = Pℓ(x1,...,xd) := +� +1≤i1<··· 0. Furthermore, by symmetry +#Dn,i = #Dn,1 for all i = 1,...,d, and +Dn,1 = {(x2,...,xd) ∈ Nd : Pℓ(1,x2,...,xd) ≤ n} += {(x2,...,xd) ∈ Nd : Pℓ(x2,...,xd) + Pℓ−1(x2,...,xd) ≤ n} +⊂ {(x2,...,xd) ∈ Nd : Pℓ(x2,...,xd) ≤ n} = Hℓ,d−1(n). +Thus, Dn,1 ⊆ Hℓ,d−1(n) and thereupon #Dn,1 ≤ #Hℓ,d−1(n). If ℓ < d − 1, then +#Hℓ,d−1(n) ∼ C(d − 1,ℓ)n(d−1)/ℓ, +n → ∞, +whereas if ℓ = d − 1, +#Hℓ,d−1(n) = #Hd−1,d−1(n) ∼ nlogd−2 n +(d − 2)! , +n → ∞. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +15 +In both cases limn→∞ #Hℓ,d−1(n)/#Dn = 0. Summarizing, Proposition 4.1 is applicable to Dn = +Hℓ,d(n). +4.2. Dilations of a convex body. +Proposition 4.6. Let D ⊂ [0,∞)d be a compact convex set with nonempty interior and an, n ∈ N, +be a sequence of positive numbers such that limn→∞ an = ∞. Then, the following sequence of sets +satisfies (7) and (13): +Dn := anD ∩ Nd. +Proof. For the proof of (7), we shall use Proposition A.3. Put +V := D ∩ (0,∞)d, +Vn := anV = anD ∩ (0,∞)d, +and note that Dn = Vn ∩ Nd. Let us check that (30) holds for the sequence Vn. First of all, since +D is compact, convex and has a non-empty interior, it holds +D = cl(Int(D)) = cl(Int(D) ∩ (0,∞)d) = cl(Int(D ∩ (0,∞)d)) = cl(V ), +and, thereupon, +∂V = cl(V ) \ Int(V ) = cl(V ) \ Int(D) = D \ Int(D) = ∂D. +Further, observe that Vol(V ) > 0 and, denoting Bdε(0) the ball {(x1,...,xd) ∈ Rd : x2 +1 + ··· + x2 +d < ε} +and A ⊕ B := {x + y : x ∈ A,y ∈ B} the Minkowski addition, +(24) +Vol(∂Vn ⊕ Bdε(0)) +Vol(Vn) += +Vol(an(∂V ⊕ Bd +ε/an(0))) +Vol(anV ) += +Vol(∂V ⊕ Bd +ε/an(0)) +Vol(V ) += +Vol(∂D ⊕ Bd +ε/an(0)) +Vol(V ) +. +Since D is a compact convex set, its boundary ∂D is (d − 1)-rectifiable subset of Rd, that is, can +be represented as the image of a Lipschitz function1 h defined on a bounded subset of Rd−1 +and taking values in Rd. Thus, by Theorem 3.2.39 in [12], +lim +n→∞anVol(∂D ⊕ Bd +ε/an(0)) = 2εHd−1(∂D) < ∞, +where Hd−1 is the (d −1)-dimensional Hausdorff measure in Rd. Summarizing, we have shown +that the right-hand side of (24) converges to zero as n → ∞. +For the proof of (13), we employ Proposition A.4 from Appendix A. +For i = 1,...,d, put +mi(D) := inf{xi ≥ 0 : (x1,...,xi,...,xd) ∈ D}, +Mi(D) := sup{xi ≥ 0 : (x1,...,xi,...,xd) ∈ D}, +1As h one can take, for example, the function ∂BR(0) ∋ x �→ πD(x), where R > 0 is such that D ⊆ BR(0) and πD(x) +is a unique closest to x point in D (metric projection on D). + +16 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +and note that 0 ≤ mi(D) < Mi(D) < ∞. Here the second inequality is strict since D has a non- +empty interior; the last inequality follows from the compactness of D. Proposition A.4 is ap- +plicable with the rectangle +Πn := + + +d +� +i=1 +� +⌊anmi(D)⌋,⌈anMi(D)⌉ +� + +� +Nd. +By construction +anD ⊂ an + + +d +� +i=1 +� +mi(D),Mi(D) +� + ⊂ + + +d +� +i=1 +� +⌊anmi(D)⌋,⌈anMi(D)⌉ +� +. +It remains to note that as n → ∞, +(25) +#Πn ∼ ad +n +d +� +i=1 +(Mi(D) − mi(D)), +and also +(26) +liminf +n→∞ +#Dn +adn +> 0, +which is a consequence of the fact that D has a non-empty interior and, therefore, contains a +small d-dimensional cube in the interior. Relations (25) and (26) imply +limsup +n→∞ +#Πn +#Dn +< ∞. +The proof of Proposition 4.6 is complete. +□ +Example 4.7 (Spherical domains). Put B := {(x1,...,xd) ∈ [0,∞)d : x2 +1 + ··· + x2 +d ≤ 1}. Then the +sequence of discrete balls +Dn = Sn := +√ +nB ∩ Nd = {(x1,...,xd) ∈ Nd : x2 +1 + ··· + x2 +d ≤ n} +satisfies (7) and (13) by Proposition 4.6. +Truncated cones, such as Weyl chambers, also satisfy (7) and (13). +Example 4.8 (Truncated Weyl chambers). Let A := {(x1,...,xd) ∈ [0,∞) : x1 ≤ ··· ≤ xd ≤ 1}. Then +the sequence of sets +Dn = An := nA ∩ Nd = {(x1,...,xd) ∈ Nd : x1 ≤ ··· ≤ xd ≤ n} +satisfies (7) and (13) by Proposition 4.6. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +17 +5. Set-theoretic operations preserving properties (7) and (13) +In this section we discuss stability properties of sets satisfying (7) and (13) with respect to +the standard set-theoretic operations. +First, immediately from the definitions, one obtains the following. +Proposition 5.1. Let D(1) +n +and D(2) +n +be two sequences of sets satisfying (7) and (13). Then the se- +quence Dn := D(1) +n ∪ D(2) +n +satisfies (7) and (13). +As far as intersections and differences of sets are concerned, additional assumptions ensur- +ing that the resulting sets are not small have to imposed. The following holds true. +Proposition 5.2. Let D(1) +n +and D(2) +n +be two sequences of sets satisfying (7). Suppose further that +(27) +D(2) +n ⊂ D(1) +n +and +limsup +n→∞ +#D(2) +n +#D(1) +n +∈ [0,1). +Then the sequence Dn := D(1) +n \ D(2) +n +satisfies (7). Moreover, if D(1) +n +satisfies (13), then so does Dn. +Proof. Using the inclusion (A \ B)∆(C \ D) ⊆ (A∆C) ∪ (B∆D) we obtain, for every fixed c ∈ Zd, +#(Dn∆(Dn + c)) +#Dn += #((D(1) +n \ D(2) +n )∆((D(1) +n + c) \ (D(2) +n + c))) +#Dn +≤ #(D(1) +n ∆(D(1) +n + c)) +#D(1) +n +#D(1) +n +#Dn ++ #(D(2) +n ∆(D(2) +n + c)) +#D(2) +n +#D(2) +n +#Dn +. +In view of (27), +(28) +0 ≤ limsup +n→∞ +#D(2) +n +#Dn +≤ limsup +n→∞ +#D(1) +n +#Dn += limsup +n→∞ +#D(1) +n +#D(1) +n − #D(2) +n +< ∞, +and we see that Dn satisfies (7). +If D(1) +n +satisfies (13), then, for every a,b ∈ N and i,j = 1,...,d, i � j, it holds that for all n ∈ N, +#(Dn ∩ Zi(a) ∩ Zj(b)) +#Dn +≤ +#(D(1) +n ∩ Zi(a) ∩ Zj(b)) +#D(1) +n +#D(1) +n +#Dn +≤ K +ab sup +n∈N +#D(1) +n +#Dn +=: K′ +ab , +where we used (28) for the last passage. +□ +With minimal changes, the above proof leads to the following. +Proposition 5.3. Let D(1) +n +and D(2) +n +be two sequences of sets satisfying (7). Suppose further that +limsup +n→∞ +#(D(1) +n ∪ D(2) +n ) +#(D(1) +n ∩ D(2) +n ) +< ∞. + +18 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Then the sequence Dn := D(1) +n ∩ D(2) +n +satisfies (7). Moreover, if D(1) +n +or D(2) +n +satisfies (13), then Dn +satisfies (13) as well. +Appendix A. On the regular growth condition for discrete domains +The following definition can be found on p. 173 in [5]. +Definition A.1. A sequence of finite sets Dn ⊂ Zd is said to be regularly growing to infinity if +as n → ∞, +(29) +#Dn → ∞ +and +#(D1n \ Dn) +#Dn +→ 0, +where for A ⊂ Zd and p ∈ N, we denote by +Ap := {x = (x1,...,xd) ∈ Zd : dist(x,A) ≤ p}, +and dist is the supremum metric on Zd. +Proposition A.2. Assume that Dn ⊂ Zd is a sequence of finite sets and #Dn → ∞ as n → ∞. The +following statements are equivalent: +(i) Condition (7) holds for all c ∈ Zd. +(ii) Condition (7) holds for c = ±ek, k = 1,...,d. +(iii) Condition (7) holds for c = ek, k = 1,...,d. +(iv) The sequence Dn is regularly growing. +Proof. Condition (i) trivially implies condition (ii), and (ii) clearly implies (iii). The fact that +(iii)=⇒(ii) follows from +#((Dn − ek)∆Dn) = #(((Dn − ek)∆Dn) + ek) = #(Dn∆(Dn + ek)) = #((Dn + ek)∆Dn). +We now prove that (ii)=⇒(iv). Note that +D1 +n = +d +� +k=1 +(Dn ± ek). +Thus, +#(D1n \ Dn) +#Dn +≤ +d +� +k=1 +#((Dn ± ek) \ Dn) +#Dn +≤ +d +� +k=1 +#((Dn ± ek)∆Dn) +#Dn +. +The right-hand side converges to 0, since by (7) every summand converges to 0. + +MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS +19 +We proceed to the proof of (iv)=⇒(i). Assume that (29) holds and fix c ∈ Zd. Using the +inclusion A \ B ⊂ (A \ C) ∪ (C \ B) which holds for any sets A,B,C, we conclude that +(Dn + c)∆Dn ⊂ +� +j +� +(Dn + uj) \ (Dn + vj) +� +, +where the union is finite and for every index j, uj − vj = ±ekj for some kj ∈ {1,...,d}. Since +#Dn = #(Dn + x) for every x ∈ Zd, it suffices to check that, for every j, +lim +n→∞ +# +� +(Dn + uj) \ (Dn + vj) +� +#(Dn + vj) += 0, +but this follows from the inclusion (Dn +uj) = (Dn +vj ±ekj) ⊂ (Dn +vj)1 and the fact that if (29) +holds for a sequence Dn, it also holds for the shifted sequence Dn+x, for every fixed x ∈ Zd. +□ +The following result is a combination of Proposition A.2 and Lemma 1.5 in [5]. In some +cases, it is useful for checking (29). +Proposition A.3. Assume that Vn, n ∈ N, is a sequence of bounded measurable subsets of Rd satis- +fying the so-called van Hove condition, meaning that for every ε > 0 +(30) +lim +n→∞ +Vol(∂Vn ⊕ Bd +ε(0)) +Vol(Vn) += 0, +where ∂Vn is the topological boundary of Vn. Then the sequence Dn := Vn ∩ Zd satisfies (29). +Our last auxiliary result provides sufficient conditions for (13). It has been used in the proof +of Proposition 4.6. +Proposition A.4. Assume that there exist two sequences (s1(n),...,sd(n))n∈N and (c1(n),...,cd(n))n∈N +of nonnegative integers such that the rectangle +Πn := + + +d +� +i=1 +[ci(n),ci(n) + si(n)] + + +� +Nd +satisfies +(31) +#Dn ⊂ Πn +and +C := sup +n∈N +#Πn +#Dn +< ∞. +Then (13) holds. More generally, if (13) holds with Dn replaced by some set Πn which satisfies (31), +then (13) holds for Dn. + +20 +ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL +Proof. Fix i,j = 1,...,d, i � j. If (31) holds, then for all n ∈ N and all a,b ∈ N it holds +#(Dn ∩ Zi(a) ∩ Zj(b)) +#Dn +≤ +#(Πn ∩ Zi(a) ∩ Zj(b)) +#Dn +≤ C +#(Πn ∩ Zi(a) ∩ Zj(b)) +#Πn +. +Since i � j, we obtain +#(Πn ∩ Zi(a) ∩ Zj(b)) +#Πn +≤ +1 +si(n) + 1 +�si(n) + 1 +a +� +1 +sj(n) + 1 +�sj(n) + 1 +b +� +≤ 1 +ab +and the desired estimate holds true with K = C. +□ +Acknowledgments +This project has received funding from the European Research Council (ERC) under the Eu- +ropean Union’s Horizon 2020 research and innovation programme under the Grant Agreement +No. 759702 and from Centre Henri Lebesgue, programme ANR-11-LABX-0020-0. ZK was sup- +ported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 +– 390685587, Mathematics M¨unster: Dynamics - Geometry - Structure. AM was supported +by UC Berkeley Economics/Haas in the framework of the U4U program. AM gratefully ac- +knowledges the financial support and hospitality of the University of Angers during his stay +in December 2022–March 2023. +References +[1] P. Billingsley (1968). Convergence of probability measures. John Wiley & Sons, Inc. +[2] P. Billingsley (1974). The probability theory of additive arithmetic functions. Ann. Probab. 5, pp. 749–791. +[3] N. H. Bingham, C. M. Goldie and J. L. Teugels (1989). Regular variation. Cambridge University Press. +[4] A. Bostan, A. Marynych and K. Raschel (2019). On the least common multiple of several random integers. J. +Number Theory 204, pp. 113–133. +[5] A. Bulinski and A. Shashkin (2007). Limit theorems for associated random fields and related systems. Advanced +Series on Statistical Science & Applied Probability, 10. World Scientific Publishing. +[6] E. Ces`aro (1885). Sur le plus grand commun diviseur de plusieurs nombres. Ann. Mat. Pura Appl. 13, pp. 291– +294. +[7] E. 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RACSAM 115, Paper No. 26. +[14] A. Iksanov, A. Marynych and K. Raschel (2022). Asymptotics of arithmetic functions of GCD and LCM of +random integers in hyperbolic regions. Results Math. 77, Paper No. 165. +[15] R. Heyman and L. T´oth (2021). On certain sums of arithmetic functions involving the GCD and LCM of two +positive integers. Results Math. 76, Paper No. 49. +[16] R. Heyman and L. T´oth (2022). Hyperbolic summation for functions of the GCD and LCM of several integers. +Ramanujan J. (to appear). https://doi.org/10.1007/s11139-022-00681-2 +[17] R. Heyman and L. T´oth (2022). Estimates for k-dimensional spherical summations of arithmetic functions of +the GCD and LCM. arXiv preprint:2204.10074. +[18] T. Hilberdink and L. T´oth (2016). On the average value of the least common multiple of k positive integers. J. +Number Theory 169 pp. 327–341. +[19] J. Kubilius (1964). Probabilistic methods in the theory of numbers. American Mathematical Society, Providence, +R.I. Vol. 11. +[20] L. T´oth (2014). Multiplicative arithmetic functions of several variables: a survey. Mathematics without bound- +aries, pp. 483–514, Springer, New York. +[21] N. Ushiroya (2012). Mean-Value Theorems for Multiplicative Arithmetic Functions of Several Variables. Inte- +gers 12, pp. 989–1002. +[22] R. Vaidyanathaswamy (1931). The theory of multiplicative arithmetic functions. Trans. Amer. Math. Soc. 33, +pp. 579–662. +Zakhar Kabluchko: Institut f¨ur Mathematische Stochastik, Westf¨alische Wilhelms-Universit¨at M¨unster, +M¨unster, Germany +Email address: zakhar.kabluchko@uni-muenster.de +Oleksandr Marynych: Faculty of Computer Science and Cybernetics, Taras Shevchenko National Univer- +sity of Kyiv, Kyiv, Ukraine +Email address: marynych@unicyb.kiev.ua +Kilian Raschel: Universit´e d’Angers, CNRS, Laboratoire Angevin de Recherche en Math´ematiques, Angers, +France +Email address: raschel@math.cnrs.fr + diff --git a/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/load_file.txt b/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f60c8f98f7f3b996655470ecd6137277be26cc3 --- /dev/null +++ b/x9FRT4oBgHgl3EQfiTdH/content/tmp_files/load_file.txt @@ -0,0 +1,1034 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf,len=1033 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='13586v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='PR] 31 Jan 2023 MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF UNIFORM RANDOM VECTORS IN LARGE INTEGER DOMAINS ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For a wide class of sequences of integer domains Dn ⊂ Nd, n ∈ N, we prove distribu- tional limit theorems for F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ), where F is a multivariate multiplicative function and (X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) is a random vector with uniform distribution on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set {X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' This generalizes previously known limit results for Dn being either a discrete cube or a discrete hyperbolic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Introduction Let F : Nd → C be an arithmetic function of d ≥ 1 integer arguments, with N = {1,2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A standard problem in analytic number theory is the estimation of the multivariate sum n1 � x1=1 ··· nd � xd=1 F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) for large values of (n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd) ∈ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A particular instance of this problem consists in establish- ing existence of the so-called mean value of F, which is defined via (1) M(f ) := lim n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd→∞ 1 n1 ···nd n1 � x1=1 ··· nd � xd=1 F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In the probabilistic language, (1) may be recast as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let (U(n1) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',U(nd) d ) be a random vector defined on some probability space (Ω,F ,P) and which has the uniform distribution on the finite rectangular set (2) Rn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd := \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed d � i=1 [1,ni] \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 � Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Primary: 11A05, 60F05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' secondary: 11N60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Distribution of arithmetic functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' greatest common divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' least common multiple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' multivariate multiplicative function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' regular growth of integer domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' van Hove condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 1 2 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL Then, with E denoting the expectation with respect to P, (3) M(F) = lim n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd→∞EF(U(n1) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',U(nd) d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A general result on existence of M(F) is due to Ushiroya [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A multivariate arithmetic function F : Nd → C is called multiplicative, see [20, 21, 22], if F(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1) = 1 and F(m1n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdnd) = F(m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',md)F(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd), for all (m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',md) ∈ Nd and (n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd) ∈ Nd such that GCD(m1 ···md,n1 ···nd) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A specialization of Ushiroya’s results from [21] to a multiplicative function F implies that under a mild summability assumption on F, the mean value M(F) exists and is equal to (4) M(F) := � p∈P � 1 − 1 p �d ∞ � i1=0 ··· ∞ � id=0 F(pi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pid) pi1+···+id , where P stands for the set of prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In the last years, there has been a lot of activity around various generalizations and exten- sions of the aforementioned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In a probabilistic direction, one may ask about the asymp- totic behavior of distributions of the random variable F(U(n1) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',U(nd) d ), as n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd → ∞ in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' This question has been addressed in [4] for a particular choice of F, namely, for F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = G(LCM(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)), with G being a univariate multiplicative arithmetic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The univari- ate case d = 1 is the classical Erd˝os-Wintner theorem, see [11], which provides necessary and sufficient conditions for the distributional convergence of F(U(n) 1 ) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In another, more analytic direction, the rectangular domains Rn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd in (2) are replaced by more sophisticated domains of summation Dn ⊂ Nd, which grow to Nd as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In particular, in the recent work [17], the case of spherical summation over the regions Sn := {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : x2 1 + ··· + x2 d ≤ n}, has been analyzed, whereas the papers [14, 15, 16] were devoted to the study of summation over hyperbolic regions Hn := {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : x1 ···xd ≤ n} and their generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' A surprising phenomenon revealed in the cited works is that the mean value M(F) given by (4) is universal for rectangular, spherical and hyperbolic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 3 More specifically, let Dn be either Rn,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',n, Sn or Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For every n ∈ N, let (X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) be a random vector defined on (Ω,F ,P) and having the uniform distribution on Dn, that is, P{(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',id)} = 1 #Dn , (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',id) ∈ Dn, where #Dn denotes the cardinality of Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then, under the same summability assumption on F as in Ushiroya’s result, we have (5) lim n→∞EF(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) = lim n→∞ 1 #Dn � (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)∈Dn F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = M(F) = � p∈P � 1 − 1 p �d ∞ � i1=0 ··· ∞ � id=0 F(pi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pid) pi1+···+id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The purpose of the present paper is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' First, we shall provide a probabilistic explana- tion which lies in the core of (5), by providing sufficient conditions on F for the distributional convergence of F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Second, we shall do this not only for the three types of regions mentioned before, but for a quite general class of integer domains Dn satisfying mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In Section 2, we formulate our standing assumptions on Dn and present our main results, which are distributional limit theorems for F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The proofs are collected in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In Section 4, we provide various examples of domains Dn satisfying our standing assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In particular, the aforementioned domains Rn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd, Sn and Hn are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In Section 5 we discuss how to construct new domains satisfying our conditions, using standard set-theoretic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Some auxiliary results are collected in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Throughout the paper we use the following standard notation: w −→ denotes the convergence in distribution (weak convergence of probability measures);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Int(A), cl(A) and ∂A are the topo- logical interior, closure and boundary of a set A ⊂ Rd, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' a(n) ∼ b(n), n → ∞, means that limn→∞(a(n)/b(n)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Main results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Throughout the paper, we assume that F is a multivariate multiplicative arithmetic function of d ≥ 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Every multivariate multiplicative function is completely determined by its values on the powers of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' More precisely, let λp(n) denote the power 4 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL of prime p ∈ P in the prime decomposition of n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then xi = � p∈P pλp(xi), i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, implies F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = � p∈P F(pλp(x1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(xd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The crucial observation for everything to follow is the representation for M(F) in (4) via independent geometric random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let (G1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',Gd(p))p∈P be an array of mutually in- dependent random variables with geometric distributions P{Gk(p) ≥ j} = 1 pj , j ∈ N0, p ∈ P , k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, where N0 := N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then M(F) = E \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � p∈P F(pG1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pGd(p)) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The main result of our paper gives sufficient conditions on F which ensure the convergence in distribution (6) F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) = � p∈P F(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) w −→ n→∞ � p∈P F(pG1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pGd(p)) =: F∞, for a general class of integer domains Dn, which we are now going to introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let (Dn)n∈N be a sequence of finite, non-empty subsets of Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that for every fixed c ∈ Zd, where Z = {0,±1,±2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='}, the following condition is fulfilled: (7) lim n→∞ #((Dn + c) ∩ Dn) #Dn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Note that (7) is equivalent to saying that for all c ∈ Zd, lim n→∞ δn(c) #Dn = 0, where, denoting ∆ the symmetric difference of two sets, (8) δn(c) := #(Dn∆(Dn + c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Condition (7) is known in the literature as the regular growth condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' see Chapter 3 in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Several equivalent versions of (7) can be found in Appendix A below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Convergence of prime powers to geometric laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Our first main result states that, solely under assumption (7), the array of random vectors (λp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',λp(X(n) d ))p∈P converges in dis- tribution to an array of independent geometric variables, thereby providing the first evidence supporting (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that (7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then � λp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',λp(X(n) d ) � p∈P w −→ n→∞ (G1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',Gd(p))p∈P , in the space (Rd)∞ endowed with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In the rectangular case Dn = Rn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 is well known in probabilistic number theory and has a long history, see, for instance, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='7) in [19] and [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Note that in this case, the components X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d are independent and X(n) j has the uniform distribution on {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nj}, for every j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Limit theorems for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' We start with finding conditions ensuring a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' finiteness of F∞ in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Recall that we assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' (20) in [4] (or just by an appeal to the Borel-Cantelli lemma), we have � p∈P 1{�d k=1 Gk(p)≥2} < ∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Furthermore, because F is multiplicative, F(1,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' finiteness of F∞ is equiva- lent to the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' convergence of the product �F∞ := � p∈P : �d k=1 Gk(p)=1 F(pG1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pGd(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, put Fi(x) := logF(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1,x,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1), where x ∈ N on the right-hand side is on the i-th position and log is the principal branch of the logarithm (a branch which satisfies log(1) = 0 and has a branch cut along (−∞,0]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' We assume that for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, there are only finitely many p ∈ P such that F(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1,p,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',1) falls in- side the branch cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Otherwise, we stipulate that the series diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' convergence of �F∞, hence of F∞, is equivalent to the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' convergence of the series (9) � p∈P \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed d � i=1 Fi(p) 1{Gi(p)=1,Gj(p)=0 for j�i} \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8, comprised of independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' An application of Kolmogorov’s three series the- orem immediately yields the following: 6 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The infinite product F∞ converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' if and only if the following series converge for every A > 0: (10) � p∈P 1 p d � i=1 1{|Fi(p)|>A}, � p∈P 1 p d � i=1 Fi(p) 1{|Fi(p)|≤A}, � p∈P 1 p d � i=1 |Fi(p)|2 1{|Fi(p)|≤A} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' It is clear that the convergence of the three series (10) is a necessary condition for (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Prov- ing (6) under (10) alone seems to be a very difficult task, even for simple regions Dn as Rn,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In this paper, we restrict our attention to a subclass of multivariate multiplicative functions satisfying (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Namely, we shall assume that, for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, (11) � p∈P 1 p 1{|Fi(p)|>A} < ∞ and � p∈P 1 p|Fi(p)| 1{|Fi(p)|≤A} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' It is obvious that (11) implies (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The difference between conditions (10) and (11) is that (11) is necessary and sufficient for the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' absolute convergence of the series (9), whereas under (10) the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' convergence of the series (9) is, in general, only conditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In order to prove (6) under (11), we shall impose a mild additional assumption on Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d and a ∈ N, put Zi(a) := {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Zd : xi is divisible by a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' As we shall see below in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1, solely under assumption (7), one has (12) lim n→∞ #(Dn ∩ Zi(a) ∩ Zj(b)) #Dn = 1 ab, for every fixed a,b ∈ N and i,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, i � j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' However, we shall need a further assumption that refines the above limit relation, providing a kind of uniformity in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Namely, we assume that there exists K > 0 such that for all i,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, i � j, a,b ∈ N and n ∈ N, (13) #(Dn ∩ Zi(a) ∩ Zj(b)) #Dn ≤ K ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Recall that (X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) is a random vector picked uniformly at random from Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Below is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that F : Nd → C is a multiplicative arithmetic function such that condi- tions (11) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let Dn, n ∈ N, be a sequence of subsets of Nd such that (7) and (13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) w −→ n→∞ � p∈P F(pG1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pGd(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 7 Examples of integer domains satisfying (7) and (13) will be presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The following functions F Nd ∋ (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) �→ GCD(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) and Nd ∋ (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) �→ LCM(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) x1 ···xd are multiplicative and satisfy Fi(x) ≡ 0 for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4 is applicable, leading to the following corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that (7) and (13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then GCD(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) w −→ n→∞ � p∈P pmink=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d Gk(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The limiting random variable has the following distribution (14) P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 � p∈P pmink=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d Gk(p) = j \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe = 1 ζ(d) 1 jd , j ∈ N, where ζ is the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that (7) and (13) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then LCM(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) X(n) 1 ···X(n) d w −→ n→∞ � p∈P pmaxk=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d Gk(p)−�d k=1 Gk(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='7 (Bibliographic comments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Below is a comparison of our results with the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Case Dn = Rn,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In this case Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='6 are known, with Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 having a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The fact that two independent random integers picked uniformly at random from {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',n} are asymptotically co-prime with probability 1/ζ(2) = 6/π2, that is lim n→∞P{GCD(X(n) 1 ,X(n) 2 ) = 1} = 6 π2 goes back to Dirichlet [10], and generalizations of this relation to d > 2 integers are due to Ces`aro [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' To the best of our knowledge, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 is due to Christopher [8], see also [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Formula (14) follows from the following chain of equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For s < d − 1, by Euler’s product formula E \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � p∈P pmink=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d Gk(p) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 s = � p∈P Epsmink=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d Gk(p) = � p∈P � 1 − 1 pd � 1 1 − ps−d = ζ(d − s) ζ(d) = 1 ζ(d) d � j=1 js jd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='6 can be extracted from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 in [18] and is given explicitly in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Further pointers to literature related to Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='6 in case Dn = Rn,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',n 8 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL can be found in the introduction [4] and in the survey [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In [4] a version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4 was proved assuming that F(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = G(LCM(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)) for some univariate multiplicative function G : N → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Asymptotics of moments accompanying the aforementioned distribu- tional convergences have been derived in [18, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Case Dn = Hn (and more general hyperbolic regions, see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In this case, Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='6 can be found in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='7 in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The corresponding asymptotics of moments has been derived in [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Case Dn = Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The distributional convergence is completely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The asymptotics of moments has been analyzed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proof of the main results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' We first need an auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Fix m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',md ∈ N and jk ∈ {0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mk − 1}, k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Put D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n := {(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',id) ∈ Dn : ik ≡ jk (modmk) for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' If (7) holds, then (15) lim n→∞ #D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n #Dn = 1 m1 ···md .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Note that (16) Dn = m1−1 � j1=0 ··· md−1 � jd=0 D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n , and the sets on the right-hand side are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Furthermore, D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n = Dn ∩(j1 +m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd +mdZ) = (j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd)+(Dn −(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd))∩(m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, ����#D(0,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',0,md) n − #D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n ���� = |#(Dn ∩ (m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdZ)) − #((Dn − (j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd)) ∩ (m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdZ))| ≤ #((Dn ∩ (m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdZ))∆((Dn − (j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd)) ∩ (m1Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',mdZ))) ≤ #(Dn∆(Dn − (j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd))), and we have proved that (with δn introduced in (8)) (17) ����#D(0,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',0,md) n − #D(j1,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd,md) n ���� ≤ δn(−(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 9 Plugging this into (16) yields ����#Dn − m1 ···md#D(0,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',0,md) n ���� ≤ m1−1 � j1=0 ··· md−1 � jd=0 δn(−(j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Dividing both sides by #Dn and sending n → ∞ implies (15) for j1 = ··· = jd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Using the estimate (17), we obtain (15) for arbitrary j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',jd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Fix pairwise distinct prime numbers p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pm ∈ P , nonnegative integers jk,t, k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',m, and write P{λpt(X(n) k ) ≥ jk,t for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d and t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',m} = P{X(n) k is divisible by pjk,t t for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d and t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',m} = P{X(n) k is divisible by m � t=1 p jk,t t =: µk for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d} = 1 #Dn ∞ � i1=1 ··· ∞ � id=1 1�(i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',id) ∈ Dn : ik ≡ 0 (modµk),k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 applied with mk = µk and jk = 0, k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, we see that the right-hand side converges to (µ1 ···µd)−1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' It remains to note that 1 µ1 ···µd = d � k=1 m � t=1 1 p jk,t t = P{Gk(pt) ≥ jk,t for all k = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d and t = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Fix a large positive constant M and note that F(X(n) 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',X(n) d ) = � p∈P F(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) = \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � p∈P ,p≤M F(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � p∈P ,p>M F(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 =: Y1(M,n)Y2(M,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1, one has Y1(M,n) w −→ n→∞ � p∈P ,p≤M F(pG1(p),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pGd(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Furthermore, the right-hand side of the latter converges a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' to F∞ as M → ∞, which is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' fi- nite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2 in [1], it remains to check that for every fixed ε > 0, (18) lim M→∞limsup n→∞ P{|Y2(M,n) − 1| ≥ ε} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 10 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL Note that (19) P{|Y2(M,n) − 1| ≥ ε} ≤ P \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3for all p ∈ P ,p > M, d � i=1 λp(X(n) i ) ≤ 1,|Y2(M,n) − 1| ≥ ε \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe + P \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3for some p ∈ P ,p > M, d � i=1 λp(X(n) i ) ≥ 2 \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The second term in (19) can be estimated as follows: P{for some p ∈ P ,p > M, d � i=1 λp(X(n) i ) ≥ 2} ≤ P{there exist p ∈ P ,p > M and i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d such that λp(X(n) i ) ≥ 2} + P{there exist p ∈ P ,p > M and i,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d,i � j such that λp(X(n) i ) ≥ 1,λp(X(n) j ) ≥ 1} = P{there exist p ∈ P ,p > M and i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d such that p2 divides X(n) i } + P{there exist p ∈ P ,p > M and i,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d,i � j such that p divides X(n) i and X(n) j } ≤ d � i=1 � p∈P ,p>M P{p2 divides X(n) i } + d � i,j=1,i�j � p∈P ,p>M P{p divides X(n) i and X(n) j } = d � i=1 � p∈P ,p>M #(Dn ∩ Zi(p2)) #Dn + d � i,j=1,i�j � p∈P ,p>M #(Dn ∩ Zi(p) ∩ Zj(p)) #Dn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The double limit (n → ∞, M → ∞) of the first term is equal to zero by an appeal to (13) with a = p2 and b = 1, since lim M→∞ � p∈P ,p>M 1 p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Similarly, the double limit of the second term is equal to zero by an appeal to (13) with a = b = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In order to deal with the first summand in (19), we first observe that on the event \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3for all p ∈ P ,p > M, d � i=1 λp(X(n) i ) ≤ 1 \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe, we may pass to the logarithm of Y2(M,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, it suffices to prove that, for every ε > 0, lim M→∞limsup n→∞ P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 for all p ∈ P ,p > M, d � i=1 λp(X(n) i ) ≤ 1, �������� � p∈P ,p>M logF(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) �������� ≥ ε \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Introduce, for n ∈ N, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d and p ∈ P , the events Cn,i,p := {λp(X(n) i ) = 1,λp(X(n) j ) = 0,j � i}, MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 11 and note that Cn,i,p ∩ Cn,j,p = ∅ as soon as i � j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' On the event Cn,i,p, we have logF(pλp(X(n) 1 ),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',pλp(X(n) d )) = Fi(p) and, therefore, it suffices to show that, for every fixed ε > 0, (20) lim M→∞limsup n→∞ P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 �������� � p∈P ,p>M d � i=1 Fi(p) 1Cn,i,p �������� ≥ ε \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Fix some A > 0 and note that, for every ε > 0, P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 �������� � p∈P ,p>M d � i=1 Fi(p) 1{|Fi(p)|>A,Cn,i,p} �������� ≥ ε \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe ≤ P{for some p ∈ P and i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, |Fi(p)| > A and Cn,i,p holds} ≤ � p∈P ,p>M d � i=1 1{|Fi(p)|>A} P{Cn,i,p} ≤ � p∈P ,p>M d � i=1 1{|Fi(p)|>A} P{λp(X(n) i ) ≥ 1} = � p∈P ,p>M d � i=1 1{|Fi(p)|>A} #(Dn ∩ Zi(p)) #Dn ≤ K � p∈P ,p>M 1 p d � i=1 1{|Fi(p)|>A}, where we used (13) with a = p and b = 1 for the last passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The right-hand side converges to zero as M → ∞, in view of the first relation in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' So, in order to prove (20), we need to check that (21) lim M→∞limsup n→∞ P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 �������� � p∈P ,p>M d � i=1 Fi(p) 1{|Fi(p)|≤A,Cn,i,p} �������� ≥ ε \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' This is accomplished by an appeal to Markov’s inequality as follows: P \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3 �������� � p∈P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p>M d � i=1 Fi(p) 1{|Fi(p)|≤A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p} �������� ≥ ε \uf8fc\uf8f4\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8f4\uf8fe ≤ 1 ε � p∈P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p>M d � i=1 |Fi(p)| 1{|Fi(p)|≤A} P{Cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p} ≤ 1 ε � p∈P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p>M d � i=1 |Fi(p)| 1{|Fi(p)|≤A} P{λp(X(n) i ) ≥ 1} = 1 ε � p∈P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p>M d � i=1 Fi(p) 1{|Fi(p)|≤A} #(Dn ∩ Zi(p)) #Dn ≤ K ε � p∈P ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='p>M 1 p d � i=1 Fi(p) 1{|Fi(p)|≤A},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 12 ZAKHAR KABLUCHKO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' OLEKSANDR MARYNYCH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' AND KILIAN RASCHEL where we have utilized (13) with a = p and b = 1 for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The proof of The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4 is complete, since the right-hand side converges to zero, as M → ∞, by the second relation in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Examples of suitable integer domains In this section we provide a series of examples of domains Dn that satisfy (7) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' In particular, we show that Rn1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',nd in (2), Sn and Hn mentioned in the introduction, are all admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, under assumption (11) on F, the distributional convergence (6) holds true for all domains listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Sublevels of monotone functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Assume that f : [1,∞)d → R is a coordinate-wise nondecreasing function such that, for every j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, lim xj→∞f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = ∞, provided xi ≥ 1, i � j, are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Put Dn := Df n = {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ≤ n} and Dn,i := Df n,i = {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xi−1,xi+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd−1 : f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xi−1,1,xi+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ≤ n}, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' If, for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, (22) lim n→∞ #Dn,i #Dn = 0, then the sequence Dn, n ∈ N, satisfies (7) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let us first verify (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' According to Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2 in Appendix A, it is sufficient to check (7) for c = ei, i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, where e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',ed denotes the standard basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Note that Dn \\ (Dn + ei) = Dn,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, (22) yields that for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, lim n→∞ #(Dn \\ (Dn + ei)) #Dn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' It remains to check that for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, (23) lim n→∞ #((Dn + ei) \\ Dn) #Dn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Without loss of generality, we shall do this for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Note that (Dn + e1) \\ Dn = {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : x1 ≥ 2,f (x1 − 1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ≤ n,f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) > n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' MULTIVARIATE MULTIPLICATIVE FUNCTIONS OF RANDOM VECTORS IN LARGE INTEGER DOMAINS 13 For every fixed collection (x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd−1 and n ∈ N, there exists at most one x1 ≥ 2, x1 ∈ N, such that f (x1 − 1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ≤ n and f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) > n, since f is monotone in x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Therefore, #((Dn + e1) \\ Dn) = ∞ � x2=1 ··· ∞ � xd=1 1{there exists x1≥2 such that f (x1−1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)≤n,f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)>n} ≤ ∞ � x2=1 ··· ∞ � xd=1 1{there exists x1≥2 such that f (x1−1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)≤n} = ∞ � x2=1 ··· ∞ � xd=1 1{f (1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)≤n} = #Dn,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' This proves (23) for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' We shall now prove that (13) holds, for all i,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, with K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' For notational simplicity, we shall do this only for i = 1 and j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The monotonicity of f implies that, for all a,b ∈ N, #Dn = a−1 � j=0 b−1 � k=0 \uf8eb \uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ∞ � x1=1 ∞ � x2=1 ··· ∞ � xd=1 1{f (ax1−j,bx2−k,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)≤n} \uf8f6 \uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≥ ab ∞ � x1=1 ∞ � x2=1 ··· ∞ � xd=1 1{f (ax1,bx2,x3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd)≤n} = ab#(Dn ∩ Z1(a) ∩ Z2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 yields the following explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='2 (Rectangular domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let f1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',fd : [1,∞) → [1,∞) be strictly increasing continu- ous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Putting f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) := max(f −1 1 (x1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',f −1 d (xd)), we obtain Dn = Rf1(n),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',fd(n) = ([1,f1(n)] × ··· × [1,fd(n)]) ∩ Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Condition (22) is fulfilled if limx→∞ fi(x) = ∞, for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='3 (Tetrahedral domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',ad > 0 be fixed positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' The sequence of tetrahedral sets Dn = Tn := {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : a1x1 + ··· + adxd ≤ n} satisfies (7) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Indeed, #Tn ∼ 1 d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='a1 ···ad nd, n → ∞, 14 ZAKHAR KABLUCHKO, OLEKSANDR MARYNYCH, AND KILIAN RASCHEL whereas, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, #Tn,i ∼ ai (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='a1···ad nd−1, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='4 (Hyperbolic domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Let f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = x1 ···xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Then the sequence of sets Dn = Hn := {(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) ∈ Nd : x1 ···xd ≤ n} satisfies (7) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Indeed, according to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' #Dn ∼ nlogd−1 n (d − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' , n → ∞, and, for every i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',d, #Dn,i ∼ nlogd−2 n (d − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' , n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Thus, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='1 is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='5 (Further hyperbolic domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Fix 2 ≤ ℓ ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=' Define the ℓ-th standard symmetric polynomial in d variables by f (x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) = Pℓ(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FRT4oBgHgl3EQfiTdH/content/2301.13586v1.pdf'} +page_content=',xd) := � 1≤i1<···